Design Optimization of Resource Allocation in OFDMA-Based Cognitive Radio-Enabled Internet of Vehicles (IoVs)

Joint optimal subcarrier and transmit power allocation with QoS guarantee for enhanced packet transmission over Cognitive Radio (CR)-Internet of Vehicles (IoVs) is a challenge. This open issue is considered in this paper. A novel SNBS-based wireless radio resource scheduling scheme in OFDMA CR-IoV network systems is proposed. This novel scheduler is termed the SNBS OFDMA-based overlay CR-Assisted Vehicular NETwork (SNO-CRAVNET) scheduling scheme. It is proposed for efficient joint transmit power and subcarrier allocation for dynamic spectral resource access in cellular OFDMA-based overlay CRAVNs in clusters. The objectives of the optimization model applied in this study include (1) maximization of the overall system throughput of the CR-IoV system, (2) avoiding harmful interference of transmissions of the shared channels’ licensed owners (or primary users (PUs)), (3) guaranteeing the proportional fairness and minimum data-rate requirement of each CR vehicular secondary user (CRV-SU), and (4) ensuring efficient transmit power allocation amongst CRV-SUs. Furthermore, a novel approach which uses Lambert-W function characteristics is introduced. Closed-form analytical solutions were obtained by applying time-sharing variable transformation. Finally, a low-complexity algorithm was developed. This algorithm overcame the iterative processes associated with searching for the optimal solution numerically through iterative programming methods. Theoretical analysis and simulation results demonstrated that, under similar conditions, the proposed solutions outperformed the reference scheduler schemes. In comparison to other scheduling schemes that are fairness-considerate, the SNO-CRAVNET scheme achieved a significantly higher overall average throughput gain. Similarly, the proposed time-sharing SNO-CRAVNET allocation based on the reformulated convex optimization problem is shown to be capable of achieving up to 99.987% for the average of the total theoretical capacity.


Introduction
In the near future, most vehicles are expected to be equipped with wireless communication technologies, such as On-Board Units (OBUs), and ultrasonic sensors to enable a variety of new services, such as safety applications, improved traffic management, and enhanced infotainment services [1]. Therefore, the vehicular ad hoc network (VANET) has gained an increased importance, receiving a great amount of attention from academia, auto-manufacturing industries, and government agencies. capable of protecting the licensed PUs from harmful interference from the CRV-SUs and the potential to satisfy the QoS requirements of CRV-SUs, especially for the communication of time-constrained safety (or emergency) messages in IoVs. The choice of cellular OFDMA-based overlay CR-Assisted Vehicular NETworks is in accordance with the current trends of developments with respect to future wireless communication systems, such as IEEE 802.16-style networks, IEEE 802.11p networks, and universal mobile telecommunication system (UMTS) long-term evolution (LTE). Moreover, multicarrier OFDMA technology can lead to improvements of the spectral efficiency, as well as the robustness needed when dealing with time-varying wireless multi-path interferences, which is likely the case with vehicular networks [14]. Tables 1 and 2 present the key mathematical notations and acronyms used in this study and their meaning, respectively. In this paper, a novel Symmetric NBS OFDMA-based overlay CR-Assisted Vehicular NETwork (SNO-CRAVNET) scheduling scheme is proposed for efficient joint transmit power and subcarrier allocation for dynamic spectral resource access in cellular OFDMA-based overlay CR-Assisted Vehicular NETworks in clusters. The joint optimal allocation strategy is determined in a simpler and faster approach by the proposed scheduler, with the help of the obtained closed-form analytical solution, as opposed to previous studies, which have adopted iterative programming methods, such as the work presented in [15][16][17][18]. Under the interweave-based CR-enabled IoV network systems, the spectrum sensing accuracy remains an open issue due to prevailing sensing errors over wireless channels. However, the scope of this study does not cover an investigation of the integration of spectrum sensing in interweave-based CR-enabled IoV network systems. The merits of the proposed novel SNO-CRAVNET scheme are confirmed through its comparison with existing approaches. Rate at which the CRV-SUs' cluster formation changes from ℒ to ℒ location Packet arrival rate Transition probability matrix Rate at which the CRV-SUs' cluster formation changes from j to j location τ Packet arrival rate P Transition probability matrix → ω Steady state probability vector of PAP [C mn ] Subcarrier allocation strategy P A Transition probability matrix of the PAP U 0 Initial utility vector Set of game theory strategies of the R CRV-SU players and utility vectors' space R N C Total possible channel assignments [R mn ] Rate allocation strategy B Minimum utility bound t mn OFDM symbol transmitted by CRV-SU n over the m th subcarrier G mn Complex circularly-symmetric Gaussian noise

System Model
The co-existence of the cellular OFDMA-based overlay CR-Assisted Vehicular NETwork with the PU network scenario as depicted in Figure 1 is considered in this paper. As demonstrated in Figure 1, the network scenario is divided into cognitive cells (CCs) [2]. The CCs consist of ℕ number of CR base stations (CR-BS) and ℝ number of CRV-SUs in vehicular cluster formations. The dynamically available channel bandwidth ( ) is evenly divided within a given CC into number of orthogonal channels. Specifically, the CR-BS in-charge of a CC receives data from CRV-SUs and efficiently performs spectral resource scheduling. Therefore, to prevent harmful interference with the transmissions of the PUs, the CR-BS controls the dynamically available resources. Figure 2 presents an illustration of the phases of the research carried out in this study. The study is divided into four main phases: The system model; proposed utility of SNO-CRAVNET and problem formulation; optimal resource scheduling strategies; and performance evaluation. The system model is further sub-divided into sub-phases, such as the vehicular cluster mobility model, activity of PUs, packet arrival process (PAP), SNO-CRAVNET architecture, and interference constraints. Each of these research phases and sub-phases are detailed in the following sections and sub-sections.

Vehicular Cluster Mobility Model
The average life span of CRV-SUs' cluster formation [1] is the total duration when all the ℝ CRV-SUs in a particular cluster (usually presumed to be exponentially distributed) maintain membership of the same CC [19]. Therefore, the CRV-SUs' cluster formation mobility can be modeled using a transition rate (i.e., speed) matrix Ɱ and given by Equation (1) where = | | denotes the number of locations in a service area, | | represents the cardinality of set , and the element ɱ( , ′ ) represents the rate at which the CRV-SUs' cluster formation changes from ℒ to ℒ ′ location. Different speeds of CRV-SUs' cluster formation can be captured in different locations by the matrix Ɱ within a service area.

Activity of PUs
Packet transmission from both CRV-SUs' cluster members (CMs) to cluster heads (CHs) and from the CHs to their respective destinations is based on the common channel shared with the licensed users. Therefore, both the CRV-SUs' CMs and the CHs must always watch the activity of the licensed users prior to accessing and using the shared channel in a fashion that does not cause harmful interference of the PUs' activity. In a shared channel, the activity of the PUs is modeled through a two-state Markov chain (see Equation (2)), such as the ON-OFF model, which corresponds to the busy and idle states, respectively. Consequently, a transition probability matrix is used to model the state transition of shared channel , as expressed below: ̃= [̃( 0, 0)̃(0, 1) (1, 0)̃(1, 1) ] ← idle ← busy (2) where 0 and 1 represent the idle and busy states, respectively. Therefore, the probability of the shared wireless channel m being in an idle state can be obtained from the expression = 1 −̃(1, 1) ((̃(0, 1) −̃(1, 1) + 1)) ⁄ .

Vehicular Cluster Mobility Model
The average life span of CRV-SUs' cluster formation [1] is the total duration when all the R CRV-SUs in a particular cluster (usually presumed to be exponentially distributed) maintain membership of the same CC [19]. Therefore, the CRV-SUs' cluster formation mobility can be modeled using a transition rate (i.e., speed) matrix multicarrier OFDMA technology can lead to improvements of the spectral efficiency, as well as the robustness needed when dealing with time-varying wireless multi-path interferences, which is likely the case with vehicular networks [14]. Tables 1 and 2 present the key mathematical notations and acronyms used in this study and their meaning, respectively.
In this paper, a novel Symmetric NBS OFDMA-based overlay CR-Assisted Vehicular NETwork (SNO-CRAVNET) scheduling scheme is proposed for efficient joint transmit power and subcarrier allocation for dynamic spectral resource access in cellular OFDMA-based overlay CR-Assisted Vehicular NETworks in clusters. The joint optimal allocation strategy is determined in a simpler and faster approach by the proposed scheduler, with the help of the obtained closed-form analytical solution, as opposed to previous studies, which have adopted iterative programming methods, such as the work presented in [15][16][17][18]. Under the interweave-based CR-enabled IoV network systems, the spectrum sensing accuracy remains an open issue due to prevailing sensing errors over wireless channels. However, the scope of this study does not cover an investigation of the integration of spectrum sensing in interweave-based CR-enabled IoV network systems. The merits of the proposed novel SNO-CRAVNET scheme are confirmed through its comparison with existing approaches. Rate at which the CRV-SUs' cluster formation changes from ℒ to ℒ location Packet arrival rate Transition probability matrix Vehicular NETworks is in accordance with the current trends of developments with respect to fut wireless communication systems, such as IEEE 802.16-style networks, IEEE 802.11p networks, a universal mobile telecommunication system (UMTS) long-term evolution (LTE). Moreov multicarrier OFDMA technology can lead to improvements of the spectral efficiency, as well as robustness needed when dealing with time-varying wireless multi-path interferences, which is lik the case with vehicular networks [14]. Tables 1 and 2 present the key mathematical notations a acronyms used in this study and their meaning, respectively. In this paper, a novel Symmetric NBS OFDMA-based overlay CR-Assisted Vehicular NETw (SNO-CRAVNET) scheduling scheme is proposed for efficient joint transmit power and subcar allocation for dynamic spectral resource access in cellular OFDMA-based overlay CR-Assis Vehicular NETworks in clusters. The joint optimal allocation strategy is determined in a simpler a faster approach by the proposed scheduler, with the help of the obtained closed-form analyt solution, as opposed to previous studies, which have adopted iterative programming methods, s as the work presented in [15][16][17][18]. Under the interweave-based CR-enabled IoV network systems, spectrum sensing accuracy remains an open issue due to prevailing sensing errors over wire channels. However, the scope of this study does not cover an investigation of the integration spectrum sensing in interweave-based CR-enabled IoV network systems. The merits of the propo novel SNO-CRAVNET scheme are confirmed through its comparison with existing approaches. Rate at which the CRV-SUs' cluster formation changes from ℒ to ℒ location Packet arrival rate Transition probability matrix

Vehicular Cluster Mobility Model
The average life span of CRV-SUs' cluster formation [1] is the total duration when all the ℝ CRV-SUs in a particular cluster (usually presumed to be exponentially distributed) maintain membership of the same CC [19]. Therefore, the CRV-SUs' cluster formation mobility can be modeled using a transition rate (i.e., speed) matrix Ɱ and given by Equation where = | | denotes the number of locations in a service area, | | represents the cardinality of set , and the element ɱ( , ) represents the rate at which the CRV-SUs' cluster formation changes from ℒ to ℒ location. Different speeds of CRV-SUs' cluster formation can be captured in different locations by the matrix Ɱ within a service area.
We represent the steady state probability vector with Ṿ ⃗ = (ℒ ) ⋯ ℒ ⋯ ℒ , where the element (ℒ) of the vector Ṿ ⃗ denotes the probability that the CRV-SUs' cluster formation occurs at the location ℒ . Therefore, by solving Ṿ ⃗ Ɱ = 0 ⃗ and Ṿ ⃗ 1 ⃗ = 1, the steady state probability vector Ṿ ⃗ can be obtained, where 1 ⃗ and 0 ⃗ represent the vectors of ones and zeros, respectively.

Activity of PUs
Packet transmission from both CRV-SUs' cluster members (CMs) to cluster heads (CHs) and from the CHs to their respective destinations is based on the common channel shared with the licensed users. Therefore, both the CRV-SUs' CMs and the CHs must always watch the activity of the licensed users prior to accessing and using the shared channel in a fashion that does not cause harmful interference of the PUs' activity. In a shared channel, the activity of the PUs is modeled through a two-state Markov chain (see Equation (2)), such as the ON-OFF model, which corresponds to the busy and idle states, respectively. Consequently, a transition probability matrix is used to model the state transition of shared channel , as expressed below: where 0 and 1 represent the idle and busy states, respectively. Therefore, the probability of the shared wireless channel m being in an idle state can be obtained from the expression = 1 − (1, 1) (0, 1) − (1, 1) + 1 .

Vehicular Cluster Mobility Model
The average life span of CRV-SUs' cluster formation [1] is the total duration when all the ℝ CRV-SUs in a particular cluster (usually presumed to be exponentially distributed) maintain membership of the same CC [19]. Therefore, the CRV-SUs' cluster formation mobility can be modeled using a transition rate (i.e., speed) matrix Ɱ and given by Equation where = | | denotes the number of locations in a service area, | | represents the cardinality of set , and the element ɱ ( , ) represents the rate at which the CRV-SUs' cluster formation changes from ℒ to ℒ location. Different speeds of CRV-SUs' cluster formation can be captured in different locations by the matrix Ɱ within a service area.
We represent the steady state probability vector with Ṿ ⃗ = (ℒ ) ⋯ ℒ ⋯ ℒ , where the element (ℒ) of the vector Ṿ ⃗ denotes the probability that the CRV-SUs' cluster formation occurs at the location ℒ . Therefore, by solving Ṿ ⃗ Ɱ = 0 ⃗ and Ṿ ⃗ 1 ⃗ = 1, the steady state probability vector Ṿ ⃗ can be obtained, where 1 ⃗ and 0 ⃗ represent the vectors of ones and zeros, respectively.

Activity of PUs
Packet transmission from both CRV-SUs' cluster members (CMs) to cluster heads (CHs) and from the CHs to their respective destinations is based on the common channel shared with the licensed users. Therefore, both the CRV-SUs' CMs and the CHs must always watch the activity of the licensed users prior to accessing and using the shared channel in a fashion that does not cause harmful interference of the PUs' activity. In a shared channel, the activity of the PUs is modeled through a two-state Markov chain (see Equation (2)), such as the ON-OFF model, which corresponds to the busy and idle states, respectively. Consequently, a transition probability matrix is used to model the state transition of shared channel , as expressed below: where 0 and 1 represent the idle and busy states, respectively. Therefore, the probability of the shared wireless channel m being in an idle state can be obtained from the expression = 1 − (1, 1) (0, 1) − (1, 1) + 1 .

Vehicular Cluster Mobility Model
The average life span of CRV-SUs' cluster formation [1] is the total duration when all the ℝ CRV-SUs in a particular cluster (usually presumed to be exponentially distributed) maintain membership of the same CC [19]. Therefore, the CRV-SUs' cluster formation mobility can be modeled using a transition rate (i.e., speed) matrix Ɱ and given by Equation where = | | denotes the number of locations in a service area, | | represents the cardinality of set , and the element ɱ ( , ) represents the rate at which the CRV-SUs' cluster formation changes from ℒ to ℒ location. Different speeds of CRV-SUs' cluster formation can be captured in different locations by the matrix Ɱ within a service area.
We represent the steady state probability vector with Ṿ ⃗ = (ℒ ) ⋯ ℒ ⋯ ℒ , where the element (ℒ) of the vector Ṿ ⃗ denotes the probability that the CRV-SUs' cluster formation occurs at the location ℒ . Therefore, by solving Ṿ ⃗ Ɱ = 0 ⃗ and Ṿ ⃗ 1 ⃗ = 1, the steady state probability vector Ṿ ⃗ can be obtained, where 1 ⃗ and 0 ⃗ represent the vectors of ones and zeros, respectively.

Activity of PUs
Packet transmission from both CRV-SUs' cluster members (CMs) to cluster heads (CHs) and from the CHs to their respective destinations is based on the common channel shared with the licensed users. Therefore, both the CRV-SUs' CMs and the CHs must always watch the activity of the licensed users prior to accessing and using the shared channel in a fashion that does not cause harmful interference of the PUs' activity. In a shared channel, the activity of the PUs is modeled through a two-state Markov chain (see Equation (2)), such as the ON-OFF model, which corresponds to the busy and idle states, respectively. Consequently, a transition probability matrix is used to model the state transition of shared channel , as expressed below: where 0 and 1 represent the idle and busy states, respectively. Therefore, the probability of the shared wireless channel m being in an idle state can be obtained from the expression = 1 − (1, 1) (0, 1) − (1, 1) + 1 .

Vehicular Cluster Mobility Model
The average life span of CRV-SUs' cluster formation [1] is the total duration when all the ℝ CRV-SUs in a particular cluster (usually presumed to be exponentially distributed) maintain membership of the same CC [19]. Therefore, the CRV-SUs' cluster formation mobility can be modeled using a transition rate (i.e., speed) matrix Ɱ and given by Equation where = | | denotes the number of locations in a service area, | | represents the cardinality of set , and the element ɱ ( , ) represents the rate at which the CRV-SUs' cluster formation changes from ℒ to ℒ location. Different speeds of CRV-SUs' cluster formation can be captured in different locations by the matrix Ɱ within a service area.
We represent the steady state probability vector with Ṿ ⃗ = (ℒ ) ⋯ ℒ ⋯ ℒ , where the element (ℒ) of the vector Ṿ ⃗ denotes the probability that the CRV-SUs' cluster formation occurs at the location ℒ . Therefore, by solving Ṿ ⃗ Ɱ = 0 ⃗ and Ṿ ⃗ 1 ⃗ = 1, the steady state probability vector Ṿ ⃗ can be obtained, where 1 ⃗ and 0 ⃗ represent the vectors of ones and zeros, respectively.

Activity of PUs
Packet transmission from both CRV-SUs' cluster members (CMs) to cluster heads (CHs) and from the CHs to their respective destinations is based on the common channel shared with the licensed users. Therefore, both the CRV-SUs' CMs and the CHs must always watch the activity of the licensed users prior to accessing and using the shared channel in a fashion that does not cause harmful interference of the PUs' activity. In a shared channel, the activity of the PUs is modeled through a two-state Markov chain (see Equation (2)), such as the ON-OFF model, which corresponds to the busy and idle states, respectively. Consequently, a transition probability matrix is used to model the state transition of shared channel , as expressed below: where 0 and 1 represent the idle and busy states, respectively. Therefore, the probability of the shared wireless channel m being in an idle state can be obtained from the expression = 1 − (1, 1) (0, 1) − (1, 1) + 1 .
( j m , j m ) where j m = |J| denotes the number of locations in a service area, |J| represents the cardinality of set J, and the element M( j, j ) represents the rate at which the CRV-SUs' cluster formation changes from L j to L j location. Different speeds of CRV-SUs' cluster formation can be captured in different locations by the matrix between the PU's band and the sub-channels. Therefore, spectral resource allocation under cellular OFDMA-based overlay CR-Assisted Vehicular NETworks promises an efficient approach that is capable of protecting the licensed PUs from harmful interference from the CRV-SUs and the potential to satisfy the QoS requirements of CRV-SUs, especially for the communication of time-constrained safety (or emergency) messages in IoVs. The choice of cellular OFDMA-based overlay CR-Assisted Vehicular NETworks is in accordance with the current trends of developments with respect to future wireless communication systems, such as IEEE 802. 16-style networks, IEEE 802.11p networks, and universal mobile telecommunication system (UMTS) long-term evolution (LTE). Moreover, multicarrier OFDMA technology can lead to improvements of the spectral efficiency, as well as the robustness needed when dealing with time-varying wireless multi-path interferences, which is likely the case with vehicular networks [14]. Tables 1 and 2 present the key mathematical notations and acronyms used in this study and their meaning, respectively. In this paper, a novel Symmetric NBS OFDMA-based overlay CR-Assisted Vehicular NETwork (SNO-CRAVNET) scheduling scheme is proposed for efficient joint transmit power and subcarrier allocation for dynamic spectral resource access in cellular OFDMA-based overlay CR-Assisted Vehicular NETworks in clusters. The joint optimal allocation strategy is determined in a simpler and faster approach by the proposed scheduler, with the help of the obtained closed-form analytical solution, as opposed to previous studies, which have adopted iterative programming methods, such as the work presented in [15][16][17][18]. Under the interweave-based CR-enabled IoV network systems, the spectrum sensing accuracy remains an open issue due to prevailing sensing errors over wireless channels. However, the scope of this study does not cover an investigation of the integration of spectrum sensing in interweave-based CR-enabled IoV network systems. The merits of the proposed novel SNO-CRAVNET scheme are confirmed through its comparison with existing approaches. Rate at which the CRV-SUs' cluster formation changes from ℒ to ℒ location Packet arrival rate Transition probability matrix t Sensors 2020, 20, x FOR PEER REVIEW between the PU's band and the sub-channels. Therefore, spectral resource allocation und OFDMA-based overlay CR-Assisted Vehicular NETworks promises an efficient approa capable of protecting the licensed PUs from harmful interference from the CRV-SUs and the to satisfy the QoS requirements of CRV-SUs, especially for the communication of time-co safety (or emergency) messages in IoVs. The choice of cellular OFDMA-based overlay CR Vehicular NETworks is in accordance with the current trends of developments with respec wireless communication systems, such as IEEE 802.16-style networks, IEEE 802.11p netw universal mobile telecommunication system (UMTS) long-term evolution (LTE). M multicarrier OFDMA technology can lead to improvements of the spectral efficiency, as w robustness needed when dealing with time-varying wireless multi-path interferences, whic the case with vehicular networks [14]. Tables 1 and 2 present the key mathematical nota acronyms used in this study and their meaning, respectively. In this paper, a novel Symmetric NBS OFDMA-based overlay CR-Assisted Vehicular (SNO-CRAVNET) scheduling scheme is proposed for efficient joint transmit power and allocation for dynamic spectral resource access in cellular OFDMA-based overlay CR Vehicular NETworks in clusters. The joint optimal allocation strategy is determined in a sim faster approach by the proposed scheduler, with the help of the obtained closed-form solution, as opposed to previous studies, which have adopted iterative programming meth as the work presented in [15][16][17][18]. Under the interweave-based CR-enabled IoV network sy spectrum sensing accuracy remains an open issue due to prevailing sensing errors ove channels. However, the scope of this study does not cover an investigation of the inte spectrum sensing in interweave-based CR-enabled IoV network systems. The merits of the novel SNO-CRAVNET scheme are confirmed through its comparison with existing approa such that the elements can be obtained as follows (see Equations (5)- (8)):

Packet Arrival Process (PAP)
A finite queue of size Q packets is used at each CRV-SU CM to buffer packets. The CRV-SU CMs fetch packets from their finite queue for onward transmission to the CRV-SU CH. A batch Markovian process (see Equation (9)) is used to model the PAP of CRV-SUs with Y phases. Specifically, Sensors 2020, 20, 6402 8 of 28 P A is used to denote the transition probability matrix of the PAP, as shown in Equation (9) below, for A ∈ {0, 1, 2, · · · , A m } arriving packets, with A m representing the maximum batch size: With respect to Equation (9) above, P A (y, y ) represents the probability that m data packets arrived at the finite queue with the phase changing from y to y . Correspondingly, the transition probability matrix P is given by P = P 0 + P 1 + P 2 + · · · + P A m . Let the steady state probability vector → ω of PAP be denoted by → ω = [ω(1) · · · ω(y) · · · ω(Y)] t . Then, the steady state probability that the phase of PAP is y is represented by the element ω(y) of the vector → ω. Therefore, by solving this steady state probability vector → ω can be obtained. Accordingly, by weighting the probability of all phases with ω(y), the packet arrival rate (PAR) is obtainable through the following expression:

SNO-CRAVNET Architecture
The model architecture and parameters of the SNO-CRAVNET scheme and a description of the initial resource allocation strategies of the scheme are presented in this subsection. Let the identically independent distributed (i.i.d.) subcarrier gain of CRV-SU n|n = 1, 2, · · · , L be represented by a mn on m th subcarrier, with m = 1, 2, · · · , N C . Let G mn represent the complex circularly-symmetric Gaussian noise, and G mn ∼ CN 0, σ 2 χ , where σ 2 χ = B(N 0 /N C ), with N 0 representing the noise density. Then, let the OFDM symbol transmitted by CRV-SU n over the m th subcarrier be denoted as t mn , so that the OFDM symbol received at the destination can be expressed as r mn = (a mn × t mn ) + G mn . In the SNO-CRAVNET scheme, matrix P N C ×R O N C ×R = [P mn ] denotes the transmit power allocation strategy, with the individual matrix elements represented by the instantaneous transmit power of CRV-SU n over channel m expressed as P mn = E[|t mn | 2 ], where E[·] stands for the expected value operator. Additionally, matrix N C ×R O N C ×R = [R mn ] represents the rate allocation strategy, with the respective elements of the matrix denoted by the instant data-rate-R mn (P mn )-showing the total number of bits actually loaded on the m th subcarrier that is allocated to the CRV-SU n. Furthermore, the Multi-level Quadrature Amplitude Modulation (M-QAM) is used for the adjustment of the transmit power level, in agreement with the combined subcarrier power gains and the total number of loaded bits. Therefore, on each allocated CRV-SU n, the bit error rate (BER) according to Chung and Goldsmith [20] can be expressed as BER mn ≈ 0.2 × exp{−1.5 × β mn /2 [R mn (P mn )−1] }, where β mn = P mn |a mn | 2 /σ 2 χ denotes the signal-to-noise ratio (SNR). By assuming, in this model, that the channel state information (CSI) [21,22] is known, we maximize the mutual information denoted as M(·) between the OFDM symbol transmitted by CRV-SU n over the m th subcarrier and the OFDM symbol received at the destination. Therefore, the maximum achievable channel capacity in a fading slot is represented as M C mn (P mn ) = maxM(t mn : r mn |a mn ) = log 2 (1 + P mn |a mn | 2 ϕ), where ϕ = −1.5/ ln(5 × BER mn ) × σ 2 χ . Considering this, transmissions can only be successful, if and only if, M C mn (P mn ) > R mn (P mn ) (i.e., the maximum achievable capacity is greater than the instantaneous specified data-rate). Contrarily, when M C mn (P mn ) = R mn (P mn ), i.e., at the maximal point, according to Shannon's theory, the feasible transmissions' maximum instantaneous data-rate can be expressed as Sensors 2020, 20, 6402 9 of 28 where B/N C represents the bandwidth of the respective dynamically available orthogonal subcarrier. Furthermore, the adaptive modulator ensures that the values of R mn (P mn ) are taken from set I = {0, 1, 2, · · · , I}, with I denoting the feasible maximum amount of information over each dynamically available orthogonal channel. Additionally, in accordance with both the transmit power and rate allocation strategy, the channel allocation strategy is denoted by matrix C N C ×R O N C ×R = [C mn ], where the channel allocation index signified by the matrix elements is represented by C mn ∈ {0, 1}. Therefore, C mn = 1 means that the dynamically available channel m is successfully allocated to CRV-SU n, and C mn = 0 means that no channel is allocated. Under SNO-CRAVNET architecture, two or more CRV-SUs cannot share a single channel at the same time. Therefore, a crucial constraint for the available channel allocation strategy is Since the conditions of the available channel are random, in this paper, the expected value operator E[·] is used to indicate the random realization of CSI's mean quantity (i.e., |a mn | 2 ). Consequently, from Equations (11) and (12), the average data-rate of CRV-SU n can be expressed as Likewise, amongst all available channels and the CRV-SUs, the overall data-rate is given by Therefore, to guarantee that the transmit power allocated to the CRV-SUs occupying every dynamically available orthogonal subcarrier does not exceed the target and is maintained below the average transmit power P Tot. , available at the CR-BS, the condition for the transmit power allocation strategy is expressed as

Interference Constraints
The regulations employed in the system model of SNO-CRAVNET to control interference against PUs' transmission from CRV-SUs' transmission are presented in this sub-section. In this model, R PUs are considered in the network (i.e., the licensed users with ownership rights over the radio spectrum). On the contrary, when the CRV-SUs exploit the identified available spectrum holes for their own transmissions, they should do so in a fashion that ensures no harmful interference with the PUs with ownership rights over the spectrum band. Therefore, to guarantee the absolute avoidance of interference towards the PUs, CRV-SUs must strictly adhere to cognitive capabilities, which include, first and foremost, reliably intelligently sensing for the availability of spectrum holes to effectively confirm whether the channel is idle or currently occupied by a licensed owner. Secondly, upon confirming the existence of spectrum holes, the CRV-SUs should intelligently change their radio parameters for efficient exploitation of the identified spectrum holes, without causing interference to any ongoing transmissions of the PUs.
In Section 2.4, it is stated that under the SNO-CRAVNET scheme, each communication channel can only be allocated to a single CRV-SU at a time. Despite the allocation of one channel to one CRV-SU at a time, the communication quality of the channel, to a large extent, also affects the communications of the CRV-SUs. Therefore, the communication quality of the channel must be maintained by ensuring that the signal-to-interference-and-noise ratio (SINR) of the CRV-SU n is not lower than a predetermined threshold value β min n . An acceptable QoS condition (Due to the orthogonality of the channels/subcarriers, the resulting interference between CRV-SUs is ignored, as is shown in Equation (12)) is obtained and expressed as Therefore, Equation (16) can be expressed in a simplified form as where Additionally, to guarantee the protection of possible transmissions from licensed users (i.e., PUs) of the spectrum band, at each n (th) PU, with n = 1, 2, 3, · · · , R , the received SNIR must be greater than β min PU , where β min PU represents the predetermined threshold value applied to protect any ongoing transmissions from PUs. Let the distance between the n th PU and CR-BS be given as d CR−BS n , so that another interference constraint to protect the PUs' transmission can be given as where υ denotes the exponent of path attenuation and P PU n is the n th PU's transmit power. d nn represents the distance between nth CRV-SU and n th PU, while N n 0 represents the noise spectral density (i.e., noise density) of the n th PU. With the help of Location-Based Systems (LBSs), for instance, the Global Positioning System (GPS), both distances, d CR−BS n and d nn can be easily obtained. In addition, information on the CRV-SU's features can be obtained by the CR-BS through feedback channels. Therefore, without a loss of generality, Equation (18) where Sensors 2020, 20, x FOR PEER REVIEW 4 of 27 Ᵽ Transmit power constraint to protect potential transmissions of PU OFDM symbol received at the destination From Equation (19), the stipulated condition guarantees that the potential transmissions of the PU are fully protected if and only if the CRV-SU n's total transmit power is constrained over channel n by the predefined threshold

The Utility of SNO-CRAVNET and Problem Formulation
The design methodology of the SNO-CRAVNET's objectives with its SNO-CRAVNET game is presented in this section in the form of a convex optimization problem, with its associated players represented by the R CRV-SUs. The design of the game bargaining scheme methodologies for the CR-enabled IoV network system is proposed in this section. We assume that each R CRV-SU, for instance, CRV-SU n, has an initial utility U 0 n ≥ 0, which represents its acceptable minimum QoS constraint with respect to the data-rate and the corresponding utility function f n . Under the symmetric Nash bargaining (SNB) theory, each utility function f n is usually designated as a convex and closed subset of F R = { }, with F R and denoting the set of game theory strategies of the R CRV-SU players and utility vectors' space, respectively. Let us assume that U 0 n is conveniently achievable for all the R CRV-SU players. Then, it follows that at least a feasible subspace 0 exists in , so that the utility vector, for instance, f (ω) = f 1 , f 2 , f 3 , · · · , f R , becomes equal or bigger in comparison to the initial utility vector, such as, U 0 = U 0 1 , U 0 2 , U 0 3 , · · · , U 0 R . Therefore, the subset 0 as the element of can be expressed as 0 = ω ∈ f (ω) ≥ U 0 . Additionally, let us suppose that the set of utility that can be achieved is denoted by achieved is denoted by Ʈ = ( )| ∈ ℧ and the category of sets of utility policies that satisfies , which is the minimum utility bound, is denoted as ℬ = Ʈ, |Ʈ ⊂ ℝ . Therefore, in accordance with the Symmetric NBS theory (see [23]), there exists a unique solution, for instance, Տ |ℬ ⟶ ℝ , which satisfies the following axioms: a) Տ (Ʈ, ) ensures a minimum utility guarantee, for instance, Տ (Ʈ, ) ∈ Ʈ , where Ʈ = ∈ Ʈ| ≥ , ∀ ; b) Տ (Ʈ, ) is the Pareto optimal, which implies that other allocations Տ (Ʈ, ) capable of guaranteeing a higher performance for all the ℝ CRV-SUs simultaneously do not exist, that is, Տ (Ʈ, ) < Տ (Ʈ, ), ∃ and Տ (Ʈ, ) ≤ Տ (Ʈ, ), ∀ ; c) Տ (Ʈ, ) guarantees symmetry, which implies that all the ℝ CRV-SUs have equal priorities, for instance, supposing that Ʈ is symmetric with regards to a sub-set ⊆ 1, 2, 3, ⋯ , , ⋯ , ℝ and ∈ Ʈ, , ∈ so that = implies that Տ (Ʈ, ) = Տ (Ʈ, ) , ≠ ; d) Տ (Ʈ, ) guarantees fairness by maintaining the independence of irrelevant alternatives, for instance, if the feasible set decreases and the solution keeps on being feasible, it follows that the solution for the lesser achievable set remains the same point. It can be expressed as ɰ ⊂ Ʈ, (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ (Ʈ, ) = Տ (ɰ, ), ∀ .
Without a loss of generality, the property of the SNO-CRAVNET is described using the following theorem.
⊂ F R . Therefore, in accordance with the Symmetric NBS theory (see [23]), there exists a unique solution, for instance, with respect to the data-rate and the bargaining (SNB) theory, each utility of ℝ = ℧ , with ℝ and ℧ denoti and utility vectors' space, respectivel ℝ CRV-SU players. Then, it follows th vector, for instance, ( ) = , , utility vector, such as, = , , expressed as ℧ = ∈ ℧| ( ) ≥ achieved is denoted by Ʈ = ( )| which is the minimum utility bound, the Symmetric NBS theory (see [23]), satisfies the following axioms: represented by the ℝ CRV-SUs. The design of the game bargaining scheme methodologies for the CRenabled IoV network system is proposed in this section. We assume that each ℝ CRV-SU, for instance, CRV-SU , has an initial utility ≥ 0, which represents its acceptable minimum QoS constraint with respect to the data-rate and the corresponding utility function . Under the symmetric Nash bargaining (SNB) theory, each utility function is usually designated as a convex and closed subset of ℝ = ℧ , with ℝ and ℧ denoting the set of game theory strategies of the ℝ CRV-SU players and utility vectors' space, respectively. Let us assume that is conveniently achievable for all the ℝ CRV-SU players. Then, it follows that at least a feasible subspace ℧ exists in ℧, so that the utility vector, for instance, ( ) = , , , ⋯ , ℝ , becomes equal or bigger in comparison to the initial utility vector, such as, = , , , ⋯ , ℝ . Therefore, the subset ℧ as the element of ℧ can be expressed as ℧ = ∈ ℧| ( ) ≥ . Additionally, let us suppose that the set of utility that can be achieved is denoted by Ʈ = ( )| ∈ ℧ and the category of sets of utility policies that satisfies , which is the minimum utility bound, is denoted as ℬ = Ʈ, |Ʈ ⊂ ℝ . Therefore, in accordance with the Symmetric NBS theory (see [23]), there exists a unique solution, for instance, Տ |ℬ ⟶ ℝ , which satisfies the following axioms: ensures a minimum utility guarantee, for instance, is the Pareto optimal, which implies that other allocations Տ (Ʈ, ) capable of guaranteeing a higher performance for all the ℝ CRV-SUs simultaneously do not exist, that is, guarantees symmetry, which implies that all the ℝ CRV-SUs have equal priorities, for instance, supposing that Ʈ is symmetric with regards to a sub-set ⊆ 1, 2, 3, ⋯ , , ⋯ , ℝ and ∈ Ʈ, , guarantees fairness by maintaining the independence of irrelevant alternatives, for instance, if the feasible set decreases and the solution keeps on being feasible, it follows that the solution for the lesser achievable set remains the same point. It can be expressed as ɰ ⊂ Ʈ, (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ (Ʈ, ) = Տ (ɰ, ), ∀ .
Without a loss of generality, the property of the SNO-CRAVNET is described using the following theorem. nbs and utility vectors' space, respectively. Let us assume that is conveniently achievable for all the ℝ CRV-SU players. Then, it follows that at least a feasible subspace ℧ exists in ℧, so that the utility vector, for instance, ( ) = , , , ⋯ , ℝ , becomes equal or bigger in comparison to the initial utility vector, such as, = , , , ⋯ , ℝ . Therefore, the subset ℧ as the element of ℧ can be expressed as ℧ = ∈ ℧| ( ) ≥ . Additionally, let us suppose that the set of utility that can be achieved is denoted by Ʈ = ( )| ∈ ℧ and the category of sets of utility policies that satisfies , which is the minimum utility bound, is denoted as ℬ = Ʈ, |Ʈ ⊂ ℝ . Therefore, in accordance with the Symmetric NBS theory (see [23]), there exists a unique solution, for instance, Տ |ℬ ⟶ ℝ , which satisfies the following axioms: ensures a minimum utility guarantee, for instance, is the Pareto optimal, which implies that other allocations Տ (Ʈ, ) capable of guaranteeing a higher performance for all the ℝ CRV-SUs simultaneously do not exist, that is, Տ (Ʈ, ) < Տ (Ʈ, ), ∃ and Տ (Ʈ, ) ≤ Տ (Ʈ, ), ∀ ; c) Տ (Ʈ, ) guarantees symmetry, which implies that all the ℝ CRV-SUs have equal priorities, for instance, supposing that Ʈ is symmetric with regards to a sub-set ⊆ 1, 2, 3, ⋯ , , ⋯ , ℝ and ∈ Ʈ, , ∈ so that = implies that Տ (Ʈ, ) = Տ (Ʈ, ) guarantees fairness by maintaining the independence of irrelevant alternatives, for instance, if the feasible set decreases and the solution keeps on being feasible, it follows that the solution for the lesser achievable set remains the same point. It can be expressed as ɰ ⊂ Ʈ, (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ (Ʈ, ) = Տ (ɰ, ), ∀ .
Without a loss of generality, the property of the SNO-CRAVNET is described using the following theorem.
Without a loss of generality, the property of the SNO-CRAVNET is described using the following theorem.
The design methodology of the SNO-CRAVNET's objectives with its SNO-CRAVNET game is presented in this section in the form of a convex optimization problem, with its associated players represented by the ℝ CRV-SUs. The design of the game bargaining scheme methodologies for the CRenabled IoV network system is proposed in this section. We assume that each ℝ CRV-SU, for instance, CRV-SU , has an initial utility ≥ 0, which represents its acceptable minimum QoS constraint with respect to the data-rate and the corresponding utility function . Under the symmetric Nash bargaining (SNB) theory, each utility function is usually designated as a convex and closed subset of ℝ = ℧ , with ℝ and ℧ denoting the set of game theory strategies of the ℝ CRV-SU players and utility vectors' space, respectively. Let us assume that is conveniently achievable for all the ℝ CRV-SU players. Then, it follows that at least a feasible subspace ℧ exists in ℧, so that the utility vector, for instance, ( ) = , , , ⋯ , ℝ , becomes equal or bigger in comparison to the initial utility vector, such as, = , , , ⋯ , ℝ . Therefore, the subset ℧ as the element of ℧ can be expressed as ℧ = ∈ ℧| ( ) ≥ . Additionally, let us suppose that the set of utility that can be achieved is denoted by Ʈ = ( )| ∈ ℧ and the category of sets of utility policies that satisfies , which is the minimum utility bound, is denoted as ℬ = Ʈ, |Ʈ ⊂ ℝ . Therefore, in accordance with the Symmetric NBS theory (see [23]), there exists a unique solution, for instance, Տ |ℬ ⟶ ℝ , which satisfies the following axioms: ensures a minimum utility guarantee, for instance, is the Pareto optimal, which implies that other allocations Տ (Ʈ, ) capable of guaranteeing a higher performance for all the ℝ CRV-SUs simultaneously do not exist, that is, guarantees symmetry, which implies that all the ℝ CRV-SUs have equal priorities, for instance, supposing that Ʈ is symmetric with regards to a sub-set ⊆ 1, 2, 3, ⋯ , , ⋯ , ℝ and ∈ Ʈ, , guarantees fairness by maintaining the independence of irrelevant alternatives, for instance, if the feasible set decreases and the solution keeps on being feasible, it follows that the solution for the lesser achievable set remains the same point. It can be expressed as ɰ ⊂ Ʈ, (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ (Ʈ, ) = Տ (ɰ, ), ∀ .
Without a loss of generality, the property of the SNO-CRAVNET is described using the following theorem. nbs with respect to the data-rate and the corresponding utility function . Under the symmetric Nash bargaining (SNB) theory, each utility function is usually designated as a convex and closed subset of ℝ = ℧ , with ℝ and ℧ denoting the set of game theory strategies of the ℝ CRV-SU players and utility vectors' space, respectively. Let us assume that is conveniently achievable for all the ℝ CRV-SU players. Then, it follows that at least a feasible subspace ℧ exists in ℧, so that the utility vector, for instance, ( ) = , , , ⋯ , ℝ , becomes equal or bigger in comparison to the initial utility vector, such as, = , , , ⋯ , ℝ . Therefore, the subset ℧ as the element of ℧ can be expressed as ℧ = ∈ ℧| ( ) ≥ . Additionally, let us suppose that the set of utility that can be achieved is denoted by Ʈ = ( )| ∈ ℧ and the category of sets of utility policies that satisfies , which is the minimum utility bound, is denoted as ℬ = Ʈ, |Ʈ ⊂ ℝ . Therefore, in accordance with the Symmetric NBS theory (see [23]), there exists a unique solution, for instance, Տ |ℬ ⟶ ℝ , which satisfies the following axioms: ensures a minimum utility guarantee, for instance, is the Pareto optimal, which implies that other allocations Տ (Ʈ, ) capable of guaranteeing a higher performance for all the ℝ CRV-SUs simultaneously do not exist, that is, Տ (Ʈ, ) < Տ (Ʈ, ), ∃ and Տ (Ʈ, ) ≤ Տ (Ʈ, ), ∀ ; guarantees symmetry, which implies that all the ℝ CRV-SUs have equal priorities, for instance, supposing that Ʈ is symmetric with regards to a sub-set ⊆ 1, 2, 3, ⋯ , , ⋯ , ℝ and ∈ Ʈ, , ∈ so that = implies that Տ (Ʈ, ) = Տ (Ʈ, ) guarantees fairness by maintaining the independence of irrelevant alternatives, for instance, if the feasible set decreases and the solution keeps on being feasible, it follows that the solution for the lesser achievable set remains the same point. It can be expressed as ɰ ⊂ Ʈ, (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ (Ʈ, ) = Տ (ɰ, ), ∀ .
Without a loss of generality, the property of the SNO-CRAVNET is described using the following theorem.
, U 0 is the Pareto optimal, which implies that other allocations The design methodology of the SNO-CR presented in this section in the form of a co represented by the ℝ CRV-SUs. The design of enabled IoV network system is proposed in th CRV-SU , has an initial utility ≥ 0, wh with respect to the data-rate and the corresp bargaining (SNB) theory, each utility function of ℝ = ℧ , with ℝ and ℧ denoting the s and utility vectors' space, respectively. Let u ℝ CRV-SU players. Then, it follows that at lea vector, for instance, ( ) = , , , ⋯ , ℝ , utility vector, such as, = , , , ⋯ , ℝ expressed as ℧ = ∈ ℧| ( ) ≥ . Addi achieved is denoted by Ʈ = ( )| ∈ ℧ an which is the minimum utility bound, is denot the Symmetric NBS theory (see [23]), there ex satisfies the following axioms: ensures a minimum utility is the Pareto optimal, whic guaranteeing a higher performance for Տ (Ʈ, ) < Տ (Ʈ, ), ∃ and Տ (Ʈ c) Տ (Ʈ, ) guarantees symmetry, whic for instance, supposing that Ʈ is symm and ∈ Ʈ, , guarantees fairness by main instance, if the feasible set decreases and solution for the lesser achievable set re (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ Without a loss of generality, the pro following theorem. nbs with respect to the data-rate and the corresp bargaining (SNB) theory, each utility function of ℝ = ℧ , with ℝ and ℧ denoting the s and utility vectors' space, respectively. Let u ℝ CRV-SU players. Then, it follows that at lea vector, for instance, ( ) = , , , ⋯ , ℝ , utility vector, such as, = , , , ⋯ , ℝ expressed as ℧ = ∈ ℧| ( ) ≥ . Addit achieved is denoted by Ʈ = ( )| ∈ ℧ an which is the minimum utility bound, is denot the Symmetric NBS theory (see [23]), there exi satisfies the following axioms: is the Pareto optimal, which guaranteeing a higher performance for Տ (Ʈ, ) < Տ (Ʈ, ), ∃ and Տ (Ʈ guarantees symmetry, which for instance, supposing that Ʈ is symm and ∈ Ʈ, , ∈ so that = d) Տ (Ʈ, ) guarantees fairness by main instance, if the feasible set decreases and solution for the lesser achievable set re (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ Without a loss of generality, the prop following theorem.
, U 0 capable of guaranteeing a higher performance for all the R CRV-SUs simultaneously do not exist, that is,

The Utility of SNO-CRAVNET and Problem Formulation
The design methodology of the SNO-CRAVNET's objectives with its SNO-CRAVNET game is presented in this section in the form of a convex optimization problem, with its associated players represented by the ℝ CRV-SUs. The design of the game bargaining scheme methodologies for the CRenabled IoV network system is proposed in this section. We assume that each ℝ CRV-SU, for instance, CRV-SU , has an initial utility ≥ 0, which represents its acceptable minimum QoS constraint with respect to the data-rate and the corresponding utility function . Under the symmetric Nash bargaining (SNB) theory, each utility function is usually designated as a convex and closed subset of ℝ = ℧ , with ℝ and ℧ denoting the set of game theory strategies of the ℝ CRV-SU players and utility vectors' space, respectively. Let us assume that is conveniently achievable for all the ℝ CRV-SU players. Then, it follows that at least a feasible subspace ℧ exists in ℧, so that the utility vector, for instance, ( ) = , , , ⋯ , ℝ , becomes equal or bigger in comparison to the initial utility vector, such as, = , , , ⋯ , ℝ . Therefore, the subset ℧ as the element of ℧ can be expressed as ℧ = ∈ ℧| ( ) ≥ . Additionally, let us suppose that the set of utility that can be achieved is denoted by Ʈ = ( )| ∈ ℧ and the category of sets of utility policies that satisfies , which is the minimum utility bound, is denoted as ℬ = Ʈ, |Ʈ ⊂ ℝ . Therefore, in accordance with the Symmetric NBS theory (see [23]), there exists a unique solution, for instance, Տ |ℬ ⟶ ℝ , which satisfies the following axioms: ensures a minimum utility guarantee, for instance, is the Pareto optimal, which implies that other allocations Տ (Ʈ, ) capable of guaranteeing a higher performance for all the ℝ CRV-SUs simultaneously do not exist, that is, guarantees symmetry, which implies that all the ℝ CRV-SUs have equal priorities, for instance, supposing that Ʈ is symmetric with regards to a sub-set ⊆ 1, 2, 3, ⋯ , , ⋯ , ℝ and ∈ Ʈ, , guarantees fairness by maintaining the independence of irrelevant alternatives, for instance, if the feasible set decreases and the solution keeps on being feasible, it follows that the solution for the lesser achievable set remains the same point. It can be expressed as ɰ ⊂ Ʈ, (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ (Ʈ, ) = Տ (ɰ, ), ∀ .
Without a loss of generality, the property of the SNO-CRAVNET is described using the following theorem. nbs enabled IoV network system is proposed in this section. We assume that each ℝ CRV-SU, for instance, CRV-SU , has an initial utility ≥ 0, which represents its acceptable minimum QoS constraint with respect to the data-rate and the corresponding utility function . Under the symmetric Nash bargaining (SNB) theory, each utility function is usually designated as a convex and closed subset of ℝ = ℧ , with ℝ and ℧ denoting the set of game theory strategies of the ℝ CRV-SU players and utility vectors' space, respectively. Let us assume that is conveniently achievable for all the ℝ CRV-SU players. Then, it follows that at least a feasible subspace ℧ exists in ℧, so that the utility vector, for instance, ( ) = , , , ⋯ , ℝ , becomes equal or bigger in comparison to the initial utility vector, such as, = , , , ⋯ , ℝ . Therefore, the subset ℧ as the element of ℧ can be expressed as ℧ = ∈ ℧| ( ) ≥ . Additionally, let us suppose that the set of utility that can be achieved is denoted by Ʈ = ( )| ∈ ℧ and the category of sets of utility policies that satisfies , which is the minimum utility bound, is denoted as ℬ = Ʈ, |Ʈ ⊂ ℝ . Therefore, in accordance with the Symmetric NBS theory (see [23]), there exists a unique solution, for instance, Տ |ℬ ⟶ ℝ , which satisfies the following axioms: ensures a minimum utility guarantee, for instance, is the Pareto optimal, which implies that other allocations Տ (Ʈ, ) capable of guaranteeing a higher performance for all the ℝ CRV-SUs simultaneously do not exist, that is, Տ (Ʈ, ) < Տ (Ʈ, ), ∃ and Տ (Ʈ, ) ≤ Տ (Ʈ, ), ∀ ; guarantees symmetry, which implies that all the ℝ CRV-SUs have equal priorities, for instance, supposing that Ʈ is symmetric with regards to a sub-set ⊆ 1, 2, 3, ⋯ , , ⋯ , ℝ and ∈ Ʈ, , ∈ so that = implies that Տ (Ʈ, ) = Տ (Ʈ, ) guarantees fairness by maintaining the independence of irrelevant alternatives, for instance, if the feasible set decreases and the solution keeps on being feasible, it follows that the solution for the lesser achievable set remains the same point. It can be expressed as ɰ ⊂ Ʈ, (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ (Ʈ, ) = Տ (ɰ, ), ∀ .
Without a loss of generality, the property of the SNO-CRAVNET is described using the following theorem.

The Utility of SNO-CRAVNET and Problem Formulation
The design methodology of the SNO-CRAVNET's objectives with its SNO-CRAVNET game is presented in this section in the form of a convex optimization problem, with its associated players represented by the ℝ CRV-SUs. The design of the game bargaining scheme methodologies for the CRenabled IoV network system is proposed in this section. We assume that each ℝ CRV-SU, for instance, CRV-SU , has an initial utility ≥ 0, which represents its acceptable minimum QoS constraint with respect to the data-rate and the corresponding utility function . Under the symmetric Nash bargaining (SNB) theory, each utility function is usually designated as a convex and closed subset of ℝ = ℧ , with ℝ and ℧ denoting the set of game theory strategies of the ℝ CRV-SU players and utility vectors' space, respectively. Let us assume that is conveniently achievable for all the ℝ CRV-SU players. Then, it follows that at least a feasible subspace ℧ exists in ℧, so that the utility vector, for instance, ( ) = , , , ⋯ , ℝ , becomes equal or bigger in comparison to the initial utility vector, such as, = , , , ⋯ , ℝ . Therefore, the subset ℧ as the element of ℧ can be expressed as ℧ = ∈ ℧| ( ) ≥ . Additionally, let us suppose that the set of utility that can be achieved is denoted by Ʈ = ( )| ∈ ℧ and the category of sets of utility policies that satisfies , which is the minimum utility bound, is denoted as ℬ = Ʈ, |Ʈ ⊂ ℝ . Therefore, in accordance with the Symmetric NBS theory (see [23]), there exists a unique solution, for instance, Տ |ℬ ⟶ ℝ , which satisfies the following axioms: ensures a minimum utility guarantee, for instance, is the Pareto optimal, which implies that other allocations Տ (Ʈ, ) capable of guaranteeing a higher performance for all the ℝ CRV-SUs simultaneously do not exist, that is, guarantees symmetry, which implies that all the ℝ CRV-SUs have equal priorities, for instance, supposing that Ʈ is symmetric with regards to a sub-set ⊆ 1, 2, 3, ⋯ , , ⋯ , ℝ and ∈ Ʈ, , guarantees fairness by maintaining the independence of irrelevant alternatives, for instance, if the feasible set decreases and the solution keeps on being feasible, it follows that the solution for the lesser achievable set remains the same point. It can be expressed as ɰ ⊂ Ʈ, (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ (Ʈ, ) = Տ (ɰ, ), ∀ .
Without a loss of generality, the property of the SNO-CRAVNET is described using the following theorem. nbs enabled IoV network system is proposed in this section. We assume that each ℝ CRV-SU, for instance, CRV-SU , has an initial utility ≥ 0, which represents its acceptable minimum QoS constraint with respect to the data-rate and the corresponding utility function . Under the symmetric Nash bargaining (SNB) theory, each utility function is usually designated as a convex and closed subset of ℝ = ℧ , with ℝ and ℧ denoting the set of game theory strategies of the ℝ CRV-SU players and utility vectors' space, respectively. Let us assume that is conveniently achievable for all the ℝ CRV-SU players. Then, it follows that at least a feasible subspace ℧ exists in ℧, so that the utility vector, for instance, ( ) = , , , ⋯ , ℝ , becomes equal or bigger in comparison to the initial utility vector, such as, = , , , ⋯ , ℝ . Therefore, the subset ℧ as the element of ℧ can be expressed as ℧ = ∈ ℧| ( ) ≥ . Additionally, let us suppose that the set of utility that can be achieved is denoted by Ʈ = ( )| ∈ ℧ and the category of sets of utility policies that satisfies , which is the minimum utility bound, is denoted as ℬ = Ʈ, |Ʈ ⊂ ℝ . Therefore, in accordance with the Symmetric NBS theory (see [23]), there exists a unique solution, for instance, Տ |ℬ ⟶ ℝ , which satisfies the following axioms: ensures a minimum utility guarantee, for instance, is the Pareto optimal, which implies that other allocations Տ (Ʈ, ) capable of guaranteeing a higher performance for all the ℝ CRV-SUs simultaneously do not exist, that is, Տ (Ʈ, ) < Տ (Ʈ, ), ∃ and Տ (Ʈ, ) ≤ Տ (Ʈ, ), ∀ ; guarantees symmetry, which implies that all the ℝ CRV-SUs have equal priorities, for instance, supposing that Ʈ is symmetric with regards to a sub-set ⊆ 1, 2, 3, ⋯ , , ⋯ , ℝ and ∈ Ʈ, , ∈ so that = implies that Տ (Ʈ, ) = Տ (Ʈ, ) guarantees fairness by maintaining the independence of irrelevant alternatives, for instance, if the feasible set decreases and the solution keeps on being feasible, it follows that the solution for the lesser achievable set remains the same point. It can be expressed as ɰ ⊂ Ʈ, (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ (Ʈ, ) = Տ (ɰ, ), ∀ .
Without a loss of generality, the property of the SNO-CRAVNET is described using the following theorem.

The Utility of SNO-CRAVNET and Problem Formulation
The design methodology of the SNO-CRAVNET's objectives with its SNO-CR presented in this section in the form of a convex optimization problem, with its as represented by the ℝ CRV-SUs. The design of the game bargaining scheme methodol enabled IoV network system is proposed in this section. We assume that each ℝ CRV CRV-SU , has an initial utility ≥ 0, which represents its acceptable minimum with respect to the data-rate and the corresponding utility function . Under the bargaining (SNB) theory, each utility function is usually designated as a convex a of ℝ = ℧ , with ℝ and ℧ denoting the set of game theory strategies of the ℝ and utility vectors' space, respectively. Let us assume that is conveniently achi ℝ CRV-SU players. Then, it follows that at least a feasible subspace ℧ exists in ℧, s vector, for instance, ( ) = , , , ⋯ , ℝ , becomes equal or bigger in comparis utility vector, such as, = , , , ⋯ , ℝ . Therefore, the subset ℧ as the elem expressed as ℧ = ∈ ℧| ( ) ≥ . Additionally, let us suppose that the set of u achieved is denoted by Ʈ = ( )| ∈ ℧ and the category of sets of utility policies which is the minimum utility bound, is denoted as ℬ = Ʈ, |Ʈ ⊂ ℝ . Therefore, in the Symmetric NBS theory (see [23]), there exists a unique solution, for instance, Տ satisfies the following axioms: ensures a minimum utility guarantee, for instance, Տ (Ʈ, ) is the Pareto optimal, which implies that other allocations Տ (Ʈ guaranteeing a higher performance for all the ℝ CRV-SUs simultaneously do Տ (Ʈ, ) < Տ (Ʈ, ), ∃ and Տ (Ʈ, ) ≤ Տ (Ʈ, ), ∀ ; c) Տ (Ʈ, ) guarantees symmetry, which implies that all the ℝ CRV-SUs have for instance, supposing that Ʈ is symmetric with regards to a sub-set ⊆ 1 and ∈ Ʈ, , guarantees fairness by maintaining the independence of irrelevant instance, if the feasible set decreases and the solution keeps on being feasible, i solution for the lesser achievable set remains the same point. It can be expr (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ (Ʈ, ) = Տ (ɰ, ), ∀ .
Without a loss of generality, the property of the SNO-CRAVNET is desc following theorem. nbs enabled IoV network system is proposed in this section. We assume that each ℝ CRV-CRV-SU , has an initial utility ≥ 0, which represents its acceptable minimum with respect to the data-rate and the corresponding utility function . Under the s bargaining (SNB) theory, each utility function is usually designated as a convex a of ℝ = ℧ , with ℝ and ℧ denoting the set of game theory strategies of the ℝ C and utility vectors' space, respectively. Let us assume that is conveniently achie ℝ CRV-SU players. Then, it follows that at least a feasible subspace ℧ exists in ℧, s vector, for instance, ( ) = , , , ⋯ , ℝ , becomes equal or bigger in comparis utility vector, such as, = , , , ⋯ , ℝ . Therefore, the subset ℧ as the elem expressed as ℧ = ∈ ℧| ( ) ≥ . Additionally, let us suppose that the set of u achieved is denoted by Ʈ = ( )| ∈ ℧ and the category of sets of utility policies which is the minimum utility bound, is denoted as ℬ = Ʈ, |Ʈ ⊂ ℝ . Therefore, in the Symmetric NBS theory (see [23]), there exists a unique solution, for instance, Տ satisfies the following axioms: ensures a minimum utility guarantee, for instance, Տ (Ʈ, ) ∈ ∈ Ʈ| ≥ is the Pareto optimal, which implies that other allocations Տ (Ʈ guaranteeing a higher performance for all the ℝ CRV-SUs simultaneously do n Տ (Ʈ, ) < Տ (Ʈ, ), ∃ and Տ (Ʈ, ) ≤ Տ (Ʈ, ), ∀ ; guarantees symmetry, which implies that all the ℝ CRV-SUs have for instance, supposing that Ʈ is symmetric with regards to a sub-set ⊆ 1 and ∈ Ʈ, , ∈ so that = implies that Տ (Ʈ, ) = Տ (Ʈ, d) Տ (Ʈ, ) guarantees fairness by maintaining the independence of irrelevant instance, if the feasible set decreases and the solution keeps on being feasible, it solution for the lesser achievable set remains the same point. It can be expr (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ (Ʈ, ) = Տ (ɰ, ), ∀ .
Without a loss of generality, the property of the SNO-CRAVNET is desc following theorem.

The Utility of SNO-CRAVNET and Problem Formulation
The design methodology of the SNO-CRAVNET's objectives wit presented in this section in the form of a convex optimization proble represented by the ℝ CRV-SUs. The design of the game bargaining sche enabled IoV network system is proposed in this section. We assume tha CRV-SU , has an initial utility ≥ 0, which represents its accepta with respect to the data-rate and the corresponding utility function bargaining (SNB) theory, each utility function is usually designated of ℝ = ℧ , with ℝ and ℧ denoting the set of game theory strate and utility vectors' space, respectively. Let us assume that is conv ℝ CRV-SU players. Then, it follows that at least a feasible subspace ℧ vector, for instance, ( ) = , , , ⋯ , ℝ , becomes equal or bigge utility vector, such as, = , , , ⋯ , ℝ . Therefore, the subset ℧ expressed as ℧ = ∈ ℧| ( ) ≥ . Additionally, let us suppose th achieved is denoted by Ʈ = ( )| ∈ ℧ and the category of sets of u which is the minimum utility bound, is denoted as ℬ = Ʈ, |Ʈ ⊂ ℝ the Symmetric NBS theory (see [23]), there exists a unique solution, for satisfies the following axioms: ensures a minimum utility guarantee, for instance, is the Pareto optimal, which implies that other allo guaranteeing a higher performance for all the ℝ CRV-SUs simul Տ (Ʈ, ) < Տ (Ʈ, ), ∃ and Տ (Ʈ, ) ≤ Տ (Ʈ, ), ∀ ; c) Տ (Ʈ, ) guarantees symmetry, which implies that all the ℝ C for instance, supposing that Ʈ is symmetric with regards to a s and ∈ Ʈ, , guarantees fairness by maintaining the independenc instance, if the feasible set decreases and the solution keeps on be solution for the lesser achievable set remains the same point. I (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ (Ʈ, ) = Տ (ɰ, ), Without a loss of generality, the property of the SNO-CRAV following theorem. nbs enabled IoV network system is proposed in this section. We assume tha CRV-SU , has an initial utility ≥ 0, which represents its accepta with respect to the data-rate and the corresponding utility function bargaining (SNB) theory, each utility function is usually designated of ℝ = ℧ , with ℝ and ℧ denoting the set of game theory strateg and utility vectors' space, respectively. Let us assume that is conv ℝ CRV-SU players. Then, it follows that at least a feasible subspace ℧ vector, for instance, ( ) = , , , ⋯ , ℝ , becomes equal or bigge utility vector, such as, = , , , ⋯ , ℝ . Therefore, the subset ℧ expressed as ℧ = ∈ ℧| ( ) ≥ . Additionally, let us suppose th achieved is denoted by Ʈ = ( )| ∈ ℧ and the category of sets of u which is the minimum utility bound, is denoted as ℬ = Ʈ, |Ʈ ⊂ ℝ . the Symmetric NBS theory (see [23]), there exists a unique solution, for satisfies the following axioms: ensures a minimum utility guarantee, for instance, is the Pareto optimal, which implies that other alloc guaranteeing a higher performance for all the ℝ CRV-SUs simult Տ (Ʈ, ) < Տ (Ʈ, ), ∃ and Տ (Ʈ, ) ≤ Տ (Ʈ, ), ∀ ; guarantees symmetry, which implies that all the ℝ C for instance, supposing that Ʈ is symmetric with regards to a s and ∈ Ʈ, , ∈ so that = implies that Տ (Ʈ, ) d) Տ (Ʈ, ) guarantees fairness by maintaining the independence instance, if the feasible set decreases and the solution keeps on be solution for the lesser achievable set remains the same point. I (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ (Ʈ, ) = Տ (ɰ, ), Without a loss of generality, the property of the SNO-CRAV following theorem.

The Utility of SNO-CRAVNET and Problem Formulation
The design methodology of the SNO-CRAVNET's objectives with its SNO-CRAVNET game is presented in this section in the form of a convex optimization problem, with its associated players represented by the ℝ CRV-SUs. The design of the game bargaining scheme methodologies for the CRenabled IoV network system is proposed in this section. We assume that each ℝ CRV-SU, for instance, CRV-SU , has an initial utility ≥ 0, which represents its acceptable minimum QoS constraint with respect to the data-rate and the corresponding utility function . Under the symmetric Nash bargaining (SNB) theory, each utility function is usually designated as a convex and closed subset of ℝ = ℧ , with ℝ and ℧ denoting the set of game theory strategies of the ℝ CRV-SU players and utility vectors' space, respectively. Let us assume that is conveniently achievable for all the ℝ CRV-SU players. Then, it follows that at least a feasible subspace ℧ exists in ℧, so that the utility vector, for instance, ( ) = , , , ⋯ , ℝ , becomes equal or bigger in comparison to the initial utility vector, such as, = , , , ⋯ , ℝ . Therefore, the subset ℧ as the element of ℧ can be expressed as ℧ = ∈ ℧| ( ) ≥ . Additionally, let us suppose that the set of utility that can be achieved is denoted by Ʈ = ( )| ∈ ℧ and the category of sets of utility policies that satisfies , which is the minimum utility bound, is denoted as ℬ = Ʈ, |Ʈ ⊂ ℝ . Therefore, in accordance with the Symmetric NBS theory (see [23]), there exists a unique solution, for instance, Տ |ℬ ⟶ ℝ , which satisfies the following axioms: ensures a minimum utility guarantee, for instance, is the Pareto optimal, which implies that other allocations Տ (Ʈ, ) capable of guaranteeing a higher performance for all the ℝ CRV-SUs simultaneously do not exist, that is, guarantees symmetry, which implies that all the ℝ CRV-SUs have equal priorities, for instance, supposing that Ʈ is symmetric with regards to a sub-set ⊆ 1, 2, 3, ⋯ , , ⋯ , ℝ and ∈ Ʈ, , guarantees fairness by maintaining the independence of irrelevant alternatives, for instance, if the feasible set decreases and the solution keeps on being feasible, it follows that the solution for the lesser achievable set remains the same point. It can be expressed as ɰ ⊂ Ʈ, (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ (Ʈ, ) = Տ (ɰ, ), ∀ .
Without a loss of generality, the property of the SNO-CRAVNET is described using the following theorem. nbs represented by the ℝ CRV-SUs. The design of the game bargaining scheme methodologies for the CRenabled IoV network system is proposed in this section. We assume that each ℝ CRV-SU, for instance, CRV-SU , has an initial utility ≥ 0, which represents its acceptable minimum QoS constraint with respect to the data-rate and the corresponding utility function . Under the symmetric Nash bargaining (SNB) theory, each utility function is usually designated as a convex and closed subset of ℝ = ℧ , with ℝ and ℧ denoting the set of game theory strategies of the ℝ CRV-SU players and utility vectors' space, respectively. Let us assume that is conveniently achievable for all the ℝ CRV-SU players. Then, it follows that at least a feasible subspace ℧ exists in ℧, so that the utility vector, for instance, ( ) = , , , ⋯ , ℝ , becomes equal or bigger in comparison to the initial utility vector, such as, = , , , ⋯ , ℝ . Therefore, the subset ℧ as the element of ℧ can be expressed as ℧ = ∈ ℧| ( ) ≥ . Additionally, let us suppose that the set of utility that can be achieved is denoted by Ʈ = ( )| ∈ ℧ and the category of sets of utility policies that satisfies , which is the minimum utility bound, is denoted as ℬ = Ʈ, |Ʈ ⊂ ℝ . Therefore, in accordance with the Symmetric NBS theory (see [23]), there exists a unique solution, for instance, Տ |ℬ ⟶ ℝ , which satisfies the following axioms: ensures a minimum utility guarantee, for instance, is the Pareto optimal, which implies that other allocations Տ (Ʈ, ) capable of guaranteeing a higher performance for all the ℝ CRV-SUs simultaneously do not exist, that is, Տ (Ʈ, ) < Տ (Ʈ, ), ∃ and Տ (Ʈ, ) ≤ Տ (Ʈ, ), ∀ ; guarantees symmetry, which implies that all the ℝ CRV-SUs have equal priorities, for instance, supposing that Ʈ is symmetric with regards to a sub-set ⊆ 1, 2, 3, ⋯ , , ⋯ , ℝ and ∈ Ʈ, , ∈ so that = implies that Տ (Ʈ, ) = Տ (Ʈ, ) guarantees fairness by maintaining the independence of irrelevant alternatives, for instance, if the feasible set decreases and the solution keeps on being feasible, it follows that the solution for the lesser achievable set remains the same point. It can be expressed as ɰ ⊂ Ʈ, (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ (Ʈ, ) = Տ (ɰ, ), ∀ .
Without a loss of generality, the property of the SNO-CRAVNET is described using the following theorem.
, U 0 guarantees symmetry, which implies that all the R CRV-SUs have equal priorities, for instance, supposing that presented in this section in the form of a convex optimization problem, with its associated pl represented by the ℝ CRV-SUs. The design of the game bargaining scheme methodologies for th enabled IoV network system is proposed in this section. We assume that each ℝ CRV-SU, for inst CRV-SU , has an initial utility ≥ 0, which represents its acceptable minimum QoS cons with respect to the data-rate and the corresponding utility function . Under the symmetric bargaining (SNB) theory, each utility function is usually designated as a convex and closed s of ℝ = ℧ , with ℝ and ℧ denoting the set of game theory strategies of the ℝ CRV-SU pl and utility vectors' space, respectively. Let us assume that is conveniently achievable for a ℝ CRV-SU players. Then, it follows that at least a feasible subspace ℧ exists in ℧, so that the u vector, for instance, ( ) = , , , ⋯ , ℝ , becomes equal or bigger in comparison to the i utility vector, such as, = , , , ⋯ , ℝ . Therefore, the subset ℧ as the element of ℧ c expressed as ℧ = ∈ ℧| ( ) ≥ . Additionally, let us suppose that the set of utility that c achieved is denoted by Ʈ = ( )| ∈ ℧ and the category of sets of utility policies that satisfie which is the minimum utility bound, is denoted as ℬ = Ʈ, |Ʈ ⊂ ℝ . Therefore, in accordance the Symmetric NBS theory (see [23]), there exists a unique solution, for instance, Տ |ℬ ⟶ ℝ , w satisfies the following axioms: ensures a minimum utility guarantee, for instance, is the Pareto optimal, which implies that other allocations Տ (Ʈ, ) capab guaranteeing a higher performance for all the ℝ CRV-SUs simultaneously do not exist, th guarantees symmetry, which implies that all the ℝ CRV-SUs have equal prio for instance, supposing that Ʈ is symmetric with regards to a sub-set ⊆ 1, 2, 3, ⋯ , , ⋯ and ∈ Ʈ, , guarantees fairness by maintaining the independence of irrelevant alternative instance, if the feasible set decreases and the solution keeps on being feasible, it follows th solution for the lesser achievable set remains the same point. It can be expressed as ɰ (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ (Ʈ, ) = Տ (ɰ, ), ∀ .
Without a loss of generality, the property of the SNO-CRAVNET is described usin following theorem.
is symmetric with regards to a sub-set Q ⊆ {1, 2, 3, · · · , n, · · · , R} and U ∈ The design methodology of the SNO-CRAVNET's objectives with its SNO-CRAVNET game is presented in this section in the form of a convex optimization problem, with its associated players represented by the ℝ CRV-SUs. The design of the game bargaining scheme methodologies for the CRenabled IoV network system is proposed in this section. We assume that each ℝ CRV-SU, for instance, CRV-SU , has an initial utility ≥ 0, which represents its acceptable minimum QoS constraint with respect to the data-rate and the corresponding utility function . Under the symmetric Nash bargaining (SNB) theory, each utility function is usually designated as a convex and closed subset of ℝ = ℧ , with ℝ and ℧ denoting the set of game theory strategies of the ℝ CRV-SU players and utility vectors' space, respectively. Let us assume that is conveniently achievable for all the ℝ CRV-SU players. Then, it follows that at least a feasible subspace ℧ exists in ℧, so that the utility vector, for instance, ( ) = , , , ⋯ , ℝ , becomes equal or bigger in comparison to the initial utility vector, such as, = , , , ⋯ , ℝ . Therefore, the subset ℧ as the element of ℧ can be expressed as ℧ = ∈ ℧| ( ) ≥ . Additionally, let us suppose that the set of utility that can be achieved is denoted by Ʈ = ( )| ∈ ℧ and the category of sets of utility policies that satisfies , which is the minimum utility bound, is denoted as ℬ = Ʈ, |Ʈ ⊂ ℝ . Therefore, in accordance with the Symmetric NBS theory (see [23]), there exists a unique solution, for instance, Տ |ℬ ⟶ ℝ , which satisfies the following axioms: ensures a minimum utility guarantee, for instance, is the Pareto optimal, which implies that other allocations Տ (Ʈ, ) capable of guaranteeing a higher performance for all the ℝ CRV-SUs simultaneously do not exist, that is, guarantees symmetry, which implies that all the ℝ CRV-SUs have equal priorities, for instance, supposing that Ʈ is symmetric with regards to a sub-set ⊆ 1, 2, 3, ⋯ , , ⋯ , ℝ and ∈ Ʈ, , guarantees fairness by maintaining the independence of irrelevant alternatives, for instance, if the feasible set decreases and the solution keeps on being feasible, it follows that the solution for the lesser achievable set remains the same point. It can be expressed as ɰ ⊂ Ʈ, (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ (Ʈ, ) = Տ (ɰ, ), ∀ .
Without a loss of generality, the property of the SNO-CRAVNET is described using the following theorem.
, n, n ∈ Q so that U 0 n = U 0 n implies that stipulated condition guarantees that the potential transmissions of only if the CRV-SU 's total transmit power is constrained ov threshold Ᵽ .

The Utility of SNO-CRAVNET and Problem Formulation
The design methodology of the SNO-CRAVNET's objectives presented in this section in the form of a convex optimization pro represented by the ℝ CRV-SUs. The design of the game bargaining s enabled IoV network system is proposed in this section. We assume CRV-SU , has an initial utility ≥ 0, which represents its acce with respect to the data-rate and the corresponding utility functio bargaining (SNB) theory, each utility function is usually designa of ℝ = ℧ , with ℝ and ℧ denoting the set of game theory str and utility vectors' space, respectively. Let us assume that is c ℝ CRV-SU players. Then, it follows that at least a feasible subspace vector, for instance, ( ) = , , , ⋯ , ℝ , becomes equal or bi utility vector, such as, = , , , ⋯ , ℝ . Therefore, the subs expressed as ℧ = ∈ ℧| ( ) ≥ . Additionally, let us suppos achieved is denoted by Ʈ = ( )| ∈ ℧ and the category of sets which is the minimum utility bound, is denoted as ℬ = Ʈ, |Ʈ ⊂ the Symmetric NBS theory (see [23]), there exists a unique solution, satisfies the following axioms: ensures a minimum utility guarantee, for instan is the Pareto optimal, which implies that other a guaranteeing a higher performance for all the ℝ CRV-SUs sim guarantees symmetry, which implies that all the ℝ for instance, supposing that Ʈ is symmetric with regards to and ∈ Ʈ, , guarantees fairness by maintaining the independe instance, if the feasible set decreases and the solution keeps on solution for the lesser achievable set remains the same poin (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ (Ʈ, ) = Տ (ɰ, Without a loss of generality, the property of the SNO-CR following theorem.

nbs
The design methodology of the SNO-CRAVNET's objectives w presented in this section in the form of a convex optimization pro represented by the ℝ CRV-SUs. The design of the game bargaining s enabled IoV network system is proposed in this section. We assume CRV-SU , has an initial utility ≥ 0, which represents its acce with respect to the data-rate and the corresponding utility functio bargaining (SNB) theory, each utility function is usually designa of ℝ = ℧ , with ℝ and ℧ denoting the set of game theory str and utility vectors' space, respectively. Let us assume that is co ℝ CRV-SU players. Then, it follows that at least a feasible subspace vector, for instance, ( ) = , , , ⋯ , ℝ , becomes equal or big utility vector, such as, = , , , ⋯ , ℝ . Therefore, the subse expressed as ℧ = ∈ ℧| ( ) ≥ . Additionally, let us suppose achieved is denoted by Ʈ = ( )| ∈ ℧ and the category of sets o which is the minimum utility bound, is denoted as ℬ = Ʈ, |Ʈ ⊂ the Symmetric NBS theory (see [23]), there exists a unique solution, f satisfies the following axioms: ensures a minimum utility guarantee, for instanc is the Pareto optimal, which implies that other a guaranteeing a higher performance for all the ℝ CRV-SUs sim guarantees symmetry, which implies that all the ℝ for instance, supposing that Ʈ is symmetric with regards to and ∈ Ʈ, , guarantees fairness by maintaining the independe instance, if the feasible set decreases and the solution keeps on solution for the lesser achievable set remains the same poin (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ (Ʈ, ) = Տ (ɰ, Without a loss of generality, the property of the SNO-CR following theorem.
, U 0 n = stipulated condition guarantees that the potential tr only if the CRV-SU 's total transmit power is threshold Ᵽ .

The Utility of SNO-CRAVNET and Problem Fo
The design methodology of the SNO-CRAVNE presented in this section in the form of a convex o represented by the ℝ CRV-SUs. The design of the gam enabled IoV network system is proposed in this secti CRV-SU , has an initial utility ≥ 0, which rep with respect to the data-rate and the corresponding bargaining (SNB) theory, each utility function is of ℝ = ℧ , with ℝ and ℧ denoting the set of g and utility vectors' space, respectively. Let us assum ℝ CRV-SU players. Then, it follows that at least a fea vector, for instance, ( ) = , , , ⋯ , ℝ , becom utility vector, such as, = , , , ⋯ , ℝ . Ther expressed as ℧ = ∈ ℧| ( ) ≥ . Additionally achieved is denoted by Ʈ = ( )| ∈ ℧ and the c which is the minimum utility bound, is denoted as ℬ the Symmetric NBS theory (see [23]), there exists a un satisfies the following axioms: is the Pareto optimal, which impl guaranteeing a higher performance for all the guarantees symmetry, which impl for instance, supposing that Ʈ is symmetric w and ∈ Ʈ, , guarantees fairness by maintaining instance, if the feasible set decreases and the so solution for the lesser achievable set remains (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ (Ʈ, Without a loss of generality, the property o following theorem.

nbs
The design methodology of the SNO-CRAVNE presented in this section in the form of a convex op represented by the ℝ CRV-SUs. The design of the gam enabled IoV network system is proposed in this secti CRV-SU , has an initial utility ≥ 0, which rep with respect to the data-rate and the corresponding bargaining (SNB) theory, each utility function is u of ℝ = ℧ , with ℝ and ℧ denoting the set of g and utility vectors' space, respectively. Let us assum ℝ CRV-SU players. Then, it follows that at least a fea vector, for instance, ( ) = , , , ⋯ , ℝ , becom utility vector, such as, = , , , ⋯ , ℝ . Ther expressed as ℧ = ∈ ℧| ( ) ≥ . Additionally achieved is denoted by Ʈ = ( )| ∈ ℧ and the c which is the minimum utility bound, is denoted as ℬ the Symmetric NBS theory (see [23]), there exists a un satisfies the following axioms: ensures a minimum utility guaran is the Pareto optimal, which impli guaranteeing a higher performance for all the Տ (Ʈ, ) < Տ (Ʈ, ), ∃ and Տ (Ʈ, ) ≤ c) Տ (Ʈ, ) guarantees symmetry, which impli for instance, supposing that Ʈ is symmetric w and ∈ Ʈ, , guarantees fairness by maintaining instance, if the feasible set decreases and the so solution for the lesser achievable set remains (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ (Ʈ, Without a loss of generality, the property o following theorem. . From Equation (19), the stipulated condition guarantees that the potential transmissions of the PU are fully protected if and only if the CRV-SU 's total transmit power is constrained over channel by the predefined threshold Ᵽ .

The Utility of SNO-CRAVNET and Problem Formulation
The design methodology of the SNO-CRAVNET's objectives with its SNO-CRAVNET game is presented in this section in the form of a convex optimization problem, with its associated players represented by the ℝ CRV-SUs. The design of the game bargaining scheme methodologies for the CRenabled IoV network system is proposed in this section. We assume that each ℝ CRV-SU, for instance, CRV-SU , has an initial utility ≥ 0, which represents its acceptable minimum QoS constraint with respect to the data-rate and the corresponding utility function . Under the symmetric Nash bargaining (SNB) theory, each utility function is usually designated as a convex and closed subset of ℝ = ℧ , with ℝ and ℧ denoting the set of game theory strategies of the ℝ CRV-SU players and utility vectors' space, respectively. Let us assume that is conveniently achievable for all the ℝ CRV-SU players. Then, it follows that at least a feasible subspace ℧ exists in ℧, so that the utility vector, for instance, ( ) = , , , ⋯ , ℝ , becomes equal or bigger in comparison to the initial utility vector, such as, = , , , ⋯ , ℝ . Therefore, the subset ℧ as the element of ℧ can be expressed as ℧ = ∈ ℧| ( ) ≥ . Additionally, let us suppose that the set of utility that can be achieved is denoted by Ʈ = ( )| ∈ ℧ and the category of sets of utility policies that satisfies , which is the minimum utility bound, is denoted as ℬ = Ʈ, |Ʈ ⊂ ℝ . Therefore, in accordance with the Symmetric NBS theory (see [23]), there exists a unique solution, for instance, Տ |ℬ ⟶ ℝ , which satisfies the following axioms: ensures a minimum utility guarantee, for instance, is the Pareto optimal, which implies that other allocations Տ (Ʈ, ) capable of guaranteeing a higher performance for all the ℝ CRV-SUs simultaneously do not exist, that is, guarantees symmetry, which implies that all the ℝ CRV-SUs have equal priorities, for instance, supposing that Ʈ is symmetric with regards to a sub-set ⊆ 1, 2, 3, ⋯ , , ⋯ , ℝ and ∈ Ʈ, , guarantees fairness by maintaining the independence of irrelevant alternatives, for instance, if the feasible set decreases and the solution keeps on being feasible, it follows that the solution for the lesser achievable set remains the same point. It can be expressed as ɰ ⊂ Ʈ, (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ (Ʈ, ) = Տ (ɰ, ), ∀ .
Without a loss of generality, the property of the SNO-CRAVNET is described using the following theorem.

The Utility of SNO-CRAVNET and Problem Formulation
The design methodology of the SNO-CRAVNET's objectives with its SNO-CRAVNET game is presented in this section in the form of a convex optimization problem, with its associated players represented by the ℝ CRV-SUs. The design of the game bargaining scheme methodologies for the CRenabled IoV network system is proposed in this section. We assume that each ℝ CRV-SU, for instance, CRV-SU , has an initial utility ≥ 0, which represents its acceptable minimum QoS constraint with respect to the data-rate and the corresponding utility function . Under the symmetric Nash bargaining (SNB) theory, each utility function is usually designated as a convex and closed subset of ℝ = ℧ , with ℝ and ℧ denoting the set of game theory strategies of the ℝ CRV-SU players and utility vectors' space, respectively. Let us assume that is conveniently achievable for all the ℝ CRV-SU players. Then, it follows that at least a feasible subspace ℧ exists in ℧, so that the utility vector, for instance, ( ) = , , , ⋯ , ℝ , becomes equal or bigger in comparison to the initial utility vector, such as, = , , , ⋯ , ℝ . Therefore, the subset ℧ as the element of ℧ can be expressed as ℧ = ∈ ℧| ( ) ≥ . Additionally, let us suppose that the set of utility that can be achieved is denoted by Ʈ = ( )| ∈ ℧ and the category of sets of utility policies that satisfies , which is the minimum utility bound, is denoted as ℬ = Ʈ, |Ʈ ⊂ ℝ . Therefore, in accordance with the Symmetric NBS theory (see [23]), there exists a unique solution, for instance, Տ |ℬ ⟶ ℝ , which satisfies the following axioms: ensures a minimum utility guarantee, for instance, is the Pareto optimal, which implies that other allocations Տ (Ʈ, ) capable of guaranteeing a higher performance for all the ℝ CRV-SUs simultaneously do not exist, that is, guarantees symmetry, which implies that all the ℝ CRV-SUs have equal priorities, for instance, supposing that Ʈ is symmetric with regards to a sub-set ⊆ 1, 2, 3, ⋯ , , ⋯ , ℝ and ∈ Ʈ, , guarantees fairness by maintaining the independence of irrelevant alternatives, for instance, if the feasible set decreases and the solution keeps on being feasible, it follows that the solution for the lesser achievable set remains the same point. It can be expressed as ɰ ⊂ Ʈ, (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ (Ʈ, ) = Տ (ɰ, ), ∀ .
Without a loss of generality, the property of the SNO-CRAVNET is described using the following theorem.
, U 0 guarantees fairness by maintaining the independence of irrelevant alternatives, for instance, if the feasible set decreases and the solution keeps on being feasible, it follows that the solution for the lesser achievable set remains the same point. It can be expressed as represented by the ℝ CRV-SUs. The design of the game bargaining scheme methodologi enabled IoV network system is proposed in this section. We assume that each ℝ CRV-SU CRV-SU , has an initial utility ≥ 0, which represents its acceptable minimum Q with respect to the data-rate and the corresponding utility function . Under the sym bargaining (SNB) theory, each utility function is usually designated as a convex and of ℝ = ℧ , with ℝ and ℧ denoting the set of game theory strategies of the ℝ CRV and utility vectors' space, respectively. Let us assume that is conveniently achieva ℝ CRV-SU players. Then, it follows that at least a feasible subspace ℧ exists in ℧, so th vector, for instance, ( ) = , , , ⋯ , ℝ , becomes equal or bigger in comparison utility vector, such as, = , , , ⋯ , ℝ . Therefore, the subset ℧ as the element expressed as ℧ = ∈ ℧| ( ) ≥ . Additionally, let us suppose that the set of utilit achieved is denoted by Ʈ = ( )| ∈ ℧ and the category of sets of utility policies tha which is the minimum utility bound, is denoted as ℬ = Ʈ, |Ʈ ⊂ ℝ . Therefore, in acc the Symmetric NBS theory (see [23]), there exists a unique solution, for instance, Տ |ℬ satisfies the following axioms: a) Տ (Ʈ, ) ensures a minimum utility guarantee, for instance, Տ (Ʈ, ) ∈ Ʈ , ∈ Ʈ| ≥ , ∀ ; b) Տ (Ʈ, ) is the Pareto optimal, which implies that other allocations Տ (Ʈ, guaranteeing a higher performance for all the ℝ CRV-SUs simultaneously do not Տ (Ʈ, ) < Տ (Ʈ, ), ∃ and Տ (Ʈ, ) ≤ Տ (Ʈ, ), ∀ ; c) Տ (Ʈ, ) guarantees symmetry, which implies that all the ℝ CRV-SUs have equ for instance, supposing that Ʈ is symmetric with regards to a sub-set ⊆ 1, 2, and ∈ Ʈ, ,

The Utility of SNO-C
The design method presented in this section represented by the ℝ CR enabled IoV network sys CRV-SU , has an initia with respect to the data bargaining (SNB) theory of ℝ = ℧ , with ℝ a and utility vectors' spac ℝ CRV-SU players. Then vector, for instance, ( utility vector, such as, expressed as ℧ = ∈ achieved is denoted by which is the minimum u the Symmetric NBS theo satisfies the following ax guaran for instance, suppo and ∈ Ʈ, , ∈ d) Տ (Ʈ, ) guaran instance, if the feasi solution for the les (ɰ, ) ∈ ℬ and Տ Without a loss of following theorem. , a convex optimization problem, with its associated players n of the game bargaining scheme methodologies for the CR-in this section. We assume that each ℝ CRV-SU, for instance, , which represents its acceptable minimum QoS constraint rresponding utility function . Under the symmetric Nash ction is usually designated as a convex and closed subset the set of game theory strategies of the ℝ CRV-SU players et us assume that is conveniently achievable for all the at least a feasible subspace ℧ exists in ℧, so that the utility , ℝ , becomes equal or bigger in comparison to the initial , ℝ . Therefore, the subset ℧ as the element of ℧ can be dditionally, let us suppose that the set of utility that can be and the category of sets of utility policies that satisfies , enoted as ℬ = Ʈ, |Ʈ ⊂ ℝ . Therefore, in accordance with re exists a unique solution, for instance, Տ |ℬ ⟶ ℝ , which tility guarantee, for instance, Տ (Ʈ, ) ∈ Ʈ , where Ʈ = which implies that other allocations Տ (Ʈ, ) capable of for all the ℝ CRV-SUs simultaneously do not exist, that is, which implies that all the ℝ CRV-SUs have equal priorities, ymmetric with regards to a sub-set ⊆ 1, 2, 3, ⋯ , , ⋯ , ℝ implies that Տ (Ʈ, ) = Տ (Ʈ, ) , ≠ ; maintaining the independence of irrelevant alternatives, for s and the solution keeps on being feasible, it follows that the et remains the same point. It can be expressed as ɰ ⊂ Ʈ, en Տ (Ʈ, ) = Տ (ɰ, ), ∀ . property of the SNO-CRAVNET is described using the , U 0 ∈ B and and expressed as where . From Equation (19), the stipulated condition guarantees that the potential transmissions of the PU are fully protected if and only if the CRV-SU 's total transmit power is constrained over channel by the predefined threshold Ᵽ .

The Utility of SNO-CRAVNET and Problem Formulation
The design methodology of the SNO-CRAVNET's objectives with its SNO-CRAVNET game is presented in this section in the form of a convex optimization problem, with its associated players represented by the ℝ CRV-SUs. The design of the game bargaining scheme methodologies for the CRenabled IoV network system is proposed in this section. We assume that each ℝ CRV-SU, for instance, CRV-SU , has an initial utility ≥ 0, which represents its acceptable minimum QoS constraint with respect to the data-rate and the corresponding utility function . Under the symmetric Nash bargaining (SNB) theory, each utility function is usually designated as a convex and closed subset of ℝ = ℧ , with ℝ and ℧ denoting the set of game theory strategies of the ℝ CRV-SU players and utility vectors' space, respectively. Let us assume that is conveniently achievable for all the ℝ CRV-SU players. Then, it follows that at least a feasible subspace ℧ exists in ℧, so that the utility vector, for instance, ( ) = , , , ⋯ , ℝ , becomes equal or bigger in comparison to the initial utility vector, such as, = , , , ⋯ , ℝ . Therefore, the subset ℧ as the element of ℧ can be expressed as ℧ = ∈ ℧| ( ) ≥ . Additionally, let us suppose that the set of utility that can be achieved is denoted by Ʈ = ( )| ∈ ℧ and the category of sets of utility policies that satisfies , which is the minimum utility bound, is denoted as ℬ = Ʈ, |Ʈ ⊂ ℝ . Therefore, in accordance with the Symmetric NBS theory (see [23]), there exists a unique solution, for instance, Տ |ℬ ⟶ ℝ , which satisfies the following axioms: ensures a minimum utility guarantee, for instance, is the Pareto optimal, which implies that other allocations Տ (Ʈ, ) capable of guaranteeing a higher performance for all the ℝ CRV-SUs simultaneously do not exist, that is, guarantees symmetry, which implies that all the ℝ CRV-SUs have equal priorities, for instance, supposing that Ʈ is symmetric with regards to a sub-set ⊆ 1, 2, 3, ⋯ , , ⋯ , ℝ and ∈ Ʈ, , guarantees fairness by maintaining the independence of irrelevant alternatives, for instance, if the feasible set decreases and the solution keeps on being feasible, it follows that the solution for the lesser achievable set remains the same point. It can be expressed as ɰ ⊂ Ʈ, (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ (Ʈ, ) = Տ (ɰ, ), ∀ .
Without a loss of generality, the property of the SNO-CRAVNET is described using the following theorem. nbs only if the CRV-SU 's total transmit power is constrained over channel by the predefined threshold Ᵽ .

The Utility of SNO-CRAVNET and Problem Formulation
The design methodology of the SNO-CRAVNET's objectives with its SNO-CRAVNET game is presented in this section in the form of a convex optimization problem, with its associated players represented by the ℝ CRV-SUs. The design of the game bargaining scheme methodologies for the CRenabled IoV network system is proposed in this section. We assume that each ℝ CRV-SU, for instance, CRV-SU , has an initial utility ≥ 0, which represents its acceptable minimum QoS constraint with respect to the data-rate and the corresponding utility function . Under the symmetric Nash bargaining (SNB) theory, each utility function is usually designated as a convex and closed subset of ℝ = ℧ , with ℝ and ℧ denoting the set of game theory strategies of the ℝ CRV-SU players and utility vectors' space, respectively. Let us assume that is conveniently achievable for all the ℝ CRV-SU players. Then, it follows that at least a feasible subspace ℧ exists in ℧, so that the utility vector, for instance, ( ) = , , , ⋯ , ℝ , becomes equal or bigger in comparison to the initial utility vector, such as, = , , , ⋯ , ℝ . Therefore, the subset ℧ as the element of ℧ can be expressed as ℧ = ∈ ℧| ( ) ≥ . Additionally, let us suppose that the set of utility that can be achieved is denoted by Ʈ = ( )| ∈ ℧ and the category of sets of utility policies that satisfies , which is the minimum utility bound, is denoted as ℬ = Ʈ, |Ʈ ⊂ ℝ . Therefore, in accordance with the Symmetric NBS theory (see [23]), there exists a unique solution, for instance, Տ |ℬ ⟶ ℝ , which satisfies the following axioms: ensures a minimum utility guarantee, for instance, is the Pareto optimal, which implies that other allocations Տ (Ʈ, ) capable of guaranteeing a higher performance for all the ℝ CRV-SUs simultaneously do not exist, that is, guarantees symmetry, which implies that all the ℝ CRV-SUs have equal priorities, for instance, supposing that Ʈ is symmetric with regards to a sub-set ⊆ 1, 2, 3, ⋯ , , ⋯ , ℝ and ∈ Ʈ, , guarantees fairness by maintaining the independence of irrelevant alternatives, for instance, if the feasible set decreases and the solution keeps on being feasible, it follows that the solution for the lesser achievable set remains the same point. It can be expressed as ɰ ⊂ Ʈ, (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ (Ʈ, ) = Տ (ɰ, ), ∀ .
Without a loss of generality, the property of the SNO-CRAVNET is described using the following theorem.
, U 0 ∈ B, then and expressed as . From stipulated condition guarantees that the potential transmissions of the PU are f only if the CRV-SU 's total transmit power is constrained over channel threshold Ᵽ .

The Utility of SNO-CRAVNET and Problem Formulation
The design methodology of the SNO-CRAVNET's objectives with its SNO presented in this section in the form of a convex optimization problem, with i represented by the ℝ CRV-SUs. The design of the game bargaining scheme meth enabled IoV network system is proposed in this section. We assume that each ℝ C CRV-SU , has an initial utility ≥ 0, which represents its acceptable minim with respect to the data-rate and the corresponding utility function . Under bargaining (SNB) theory, each utility function is usually designated as a conv of ℝ = ℧ , with ℝ and ℧ denoting the set of game theory strategies of th and utility vectors' space, respectively. Let us assume that is conveniently ℝ CRV-SU players. Then, it follows that at least a feasible subspace ℧ exists in vector, for instance, ( ) = , , , ⋯ , ℝ , becomes equal or bigger in com utility vector, such as, = , , , ⋯ , ℝ . Therefore, the subset ℧ as the expressed as ℧ = ∈ ℧| ( ) ≥ . Additionally, let us suppose that the set achieved is denoted by Ʈ = ( )| ∈ ℧ and the category of sets of utility poli which is the minimum utility bound, is denoted as ℬ = Ʈ, |Ʈ ⊂ ℝ . Therefor the Symmetric NBS theory (see [23]), there exists a unique solution, for instance, satisfies the following axioms: ensures a minimum utility guarantee, for instance, is the Pareto optimal, which implies that other allocations Տ guaranteeing a higher performance for all the ℝ CRV-SUs simultaneously Տ (Ʈ, ) < Տ (Ʈ, ), ∃ and Տ (Ʈ, ) ≤ Տ (Ʈ, ), ∀ ; c) Տ (Ʈ, ) guarantees symmetry, which implies that all the ℝ CRV-SUs h for instance, supposing that Ʈ is symmetric with regards to a sub-set and ∈ Ʈ, , guarantees fairness by maintaining the independence of irrele instance, if the feasible set decreases and the solution keeps on being feasib solution for the lesser achievable set remains the same point. It can be (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ (Ʈ, ) = Տ (ɰ, ), ∀ .
Without a loss of generality, the property of the SNO-CRAVNET is following theorem. nbs only if the CRV-SU 's total transmit power is constrained over channel threshold Ᵽ .

The Utility of SNO-CRAVNET and Problem Formulation
The design methodology of the SNO-CRAVNET's objectives with its SNO presented in this section in the form of a convex optimization problem, with i represented by the ℝ CRV-SUs. The design of the game bargaining scheme metho enabled IoV network system is proposed in this section. We assume that each ℝ C CRV-SU , has an initial utility ≥ 0, which represents its acceptable minim with respect to the data-rate and the corresponding utility function . Under bargaining (SNB) theory, each utility function is usually designated as a conv of ℝ = ℧ , with ℝ and ℧ denoting the set of game theory strategies of th and utility vectors' space, respectively. Let us assume that is conveniently a ℝ CRV-SU players. Then, it follows that at least a feasible subspace ℧ exists in vector, for instance, ( ) = , , , ⋯ , ℝ , becomes equal or bigger in comp utility vector, such as, = , , , ⋯ , ℝ . Therefore, the subset ℧ as the expressed as ℧ = ∈ ℧| ( ) ≥ . Additionally, let us suppose that the set achieved is denoted by Ʈ = ( )| ∈ ℧ and the category of sets of utility poli which is the minimum utility bound, is denoted as ℬ = Ʈ, |Ʈ ⊂ ℝ . Therefore the Symmetric NBS theory (see [23]), there exists a unique solution, for instance, satisfies the following axioms: ensures a minimum utility guarantee, for instance, Տ (Ʈ, is the Pareto optimal, which implies that other allocations Տ guaranteeing a higher performance for all the ℝ CRV-SUs simultaneously guarantees symmetry, which implies that all the ℝ CRV-SUs h for instance, supposing that Ʈ is symmetric with regards to a sub-set and ∈ Ʈ, , guarantees fairness by maintaining the independence of irrelev instance, if the feasible set decreases and the solution keeps on being feasib solution for the lesser achievable set remains the same point. It can be e (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ (Ʈ, ) = Տ (ɰ, ), ∀ .
Without a loss of generality, the property of the SNO-CRAVNET is d following theorem.
, U 0 = and expressed as stipulated condition guarantees that the potential transmissions o only if the CRV-SU 's total transmit power is constrained o threshold Ᵽ .

The Utility of SNO-CRAVNET and Problem Formulation
The design methodology of the SNO-CRAVNET's objective presented in this section in the form of a convex optimization p represented by the ℝ CRV-SUs. The design of the game bargaining enabled IoV network system is proposed in this section. We assum CRV-SU , has an initial utility ≥ 0, which represents its ac with respect to the data-rate and the corresponding utility funct bargaining (SNB) theory, each utility function is usually design of ℝ = ℧ , with ℝ and ℧ denoting the set of game theory s and utility vectors' space, respectively. Let us assume that is ℝ CRV-SU players. Then, it follows that at least a feasible subspac vector, for instance, ( ) = , , , ⋯ , ℝ , becomes equal or b utility vector, such as, = , , , ⋯ , ℝ . Therefore, the sub expressed as ℧ = ∈ ℧| ( ) ≥ . Additionally, let us suppo achieved is denoted by Ʈ = ( )| ∈ ℧ and the category of set which is the minimum utility bound, is denoted as ℬ = Ʈ, |Ʈ ⊂ the Symmetric NBS theory (see [23]), there exists a unique solution satisfies the following axioms: ensures a minimum utility guarantee, for insta is the Pareto optimal, which implies that other guaranteeing a higher performance for all the ℝ CRV-SUs s Տ (Ʈ, ) < Տ (Ʈ, ), ∃ and Տ (Ʈ, ) ≤ Տ (Ʈ, ), c) Տ (Ʈ, ) guarantees symmetry, which implies that all the for instance, supposing that Ʈ is symmetric with regards t and ∈ Ʈ, , guarantees fairness by maintaining the independ instance, if the feasible set decreases and the solution keeps o solution for the lesser achievable set remains the same po (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ (Ʈ, ) = Տ (ɰ Without a loss of generality, the property of the SNO-C following theorem. nbs presented in this section in the form of a convex optimization problem, with its associated players represented by the ℝ CRV-SUs. The design of the game bargaining scheme methodologies for the CR-enabled IoV network system is proposed in this section. We assume that each ℝ CRV-SU, for instance, CRV-SU , has an initial utility ≥ 0, which represents its acceptable minimum QoS constraint with respect to the data-rate and the corresponding utility function . Under the symmetric Nash bargaining (SNB) theory, each utility function is usually designated as a convex and closed subset of ℝ = ℧ , with ℝ and ℧ denoting the set of game theory strategies of the ℝ CRV-SU players and utility vectors' space, respectively. Let us assume that is conveniently achievable for all the ℝ CRV-SU players. Then, it follows that at least a feasible subspace ℧ exists in ℧, so that the utility vector, for instance, ( ) = , , , ⋯ , ℝ , becomes equal or bigger in comparison to the initial utility vector, such as, = , , , ⋯ , ℝ . Therefore, the subset ℧ as the element of ℧ can be expressed as ℧ = ∈ ℧| ( ) ≥ . Additionally, let us suppose that the set of utility that can be achieved is denoted by Ʈ = ( )| ∈ ℧ and the category of sets of utility policies that satisfies , which is the minimum utility bound, is denoted as ℬ = Ʈ, |Ʈ ⊂ ℝ . Therefore, in accordance with the Symmetric NBS theory (see [23]), there exists a unique solution, for instance, Տ |ℬ ⟶ ℝ , which satisfies the following axioms: a) Տ (Ʈ, ) ensures a minimum utility guarantee, for instance, is the Pareto optimal, which implies that other allocations Տ (Ʈ, ) capable of guaranteeing a higher performance for all the ℝ CRV-SUs simultaneously do not exist, that is, c) Տ (Ʈ, ) guarantees symmetry, which implies that all the ℝ CRV-SUs have equal priorities, for instance, supposing that Ʈ is symmetric with regards to a sub-set ⊆ 1, 2, 3, ⋯ , , ⋯ , ℝ and ∈ Ʈ, , ∈ so that = implies that Տ (Ʈ, ) = Տ (Ʈ, ) , ≠ ; d) Տ (Ʈ, ) guarantees fairness by maintaining the independence of irrelevant alternatives, for instance, if the feasible set decreases and the solution keeps on being feasible, it follows that the solution for the lesser achievable set remains the same point. It can be expressed as ɰ ⊂ Ʈ, Without a loss of generality, the property of the SNO-CRAVNET is described using the following theorem.
Without a loss of generality, the property of the SNO-CRAVNET is described using the following theorem. Theorem 1. It is assumed that the utility function defined by is convex upper bounded. Therefore, is convex and equal to ⊆ F R . Then, it is supposed that Sensors 2020, 20, x FOR PEER REVIEW 11 of 27 Theorem 1. It is assumed that the utility function defined by ℧ is convex upper bounded. Therefore, ℧ is convex and equal to ℧ ⊆ ℝ . Then, it is supposed that Ɲ is the set of indices of ℝ CRV-SUs that are capable of achieving a strictly superior performance in comparison to their initial performance. Therefore, it follows that there exists a symmetric Nash bargaining point , which confirms ( ) ≥ , ∈ Ɲ and consists of a unique solution for the maximization problem expressed below: Proof. Theorem 1's Proof is similar to that of the SNBS feature provided in [24] (Proof omitted here because it is similar to the one in [24] and also due to space limitations). □ Irrespective of the fact that the logarithmic basis of the optimization objective in Equation (20) stands, it is observed that resource allocation mechanisms (i.e., allocation problems) which depend on Theorem 1 are not typically convex over given convex sets. In particular, with such allocation problems under certain constraints, the convexity and existence of the feasible set which can satisfy the objective and all the constraints have to be thoroughly investigated. For instance, with respect to the CR constraints on transmit power allocation policy, channel selection, stability (i.e., protection of PU's communication), and SNIR, as shown in Equations (12), (15), (17) and (19), the throughput definition given by Equation (13) can be adopted as the optimization objective in Equation (20) above. Therefore, an initial SNO-CRAVNET problem can be expressed as follows: Find the joint optimal transmit power and subcarrier allocation strategies ℂ ×ℝ ×ℝ and ℙ ×ℝ ×ℝ .
subject to is the set of indices of R CRV-SUs that are capable of achieving a strictly superior performance in comparison to their initial performance. Therefore, it follows that there exists a symmetric Nash bargaining point ω, which confirms f n (ω) ≥ U 0 n , n ∈ Sensors 2020, 20, x FOR PEER REVIEW 11 of 27 Theorem 1. It is assumed that the utility function defined by ℧ is convex upper bounded. Therefore, ℧ is convex and equal to ℧ ⊆ ℝ . Then, it is supposed that Ɲ is the set of indices of ℝ CRV-SUs that are capable of achieving a strictly superior performance in comparison to their initial performance. Therefore, it follows that there exists a symmetric Nash bargaining point , which confirms ( ) ≥ , ∈ Ɲ and consists of a unique solution for the maximization problem expressed below: Proof. Theorem 1's Proof is similar to that of the SNBS feature provided in [24] (Proof omitted here because it is similar to the one in [24] and also due to space limitations). □ Irrespective of the fact that the logarithmic basis of the optimization objective in Equation (20) stands, it is observed that resource allocation mechanisms (i.e., allocation problems) which depend on Theorem 1 are not typically convex over given convex sets. In particular, with such allocation problems under certain constraints, the convexity and existence of the feasible set which can satisfy the objective and all the constraints have to be thoroughly investigated. For instance, with respect to the CR constraints on transmit power allocation policy, channel selection, stability (i.e., protection of PU's communication), and SNIR, as shown in Equations (12), (15), (17) and (19), the throughput definition given by Equation (13) can be adopted as the optimization objective in Equation (20) above. Therefore, an initial SNO-CRAVNET problem can be expressed as follows: Find the joint optimal transmit power and subcarrier allocation strategies ℂ ×ℝ ×ℝ and ℙ ×ℝ ×ℝ .
subject to ∈ 0, 1 , ∀m, , and consists of a unique solution for the maximization problem expressed below: Proof. Theorem 1's Proof is similar to that of the SNBS feature provided in [24] (Proof omitted here because it is similar to the one in [24] and also due to space limitations).
Irrespective of the fact that the logarithmic basis of the optimization objective in Equation (20) stands, it is observed that resource allocation mechanisms (i.e., allocation problems) which depend on Theorem 1 are not typically convex over given convex sets. In particular, with such allocation problems under certain constraints, the convexity and existence of the feasible set which can satisfy the objective and all the constraints have to be thoroughly investigated. For instance, with respect to the CR constraints on transmit power allocation policy, channel selection, stability (i.e., protection of PU's communication), and SNIR, as shown in Equations (12), (15), (17) and (19), the throughput definition given by Equation (13) can be adopted as the optimization objective in Equation (20) above. Therefore, an initial SNO-CRAVNET problem can be expressed as follows: Find the joint optimal transmit power and subcarrier allocation strategies subject to The problem shown in Equation (21) and in the constraints (22)-(27) is a mixed combinatorial problem because it includes both a discrete variable {C mn } and continuous variable {P mn }. Generally, the conventional approach normally adopted to solve such a mixed combinatorial problem is usually applied by performing an exhaustive search method [25] over the R CRV-SUs and N C number of dynamically available channels. Therefore, there are a total of R N C possible channel assignments. To guarantee that the individual requirement for each of the R CRV-SUs is satisfied for each of the R N C possible channel assignments, the total transmit power P Tot is allocated and, at the same time, summation of the SNO-CRAVNET data-rate of each of the R CRV-SUs is equally maximized, accordingly.
Consequently, while all the constraints in Equations (22)- (27) are satisfied, the assignment of the dynamically available channels, together with their corresponding total transmit power allocation P Tot. which leads to the biggest summation of the data-rate, becomes the overall optimal solution. However, because of the high computational complexity of this method [25], together with the known limited computation, bandwidth, and storage resources in vehicular communication networks [26,27], extremely complex algorithms cannot be the best alternative for implementation in CR-enabled vehicular networks.
To overcome this challenge, the mixed combinatorial problem seen in Equation (21) and in the constraints shown in Equations (22)-(27) is methodically transformed to a convex optimization problem.
The key aim of this transformation of the mixed combinatorial problem into a convex optimization problem is to make sure that the outcome of the transformation process must be a new problem that can symmetrically embrace the property of the proposed SNO-CRAVNET under the regulations of the emerging CR system. Furthermore, the new convex optimization problem must be defined over a feasible set that maintains its convexity and, at the same time, ensures that all the involved constraints are satisfied. The process of the transformation is as shown below. Firstly, as presented in Section 2.4, the requirement R mn (P mn ) ∈ I is relaxed into R mn (P mn ) ∈ [0, I], so that R mn (P mn ) can become a real number between the interval [0, I]. From Equation (12) factor of the mth subcarrier, which shows the period of time that subcarrier m is allocated to CRV-SU n over every one of the transmission frames. Then, with the aid of the time-sharing transformation, the objective T( ∼ C mn , P mn ) can be defined as convex over ∼ C mn , though it still remains non-convex over ( ∼ C mn , P mn ). Secondly, with the help of the same time-sharing approach, P mn is transformed into a continuous variable . Therefore, with ∼ C mn and ∼ P mn , the reformulated convex optimization problem can now easily be formulated: Find the optimal joint channel and transmit power allocation strategies subject to Between Equations (29)-(34), the constraints presented in Equations (29) and (30) guarantee that, at a given time-share, only one CRV-SU can be allocated a channel and must adhere to the properties of Equation (12) (see Section 2.4). The constraint in Equation (31) guarantees that the allocated transmit power must not be negative, while the constraint provided in Equation (32) maintains the transmitted power, in order to ensure that the transmit power allocated to the CRV-SU n occupying all the dynamically available orthogonal channels is maintained below the total transmit power P Tot. available at CR-BS, as defined in Equation (15). Lastly, as illustrated in Equations (17) and (19), transmit power constraints for each R CRV-SU and PU are guaranteed by constraints presented in Equations (33) and (34), respectively. Proposition 1. In the above stated optimal joint subcarrier and transmit power allocation strategies, the problem defined in Equation (28) and in the constraints presented in Equations (29)-(34) is a convex optimization problem.

Proof. Proposition 1's Proof is shown in Appendix A.
With regard to Proposition 1, it is established that the problem defined in Equation (28) and in the constraints presented in Equations (29)-(34) is clearly convex over a given convex set. Therefore, there exists a unique optimal solution that can be achieved within the polynomial time [25].
Without a loss of generality, the property of the SNO-CRAVNET is d following theorem.
( ∼ C mn , ∼ P mn ), as shown in Equation (28), can stringently increase for all ∼ C mn , thereby satisfying Proof. Appendix A presents the Proof of Proposition 2.
Proposition 2 certifies that the transformation of the objective in Equation (20), as well as in Equation (21) to ℝ CRV-SU players. Then, it follows that at least a feasible subspace ℧ exists in ℧, so that the utility vector, for instance, ( ) = , , , ⋯ , ℝ , becomes equal or bigger in comparison to the initial utility vector, such as, = , , , ⋯ , ℝ . Therefore, the subset ℧ as the element of ℧ can be expressed as ℧ = ∈ ℧| ( ) ≥ . Additionally, let us suppose that the set of utility that can be achieved is denoted by Ʈ = ( )| ∈ ℧ and the category of sets of utility policies that satisfies , which is the minimum utility bound, is denoted as ℬ = Ʈ, |Ʈ ⊂ ℝ . Therefore, in accordance with the Symmetric NBS theory (see [23]), there exists a unique solution, for instance, Տ |ℬ ⟶ ℝ , which satisfies the following axioms: ensures a minimum utility guarantee, for instance, is the Pareto optimal, which implies that other allocations Տ (Ʈ, ) capable of guaranteeing a higher performance for all the ℝ CRV-SUs simultaneously do not exist, that is, guarantees symmetry, which implies that all the ℝ CRV-SUs have equal priorities, for instance, supposing that Ʈ is symmetric with regards to a sub-set ⊆ 1, 2, 3, ⋯ , , ⋯ , ℝ and ∈ Ʈ, , guarantees fairness by maintaining the independence of irrelevant alternatives, for instance, if the feasible set decreases and the solution keeps on being feasible, it follows that the solution for the lesser achievable set remains the same point. It can be expressed as ɰ ⊂ Ʈ, (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ (Ʈ, ) = Տ (ɰ, ), ∀ .
Without a loss of generality, the property of the SNO-CRAVNET is described using the following theorem. (28), can be achieved by exploiting the firmly increasing property of the logarithm function. Proposition 3. The utility function bargaining (SNB) theory, each utility function is usually designated as a convex and closed of ℝ = ℧ , with ℝ and ℧ denoting the set of game theory strategies of the ℝ CRV-SU and utility vectors' space, respectively. Let us assume that is conveniently achievable fo ℝ CRV-SU players. Then, it follows that at least a feasible subspace ℧ exists in ℧, so that th vector, for instance, ( ) = , , , ⋯ , ℝ , becomes equal or bigger in comparison to th utility vector, such as, = , , , ⋯ , ℝ . Therefore, the subset ℧ as the element of ℧ expressed as ℧ = ∈ ℧| ( ) ≥ . Additionally, let us suppose that the set of utility tha achieved is denoted by Ʈ = ( )| ∈ ℧ and the category of sets of utility policies that satis which is the minimum utility bound, is denoted as ℬ = Ʈ, |Ʈ ⊂ ℝ . Therefore, in accordan the Symmetric NBS theory (see [23]), there exists a unique solution, for instance, Տ |ℬ ⟶ ℝ satisfies the following axioms: ensures a minimum utility guarantee, for instance, is the Pareto optimal, which implies that other allocations Տ (Ʈ, ) cap guaranteeing a higher performance for all the ℝ CRV-SUs simultaneously do not exist, guarantees symmetry, which implies that all the ℝ CRV-SUs have equal pr for instance, supposing that Ʈ is symmetric with regards to a sub-set ⊆ 1, 2, 3, ⋯ , and ∈ Ʈ, , guarantees fairness by maintaining the independence of irrelevant alternati instance, if the feasible set decreases and the solution keeps on being feasible, it follows solution for the lesser achievable set remains the same point. It can be expressed as (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ (Ʈ, ) = Տ (ɰ, ), ∀ .
Without a loss of generality, the property of the SNO-CRAVNET is described us following theorem. (28) is Nash bargaining theorem compliant and, at the same time, satisfies the proportional fairness metric.
Proof. Appendix A presents the Proof of Proposition 3.
In our case, Proposition 3 shows that, for the data-rate allocation, a unique Nash bargaining equilibrium can be obtained. Likewise, as a special case of the NBS fairness [29], proportional fairness can be achieved when U 0 n = 0, ∀n.

Optimal Resource Scheduling Strategies
The convex optimization problem's optimal solution, which is presented in Equation (28) and in the constraints expressed in Equations (29)-(34), is derived in this section. Additionally, a simple and efficient strategy, which supports an iteration-independent joint transmit power and subcarrier scheduling, is proposed. The optimal subcarrier allocation ∼ C mn with a consideration of the time-sharing approach is a real number implying the fraction of time which subcarrier m requires for the transmission of a given amount of information. Firstly, uniform transmit power scheduling, that is, ∼ P mn = P Tot. /(N C ·B), is performed for all the available subcarriers. Then, an equal amount of information is transferred over all the available subcarriers. Secondly, based on the study carried out by Hahne [29], the optimal time-sharing subcarrier scheduling strategy is obtained. Theorem 2. The SNO-CRAVNET optimal time-sharing subcarrier scheduling strategy is given as mn , and the individual matrix elements are expressed as ∼ C * mn = problem defined in Equation (28) and in the constraints presented in Equations (29)-(34) is a convex optimization problem.
Proof. Proposition 1's Proof is shown in Appendix A. □ With regard to Proposition 1, it is established that the problem defined in Equation (28) and in the constraints presented in Equations (29)-(34) is clearly convex over a given convex set. Therefore, there exists a unique optimal solution that can be achieved within the polynomial time [25].
Proof. Appendix A presents the Proof of Proposition 2. □ Proposition 2 certifies that the transformation of the objective in Equation (20), as well as in Equation (21) to Ʈ , in Equation (28), can be achieved by exploiting the firmly increasing property of the logarithm function. Proof. Appendix A presents the Proof of Proposition 3. □ In our case, Proposition 3 shows that, for the data-rate allocation, a unique Nash bargaining equilibrium can be obtained. Likewise, as a special case of the NBS fairness [29], proportional fairness can be achieved when = 0, ∀ .

Optimal Resource Scheduling Strategies
The convex optimization problem's optimal solution, which is presented in Equation (28) and in the constraints expressed in Equations (29)-(34), is derived in this section. Additionally, a simple and efficient strategy, which supports an iteration-independent joint transmit power and subcarrier scheduling, is proposed. The optimal subcarrier allocation with a consideration of the timesharing approach is a real number implying the fraction of time which subcarrier requires for the transmission of a given amount of information. Firstly, uniform transmit power scheduling, that is, , is performed for all the available subcarriers. Then, an equal amount of information is transferred over all the available subcarriers. Secondly, based on the study carried out by Hahne [29], the optimal time-sharing subcarrier scheduling strategy is obtained. Theorem 2. The SNO-CRAVNET optimal time-sharing subcarrier scheduling strategy is given as ℂ ×ℝ * ×ℝ = ℂ * , and the individual matrix elements are expressed as where ₼ * = ɸ (1), ∀ .
Proof. Appendix B presents the Proof of Theorem 2. □ −1 mn ( problem defined in Equation (28) and in the constraints presented in Equations (29)-(34) is a convex optimization problem.
Proof. Proposition 1's Proof is shown in Appendix A. □ With regard to Proposition 1, it is established that the problem defined in Equation (28) and in the constraints presented in Equations (29)-(34) is clearly convex over a given convex set. Therefore, there exists a unique optimal solution that can be achieved within the polynomial time [25].
Proof. Appendix A presents the Proof of Proposition 2. □ Proposition 2 certifies that the transformation of the objective in Equation (20), as well as in Equation (21) to Ʈ , in Equation (28), can be achieved by exploiting the firmly increasing property of the logarithm function. Proof. Appendix A presents the Proof of Proposition 3. □ In our case, Proposition 3 shows that, for the data-rate allocation, a unique Nash bargaining equilibrium can be obtained. Likewise, as a special case of the NBS fairness [29], proportional fairness can be achieved when = 0, ∀ .

Optimal Resource Scheduling Strategies
The convex optimization problem's optimal solution, which is presented in Equation (28) and in the constraints expressed in Equations (29)-(34), is derived in this section. Additionally, a simple and efficient strategy, which supports an iteration-independent joint transmit power and subcarrier scheduling, is proposed. The optimal subcarrier allocation with a consideration of the timesharing approach is a real number implying the fraction of time which subcarrier requires for the transmission of a given amount of information. Firstly, uniform transmit power scheduling, that is, , is performed for all the available subcarriers. Then, an equal amount of information is transferred over all the available subcarriers. Secondly, based on the study carried out by Hahne [29], the optimal time-sharing subcarrier scheduling strategy is obtained. Theorem 2. The SNO-CRAVNET optimal time-sharing subcarrier scheduling strategy is given as ℂ ×ℝ * ×ℝ = ℂ * , and the individual matrix elements are expressed as where ₼ * = ɸ (1), ∀ .
ted optimal joint subcarrier and transmit power allocation strategies, the ) and in the constraints presented in Equations (29)-(34) is a convex hown in Appendix A. □ n 1, it is established that the problem defined in Equation (28) and in uations (29)-(34) is clearly convex over a given convex set. Therefore, lution that can be achieved within the polynomial time [25].
e Proof of Proposition 2. □ t the transformation of the objective in Equation (20), as well as in n Equation (28), can be achieved by exploiting the firmly increasing ion.
ion Ʈ , proposed in Equation (28) is Nash bargaining theorem atisfies the proportional fairness metric.
e Proof of Proposition 3. □ shows that, for the data-rate allocation, a unique Nash bargaining kewise, as a special case of the NBS fairness [29], proportional fairness ∀ .
ng Strategies roblem's optimal solution, which is presented in Equation (28) and in uations (29)- (34), is derived in this section. Additionally, a simple and orts an iteration-independent joint transmit power and subcarrier ptimal subcarrier allocation with a consideration of the timeber implying the fraction of time which subcarrier requires for the t of information. Firstly, uniform transmit power scheduling, that is, med for all the available subcarriers. Then, an equal amount of all the available subcarriers. Secondly, based on the study carried out e-sharing subcarrier scheduling strategy is obtained.
ET optimal time-sharing subcarrier scheduling strategy is given as individual matrix elements are expressed as Proof of Theorem 2. □ * m = Proposition 1. In the above stated optimal joint subcarrier and transmit power allocation strategies, the problem defined in Equation (28) and in the constraints presented in Equations (29)-(34) is a convex optimization problem.
Proof. Proposition 1's Proof is shown in Appendix A. □ With regard to Proposition 1, it is established that the problem defined in Equation (28) and in the constraints presented in Equations (29)-(34) is clearly convex over a given convex set. Therefore, there exists a unique optimal solution that can be achieved within the polynomial time [25].
Proof. Appendix A presents the Proof of Proposition 2. □ Proposition 2 certifies that the transformation of the objective in Equation (20), as well as in Equation (21) to Ʈ , in Equation (28), can be achieved by exploiting the firmly increasing property of the logarithm function. Proof. Appendix A presents the Proof of Proposition 3. □ In our case, Proposition 3 shows that, for the data-rate allocation, a unique Nash bargaining equilibrium can be obtained. Likewise, as a special case of the NBS fairness [29], proportional fairness can be achieved when = 0, ∀ .

Optimal Resource Scheduling Strategies
The convex optimization problem's optimal solution, which is presented in Equation (28) and in the constraints expressed in Equations (29)- (34), is derived in this section. Additionally, a simple and efficient strategy, which supports an iteration-independent joint transmit power and subcarrier scheduling, is proposed. The optimal subcarrier allocation with a consideration of the timesharing approach is a real number implying the fraction of time which subcarrier requires for the transmission of a given amount of information. Firstly, uniform transmit power scheduling, that is, , is performed for all the available subcarriers. Then, an equal amount of information is transferred over all the available subcarriers. Secondly, based on the study carried out by Hahne [29], the optimal time-sharing subcarrier scheduling strategy is obtained. Theorem 2. The SNO-CRAVNET optimal time-sharing subcarrier scheduling strategy is given as ℂ ×ℝ * ×ℝ = ℂ * , and the individual matrix elements are expressed as where ₼ * = ɸ (1), ∀ .
Proof. Appendix B presents the Proof of Theorem 2.
Based on Equation (35), the matrix mn , which illustrates the time-sharing scheduling of each subcarrier for all the R CRV-SUs, is determined. Additionally, this further helps in determining the quality of each subcarrier, that is, based on Equation (35), if it is observed that ∼ C * mn < ∼ C * m n , this indicates that even though both subcarriers were allocated an equal amount of transmit power, subcarrier m requires less time than subcarrier m for the transfer of an equal amount of information by the same CRV-SU n. Therefore, subcarrier m is in better conditions, i.e., has a higher quality in comparison to subcarrier m . Furthermore, accounting for ∼ C * N C ×R O N C ×R , the optimal transmit power scheduling strategy is defined as shown here.
scheduling of each subcarrier for all the ℝ CRV-SUs, is determined. Additionally, this further helps in determining the quality of each subcarrier, that is, based on Equation (35), if it is observed that ℂ * < ℂ * , this indicates that even though both subcarriers were allocated an equal amount of transmit power, subcarrier requires less time than subcarrier for the transfer of an equal amount of information by the same CRV-SU . Therefore, subcarrier is in better conditions, i.e., has a higher quality in comparison to subcarrier . Furthermore, accounting for ℂ ×ℝ * ×ℝ , the optimal transmit power scheduling strategy is defined as shown here. Theorem 3. The SNO-CRAVNET optimal transmit power scheduling strategy is given by ℙ ×ℝ * ×ℝ = * and the individual matrix elements are given by * where (•) represents the Lambert -function. The definitions of both Þ and are contained in Appendix B, and the symbol ( ) represents (0, ).
Proof. Appendix B presents the Proof of Theorem 3. □ The optimal transmit power scheduling for CRV-SU on every subcarrier is obtained from Theorem 3. In other words, the optimal transmit power that subcarrier requires to be able to transmit a given amount of information based on the licensed PU's protection parameters, the associated subcarrier's conditions, and the characteristics of CRV-SU is denoted by * . Consequently, through a linear search of the subcarriers, efficient resource scheduling can be performed, for instance, for = 1 to , find the optimal CRV-SU * = arg min ℂ * . Then, allocate the corresponding transmit power as defined in Equation (36) to all the selected CRV-SUs * s. Despite the fact that the procedure derives ℂ * s accounting for the subcarrier scheduling constraints presented in Equations (29) and (30) under the symmetric NBS's rule, the procedure indirectly considers the transmit power constraints presented in Equations (31)- (34). Then, with high QoS heterogeneity amongst the ℝ CRV-SUs (i.e., concerning the subcarrier's stringent QoS requirements and interference conditions), ℂ * < ℂ * does not necessarily indicate that * < * , and vice versa, which leads to transmit power inefficiency. Therefore, to overcome this, the following optimal transmit power scheduling method is introduced, in order to increase the transmit power efficiency.
where * represents the optimal CRV-SU.
Proof. Appendix B presents the Proof of Theorem 4. □ Using Equation (37), matrix ℙ ×ℝ * ×ℝ is obtained, which indicates that the optimal transmit power is allocated to the optimal CRV-SU * on subcarrier . Therefore, the optimal transmit power of each CRV-SU can be determined through * = ∑ * , ∀ . Likewise, accounting for ℙ ×ℝ * ×ℝ , the optimal subcarrier allocation can be defined as shown here.
Theorem 5. The SNO-CRAVNET optimal subcarrier allocation strategy is given by ℂ ×ℝ * ×ℝ = ℂ * , where the individual matrix elements are determined by scheduling of each subcarrier for all the ℝ CRV-SUs, is determined. Additionally, this further in determining the quality of each subcarrier, that is, based on Equation (35), if it is observe ℂ * < ℂ * , this indicates that even though both subcarriers were allocated an equal amou transmit power, subcarrier requires less time than subcarrier for the transfer of an amount of information by the same CRV-SU . Therefore, subcarrier is in better condition has a higher quality in comparison to subcarrier . Furthermore, accounting for ℂ ×ℝ * ×ℝ optimal transmit power scheduling strategy is defined as shown here. Proof. Appendix B presents the Proof of Theorem 3. □ The optimal transmit power scheduling for CRV-SU on every subcarrier is obtained Theorem 3. In other words, the optimal transmit power that subcarrier requires to be a transmit a given amount of information based on the licensed PU's protection parameter associated subcarrier's conditions, and the characteristics of CRV-SU is denoted by * . Consequently, through a linear search of the subcarriers, efficient resource schedulin be performed, for instance, for = 1 to , find the optimal CRV-SU * = arg min ℂ * . allocate the corresponding transmit power as defined in Equation (36) to all the selected CRV * s. Despite the fact that the procedure derives ℂ * s accounting for the subcarrier sched constraints presented in Equations (29) and (30) under the symmetric NBS's rule, the proc indirectly considers the transmit power constraints presented in Equations (31)- (34). Then, with QoS heterogeneity amongst the ℝ CRV-SUs (i.e., concerning the subcarrier's stringent requirements and interference conditions), ℂ * < ℂ * does not necessarily indicate that * , and vice versa, which leads to transmit power inefficiency. Therefore, to overcome th following optimal transmit power scheduling method is introduced, in order to increase the tra power efficiency.
where w(·) represents the Lambert w -function. The definitions of both , which illustrates the time-s scheduling of each subcarrier for all the ℝ CRV-SUs, is determined. Additionally, this furthe in determining the quality of each subcarrier, that is, based on Equation (35), if it is observe ℂ * < ℂ * , this indicates that even though both subcarriers were allocated an equal amo transmit power, subcarrier requires less time than subcarrier for the transfer of an amount of information by the same CRV-SU . Therefore, subcarrier is in better conditio has a higher quality in comparison to subcarrier . Furthermore, accounting for ℂ ×ℝ * × optimal transmit power scheduling strategy is defined as shown here. Proof. Appendix B presents the Proof of Theorem 3. □ The optimal transmit power scheduling for CRV-SU on every subcarrier is obtaine Theorem 3. In other words, the optimal transmit power that subcarrier requires to be transmit a given amount of information based on the licensed PU's protection paramete associated subcarrier's conditions, and the characteristics of CRV-SU is denoted by * . Consequently, through a linear search of the subcarriers, efficient resource scheduli be performed, for instance, for = 1 to , find the optimal CRV-SU * = arg min ℂ * . allocate the corresponding transmit power as defined in Equation (36) to all the selected CR * s. Despite the fact that the procedure derives ℂ * s accounting for the subcarrier sche constraints presented in Equations (29) and (30) under the symmetric NBS's rule, the pro indirectly considers the transmit power constraints presented in Equations (31)- (34). Then, wit QoS heterogeneity amongst the ℝ CRV-SUs (i.e., concerning the subcarrier's stringen requirements and interference conditions), ℂ * < ℂ * does not necessarily indicate that * , and vice versa, which leads to transmit power inefficiency. Therefore, to overcome th following optimal transmit power scheduling method is introduced, in order to increase the tr power efficiency. The optimal transmit power scheduling for CRV-SU n on every subcarrier m is obtained from Theorem 3. In other words, the optimal transmit power that subcarrier m requires to be able to transmit a given amount of information based on the licensed PU's protection parameters, the associated subcarrier's conditions, and the characteristics of CRV-SU n is denoted by ∼ P * mn . Consequently, through a linear search of the N C subcarriers, efficient resource scheduling can be performed, for instance, for m = 1 to N C , find the optimal CRV-SU n * = arg min m n , and vice versa, which leads to transmit power inefficiency. Therefore, to overcome this, the following optimal transmit power scheduling method is introduced, in order to increase the transmit power efficiency.
where n * represents the optimal CRV-SU.
Proof. Appendix B presents the Proof of Theorem 4.
Using Equation (37), matrix P * N C ×R O N C ×R is obtained, which indicates that the optimal transmit power is allocated to the optimal CRV-SU n * on subcarrier m. Therefore, the optimal transmit power of each CRV-SU can be determined through P * n = N C m=1 P * mn , ∀n. Likewise, accounting for P * N C ×R O N C ×R , the optimal subcarrier allocation can be defined as shown here.
Proof. Appendix B presents the Proof of Theorem 5.
Accordingly, from Equation (38), the optimal subcarrier scheduling matrix C * N C ×R O N C ×R is obtained. Consequently, by using Equations (35)-(38), the joint transmit power and subcarrier allocation strategy for CRV-SU systems is determined as illustrated, with the aid of the pseudo-code, in Algorithm 1.
Additionally, from the combination of Equations (11) and (36)-(38), the optimal rate scheduling mn is obtained, where the instantaneous optimal data-rate as the individual matrix elements is determined by Based on Equation (35), the matrix ℂ ×ℝ * ×ℝ = ℂ * , which illustrates the time-sharing scheduling of each subcarrier for all the ℝ CRV-SUs, is determined. Additionally, this further helps in determining the quality of each subcarrier, that is, based on Equation (35), if it is observed that ℂ * < ℂ * , this indicates that even though both subcarriers were allocated an equal amount of transmit power, subcarrier requires less time than subcarrier for the transfer of an equal amount of information by the same CRV-SU . Therefore, subcarrier is in better conditions, i.e., has a higher quality in comparison to subcarrier . Furthermore, accounting for ℂ ×ℝ * ×ℝ , the optimal transmit power scheduling strategy is defined as shown here.
where (•) represents the Lambert -function. The definitions of both Þ and are contained in Appendix B, and the symbol ( ) represents (0, ).

Proof. Appendix B presents the Proof of Theorem 3. □
The optimal transmit power scheduling for CRV-SU on every subcarrier is obtained from Theorem 3. In other words, the optimal transmit power that subcarrier requires to be able to transmit a given amount of information based on the licensed PU's protection parameters, the associated subcarrier's conditions, and the characteristics of CRV-SU is denoted by * . Consequently, through a linear search of the subcarriers, efficient resource scheduling can be performed, for instance, for = 1 to , find the optimal CRV-SU * = arg min ℂ * . Then, allocate the corresponding transmit power as defined in Equation (36) to all the selected CRV-SUs * s. Despite the fact that the procedure derives ℂ * s accounting for the subcarrier scheduling constraints presented in Equations (29) and (30) under the symmetric NBS's rule, the procedure indirectly considers the transmit power constraints presented in Equations (31)- (34). Then, with high QoS heterogeneity amongst the ℝ CRV-SUs (i.e., concerning the subcarrier's stringent QoS requirements and interference conditions), ℂ * < ℂ * does not necessarily indicate that * < * , and vice versa, which leads to transmit power inefficiency. Therefore, to overcome this, the following optimal transmit power scheduling method is introduced, in order to increase the transmit power efficiency.
where * represents the optimal CRV-SU.
Proof. Appendix B presents the Proof of Theorem 4. □ Using Equation (37), matrix ℙ ×ℝ * ×ℝ is obtained, which indicates that the optimal transmit power is allocated to the optimal CRV-SU * on subcarrier . Therefore, the optimal transmit power of each CRV-SU can be determined through * = ∑ * , ∀ . Likewise, accounting for ℙ ×ℝ * ×ℝ , the optimal subcarrier allocation can be defined as shown here.
Theorem 5. The SNO-CRAVNET optimal subcarrier allocation strategy is given by ℂ ×ℝ * ×ℝ = ℂ * , where the individual matrix elements are determined by Sensors 2020, 20, x FOR PEER REVIEW Based on Equation (35), the matrix ℂ ×ℝ * ×ℝ = ℂ * , which illustrates the ti scheduling of each subcarrier for all the ℝ CRV-SUs, is determined. Additionally, this fu in determining the quality of each subcarrier, that is, based on Equation (35), if it is ob ℂ * < ℂ * , this indicates that even though both subcarriers were allocated an equal transmit power, subcarrier requires less time than subcarrier for the transfer o amount of information by the same CRV-SU . Therefore, subcarrier is in better con has a higher quality in comparison to subcarrier . Furthermore, accounting for ℂ ×ℝ * optimal transmit power scheduling strategy is defined as shown here. Proof. Appendix B presents the Proof of Theorem 3. □ The optimal transmit power scheduling for CRV-SU on every subcarrier is obt Theorem 3. In other words, the optimal transmit power that subcarrier requires to transmit a given amount of information based on the licensed PU's protection para associated subcarrier's conditions, and the characteristics of CRV-SU is denoted by Consequently, through a linear search of the subcarriers, efficient resource sche be performed, for instance, for = 1 to , find the optimal CRV-SU * = arg min allocate the corresponding transmit power as defined in Equation (36) to all the selecte * s. Despite the fact that the procedure derives ℂ * s accounting for the subcarrier constraints presented in Equations (29) and (30) under the symmetric NBS's rule, the indirectly considers the transmit power constraints presented in Equations (31)- (34). Then QoS heterogeneity amongst the ℝ CRV-SUs (i.e., concerning the subcarrier's stri requirements and interference conditions), ℂ * < ℂ * does not necessarily indicate * , and vice versa, which leads to transmit power inefficiency. Therefore, to overcom following optimal transmit power scheduling method is introduced, in order to increase t power efficiency. is obtained, which indicates that t transmit power is allocated to the optimal CRV-SU * on subcarrier . Therefore, t transmit power of each CRV-SU can be determined through * = ∑ * , ∀ accounting for ℙ ×ℝ * ×ℝ , the optimal subcarrier allocation can be defined as shown Therefore, from Equations (14) and (39), the overall optimal throughput of the SNO-CRAVNET system amongst all the R CRV-SUs and subcarriers can be obtained, for instance, mn , P * mn . Furthermore, by substituting C * mn and P * mn in the transformed convex optimization problem presented in Equations (28)-(34), it can be observed that an upper-bound of the maximum SNO-CRAVNET overall system throughput defined as E * can be obtained, where * represents the reachable (i.e., maximum) SNO-CRAVNET overall system throughput of subcarrier n * . On the contrary, by substituting C * mn and P * mn in the original optimization problem presented in Equations (21)-(27), a lower-bound of the reachable data-rate defined as E * can be obtained. Without a loss of generality, let Tot. be the total reachable data-rate obtained by a combinatorial search of the original optimization problem presented in Equations (21)- (27); then, E * ≥ E Tot. ≥ E * . Therefore, the difference that exists between the lower-bound E * and the upper-bound E * of the maximum SNO-CRAVNET overall system throughput indicates how far apart the proposed scheme is from the actual optimal solution. Consequently, based on the experimental results shown in Section 5.2.1, it is shown that, in the case of the proposed scheme, the gap that exists between the lower-bound E * and the upper-bound E * of the maximum SNO-CRAVNET overall system throughput is insignificantly small, for instance, smaller than 0.016%.

Simulation Settings
As depicted in Figure 1, the co-existence of the cellular OFDMA-based overlay CR-Assisted Vehicular NETwork with the PU network scenario of seven CCs (i.e., J = {L 1 , L 2 , · · · , L 7 }) is considered. In each of the seven CCs (i.e., locations), there are two shared wireless channels. As defined in Equation (4), in the shared wireless channel m, the activity of PUs is modeled by T m for m = 1, 2.
The simulation experiments use a system which consists of R = 10 CRV-SUs, R = 20 CRV-SUs, and N C = 64, with a total power P Tot. = 3 W. In each CRV-SU node, the duration of the time-slot is 20 ms and the queue size Q is 20 packets. The radius of the CCs is 5 km and the average packet arrival rate follows a Poisson process, with τ = 0.5 packets per time-slot. The average speed of the CRV-SUs is 50 km/h. The frequency selective fading subcarrier involves six independent Rayleigh fading multipaths with an exponential power delay profile (PDP) of 100 ns. The transmit power for each of the R CRV-SUs is constrained by the threshold P min n = 0.5 W. The rest of the parameters used in the simulations with their set values are shown in Table 3.

Discussion of the Results
The performance evaluation of SNO-CRAVNET was carried out in comparison with existing relevant scheduling schemes for CR-Assisted Vehicular NETwork systems. The relevant reference schemes selected for the purpose of performance evaluation against SNO-CRAVNET were the Dependent Rounding-based Scheme (DR) [30], Pure Nash Equilibrium Search scheme (PNE-S) [31], and Cuckoo Search scheme (CS) with Multi-objective Optimization based on the Decomposition scheme (MOCS/D) [32]. To ensure that consistency and fairness were maintained regarding the comparisons of the proposed SNO-CRAVNET and reference schemes, derivation of the optimal strategies of DR, PNE-S, and MOCS/D was achieved through optimization problems involving the constraints of CR, as shown in Equations (7) and (9), and further system throughput optimization constraints for the minimal CRV-SU's utility requirement U 0 n . For example, U 0 mn , P * mn for PNE-S and MOCS/D. The cost functions which correspond to each of the scenarios are shown below:

System Throughput Evaluation
The performance of the proposed SNO-CRAVNET is depicted in Figure 3 through a comparison of the reference schemes DR, PNE-S, and MOCS/D, using the overall achieved system throughput measured against the overall supplied transmit power. The overall achieved average system throughput of each of the schemes, as expected, sharply increases with a corresponding increase in the total supplied transmit power. As can been seen in Figure 3 overleaf, the slightly higher overall achieved average system throughput of the PNE-S in comparison to the proposed SNO-CRAVNET is because PNE-S does not take into account the resource allocation fairness among the CRV-SUs, as opposed to SNO-CRAVNET, DR, and MOCS/D. On the other hand, although the performance of DR is nearly equal to that of the proposed SNO-CRAVNET and a little above the performance of MOCS/D, Figure 3 clearly shows that SNO-CRAVNET outperforms both. The same occurs in Figure 4, where the performance of all the schemes is seen to increase accordingly with a further increase in the number of CRV-SUs from 7 to 14. As is the case in Figure 3, SNO-CRAVNET still outperforms both DR and MOCS/D. This could be explained by the fact that DR requires additional transmit power in comparison to SNO-CRAVNET, whereas MOCS/D fails to utilize the subcarrier resources opportunistically, thereby resulting in a lower overall average system throughput performance, as can be seen in both Figures 3 and 4

System Throughput Evaluation
The performance of the proposed SNO-CRAVNET is depicted in Figure 3 through a comparison of the reference schemes DR, PNE-S, and MOCS/D, using the overall achieved system throughput measured against the overall supplied transmit power. The overall achieved average system throughput of each of the schemes, as expected, sharply increases with a corresponding increase in the total supplied transmit power. As can been seen in Figure 3 overleaf, the slightly higher overall achieved average system throughput of the PNE-S in comparison to the proposed SNO-CRAVNET is because PNE-S does not take into account the resource allocation fairness among the CRV-SUs, as opposed to SNO-CRAVNET, DR, and MOCS/D. On the other hand, although the performance of DR is nearly equal to that of the proposed SNO-CRAVNET and a little above the performance of MOCS/D, Figure 3 clearly shows that SNO-CRAVNET outperforms both. The same occurs in Figure 4, where the performance of all the schemes is seen to increase accordingly with a further increase in the number of CRV-SUs from 7 to 14. As is the case in Figure 3, SNO-CRAVNET still outperforms both DR and MOCS/D. This could be explained by the fact that DR requires additional transmit power in comparison to SNO-CRAVNET, whereas MOCS/D fails to utilize the subcarrier resources opportunistically, thereby resulting in a lower overall average system throughput performance, as can be seen in both Figures 3 and 4 below.

Average Throughput Gain Evaluation
In Figure 5 below, the performance evaluation of SNO-CRAVNET in comparison to DR, PNE-S, and MOCS/D using the overall achieved average throughput gain measured against the varying number of CRV-SUs is presented. Supposing the scheduler allocated an ℜ data-rate, the Sensors 2020, 20, x FOR PEER REVIEW 17 of 27

System Throughput Evaluation
The performance of the proposed SNO-CRAVNET is depicted in Figure 3 through a comparison of the reference schemes DR, PNE-S, and MOCS/D, using the overall achieved system throughput measured against the overall supplied transmit power. The overall achieved average system throughput of each of the schemes, as expected, sharply increases with a corresponding increase in the total supplied transmit power. As can been seen in Figure 3 overleaf, the slightly higher overall achieved average system throughput of the PNE-S in comparison to the proposed SNO-CRAVNET is because PNE-S does not take into account the resource allocation fairness among the CRV-SUs, as opposed to SNO-CRAVNET, DR, and MOCS/D. On the other hand, although the performance of DR is nearly equal to that of the proposed SNO-CRAVNET and a little above the performance of MOCS/D, Figure 3 clearly shows that SNO-CRAVNET outperforms both. The same occurs in Figure 4, where the performance of all the schemes is seen to increase accordingly with a further increase in the number of CRV-SUs from 7 to 14. As is the case in Figure 3, SNO-CRAVNET still outperforms both DR and MOCS/D. This could be explained by the fact that DR requires additional transmit power in comparison to SNO-CRAVNET, whereas MOCS/D fails to utilize the subcarrier resources opportunistically, thereby resulting in a lower overall average system throughput performance, as can be seen in both Figures 3 and 4 below.

Average Throughput Gain Evaluation
In Figure 5 below, the performance evaluation of SNO-CRAVNET in comparison to DR, PNE-S, and MOCS/D using the overall achieved average throughput gain measured against the varying number of CRV-SUs is presented. Supposing the scheduler allocated an ℜ data-rate, the

Average Throughput Gain Evaluation
In Figure 5 below, the performance evaluation of SNO-CRAVNET in comparison to DR, PNE-S, and MOCS/D using the overall achieved average throughput gain measured against the varying number of CRV-SUs is presented. Supposing the scheduler allocated an Allocated data-rate, the overall achieved average throughput gain could be calculated as Allocated − R n=1 U 0 n . As shown in Figure 5, the PNE-S obtained a slightly higher overall average throughput gain because of its non-fairness consideration among the R CRV-SUs. In the case of SNO-CRAVNET, PNE-S only had a relatively marginal higher average throughput gain. For instance, when R = 6, the PNE Search scheme achieved an average throughput gain of 0.48 bits/s/Hz, whereas the proposed SNO-CRAVNET achieved 0.46 bits/s/Hz. Therefore, even though PNE-S does not consider fairness, it only outperformed the proposed fairness-considerate SNO-CRAVNET by 0.02 bits/s/Hz. However, in comparison to other scheduling schemes such as DR and MOCS/D that consider fairness, the proposed SNO-CRAVNET recorded a significantly higher overall average throughput gain, as can be seen in Figure 5. For example, when R = 14, SNO-CRAVNET achieved a value that was 0.5 bits/s/Hz and 1.6 bits/s/Hz higher than that of DR and MOCS/D, respectively. . As shown in Figure 5, the PNE-S obtained a slightly higher overall average throughput gain because of its nonfairness consideration among the ℝ CRV-SUs. In the case of SNO-CRAVNET, PNE-S only had a relatively marginal higher average throughput gain. For instance, when ℝ = 6, the PNE Search scheme achieved an average throughput gain of 0.48 bits/s/Hz, whereas the proposed SNO-CRAVNET achieved 0.46 bits/s/Hz. Therefore, even though PNE-S does not consider fairness, it only outperformed the proposed fairness-considerate SNO-CRAVNET by 0.02 bits/s/Hz. However, in comparison to other scheduling schemes such as DR and MOCS/D that consider fairness, the proposed SNO-CRAVNET recorded a significantly higher overall average throughput gain, as can be seen in Figure 5. For example, when ℝ = 14, SNO-CRAVNET achieved a value that was 0.5 bits/s/Hz and 1.6 bits/s/Hz higher than that of DR and MOCS/D, respectively.  Figure 6 demonstrates the performance evaluation of the proposed SNO-CRAVNET against the existing related schemes using the total transmit power gain measured against a varying number of CRV-SUs. Based on the assumption that and Ᵽ are the minimum power required by a CRV-SU and the minimum power required by a scheduler to guarantee the QoS requirements of each ℝ CRV-SU, the total transmit power gained is obtained as Ᵽ − ∑ ℝ

=1
. Figure 6 shows that SNO-CRAVNET achieves remarkably higher transmit power gain as the number of CRV-SUs increases compared with DR and MOCS/D. For instance, when ℝ = 8, SNO-CRAVNET achieved 0.01 W and 0.04 W of total transmit power gain more than DR and MOCS/D, respectively. Similarly, when ℝ = 14, SNO-CRAVNET achieved 0.02 W and 0.08 W of total transmit power gain more than DR and MOCS/D, respectively.  Figure 6 demonstrates the performance evaluation of the proposed SNO-CRAVNET against the existing related schemes using the total transmit power gain measured against a varying number of CRV-SUs. Based on the assumption that P min n and Allocated are the minimum power required by a CRV-SU and the minimum power required by a scheduler to guarantee the QoS requirements of each R CRV-SU, the total transmit power gained is obtained as Allocated − R n=1 P min n . Figure 6 shows that SNO-CRAVNET achieves remarkably higher transmit power gain as the number of CRV-SUs increases compared with DR and MOCS/D. For instance, when R = 8, SNO-CRAVNET achieved 0.01 W and 0.04 W of total transmit power gain more than DR and MOCS/D, respectively. Similarly, when R = 14, SNO-CRAVNET achieved 0.02 W and 0.08 W of total transmit power gain more than DR and MOCS/D, respectively.

Jain's Fairness Index (JFI) Evaluation
The resource fairness provision was investigated, as depicted in Figure 7, through performance evaluation using JFI measured against a varying number of CRV-SUs. According to [2,33], the JFI is expressed as

Jain's Fairness Index (JFI) Evaluation
The resource fairness provision was investigated, as depicted in Figure 7, through performance evaluation using JFI measured against a varying number of CRV-SUs. According to [2,33], the JFI is expressed as where n denotes CRV-SU n's rate allocation. Consequently, JFI = 1 indicates perfectly fair resource allocation by the scheduler. Conversely, JFI reduces towards 0 with an increase in the CRV-SUs rate's disparity. In Figure 7, it can be seen that MOCS/D achieved perfectly fair resource allocation (i.e., JFI = 1) due to non-opportunistic scheduling, but recorded a low overall average system throughput performance, as can be observed in both Figures 3 and 4, and high transmit power demands. In contrast, as expected, the fairness inconsiderate PNE-S is the most unfair amongst the schemes and achieved the most imperfectly fair resource allocation (i.e., JFI = 0) when 2 ≤ R ≤ 14. However, Figure 7 demonstrates that both SNO-CRAVNET and DR can achieve fair resource allocation.
Although it can be observed that their performances decrease with an increase in the number of CRV-SUs, SNO-CRAVNET continuously outperformed DR in all cases, (i.e., 2 ≤ R ≤ 14). Accordingly, DR is forced to allocate resources less fairly in comparison to SNO-CRAVNET due to resource starvation. In general, SNO-CRAVNET shows a performance gain (nearly 5% improvement) over DR, as is evident in Figure 7, in terms of the percentage of JFI achieved. Obviously, in the case of both SNO-CRAVNET and DR, when the allocated transmit power is insufficient, the JFI value decreases.

Accuracy of the Proposed Method
The accuracy of the proposed time-sharing approach is studied in Figures 8 and 9, which depict comparisons of the derived optimal strategies through a combinatorial search within the original optimization problem expressed in Equations (11) and (22)- (27) and the reformulated convex optimization problem presented in Equations (13) and (29)-(34) as discussed in Section 3, respectively. The comparisons were performed with respect to (wrt) the optimal supplied transmit power and achievable optimal throughput of CRV-SU 1 (see Figure 8) and CRV-SU 2 (see Figure 9). Furthermore, for other CRV-SUs, similar results were obtained. As clearly demonstrated in Figure 8 and Figure 9, the transmit power and the achievable optimal throughput values obtained in the case of CRV-SU 1 (see Figure 8) and CRV-SU 2 (see Figure 9) are nearly the same and the performance gaps in both cases are infinitesimally trivial, such as 0.014%. Therefore, both Figures 8 and 9 show that the proposed time-sharing SNO-CRAVNET allocation based on Equations (13) and (29)-(34) is capable of achieving up to an average of 99.987% for the total theoretical capacity.

Accuracy of the Proposed Method
The accuracy of the proposed time-sharing approach is studied in Figures 8 and 9, which depict comparisons of the derived optimal strategies through a combinatorial search within the original optimization problem expressed in Equations (11) and (22)- (27) and the reformulated convex optimization problem presented in Equations (13) and (29)-(34) as discussed in Section 3, respectively. The comparisons were performed with respect to (wrt) the optimal supplied transmit power and achievable optimal throughput of CRV-SU 1 (see Figure 8) and CRV-SU 2 (see Figure 9). Furthermore, for other CRV-SUs, similar results were obtained. As clearly demonstrated in Figures 8 and 9, the transmit power and the achievable optimal throughput values obtained in the case of CRV-SU 1 (see Figure 8) and CRV-SU 2 (see Figure 9) are nearly the same and the performance gaps in both cases are infinitesimally trivial, such as 0.014%. Therefore, both Figures 8 and 9 show that the proposed optimization problem presented in Equations (13) and (29)-(34) as discussed in Section 3, respectively. The comparisons were performed with respect to (wrt) the optimal supplied transmit power and achievable optimal throughput of CRV-SU 1 (see Figure 8) and CRV-SU 2 (see Figure 9). Furthermore, for other CRV-SUs, similar results were obtained. As clearly demonstrated in Figure 8 and Figure 9, the transmit power and the achievable optimal throughput values obtained in the case of CRV-SU 1 (see Figure 8) and CRV-SU 2 (see Figure 9) are nearly the same and the performance gaps in both cases are infinitesimally trivial, such as 0.014%. Therefore, both Figures 8 and 9 show that the proposed time-sharing SNO-CRAVNET allocation based on Equations (13) and (29)-(34) is capable of achieving up to an average of 99.987% for the total theoretical capacity. Figure 8. Performance evaluation using optimal throughput measured against the optimal supplied transmit power for CRV-SU 1. . Performance evaluation using optimal throughput measured against the optimal supplied transmit power for CRV-SU 2.

Conclusions
This paper has presented an efficient joint optimal subcarrier and transmit power allocation framework with QoS guarantee to support enhanced packet transmission over a Cognitive Radioenabled IoV network system. The study proposed a novel SNBS-based wireless radio resource scheduling scheme in an OFDMA CR-enabled IoV network system. The CRV-SUs form clusters, leading to an improved CR-enabled IoV communication efficiency in a network system over the shared wireless radio channels (i.e., the channels that belong to licensed PUs). Although the shared wireless radio channels are primarily allocated to the PUs, the same channels can be opportunistically accessed by the CRV-SUs on the condition that the SINR with the PUs is maintained below the threshold level. Furthermore, a convex optimization problem was formulated by applying a timesharing technique. The formulated convex optimization problem involves constraints on CR technology regulations, joint optimal subcarrier, and transmit power allocation. Then, the optimal subcarrier and transmit power allocation strategies were derived via mathematical analysis. The developed iteration-independent and low-complexity algorithm ensures easy convergence to Pareto optimality. Theoretical analysis and simulation results show that the proposed SNO-CRAVNET outperformed the reference scheduler schemes. In comparison to other scheduling schemes that are fairness-considerate, the proposed SNO-CRAVNET recorded a significantly higher overall average throughput gain, as is shown in Figure 5. Similarly, the accuracy of the proposed time-sharing method wrt the optimal transmit power and the achievable optimal throughput of CRV-SU 1 and CRV-SU 2 was investigated. It is shown in Figure 6 that the proposed time-sharing SNO-CRAVNET allocation based on the reformulated convex optimization problem is capable of achieving up to an Figure 9. Performance evaluation using optimal throughput measured against the optimal supplied transmit power for CRV-SU 2.

Conclusions
This paper has presented an efficient joint optimal subcarrier and transmit power allocation framework with QoS guarantee to support enhanced packet transmission over a Cognitive Radio-enabled IoV network system. The study proposed a novel SNBS-based wireless radio resource scheduling scheme in an OFDMA CR-enabled IoV network system. The CRV-SUs form clusters, leading to an improved CR-enabled IoV communication efficiency in a network system over the shared wireless radio channels (i.e., the channels that belong to licensed PUs). Although the shared wireless radio channels are primarily allocated to the PUs, the same channels can be opportunistically accessed by the CRV-SUs on the condition that the SINR with the PUs is maintained below the threshold level. Furthermore, a convex optimization problem was formulated by applying a time-sharing technique. The formulated convex optimization problem involves constraints on CR technology regulations, joint optimal subcarrier, and transmit power allocation. Then, the optimal subcarrier and transmit power allocation strategies were derived via mathematical analysis. The developed iteration-independent and low-complexity algorithm ensures easy convergence to Pareto optimality. Theoretical analysis and simulation results show that the proposed SNO-CRAVNET outperformed the reference scheduler schemes. In comparison to other scheduling schemes that are fairness-considerate, the proposed SNO-CRAVNET recorded a significantly higher overall average throughput gain, as is shown in Figure 5. Similarly, the accuracy of the proposed time-sharing method wrt the optimal transmit power and the achievable optimal throughput of CRV-SU 1 and CRV-SU 2 was investigated. It is shown in Figure 6 that the proposed time-sharing SNO-CRAVNET allocation based on the reformulated convex optimization problem is capable of achieving up to an average of 99.987% for the total theoretical capacity. In the same vein, the proposed SNO-CRAVNET scheme outperformed the other reference scheduling schemes in terms of fair resource allocation, which further emphasizes that the open issue of joint optimal subcarrier and transmit power allocation with QoS guarantee for enhanced data transmission over CR-IoVs was achieved.
An investigation of the integration of spectrum sensing in interweave-based CR-enabled IoV network systems represents an interesting possible future research direction. Under the interweave-based CR-enabled IoV network systems, the spectrum sensing accuracy remains an open issue due to prevailing sensing errors over wireless channels. Additionally, as part of future work, a hidden CRV-SU problem will be considered in deriving the transition probability matrix, in order to further understand how the presence of hidden CRV-SUs may affect the transition probability matrix and transmit power allocation.

Acknowledgments:
The authors wish to thank the reviewers for their comments and suggestions, which have greatly helped in improving the quality of the paper.

Conflicts of Interest:
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.
In Equation (28), represents the distance between nth CRV-SU and th PU, while density (i.e., noise density) of the th PU. With the help of Loca instance, the Global Positioning System (GPS), both distances, obtained. In addition, information on the CRV-SU's features can be feedback channels. Therefore, without a loss of generality, Equatio and expressed as stipulated condition guarantees that the potential transmissions of t only if the CRV-SU 's total transmit power is constrained over threshold Ᵽ .

The Utility of SNO-CRAVNET and Problem Formulation
The design methodology of the SNO-CRAVNET's objectives w presented in this section in the form of a convex optimization prob represented by the ℝ CRV-SUs. The design of the game bargaining sch enabled IoV network system is proposed in this section. We assume th CRV-SU , has an initial utility ≥ 0, which represents its accep with respect to the data-rate and the corresponding utility function bargaining (SNB) theory, each utility function is usually designate of ℝ = ℧ , with ℝ and ℧ denoting the set of game theory strat and utility vectors' space, respectively. Let us assume that is con ℝ CRV-SU players. Then, it follows that at least a feasible subspace ℧ vector, for instance, ( ) = , , , ⋯ , ℝ , becomes equal or bigg utility vector, such as, = , , , ⋯ , ℝ . Therefore, the subset expressed as ℧ = ∈ ℧| ( ) ≥ . Additionally, let us suppose achieved is denoted by Ʈ = ( )| ∈ ℧ and the category of sets of which is the minimum utility bound, is denoted as ℬ = Ʈ, |Ʈ ⊂ ℝ the Symmetric NBS theory (see [23]), there exists a unique solution, fo satisfies the following axioms: a) Տ (Ʈ, ) ensures a minimum utility guarantee, for instance ∈ Ʈ| ≥ , ∀ ; b) Տ (Ʈ, ) is the Pareto optimal, which implies that other all guaranteeing a higher performance for all the ℝ CRV-SUs simu Տ (Ʈ, ) < Տ (Ʈ, ), ∃ and Տ (Ʈ, ) ≤ Տ (Ʈ, ), ∀ ; c) Տ (Ʈ, ) guarantees symmetry, which implies that all the ℝ for instance, supposing that Ʈ is symmetric with regards to a and ∈ Ʈ, , ∈ so that = implies that Տ (Ʈ, d) Տ (Ʈ, ) guarantees fairness by maintaining the independen instance, if the feasible set decreases and the solution keeps on b solution for the lesser achievable set remains the same point.
where denotes the exponent of path attenuation and is the th PU's transmit power. represents the distance between nth CRV-SU and th PU, while represents the noise spectral density (i.e., noise density) of the th PU. With the help of Location-Based Systems (LBSs), for instance, the Global Positioning System (GPS), both distances, and can be easily obtained. In addition, information on the CRV-SU's features can be obtained by the CR-BS through feedback channels. Therefore, without a loss of generality, Equation (18) can be further simplified and expressed as where . From Equation (19), the stipulated condition guarantees that the potential transmissions of the PU are fully protected if and only if the CRV-SU 's total transmit power is constrained over channel by the predefined threshold Ᵽ .

The Utility of SNO-CRAVNET and Problem Formulation
The design methodology of the SNO-CRAVNET's objectives with its SNO-CRAVNET game is presented in this section in the form of a convex optimization problem, with its associated players represented by the ℝ CRV-SUs. The design of the game bargaining scheme methodologies for the CRenabled IoV network system is proposed in this section. We assume that each ℝ CRV-SU, for instance, CRV-SU , has an initial utility ≥ 0, which represents its acceptable minimum QoS constraint with respect to the data-rate and the corresponding utility function . Under the symmetric Nash bargaining (SNB) theory, each utility function is usually designated as a convex and closed subset of ℝ = ℧ , with ℝ and ℧ denoting the set of game theory strategies of the ℝ CRV-SU players and utility vectors' space, respectively. Let us assume that is conveniently achievable for all the ℝ CRV-SU players. Then, it follows that at least a feasible subspace ℧ exists in ℧, so that the utility vector, for instance, ( ) = , , , ⋯ , ℝ , becomes equal or bigger in comparison to the initial utility vector, such as, = , , , ⋯ , ℝ . Therefore, the subset ℧ as the element of ℧ can be expressed as ℧ = ∈ ℧| ( ) ≥ . Additionally, let us suppose that the set of utility that can be achieved is denoted by Ʈ = ( )| ∈ ℧ and the category of sets of utility policies that satisfies , which is the minimum utility bound, is denoted as ℬ = Ʈ, |Ʈ ⊂ ℝ . Therefore, in accordance with the Symmetric NBS theory (see [23]), there exists a unique solution, for instance, Տ |ℬ ⟶ ℝ , which satisfies the following axioms: a) Տ (Ʈ, ) ensures a minimum utility guarantee, for instance, Տ (Ʈ, ) ∈ Ʈ , where Ʈ = ∈ Ʈ| ≥ , ∀ ; b) Տ (Ʈ, ) is the Pareto optimal, which implies that other allocations Տ (Ʈ, ) capable of guaranteeing a higher performance for all the ℝ CRV-SUs simultaneously do not exist, that is, Տ (Ʈ, ) < Տ (Ʈ, ), ∃ and Տ (Ʈ, ) ≤ Տ (Ʈ, ), ∀ ; c) Տ (Ʈ, ) guarantees symmetry, which implies that all the ℝ CRV-SUs have equal priorities, for instance, supposing that Ʈ is symmetric with regards to a sub-set ⊆ 1, 2, 3, ⋯ , , ⋯ , ℝ and ∈ Ʈ, , ∈ so that = implies that Տ (Ʈ, ) = Տ (Ʈ, ) , ≠ ; d) Տ (Ʈ, ) guarantees fairness by maintaining the independence of irrelevant alternatives, for instance, if the feasible set decreases and the solution keeps on being feasible, it follows that the solution for the lesser achievable set remains the same point. It can be expressed as ɰ ⊂ Ʈ, (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ (Ʈ, ) = Տ (ɰ, ), ∀ .
Without a loss of generality, the property of the SNO-CRAVNET is described using the following theorem. ongoing transmissions from PUs. Let the distance between the th PU and CR-BS be given as , so that another interference constraint to protect the PUs' transmission can be given as where denotes the exponent of path attenuation and is the th PU's transmit power. represents the distance between nth CRV-SU and th PU, while represents the noise spectral density (i.e., noise density) of the th PU. With the help of Location-Based Systems (LBSs), for instance, the Global Positioning System (GPS), both distances, and can be easily obtained. In addition, information on the CRV-SU's features can be obtained by the CR-BS through feedback channels. Therefore, without a loss of generality, Equation (18) can be further simplified and expressed as where . From Equation (19), the stipulated condition guarantees that the potential transmissions of the PU are fully protected if and only if the CRV-SU 's total transmit power is constrained over channel by the predefined threshold Ᵽ .

The Utility of SNO-CRAVNET and Problem Formulation
The design methodology of the SNO-CRAVNET's objectives with its SNO-CRAVNET game is presented in this section in the form of a convex optimization problem, with its associated players represented by the ℝ CRV-SUs. The design of the game bargaining scheme methodologies for the CRenabled IoV network system is proposed in this section. We assume that each ℝ CRV-SU, for instance, CRV-SU , has an initial utility ≥ 0, which represents its acceptable minimum QoS constraint with respect to the data-rate and the corresponding utility function . Under the symmetric Nash bargaining (SNB) theory, each utility function is usually designated as a convex and closed subset of ℝ = ℧ , with ℝ and ℧ denoting the set of game theory strategies of the ℝ CRV-SU players and utility vectors' space, respectively. Let us assume that is conveniently achievable for all the ℝ CRV-SU players. Then, it follows that at least a feasible subspace ℧ exists in ℧, so that the utility vector, for instance, ( ) = , , , ⋯ , ℝ , becomes equal or bigger in comparison to the initial utility vector, such as, = , , , ⋯ , ℝ . Therefore, the subset ℧ as the element of ℧ can be expressed as ℧ = ∈ ℧| ( ) ≥ . Additionally, let us suppose that the set of utility that can be achieved is denoted by Ʈ = ( )| ∈ ℧ and the category of sets of utility policies that satisfies , which is the minimum utility bound, is denoted as ℬ = Ʈ, |Ʈ ⊂ ℝ . Therefore, in accordance with the Symmetric NBS theory (see [23]), there exists a unique solution, for instance, Տ |ℬ ⟶ ℝ , which satisfies the following axioms: ensures a minimum utility guarantee, for instance, is the Pareto optimal, which implies that other allocations Տ (Ʈ, ) capable of guaranteeing a higher performance for all the ℝ CRV-SUs simultaneously do not exist, that is, Տ (Ʈ, ) < Տ (Ʈ, ), ∃ and Տ (Ʈ, ) ≤ Տ (Ʈ, ), ∀ ; c) Տ (Ʈ, ) guarantees symmetry, which implies that all the ℝ CRV-SUs have equal priorities, for instance, supposing that Ʈ is symmetric with regards to a sub-set ⊆ 1, 2, 3, ⋯ , , ⋯ , ℝ and ∈ Ʈ, , ∈ so that = implies that Տ (Ʈ, ) = Տ (Ʈ, ) , ≠ ; d) Տ (Ʈ, ) guarantees fairness by maintaining the independence of irrelevant alternatives, for instance, if the feasible set decreases and the solution keeps on being feasible, it follows that the solution for the lesser achievable set remains the same point. It can be expressed as ɰ ⊂ Ʈ, (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ (Ʈ, ) = Տ (ɰ, ), ∀ .
Without a loss of generality, the property of the SNO-CRAVNET is described using the following theorem. ∼ C mn , ∼ P mn is a concave function, given that any non-negative definite combination of concave functions is likewise concave. Additionally, over a convex set, the convex optimization problem expressed in Equations (28)-(34) is determined. Given that each constraint as shown in Equations (29)-(34), according to its affinity determining a convex set, the set defined by each of the constraints is convex, since the intersection of convex sets is convex according to [32]. The Proof of Proposition 1 is completed by the above presented argument. Proof of Proposition 2. Let the first order derivative of satisfies the following axioms: a) Տ (Ʈ, ) ensures a minimum utility guarantee, for instan ∈ Ʈ| ≥ , ∀ ; b) Տ (Ʈ, ) is the Pareto optimal, which implies that other a guaranteeing a higher performance for all the ℝ CRV-SUs sim Տ (Ʈ, ) < Տ (Ʈ, ), ∃ and Տ (Ʈ, ) ≤ Տ (Ʈ, ), ∀ c) Տ (Ʈ, ) guarantees symmetry, which implies that all the for instance, supposing that Ʈ is symmetric with regards to and ∈ Ʈ, , ∈ so that = implies that Տ (Ʈ, d) Տ (Ʈ, ) guarantees fairness by maintaining the independ instance, if the feasible set decreases and the solution keeps on solution for the lesser achievable set remains the same poin (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ (Ʈ, ) = Տ (ɰ, Without a loss of generality, the property of the SNO-CR following theorem. Additionally, let us suppose that the s achieved is denoted by Ʈ = ( )| ∈ ℧ and the category of sets of utility po which is the minimum utility bound, is denoted as ℬ = Ʈ, |Ʈ ⊂ ℝ . Therefo the Symmetric NBS theory (see [23]), there exists a unique solution, for instance satisfies the following axioms: ensures a minimum utility guarantee, for instance, Տ (Ʈ, is the Pareto optimal, which implies that other allocations guaranteeing a higher performance for all the ℝ CRV-SUs simultaneous Տ (Ʈ, ) < Տ (Ʈ, ), ∃ and Տ (Ʈ, ) ≤ Տ (Ʈ, ), ∀ ; c) Տ (Ʈ, ) guarantees symmetry, which implies that all the ℝ CRV-SUs for instance, supposing that Ʈ is symmetric with regards to a sub-set and ∈ Ʈ, , guarantees fairness by maintaining the independence of irre instance, if the feasible set decreases and the solution keeps on being feas solution for the lesser achievable set remains the same point. It can b (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ (Ʈ, ) = Տ (ɰ, ), ∀ .
Without a loss of generality, the property of the SNO-CRAVNET is following theorem. (A1) With all the the numerator can be expressed using the function H (y) = log 2 ((1 + y) − y)/ ln(2(1 + y)) , where y > 0. Hence, when y > 0, it can be easily verified that the first order derivative of H (y) over y is non-negative, that is, > 0. Therefore, it follows that H (y) is a strictly increasing function, which can be determined through H (y) > H (0) = 0. Therefore, increase for all , thereby satisfying ℛ , > .
Proof. Appendix A presents the Proof of Proposition 2. □ Proposition 2 certifies that the transformation of the objective in Equation (20), as wel Equation (21) to Ʈ , in Equation (28), can be achieved by exploiting the firmly incr property of the logarithm function.

Proposition 3. The utility function Ʈ , proposed in Equation (28) is Nash bargaining t compliant and, at the same time, satisfies the proportional fairness metric.
Proof. Appendix A presents the Proof of Proposition 3. □ In our case, Proposition 3 shows that, for the data-rate allocation, a unique Nash barg equilibrium can be obtained. Likewise, as a special case of the NBS fairness [29], proportional fa can be achieved when = 0, ∀ .

Optimal Resource Scheduling Strategies
The convex optimization problem's optimal solution, which is presented in Equation (28) the constraints expressed in Equations (29)- (34), is derived in this section. Additionally, a simp efficient strategy, which supports an iteration-independent joint transmit power and subc scheduling, is proposed. The optimal subcarrier allocation with a consideration of the sharing approach is a real number implying the fraction of time which subcarrier requires transmission of a given amount of information. Firstly, uniform transmit power scheduling, t = . ( • ) ⁄ , is performed for all the available subcarriers. Then, an equal amou information is transferred over all the available subcarriers. Secondly, based on the study carri by Hahne [29], the optimal time-sharing subcarrier scheduling strategy is obtained.
where Ᵽ = × ⁄ − × ( ) . From Equation (19), the stipulated condition guarantees that the potential transmissions of the PU are fully protected if and only if the CRV-SU 's total transmit power is constrained over channel by the predefined threshold Ᵽ .

The Utility of SNO-CRAVNET and Problem Formulation
The design methodology of the SNO-CRAVNET's objectives with its SNO-CRAVNET game is presented in this section in the form of a convex optimization problem, with its associated players represented by the ℝ CRV-SUs. The design of the game bargaining scheme methodologies for the CRenabled IoV network system is proposed in this section. We assume that each ℝ CRV-SU, for instance, CRV-SU , has an initial utility ≥ 0, which represents its acceptable minimum QoS constraint with respect to the data-rate and the corresponding utility function . Under the symmetric Nash bargaining (SNB) theory, each utility function is usually designated as a convex and closed subset of ℝ = ℧ , with ℝ and ℧ denoting the set of game theory strategies of the ℝ CRV-SU players and utility vectors' space, respectively. Let us assume that is conveniently achievable for all the ℝ CRV-SU players. Then, it follows that at least a feasible subspace ℧ exists in ℧, so that the utility vector, for instance, ( ) = , , , ⋯ , ℝ , becomes equal or bigger in comparison to the initial utility vector, such as, = , , , ⋯ , ℝ . Therefore, the subset ℧ as the element of ℧ can be expressed as ℧ = ∈ ℧| ( ) ≥ . Additionally, let us suppose that the set of utility that can be achieved is denoted by Ʈ = ( )| ∈ ℧ and the category of sets of utility policies that satisfies , which is the minimum utility bound, is denoted as ℬ = Ʈ, |Ʈ ⊂ ℝ . Therefore, in accordance with the Symmetric NBS theory (see [23]), there exists a unique solution, for instance, Տ |ℬ ⟶ ℝ , which satisfies the following axioms: ensures a minimum utility guarantee, for instance, is the Pareto optimal, which implies that other allocations Տ (Ʈ, ) capable of guaranteeing a higher performance for all the ℝ CRV-SUs simultaneously do not exist, that is, Տ (Ʈ, ) < Տ (Ʈ, ), ∃ and Տ (Ʈ, ) ≤ Տ (Ʈ, ), ∀ ; c) Տ (Ʈ, ) guarantees symmetry, which implies that all the ℝ CRV-SUs have equal priorities, for instance, supposing that Ʈ is symmetric with regards to a sub-set ⊆ 1, 2, 3, ⋯ , , ⋯ , ℝ and ∈ Ʈ, , ∈ so that = implies that Տ (Ʈ, ) = Տ (Ʈ, ) , ≠ ; d) Տ (Ʈ, ) guarantees fairness by maintaining the independence of irrelevant alternatives, for instance, if the feasible set decreases and the solution keeps on being feasible, it follows that the solution for the lesser achievable set remains the same point. It can be expressed as ɰ ⊂ Ʈ, (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ (Ʈ, ) = Տ (ɰ, ), ∀ .
Without a loss of generality, the property of the SNO-CRAVNET is described using the following theorem. Proof of Proposition 3. The utility function can be expressed with regards to the data-rate vector, that is,

rewritten as
obtained. In addition, information o feedback channels. Therefore, with and expressed as where Ᵽ = × stipulated condition guarantees tha only if the CRV-SU 's total tran threshold Ᵽ .

The Utility of SNO-CRAVNET a
The design methodology of the presented in this section in the form represented by the ℝ CRV-SUs. The enabled IoV network system is prop CRV-SU , has an initial utility with respect to the data-rate and th bargaining (SNB) theory, each utility of ℝ = ℧ , with ℝ and ℧ deno and utility vectors' space, respectiv ℝ CRV-SU players. Then, it follows vector, for instance, ( ) = , , utility vector, such as, = , , expressed as ℧ = ∈ ℧| ( ) ≥ achieved is denoted by Ʈ = ( )| which is the minimum utility bound the Symmetric NBS theory (see [23]) satisfies the following axioms: is the Pareto optim guaranteeing a higher perform Տ (Ʈ, ) < Տ (Ʈ, ), ∃ an c) Տ (Ʈ, ) guarantees symme for instance, supposing that Ʈ and ∈ Ʈ, , ∈ so that d) Տ (Ʈ, ) guarantees fairness instance, if the feasible set decr solution for the lesser achieva (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ Without a loss of generality, following theorem.
H represents the vector transpose.
In Equation (28), the disagreement points of represents the distance between nth CRV-SU and th PU, while represents the noise spectral density (i.e., noise density) of the th PU. With the help of Location-Based Systems (LBSs), for instance, the Global Positioning System (GPS), both distances, and can be easily obtained. In addition, information on the CRV-SU's features can be obtained by the CR-BS through feedback channels. Therefore, without a loss of generality, Equation (18) can be further simplified and expressed as where . From Equation (19), the stipulated condition guarantees that the potential transmissions of the PU are fully protected if and only if the CRV-SU 's total transmit power is constrained over channel by the predefined threshold Ᵽ .

The Utility of SNO-CRAVNET and Problem Formulation
The design methodology of the SNO-CRAVNET's objectives with its SNO-CRAVNET game is presented in this section in the form of a convex optimization problem, with its associated players represented by the ℝ CRV-SUs. The design of the game bargaining scheme methodologies for the CRenabled IoV network system is proposed in this section. We assume that each ℝ CRV-SU, for instance, CRV-SU , has an initial utility ≥ 0, which represents its acceptable minimum QoS constraint with respect to the data-rate and the corresponding utility function . Under the symmetric Nash bargaining (SNB) theory, each utility function is usually designated as a convex and closed subset of ℝ = ℧ , with ℝ and ℧ denoting the set of game theory strategies of the ℝ CRV-SU players and utility vectors' space, respectively. Let us assume that is conveniently achievable for all the ℝ CRV-SU players. Then, it follows that at least a feasible subspace ℧ exists in ℧, so that the utility vector, for instance, ( ) = , , , ⋯ , ℝ , becomes equal or bigger in comparison to the initial utility vector, such as, = , , , ⋯ , ℝ . Therefore, the subset ℧ as the element of ℧ can be expressed as ℧ = ∈ ℧| ( ) ≥ . Additionally, let us suppose that the set of utility that can be achieved is denoted by Ʈ = ( )| ∈ ℧ and the category of sets of utility policies that satisfies , which is the minimum utility bound, is denoted as ℬ = Ʈ, |Ʈ ⊂ ℝ . Therefore, in accordance with the Symmetric NBS theory (see [23]), there exists a unique solution, for instance, Տ |ℬ ⟶ ℝ , which satisfies the following axioms: ensures a minimum utility guarantee, for instance, is the Pareto optimal, which implies that other allocations Տ (Ʈ, ) capable of guaranteeing a higher performance for all the ℝ CRV-SUs simultaneously do not exist, that is, Տ (Ʈ, ) < Տ (Ʈ, ), ∃ and Տ (Ʈ, ) ≤ Տ (Ʈ, ), ∀ ; c) Տ (Ʈ, ) guarantees symmetry, which implies that all the ℝ CRV-SUs have equal priorities, for instance, supposing that Ʈ is symmetric with regards to a sub-set ⊆ 1, 2, 3, ⋯ , , ⋯ , ℝ and ∈ Ʈ, , ∈ so that = implies that Տ (Ʈ, ) = Տ (Ʈ, ) , ≠ ; d) Տ (Ʈ, ) guarantees fairness by maintaining the independence of irrelevant alternatives, for instance, if the feasible set decreases and the solution keeps on being feasible, it follows that the solution for the lesser achievable set remains the same point. It can be expressed as ɰ ⊂ Ʈ, (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ (Ʈ, ) = Տ (ɰ, ), ∀ .
Without a loss of generality, the property of the SNO-CRAVNET is described using the following theorem.
( ∼ C mn , ∼ P mn ), for equilibrium analysis, are set to U 0 n = 0. Therefore, represents the distance between nt density (i.e., noise density) of the instance, the Global Positioning obtained. In addition, information feedback channels. Therefore, with and expressed as where Ᵽ = stipulated condition guarantees th only if the CRV-SU 's total tra threshold Ᵽ .

The Utility of SNO-CRAVNET
The design methodology of th presented in this section in the for represented by the ℝ CRV-SUs. The enabled IoV network system is pro CRV-SU , has an initial utility with respect to the data-rate and t bargaining (SNB) theory, each utili of ℝ = ℧ , with ℝ and ℧ den and utility vectors' space, respectiv ℝ CRV-SU players. Then, it follows vector, for instance, ( ) = , utility vector, such as, = , expressed as ℧ = ∈ ℧| ( ) ≥ achieved is denoted by Ʈ = ( ) which is the minimum utility boun the Symmetric NBS theory (see [23] satisfies the following axioms: is the Pareto opti guaranteeing a higher perform Տ (Ʈ, ) < Տ (Ʈ, ), ∃ a c) Տ (Ʈ, ) guarantees symm for instance, supposing that and ∈ Ʈ, , ∈ so that d) Տ (Ʈ, ) guarantees fairnes instance, if the feasible set dec solution for the lesser achiev (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ Without a loss of generality following theorem.

(R) is minimized to
where denotes the exponent of path attenuation and is the th PU's transmit power. represents the distance between nth CRV-SU and th PU, while represents the noise spectral density (i.e., noise density) of the th PU. With the help of Location-Based Systems (LBSs), for instance, the Global Positioning System (GPS), both distances, and can be easily obtained. In addition, information on the CRV-SU's features can be obtained by the CR-BS through feedback channels. Therefore, without a loss of generality, Equation (18) can be further simplified and expressed as where . From Equation (19), the stipulated condition guarantees that the potential transmissions of the PU are fully protected if and only if the CRV-SU 's total transmit power is constrained over channel by the predefined threshold Ᵽ .

The Utility of SNO-CRAVNET and Problem Formulation
The design methodology of the SNO-CRAVNET's objectives with its SNO-CRAVNET game is presented in this section in the form of a convex optimization problem, with its associated players represented by the ℝ CRV-SUs. The design of the game bargaining scheme methodologies for the CRenabled IoV network system is proposed in this section. We assume that each ℝ CRV-SU, for instance, CRV-SU , has an initial utility ≥ 0, which represents its acceptable minimum QoS constraint with respect to the data-rate and the corresponding utility function . Under the symmetric Nash bargaining (SNB) theory, each utility function is usually designated as a convex and closed subset of ℝ = ℧ , with ℝ and ℧ denoting the set of game theory strategies of the ℝ CRV-SU players and utility vectors' space, respectively. Let us assume that is conveniently achievable for all the ℝ CRV-SU players. Then, it follows that at least a feasible subspace ℧ exists in ℧, so that the utility vector, for instance, ( ) = , , , ⋯ , ℝ , becomes equal or bigger in comparison to the initial utility vector, such as, = , , , ⋯ , ℝ . Therefore, the subset ℧ as the element of ℧ can be expressed as ℧ = ∈ ℧| ( ) ≥ . Additionally, let us suppose that the set of utility that can be achieved is denoted by Ʈ = ( )| ∈ ℧ and the category of sets of utility policies that satisfies , which is the minimum utility bound, is denoted as ℬ = Ʈ, |Ʈ ⊂ ℝ . Therefore, in accordance with the Symmetric NBS theory (see [23]), there exists a unique solution, for instance, Տ |ℬ ⟶ ℝ , which satisfies the following axioms: ensures a minimum utility guarantee, for instance, is the Pareto optimal, which implies that other allocations Տ (Ʈ, ) capable of guaranteeing a higher performance for all the ℝ CRV-SUs simultaneously do not exist, that is, Տ (Ʈ, ) < Տ (Ʈ, ), ∃ and Տ (Ʈ, ) ≤ Տ (Ʈ, ), ∀ ; c) Տ (Ʈ, ) guarantees symmetry, which implies that all the ℝ CRV-SUs have equal priorities, for instance, supposing that Ʈ is symmetric with regards to a sub-set ⊆ 1, 2, 3, ⋯ , , ⋯ , ℝ and ∈ Ʈ, , ∈ so that = implies that Տ (Ʈ, ) = Տ (Ʈ, ) guarantees fairness by maintaining the independence of irrelevant alternatives, for instance, if the feasible set decreases and the solution keeps on being feasible, it follows that the solution for the lesser achievable set remains the same point. It can be expressed as ɰ ⊂ Ʈ, (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ (Ʈ, ) = Տ (ɰ, ), ∀ .
Without a loss of generality, the property of the SNO-CRAVNET is described using the following theorem.
. The objective function with regards to Proposition 2 can be expressed as ln where denotes the exponent of path attenuation and is the th PU's transmit power. represents the distance between nth CRV-SU and th PU, while represents the noise spectral density (i.e., noise density) of the th PU. With the help of Location-Based Systems (LBSs), for instance, the Global Positioning System (GPS), both distances, and can be easily obtained. In addition, information on the CRV-SU's features can be obtained by the CR-BS through feedback channels. Therefore, without a loss of generality, Equation (18) can be further simplified and expressed as where . From Equation (19), the stipulated condition guarantees that the potential transmissions of the PU are fully protected if and only if the CRV-SU 's total transmit power is constrained over channel by the predefined threshold Ᵽ .

The Utility of SNO-CRAVNET and Problem Formulation
The design methodology of the SNO-CRAVNET's objectives with its SNO-CRAVNET game is presented in this section in the form of a convex optimization problem, with its associated players represented by the ℝ CRV-SUs. The design of the game bargaining scheme methodologies for the CRenabled IoV network system is proposed in this section. We assume that each ℝ CRV-SU, for instance, CRV-SU , has an initial utility ≥ 0, which represents its acceptable minimum QoS constraint with respect to the data-rate and the corresponding utility function . Under the symmetric Nash bargaining (SNB) theory, each utility function is usually designated as a convex and closed subset of ℝ = ℧ , with ℝ and ℧ denoting the set of game theory strategies of the ℝ CRV-SU players and utility vectors' space, respectively. Let us assume that is conveniently achievable for all the ℝ CRV-SU players. Then, it follows that at least a feasible subspace ℧ exists in ℧, so that the utility vector, for instance, ( ) = , , , ⋯ , ℝ , becomes equal or bigger in comparison to the initial utility vector, such as, = , , , ⋯ , ℝ . Therefore, the subset ℧ as the element of ℧ can be expressed as ℧ = ∈ ℧| ( ) ≥ . Additionally, let us suppose that the set of utility that can be achieved is denoted by Ʈ = ( )| ∈ ℧ and the category of sets of utility policies that satisfies , which is the minimum utility bound, is denoted as ℬ = Ʈ, |Ʈ ⊂ ℝ . Therefore, in accordance with the Symmetric NBS theory (see [23]), there exists a unique solution, for instance, Տ |ℬ ⟶ ℝ , which satisfies the following axioms: ensures a minimum utility guarantee, for instance, is the Pareto optimal, which implies that other allocations Տ (Ʈ, ) capable of guaranteeing a higher performance for all the ℝ CRV-SUs simultaneously do not exist, that is, Տ (Ʈ, ) < Տ (Ʈ, ), ∃ and Տ (Ʈ, ) ≤ Տ (Ʈ, ), ∀ ; c) Տ (Ʈ, ) guarantees symmetry, which implies that all the ℝ CRV-SUs have equal priorities, for instance, supposing that Ʈ is symmetric with regards to a sub-set ⊆ 1, 2, 3, ⋯ , , ⋯ , ℝ and ∈ Ʈ, , ∈ so that = implies that Տ (Ʈ, ) = Տ (Ʈ, ) guarantees fairness by maintaining the independence of irrelevant alternatives, for instance, if the feasible set decreases and the solution keeps on being feasible, it follows that the solution for the lesser achievable set remains the same point. It can be expressed as ɰ ⊂ Ʈ, (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ (Ʈ, ) = Տ (ɰ, ), ∀ .
Without a loss of generality, the property of the SNO-CRAVNET is described using the following theorem.
and ∼ P mn are the optimal channel, achievable data-rate, and transmit power allocation, respectively, then it follows that, at R = R * , the following condition stands: Sensors 2020, 20, x FOR PEER REVIEW ongoing transmissions from PUs. Let the distance between the th PU and CR-BS be giv , so that another interference constraint to protect the PUs' transmission can be given as where denotes the exponent of path attenuation and is the th PU's transmit power represents the distance between nth CRV-SU and th PU, while represents the noise sp density (i.e., noise density) of the th PU. With the help of Location-Based Systems (LBS instance, the Global Positioning System (GPS), both distances, and can be obtained. In addition, information on the CRV-SU's features can be obtained by the CR-BS th feedback channels. Therefore, without a loss of generality, Equation (18) can be further sim and expressed as . From Equation (19 stipulated condition guarantees that the potential transmissions of the PU are fully protected only if the CRV-SU 's total transmit power is constrained over channel by the pred threshold Ᵽ .

The Utility of SNO-CRAVNET and Problem Formulation
The design methodology of the SNO-CRAVNET's objectives with its SNO-CRAVNET g presented in this section in the form of a convex optimization problem, with its associated p represented by the ℝ CRV-SUs. The design of the game bargaining scheme methodologies for th enabled IoV network system is proposed in this section. We assume that each ℝ CRV-SU, for ins CRV-SU , has an initial utility ≥ 0, which represents its acceptable minimum QoS con with respect to the data-rate and the corresponding utility function . Under the symmetric bargaining (SNB) theory, each utility function is usually designated as a convex and closed of ℝ = ℧ , with ℝ and ℧ denoting the set of game theory strategies of the ℝ CRV-SU p and utility vectors' space, respectively. Let us assume that is conveniently achievable for ℝ CRV-SU players. Then, it follows that at least a feasible subspace ℧ exists in ℧, so that the vector, for instance, ( ) = , , , ⋯ , ℝ , becomes equal or bigger in comparison to the utility vector, such as, = , , , ⋯ , ℝ . Therefore, the subset ℧ as the element of ℧ expressed as ℧ = ∈ ℧| ( ) ≥ . Additionally, let us suppose that the set of utility that achieved is denoted by Ʈ = ( )| ∈ ℧ and the category of sets of utility policies that satisfi which is the minimum utility bound, is denoted as ℬ = Ʈ, |Ʈ ⊂ ℝ . Therefore, in accordanc the Symmetric NBS theory (see [23]), there exists a unique solution, for instance, Տ |ℬ ⟶ ℝ , satisfies the following axioms: ensures a minimum utility guarantee, for instance, is the Pareto optimal, which implies that other allocations Տ (Ʈ, ) capa guaranteeing a higher performance for all the ℝ CRV-SUs simultaneously do not exist, t Տ (Ʈ, ) < Տ (Ʈ, ), ∃ and Տ (Ʈ, ) ≤ Տ (Ʈ, ), ∀ ; guarantees symmetry, which implies that all the ℝ CRV-SUs have equal prio for instance, supposing that Ʈ is symmetric with regards to a sub-set ⊆ 1, 2, 3, ⋯ , , and ∈ Ʈ, , ∈ so that = implies that Տ (Ʈ, ) = Տ (Ʈ, ) , ≠ d) Տ (Ʈ, ) guarantees fairness by maintaining the independence of irrelevant alternativ instance, if the feasible set decreases and the solution keeps on being feasible, it follows th solution for the lesser achievable set remains the same point. It can be expressed as ɰ (ɰ, ) ∈ ℬ and Տ (Ʈ, ) ∈ ℬ, then Տ (Ʈ, ) = Տ (ɰ, ), ∀ .
Without a loss of generality, the property of the SNO-CRAVNET is described usin following theorem.
In Equation (A2), the condition ensures proportional fairness [34], since the objective function cannot be improved by each movement in the line of R − R * at the optimal data-rate vector R * . Therefore, Sensors 2020, 20, 6402 24 of 28 proportionally fair resource allocation is guaranteed by the optimal solution. Proof of Proposition 3 is completed.
, , ℛ , , ⋯ , ℛ ℝ ℝ , ℝ ℋ , and are the optimal channel, achievable datarate, and transmit power allocation, respectively, then it follows that, at ℛ = ℛ * , the following condition stands: In Equation (A2), the condition ensures proportional fairness [34], since the objective function cannot be improved by each movement in the line of ℛ − ℛ * at the optimal data-rate vector ℛ * . Therefore, proportionally fair resource allocation is guaranteed by the optimal solution. Proof of Proposition 3 is completed. □ ℓ , , , ₼ , , =
endix B presents the Proof of Theorem 2. □ n , ℛ , , ℛ , , ⋯ , ℛ ℝ ℝ , ℝ ℋ , and are the optimal channel, achievable datarate, and transmit power allocation, respectively, then it follows that, at ℛ = ℛ * , the following condition stands: In Equation (A2), the condition ensures proportional fairness [34], since the objective function cannot be improved by each movement in the line of ℛ − ℛ * at the optimal data-rate vector ℛ * . Therefore, proportionally fair resource allocation is guaranteed by the optimal solution. Proof of Proposition 3 is completed. □ ℓ , , , ₼ , =

Proofs of Theorems 2, 3, 4, and 5
The convex optimization problem's Lagrangian function ℓ as expressed in Equations (28)-(34) is defined as shown in Equation (A3) above, where ≥ 0, ₼ ≥ 0, ≥ 0, and ≥ 0 denote the four constraints' (i.e., Equation (30) and (32)-(34)) Lagrangian multipliers, respectively. By applying the Karush-Kuhn-Tucker (KKT) conditions according to [35], the optimal subcarrier allocation index ℂ * is obtained through the differentiation of ℓ over in Equation (A3) above, so that, Proof. Recall that ∂ƛ ℂ * ∂ℂ * < 0 ⁄ . Then, it follows that Therefore, ɸ(₼ * ) is an entirely strictly non-increasing function. Proof of Proposition 4 is completed. The optimum Lagrangian multiplier, ₼ * , is obtained from Proposition 4, which is given in Equation (A4) below: The convex optimization problem's optimal solution, which is presented in Equation (28) and in the constraints expressed in Equations (29)- (34), is derived in this section. Additionally, a simple and efficient strategy, which supports an iteration-independent joint transmit power and subcarrier scheduling, is proposed. The optimal subcarrier allocation with a consideration of the timesharing approach is a real number implying the fraction of time which subcarrier requires for the transmission of a given amount of information. Firstly, uniform transmit power scheduling, that is, = , is performed for all the available subcarriers. Then, an equal amount of information is transferred over all the available subcarriers. Secondly, based on the study carried out by Hahne [29], the optimal time-sharing subcarrier scheduling strategy is obtained. Theorem 2. The SNO-CRAVNET optimal time-sharing subcarrier scheduling strategy is given as ℂ ×ℝ * ×ℝ = ℂ * , and the individual matrix elements are expressed as ℂ * = ƛ (₼ * ), where ₼ * = ɸ (1), ∀ .
Proof. Appendix B presents the Proof of Theorem 2. □ are the optimal channel, achievable datarate, and transmit power allocation, respectively, then it follows that, at ℛ = ℛ * , the following condition stands: In Equation (A2), the condition ensures proportional fairness [34], since the objective function cannot be improved by each movement in the line of ℛ − ℛ * at the optimal data-rate vector ℛ * . Therefore, proportionally fair resource allocation is guaranteed by the optimal solution. Proof of Proposition 3 is completed. □ ℓ , , , ₼ ,

Appendix B
Proofs of Theorems 2, 3, 4, and 5 The convex optimization problem's Lagrangian function ∼ as expressed in Equations (28) , and are the optimal channel, achievable rate, and transmit power allocation, respectively, then it follows that, at ℛ = ℛ * , the foll condition stands: In Equation (A2), the condition ensures proportional fairness [34], since the objective fu cannot be improved by each movement in the line of ℛ − ℛ * at the optimal data-rate vect Therefore, proportionally fair resource allocation is guaranteed by the optimal solution. Pr Proposition 3 is completed. □ ℓ , , , ₼ , , =

Optimal Resource Scheduling Strategies
The convex optimization problem's optimal solution, which is presented in Equation (28) and in the constraints expressed in Equations (29)- (34), is derived in this section. Additionally, a simple and efficient strategy, which supports an iteration-independent joint transmit power and subcarrier scheduling, is proposed. The optimal subcarrier allocation with a consideration of the timesharing approach is a real number implying the fraction of time which subcarrier requires for the transmission of a given amount of information. Firstly, uniform transmit power scheduling, that is, = , is performed for all the available subcarriers. Then, an equal amount of information is transferred over all the available subcarriers. Secondly, based on the study carried out by Hahne [29], the optimal time-sharing subcarrier scheduling strategy is obtained.
where ₼ * = ɸ (1), ∀ . , and are the optimal channel, ac rate, and transmit power allocation, respectively, then it follows that, at ℛ = ℛ * , condition stands: In Equation (A2), the condition ensures proportional fairness [34], since the obje cannot be improved by each movement in the line of ℛ − ℛ * at the optimal data-r Therefore, proportionally fair resource allocation is guaranteed by the optimal solu Proposition 3 is completed. □ ℓ , , , ₼ , =

The utility function Ʈ
, proposed in Equation (28) is Nash bargaining theorem d, at the same time, satisfies the proportional fairness metric. endix A presents the Proof of Proposition 3. □ case, Proposition 3 shows that, for the data-rate allocation, a unique Nash bargaining can be obtained. Likewise, as a special case of the NBS fairness [29], proportional fairness ved when = 0, ∀ .

Resource Scheduling Strategies
nvex optimization problem's optimal solution, which is presented in Equation (28) and in nts expressed in Equations (29)- (34), is derived in this section. Additionally, a simple and ategy, which supports an iteration-independent joint transmit power and subcarrier is proposed. The optimal subcarrier allocation with a consideration of the timeroach is a real number implying the fraction of time which subcarrier requires for the n of a given amount of information. Firstly, uniform transmit power scheduling, that is, , is performed for all the available subcarriers. Then, an equal amount of is transferred over all the available subcarriers. Secondly, based on the study carried out 9], the optimal time-sharing subcarrier scheduling strategy is obtained.
. The SNO-CRAVNET optimal time-sharing subcarrier scheduling strategy is given as ℝ = ℂ * , and the individual matrix elements are expressed as ℂ * = ƛ (₼ * ), are the optimal channel, achievable datarate, and transmit power allocation, respectively, then it follows that, at ℛ = ℛ * , the following condition stands: In Equation (A2), the condition ensures proportional fairness [34], since the objective function cannot be improved by each movement in the line of ℛ − ℛ * at the optimal data-rate vector ℛ * . Therefore, proportionally fair resource allocation is guaranteed by the optimal solution. Proof of Proposition 3 is completed. □ ℓ , , , ₼ , , =

Optimal Resource Scheduling Strategies
The convex optimization problem's optimal solution, which is presented in Equation (28) and in the constraints expressed in Equations (29)- (34), is derived in this section. Additionally, a simple and efficient strategy, which supports an iteration-independent joint transmit power and subcarrier scheduling, is proposed. The optimal subcarrier allocation with a consideration of the timesharing approach is a real number implying the fraction of time which subcarrier requires for the transmission of a given amount of information. Firstly, uniform transmit power scheduling, that is, = , is performed for all the available subcarriers. Then, an equal amount of information is transferred over all the available subcarriers. Secondly, based on the study carried out by Hahne [29], the optimal time-sharing subcarrier scheduling strategy is obtained.
where ₼ * = ɸ (1), ∀ . , and are the optimal channel, achievable rate, and transmit power allocation, respectively, then it follows that, at ℛ = ℛ * , the follo condition stands: In Equation (A2), the condition ensures proportional fairness [34], since the objective fun cannot be improved by each movement in the line of ℛ − ℛ * at the optimal data-rate vecto Therefore, proportionally fair resource allocation is guaranteed by the optimal solution. Pro Proposition 3 is completed. □ ℓ , , , ₼ , , =

Optimal Resource Scheduling Strategies
The convex optimization problem's optimal solution, which is presented in Equation (28 the constraints expressed in Equations (29)- (34), is derived in this section. Additionally, a sim efficient strategy, which supports an iteration-independent joint transmit power and su scheduling, is proposed. The optimal subcarrier allocation with a consideration of th sharing approach is a real number implying the fraction of time which subcarrier require transmission of a given amount of information. Firstly, uniform transmit power scheduling = , is performed for all the available subcarriers. Then, an equal am information is transferred over all the available subcarriers. Secondly, based on the study car by Hahne [29], the optimal time-sharing subcarrier scheduling strategy is obtained.  Proof. Appendix A presents the Proof of Proposition 2. □ Proposition 2 certifies that the transformation of the objective in Equation (20), as well as in Equation (21) to Ʈ , in Equation (28), can be achieved by exploiting the firmly increasing property of the logarithm function. (28) is Nash bargaining theorem compliant and, at the same time, satisfies the proportional fairness metric.

Proposition 3. The utility function Ʈ , proposed in Equation
Proof. Appendix A presents the Proof of Proposition 3. □ In our case, Proposition 3 shows that, for the data-rate allocation, a unique Nash bargaining equilibrium can be obtained. Likewise, as a special case of the NBS fairness [29], proportional fairness can be achieved when = 0, ∀ .

Optimal Resource Scheduling Strategies
The convex optimization problem's optimal solution, which is presented in Equation (28) and in the constraints expressed in Equations (29)- (34), is derived in this section. Additionally, a simple and efficient strategy, which supports an iteration-independent joint transmit power and subcarrier scheduling, is proposed. The optimal subcarrier allocation with a consideration of the timesharing approach is a real number implying the fraction of time which subcarrier requires for the transmission of a given amount of information. Firstly, uniform transmit power scheduling, that is, , is performed for all the available subcarriers. Then, an equal amount of information is transferred over all the available subcarriers. Secondly, based on the study carried out by Hahne [29], the optimal time-sharing subcarrier scheduling strategy is obtained. Theorem 2. The SNO-CRAVNET optimal time-sharing subcarrier scheduling strategy is given as ℂ ×ℝ * ×ℝ = ℂ * , and the individual matrix elements are expressed as where ₼ * = ɸ (1), ∀ .  (20), as w Equation (21) to Ʈ , in Equation (28), can be achieved by exploiting the firmly in property of the logarithm function. (28) is Nash bargainin compliant and, at the same time, satisfies the proportional fairness metric.

Proposition 3. The utility function Ʈ , proposed in Equation
Proof. Appendix A presents the Proof of Proposition 3. □ In our case, Proposition 3 shows that, for the data-rate allocation, a unique Nash ba equilibrium can be obtained. Likewise, as a special case of the NBS fairness [29], proportiona can be achieved when = 0, ∀ .

Optimal Resource Scheduling Strategies
The convex optimization problem's optimal solution, which is presented in Equation (2 the constraints expressed in Equations (29)- (34), is derived in this section. Additionally, a si efficient strategy, which supports an iteration-independent joint transmit power and s scheduling, is proposed. The optimal subcarrier allocation with a consideration of sharing approach is a real number implying the fraction of time which subcarrier requir transmission of a given amount of information. Firstly, uniform transmit power schedulin , is performed for all the available subcarriers. Then, an equal am information is transferred over all the available subcarriers. Secondly, based on the study ca by Hahne [29], the optimal time-sharing subcarrier scheduling strategy is obtained. there exists a unique optimal solution that can be achieved within the polynomial time [25].
Proof. Appendix A presents the Proof of Proposition 2. □ Proposition 2 certifies that the transformation of the objective in Equation (20), as well as in Equation (21) to Ʈ , in Equation (28), can be achieved by exploiting the firmly increasing property of the logarithm function. Proof. Appendix A presents the Proof of Proposition 3. □ In our case, Proposition 3 shows that, for the data-rate allocation, a unique Nash bargaining equilibrium can be obtained. Likewise, as a special case of the NBS fairness [29], proportional fairness can be achieved when = 0, ∀ .

Optimal Resource Scheduling Strategies
The convex optimization problem's optimal solution, which is presented in Equation (28) and in the constraints expressed in Equations (29)- (34), is derived in this section. Additionally, a simple and efficient strategy, which supports an iteration-independent joint transmit power and subcarrier scheduling, is proposed. The optimal subcarrier allocation with a consideration of the time-sharing approach is a real number implying the fraction of time which subcarrier requires for the transmission of a given amount of information. Firstly, uniform transmit power scheduling, that is, , is performed for all the available subcarriers. Then, an equal amount of information is transferred over all the available subcarriers. Secondly, based on the study carried out by Hahne [29], the optimal time-sharing subcarrier scheduling strategy is obtained.
Proof. Appendix B presents the Proof of Theorem 2. □ mn ( ∼ C * mn ) is strictly non-increasing, while its inverse function, that is, in Equations (29)-(34) is clearly convex over a given convex set. Therefore, imal solution that can be achieved within the polynomial time [25]. me that > 0. Then, Ʈ , , as shown in Equation (28), can stringently y satisfying ℛ , > . nts the Proof of Proposition 2. □ ies that the transformation of the objective in Equation (20), as well as in in Equation (28), can be achieved by exploiting the firmly increasing function.
y function Ʈ , proposed in Equation (28) is Nash bargaining theorem time, satisfies the proportional fairness metric. nts the Proof of Proposition 3. □ ition 3 shows that, for the data-rate allocation, a unique Nash bargaining ed. Likewise, as a special case of the NBS fairness [29], proportional fairness = 0, ∀ .
eduling Strategies ation problem's optimal solution, which is presented in Equation (28) and in in Equations (29)- (34), is derived in this section. Additionally, a simple and supports an iteration-independent joint transmit power and subcarrier . The optimal subcarrier allocation with a consideration of the time-l number implying the fraction of time which subcarrier requires for the mount of information. Firstly, uniform transmit power scheduling, that is, performed for all the available subcarriers. Then, an equal amount of d over all the available subcarriers. Secondly, based on the study carried out al time-sharing subcarrier scheduling strategy is obtained.
CRAVNET optimal time-sharing subcarrier scheduling strategy is given as nd the individual matrix elements are expressed as ℂ * = ƛ (₼ * ), . nts the Proof of Theorem 2. □ the constraints presented in Equations (29)-(34) is clearly convex over a given con there exists a unique optimal solution that can be achieved within the polynomial  Proof. Appendix A presents the Proof of Proposition 3. □ In our case, Proposition 3 shows that, for the data-rate allocation, a uniqu equilibrium can be obtained. Likewise, as a special case of the NBS fairness [29], pr can be achieved when = 0, ∀ .

Optimal Resource Scheduling Strategies
The convex optimization problem's optimal solution, which is presented in E the constraints expressed in Equations (29)- (34), is derived in this section. Additio efficient strategy, which supports an iteration-independent joint transmit pow scheduling, is proposed. The optimal subcarrier allocation with a conside sharing approach is a real number implying the fraction of time which subcarrier transmission of a given amount of information. Firstly, uniform transmit power = , is performed for all the available subcarriers. Then, an information is transferred over all the available subcarriers. Secondly, based on th by Hahne [29], the optimal time-sharing subcarrier scheduling strategy is obtaine With regard to Proposition 1, it is established that the problem defined in Equation (28) and in constraints presented in Equations (29)-(34) is clearly convex over a given convex set. Therefore, re exists a unique optimal solution that can be achieved within the polynomial time [25].  (20), as well as in ation (21) to Ʈ , in Equation (28), can be achieved by exploiting the firmly increasing perty of the logarithm function. position 3. The utility function Ʈ , proposed in Equation (28) is Nash bargaining theorem pliant and, at the same time, satisfies the proportional fairness metric.
of. Appendix A presents the Proof of Proposition 3. □ In our case, Proposition 3 shows that, for the data-rate allocation, a unique Nash bargaining ilibrium can be obtained. Likewise, as a special case of the NBS fairness [29], proportional fairness be achieved when = 0, ∀ .

ptimal Resource Scheduling Strategies
The convex optimization problem's optimal solution, which is presented in Equation (28) and in constraints expressed in Equations (29)- (34), is derived in this section. Additionally, a simple and cient strategy, which supports an iteration-independent joint transmit power and subcarrier eduling, is proposed. The optimal subcarrier allocation with a consideration of the time-ring approach is a real number implying the fraction of time which subcarrier requires for the smission of a given amount of information. Firstly, uniform transmit power scheduling, that is, , is performed for all the available subcarriers. Then, an equal amount of rmation is transferred over all the available subcarriers. Secondly, based on the study carried out Hahne [29], the optimal time-sharing subcarrier scheduling strategy is obtained. With regard to Proposition 1, it is established that the problem defined in Equation (28) and in constraints presented in Equations (29)-(34) is clearly convex over a given convex set. Therefore, e exists a unique optimal solution that can be achieved within the polynomial time [25].  (20), as well as in ation (21) to Ʈ , in Equation (28), can be achieved by exploiting the firmly increasing perty of the logarithm function. In our case, Proposition 3 shows that, for the data-rate allocation, a unique Nash bargaining ilibrium can be obtained. Likewise, as a special case of the NBS fairness [29], proportional fairness be achieved when = 0, ∀ .

ptimal Resource Scheduling Strategies
The convex optimization problem's optimal solution, which is presented in Equation (28) and in constraints expressed in Equations (29)- (34), is derived in this section. Additionally, a simple and cient strategy, which supports an iteration-independent joint transmit power and subcarrier eduling, is proposed. The optimal subcarrier allocation with a consideration of the timering approach is a real number implying the fraction of time which subcarrier requires for the smission of a given amount of information. Firstly, uniform transmit power scheduling, that is, = , is performed for all the available subcarriers. Then, an equal amount of rmation is transferred over all the available subcarriers. Secondly, based on the study carried out ahne [29], the optimal time-sharing subcarrier scheduling strategy is obtained.
Proof. Appendix A presents the Proof of Proposition 3. □ In our case, Proposition 3 shows that, for the data-rate allocation, a unique Nash equilibrium can be obtained. Likewise, as a special case of the NBS fairness [29], proportio can be achieved when = 0, ∀ .

Optimal Resource Scheduling Strategies
The convex optimization problem's optimal solution, which is presented in Equation the constraints expressed in Equations (29)- (34), is derived in this section. Additionally, a efficient strategy, which supports an iteration-independent joint transmit power and scheduling, is proposed. The optimal subcarrier allocation with a consideration o sharing approach is a real number implying the fraction of time which subcarrier requ transmission of a given amount of information. Firstly, uniform transmit power schedu , is performed for all the available subcarriers. Then, an equal information is transferred over all the available subcarriers. Secondly, based on the study by Hahne [29], the optimal time-sharing subcarrier scheduling strategy is obtained.  (20), as Equation (21) to Ʈ , in Equation (28), can be achieved by exploiting the firmly property of the logarithm function. Proof. Appendix A presents the Proof of Proposition 3. □ In our case, Proposition 3 shows that, for the data-rate allocation, a unique Nash equilibrium can be obtained. Likewise, as a special case of the NBS fairness [29], proportio can be achieved when = 0, ∀ .

Optimal Resource Scheduling Strategies
The convex optimization problem's optimal solution, which is presented in Equation the constraints expressed in Equations (29)-(34), is derived in this section. Additionally, a efficient strategy, which supports an iteration-independent joint transmit power and scheduling, is proposed. The optimal subcarrier allocation with a consideration o sharing approach is a real number implying the fraction of time which subcarrier requ transmission of a given amount of information. Firstly, uniform transmit power schedul = , is performed for all the available subcarriers. Then, an equal information is transferred over all the available subcarriers. Secondly, based on the study by Hahne [29], the optimal time-sharing subcarrier scheduling strategy is obtained.

Proposition A1. Let
With regard to Proposition 1, it is established that the problem defined in Equation (28) and in the constraints presented in Equations (29)-(34) is clearly convex over a given convex set. Therefore, there exists a unique optimal solution that can be achieved within the polynomial time [25].  (20), as well as in Equation (21) to Ʈ , in Equation (28), can be achieved by exploiting the firmly increasing property of the logarithm function. Proposition 3. The utility function Ʈ , proposed in Equation (28) is Nash bargaining theorem compliant and, at the same time, satisfies the proportional fairness metric.
Proof. Appendix A presents the Proof of Proposition 3. □ In our case, Proposition 3 shows that, for the data-rate allocation, a unique Nash bargaining equilibrium can be obtained. Likewise, as a special case of the NBS fairness [29], proportional fairness can be achieved when = 0, ∀ .

Optimal Resource Scheduling Strategies
The convex optimization problem's optimal solution, which is presented in Equation (28) and in the constraints expressed in Equations (29)-(34), is derived in this section. Additionally, a simple and efficient strategy, which supports an iteration-independent joint transmit power and subcarrier scheduling, is proposed. The optimal subcarrier allocation with a consideration of the timesharing approach is a real number implying the fraction of time which subcarrier requires for the transmission of a given amount of information. Firstly, uniform transmit power scheduling, that is, , is performed for all the available subcarriers. Then, an equal amount of information is transferred over all the available subcarriers. Secondly, based on the study carried out by Hahne [29], the optimal time-sharing subcarrier scheduling strategy is obtained. Theorem 2. The SNO-CRAVNET optimal time-sharing subcarrier scheduling strategy is given as ℂ ×ℝ * ×ℝ = ℂ * , and the individual matrix elements are expressed as where ₼ * = ɸ (1), ∀ .
Proof. Appendix B presents the Proof of Theorem 2. □ ( ition 1's Proof is shown in Appendix A. □ rd to Proposition 1, it is established that the problem defined in Equation (28) and in presented in Equations (29)-(34) is clearly convex over a given convex set. Therefore, nique optimal solution that can be achieved within the polynomial time [25].
ix A presents the Proof of Proposition 2. □ on 2 certifies that the transformation of the objective in Equation (20), as well as in to Ʈ , in Equation (28), can be achieved by exploiting the firmly increasing e logarithm function.
. The utility function Ʈ , proposed in Equation (28) is Nash bargaining theorem at the same time, satisfies the proportional fairness metric.
ix A presents the Proof of Proposition 3. □ se, Proposition 3 shows that, for the data-rate allocation, a unique Nash bargaining n be obtained. Likewise, as a special case of the NBS fairness [29], proportional fairness d when = 0, ∀ .
source Scheduling Strategies ex optimization problem's optimal solution, which is presented in Equation (28) and in expressed in Equations (29)- (34), is derived in this section. Additionally, a simple and gy, which supports an iteration-independent joint transmit power and subcarrier proposed. The optimal subcarrier allocation with a consideration of the timeach is a real number implying the fraction of time which subcarrier requires for the f a given amount of information. Firstly, uniform transmit power scheduling, that is, • ) , is performed for all the available subcarriers. Then, an equal amount of transferred over all the available subcarriers. Secondly, based on the study carried out , the optimal time-sharing subcarrier scheduling strategy is obtained.
The SNO-CRAVNET optimal time-sharing subcarrier scheduling strategy is given as = ℂ * , and the individual matrix elements are expressed as ℂ * = ƛ (₼ * ), Proof. Proposition 1's Proof is shown in Appendix A. □ With regard to Proposition 1, it is established that the problem defined in Equation (28) and in the constraints presented in Equations (29)-(34) is clearly convex over a given convex set. Therefore, there exists a unique optimal solution that can be achieved within the polynomial time [25].  (20), as well as in Equation (21) to Ʈ , in Equation (28), can be achieved by exploiting the firmly increasing property of the logarithm function. Proof. Appendix A presents the Proof of Proposition 3. □ In our case, Proposition 3 shows that, for the data-rate allocation, a unique Nash bargaining equilibrium can be obtained. Likewise, as a special case of the NBS fairness [29], proportional fairness can be achieved when = 0, ∀ .

Optimal Resource Scheduling Strategies
The convex optimization problem's optimal solution, which is presented in Equation (28) and in the constraints expressed in Equations (29)- (34), is derived in this section. Additionally, a simple and efficient strategy, which supports an iteration-independent joint transmit power and subcarrier scheduling, is proposed. The optimal subcarrier allocation with a consideration of the time-sharing approach is a real number implying the fraction of time which subcarrier requires for the transmission of a given amount of information. Firstly, uniform transmit power scheduling, that is, , is performed for all the available subcarriers. Then, an equal amount of information is transferred over all the available subcarriers. Secondly, based on the study carried out by Hahne [29], the optimal time-sharing subcarrier scheduling strategy is obtained.
where ₼ * = ɸ (1), ∀ . With regard to Proposition 1, it is established that the problem defined in Equation (28) and in the constraints presented in Equations (29)-(34) is clearly convex over a given convex set. Therefore, there exists a unique optimal solution that can be achieved within the polynomial time [25].  (20), as well as in Equation (21) to Ʈ , in Equation (28), can be achieved by exploiting the firmly increasing property of the logarithm function. Proof. Appendix A presents the Proof of Proposition 3. □ In our case, Proposition 3 shows that, for the data-rate allocation, a unique Nash bargaining equilibrium can be obtained. Likewise, as a special case of the NBS fairness [29], proportional fairness can be achieved when = 0, ∀ .

Optimal Resource Scheduling Strategies
The convex optimization problem's optimal solution, which is presented in Equation (28) and in the constraints expressed in Equations (29)- (34), is derived in this section. Additionally, a simple and efficient strategy, which supports an iteration-independent joint transmit power and subcarrier scheduling, is proposed. The optimal subcarrier allocation with a consideration of the timesharing approach is a real number implying the fraction of time which subcarrier requires for the transmission of a given amount of information. Firstly, uniform transmit power scheduling, that is, = , is performed for all the available subcarriers. Then, an equal amount of information is transferred over all the available subcarriers. Secondly, based on the study carried out by Hahne [29], the optimal time-sharing subcarrier scheduling strategy is obtained.

Optimal Resource Scheduling Strategies
The convex optimization problem's optim the constraints expressed in Equations (29)-(34 efficient strategy, which supports an iteratio scheduling, is proposed. The optimal subcarr sharing approach is a real number implying th transmission of a given amount of information , is performed for all th information is transferred over all the available by Hahne [29], the optimal time-sharing subca  With regard to Proposition 1, it is established that the problem defined in Equat the constraints presented in Equations (29)-(34) is clearly convex over a given convex there exists a unique optimal solution that can be achieved within the polynomial tim  (28), can be achieved by exploiting the fir property of the logarithm function. Proof. Appendix A presents the Proof of Proposition 3. □ In our case, Proposition 3 shows that, for the data-rate allocation, a unique Na equilibrium can be obtained. Likewise, as a special case of the NBS fairness [29], propo can be achieved when = 0, ∀ .

Optimal Resource Scheduling Strategies
The convex optimization problem's optimal solution, which is presented in Equa the constraints expressed in Equations (29)- (34), is derived in this section. Additionall efficient strategy, which supports an iteration-independent joint transmit power scheduling, is proposed. The optimal subcarrier allocation with a consideratio sharing approach is a real number implying the fraction of time which subcarrier r transmission of a given amount of information. Firstly, uniform transmit power sche = , is performed for all the available subcarriers. Then, an equ information is transferred over all the available subcarriers. Secondly, based on the stu by Hahne [29], the optimal time-sharing subcarrier scheduling strategy is obtained. Proof. Recall that ∂ ed in Equation (28) and in the constraints presented in Equations (29) (20), as well as in ) to Ʈ , in Equation (28), can be achieved by exploiting the firmly increasing he logarithm function.

The utility function Ʈ
, proposed in Equation (28) is Nash bargaining theorem , at the same time, satisfies the proportional fairness metric. dix A presents the Proof of Proposition 3. □ ase, Proposition 3 shows that, for the data-rate allocation, a unique Nash bargaining an be obtained. Likewise, as a special case of the NBS fairness [29], proportional fairness ed when = 0, ∀ .

esource Scheduling Strategies
vex optimization problem's optimal solution, which is presented in Equation (28) and in ts expressed in Equations (29)- (34), is derived in this section. Additionally, a simple and tegy, which supports an iteration-independent joint transmit power and subcarrier s proposed. The optimal subcarrier allocation with a consideration of the time-oach is a real number implying the fraction of time which subcarrier requires for the of a given amount of information. Firstly, uniform transmit power scheduling, that is, ( • ) , is performed for all the available subcarriers. Then, an equal amount of s transferred over all the available subcarriers. Secondly, based on the study carried out ], the optimal time-sharing subcarrier scheduling strategy is obtained.
The SNO-CRAVNET optimal time-sharing subcarrier scheduling strategy is given as = ℂ * , and the individual matrix elements are expressed as ℂ * = ƛ (₼ * ), In the above stated optimal joint subcarrier and transmit power allocation strategies, the problem defined in Equation (28) and in the constraints presented in Equations (29)-(34) is a convex optimization problem.
Proof. Proposition 1's Proof is shown in Appendix A. □ With regard to Proposition 1, it is established that the problem defined in Equation (28) and in the constraints presented in Equations (29)-(34) is clearly convex over a given convex set. Therefore, there exists a unique optimal solution that can be achieved within the polynomial time [25].  (20), as well as in Equation (21) to Ʈ , in Equation (28), can be achieved by exploiting the firmly increasing property of the logarithm function. Proof. Appendix A presents the Proof of Proposition 3. □ In our case, Proposition 3 shows that, for the data-rate allocation, a unique Nash bargaining equilibrium can be obtained. Likewise, as a special case of the NBS fairness [29], proportional fairness can be achieved when = 0, ∀ .

Optimal Resource Scheduling Strategies
The convex optimization problem's optimal solution, which is presented in Equation (28) and in the constraints expressed in Equations (29)- (34), is derived in this section. Additionally, a simple and efficient strategy, which supports an iteration-independent joint transmit power and subcarrier scheduling, is proposed. The optimal subcarrier allocation with a consideration of the timesharing approach is a real number implying the fraction of time which subcarrier requires for the transmission of a given amount of information. Firstly, uniform transmit power scheduling, that is, , is performed for all the available subcarriers. Then, an equal amount of information is transferred over all the available subcarriers. Secondly, based on the study carried out by Hahne [29], the optimal time-sharing subcarrier scheduling strategy is obtained.
where ₼ * = ɸ (1), ∀ . n 1. In the above stated optimal joint subcarrier and transmit power allocation strategies, the ined in Equation (28) and in the constraints presented in Equations (29) (20), as well as in 21) to Ʈ , in Equation (28), can be achieved by exploiting the firmly increasing f the logarithm function.  (28) is Nash bargaining theorem nd, at the same time, satisfies the proportional fairness metric. endix A presents the Proof of Proposition 3. □ case, Proposition 3 shows that, for the data-rate allocation, a unique Nash bargaining can be obtained. Likewise, as a special case of the NBS fairness [29], proportional fairness eved when = 0, ∀ .

Resource Scheduling Strategies
nvex optimization problem's optimal solution, which is presented in Equation (28) and in ints expressed in Equations (29)- (34), is derived in this section. Additionally, a simple and rategy, which supports an iteration-independent joint transmit power and subcarrier , is proposed. The optimal subcarrier allocation with a consideration of the timeproach is a real number implying the fraction of time which subcarrier requires for the n of a given amount of information. Firstly, uniform transmit power scheduling, that is, , is performed for all the available subcarriers. Then, an equal amount of n is transferred over all the available subcarriers. Secondly, based on the study carried out 29], the optimal time-sharing subcarrier scheduling strategy is obtained.

2.
The SNO-CRAVNET optimal time-sharing subcarrier scheduling strategy is given as ×ℝ = ℂ * , and the individual matrix elements are expressed as ℂ * = ƛ (₼ * ),  (33) and (34), respectively. Proposition 1. In the above stated optimal joint subcarrier and transmit power allocation strategies, the problem defined in Equation (28) and in the constraints presented in Equations (29)-(34) is a convex optimization problem.

Proof. Proposition 1's Proof is shown in Appendix A. □
With regard to Proposition 1, it is established that the problem defined in Equation (28) and in the constraints presented in Equations (29)-(34) is clearly convex over a given convex set. Therefore, there exists a unique optimal solution that can be achieved within the polynomial time [25].  (20), as well as in Equation (21) to Ʈ , in Equation (28), can be achieved by exploiting the firmly increasing property of the logarithm function. Proof. Appendix A presents the Proof of Proposition 3. □ In our case, Proposition 3 shows that, for the data-rate allocation, a unique Nash bargaining equilibrium can be obtained. Likewise, as a special case of the NBS fairness [29], proportional fairness can be achieved when = 0, ∀ .

Optimal Resource Scheduling Strategies
The convex optimization problem's optimal solution, which is presented in Equation (28) and in the constraints expressed in Equations (29)- (34), is derived in this section. Additionally, a simple and efficient strategy, which supports an iteration-independent joint transmit power and subcarrier scheduling, is proposed. The optimal subcarrier allocation with a consideration of the time-sharing approach is a real number implying the fraction of time which subcarrier requires for the transmission of a given amount of information. Firstly, uniform transmit power scheduling, that is, , is performed for all the available subcarriers. Then, an equal amount of information is transferred over all the available subcarriers. Secondly, based on the study carried out by Hahne [29], the optimal time-sharing subcarrier scheduling strategy is obtained.
Proof. Appendix B presents the Proof of Theorem 2. □ −1 mn ( Equations (33) and (34), respectively. Proposition 1. In the above stated optimal joint subcarrier and transmit power allocation strategies, the problem defined in Equation (28) and in the constraints presented in Equations (29)-(34) is a convex optimization problem.

Proof. Proposition 1's Proof is shown in Appendix A. □
With regard to Proposition 1, it is established that the problem defined in Equation (28) and in the constraints presented in Equations (29)-(34) is clearly convex over a given convex set. Therefore, there exists a unique optimal solution that can be achieved within the polynomial time [25].  (20), as well as in Equation (21) to Ʈ , in Equation (28), can be achieved by exploiting the firmly increasing property of the logarithm function. Proof. Appendix A presents the Proof of Proposition 3. □ In our case, Proposition 3 shows that, for the data-rate allocation, a unique Nash bargaining equilibrium can be obtained. Likewise, as a special case of the NBS fairness [29], proportional fairness can be achieved when = 0, ∀ .

Optimal Resource Scheduling Strategies
The convex optimization problem's optimal solution, which is presented in Equation (28) and in the constraints expressed in Equations (29)- (34), is derived in this section. Additionally, a simple and efficient strategy, which supports an iteration-independent joint transmit power and subcarrier scheduling, is proposed. The optimal subcarrier allocation with a consideration of the timesharing approach is a real number implying the fraction of time which subcarrier requires for the transmission of a given amount of information. Firstly, uniform transmit power scheduling, that is, = , is performed for all the available subcarriers. Then, an equal amount of information is transferred over all the available subcarriers. Secondly, based on the study carried out by Hahne [29], the optimal time-sharing subcarrier scheduling strategy is obtained.
Proof. Appendix B presents the Proof of Theorem 2. □ * m ) ∂ transmit power constraints for each ℝ CRV-SU and PU are guaranteed by constraints presented in Equations (33) and (34), respectively. Proposition 1. In the above stated optimal joint subcarrier and transmit power allocation strategies, the problem defined in Equation (28) and in the constraints presented in Equations (29)-(34) is a convex optimization problem.

Proof. Proposition 1's Proof is shown in Appendix A. □
With regard to Proposition 1, it is established that the problem defined in Equation (28) and in the constraints presented in Equations (29)-(34) is clearly convex over a given convex set. Therefore, there exists a unique optimal solution that can be achieved within the polynomial time [25].
Proof. Appendix A presents the Proof of Proposition 2. □ Proposition 2 certifies that the transformation of the objective in Equation (20), as well as in Equation (21) to Ʈ , in Equation (28), can be achieved by exploiting the firmly increasing property of the logarithm function. Proof. Appendix A presents the Proof of Proposition 3. □ In our case, Proposition 3 shows that, for the data-rate allocation, a unique Nash bargaining equilibrium can be obtained. Likewise, as a special case of the NBS fairness [29], proportional fairness can be achieved when = 0, ∀ .

Optimal Resource Scheduling Strategies
The convex optimization problem's optimal solution, which is presented in Equation (28) and in the constraints expressed in Equations (29)- (34), is derived in this section. Additionally, a simple and efficient strategy, which supports an iteration-independent joint transmit power and subcarrier scheduling, is proposed. The optimal subcarrier allocation with a consideration of the timesharing approach is a real number implying the fraction of time which subcarrier requires for the transmission of a given amount of information. Firstly, uniform transmit power scheduling, that is, = , is performed for all the available subcarriers. Then, an equal amount of information is transferred over all the available subcarriers. Secondly, based on the study carried out by Hahne [29], the optimal time-sharing subcarrier scheduling strategy is obtained.
Proof. Appendix B presents the Proof of Theorem 2. □ * m transmit power constraints for each ℝ CRV-SU and PU are guaranteed by constraints presented in Equations (33) and (34), respectively. Proposition 1. In the above stated optimal joint subcarrier and transmit power allocation strategies, the problem defined in Equation (28) and in the constraints presented in Equations (29)-(34) is a convex optimization problem.

Proof. Proposition 1's Proof is shown in Appendix A. □
With regard to Proposition 1, it is established that the problem defined in Equation (28) and in the constraints presented in Equations (29)-(34) is clearly convex over a given convex set. Therefore, there exists a unique optimal solution that can be achieved within the polynomial time [25].
Proof. Appendix A presents the Proof of Proposition 2. □ Proposition 2 certifies that the transformation of the objective in Equation (20), as well as in Equation (21) to Ʈ , in Equation (28), can be achieved by exploiting the firmly increasing property of the logarithm function.

Proposition 3. The utility function Ʈ
, proposed in Equation (28) is Nash bargaining theorem compliant and, at the same time, satisfies the proportional fairness metric.
Proof. Appendix A presents the Proof of Proposition 3. □ In our case, Proposition 3 shows that, for the data-rate allocation, a unique Nash bargaining equilibrium can be obtained. Likewise, as a special case of the NBS fairness [29], proportional fairness can be achieved when = 0, ∀ .

Optimal Resource Scheduling Strategies
The convex optimization problem's optimal solution, which is presented in Equation (28) and in the constraints expressed in Equations (29)- (34), is derived in this section. Additionally, a simple and efficient strategy, which supports an iteration-independent joint transmit power and subcarrier scheduling, is proposed. The optimal subcarrier allocation with a consideration of the time-sharing approach is a real number implying the fraction of time which subcarrier requires for the transmission of a given amount of information. Firstly, uniform transmit power scheduling, that is, , is performed for all the available subcarriers. Then, an equal amount of information is transferred over all the available subcarriers. Secondly, based on the study carried out by Hahne [29], the optimal time-sharing subcarrier scheduling strategy is obtained.
Proof. Appendix B presents the Proof of Theorem 2. □ Therefore, all the dynamically available orthogonal channels is maintained below the total transmit power . available at CR-BS, as defined in Equation (15). Lastly, as illustrated in Equations (17) and (19), transmit power constraints for each ℝ CRV-SU and PU are guaranteed by constraints presented in Equations (33) and (34), respectively. Proposition 1. In the above stated optimal joint subcarrier and transmit power allocation strategies, the problem defined in Equation (28) and in the constraints presented in Equations (29)-(34) is a convex optimization problem.

Proof. Proposition 1's Proof is shown in Appendix A. □
With regard to Proposition 1, it is established that the problem defined in Equation (28) and in the constraints presented in Equations (29)-(34) is clearly convex over a given convex set. Therefore, there exists a unique optimal solution that can be achieved within the polynomial time [25].
Proof. Appendix A presents the Proof of Proposition 2. □ Proposition 2 certifies that the transformation of the objective in Equation (20), as well as in Equation (21) to Ʈ , in Equation (28), can be achieved by exploiting the firmly increasing property of the logarithm function.

Proposition 3. The utility function Ʈ
, proposed in Equation (28) is Nash bargaining theorem compliant and, at the same time, satisfies the proportional fairness metric.
Proof. Appendix A presents the Proof of Proposition 3. □ In our case, Proposition 3 shows that, for the data-rate allocation, a unique Nash bargaining equilibrium can be obtained. Likewise, as a special case of the NBS fairness [29], proportional fairness can be achieved when = 0, ∀ .

Optimal Resource Scheduling Strategies
The convex optimization problem's optimal solution, which is presented in Equation (28) and in the constraints expressed in Equations (29)- (34), is derived in this section. Additionally, a simple and efficient strategy, which supports an iteration-independent joint transmit power and subcarrier scheduling, is proposed. The optimal subcarrier allocation with a consideration of the timesharing approach is a real number implying the fraction of time which subcarrier requires for the transmission of a given amount of information. Firstly, uniform transmit power scheduling, that is, = . (

• ) ⁄
, is performed for all the available subcarriers. Then, an equal amount of information is transferred over all the available subcarriers. Secondly, based on the study carried out by Hahne [29], the optimal time-sharing subcarrier scheduling strategy is obtained.
Proof. Appendix B presents the Proof of Theorem 2. □ ( PEER REVIEW 13 of 27 y available orthogonal channels is maintained below the total transmit power . S, as defined in Equation (15). Lastly, as illustrated in Equations (17) and (19), nstraints for each ℝ CRV-SU and PU are guaranteed by constraints presented in (34), respectively.
the above stated optimal joint subcarrier and transmit power allocation strategies, the Equation (28)

and in the constraints presented in Equations (29)-(34) is a convex .
1's Proof is shown in Appendix A. □ o Proposition 1, it is established that the problem defined in Equation (28) and in sented in Equations (29)-(34) is clearly convex over a given convex set. Therefore, ue optimal solution that can be achieved within the polynomial time [25].
presents the Proof of Proposition 2. □ certifies that the transformation of the objective in Equation (20), as well as in , in Equation (28), can be achieved by exploiting the firmly increasing arithm function. e utility function Ʈ , proposed in Equation (28) is Nash bargaining theorem e same time, satisfies the proportional fairness metric.
presents the Proof of Proposition 3. □ roposition 3 shows that, for the data-rate allocation, a unique Nash bargaining obtained. Likewise, as a special case of the NBS fairness [29], proportional fairness hen = 0, ∀ .

rce Scheduling Strategies
ptimization problem's optimal solution, which is presented in Equation (28) and in ressed in Equations (29)- (34), is derived in this section. Additionally, a simple and which supports an iteration-independent joint transmit power and subcarrier posed. The optimal subcarrier allocation with a consideration of the timeis a real number implying the fraction of time which subcarrier requires for the given amount of information. Firstly, uniform transmit power scheduling, that is, ) , is performed for all the available subcarriers. Then, an equal amount of sferred over all the available subcarriers. Secondly, based on the study carried out optimal time-sharing subcarrier scheduling strategy is obtained.
presents the Proof of Theorem 2. □

Optimal Resource Scheduling Strategies
The convex optimization problem's optimal solution, which is presented in Equation (28) and in the constraints expressed in Equations (29)- (34), is derived in this section. Additionally, a simple and efficient strategy, which supports an iteration-independent joint transmit power and subcarrier scheduling, is proposed. The optimal subcarrier allocation with a consideration of the timesharing approach is a real number implying the fraction of time which subcarrier requires for the transmission of a given amount of information. Firstly, uniform transmit power scheduling, that is, = , is performed for all the available subcarriers. Then, an equal amount of information is transferred over all the available subcarriers. Secondly, based on the study carried out by Hahne [29], the optimal time-sharing subcarrier scheduling strategy is obtained.
Proof. Appendix B presents the Proof of Theorem 2. □ * m = Sensors 2020, 20, x FOR PEER REVIEW all the dynamically available orthogonal channels is maintained below the total tran available at CR-BS, as defined in Equation (15). Lastly, as illustrated in Equatio transmit power constraints for each ℝ CRV-SU and PU are guaranteed by constra Equations (33) and (34), respectively. Proposition 1. In the above stated optimal joint subcarrier and transmit power allocat problem defined in Equation (28) and in the constraints presented in Equations (29) optimization problem.
Proof. Proposition 1's Proof is shown in Appendix A. □ With regard to Proposition 1, it is established that the problem defined in Equ the constraints presented in Equations (29)-(34) is clearly convex over a given conv there exists a unique optimal solution that can be achieved within the polynomial t  (28), can be achieved by exploiting the property of the logarithm function. Proof. Appendix A presents the Proof of Proposition 3. □ In our case, Proposition 3 shows that, for the data-rate allocation, a unique equilibrium can be obtained. Likewise, as a special case of the NBS fairness [29], prop can be achieved when = 0, ∀ .

Optimal Resource Scheduling Strategies
The convex optimization problem's optimal solution, which is presented in Equ the constraints expressed in Equations (29)- (34), is derived in this section. Additiona efficient strategy, which supports an iteration-independent joint transmit powe scheduling, is proposed. The optimal subcarrier allocation with a considera sharing approach is a real number implying the fraction of time which subcarrier transmission of a given amount of information. Firstly, uniform transmit power sc = . (
(A14) Therefore, by relating Equations (A7) and (A14), the optimal transmit power allocation * of the proposed novel SNO-CRAVNET is determined to in Equation (36). Proof of Theorem 3 is completed. □ , which illustrates the time-sharing scheduling of each subcarrier for all the ℝ CRV-SUs, is determined. Additionally, this further helps in determining the quality of each subcarrier, that is, based on Equation (35), if it is observed that ℂ * < ℂ * , this indicates that even though both subcarriers were allocated an equal amount of transmit power, subcarrier requires less time than subcarrier for the transfer of an equal amount of information by the same CRV-SU . Therefore, subcarrier is in better conditions, i.e., has a higher quality in comparison to subcarrier . Furthermore, accounting for ℂ ×ℝ * ×ℝ , the optimal transmit power scheduling strategy is defined as shown here.
where (•) represents the Lambert -function. The definitions of both Þ and are contained in Appendix B, and the symbol ( ) represents (0, ).
Proof. Appendix B presents the Proof of Theorem 3. □ The optimal transmit power scheduling for CRV-SU on every subcarrier is obtained from Theorem 3. In other words, the optimal transmit power that subcarrier requires to be able to transmit a given amount of information based on the licensed PU's protection parameters, the mn) ) ))} .
(A14) Therefore, by relating Equations (A7) and (A14), the optimal transmit power allocation ∼ P * mn of the proposed novel SNO-CRAVNET is determined to in Equation (36). Proof of Theorem 3 is completed.
Proof. The optimal transmit power allocation ∼ P * mn as contained in Equation (36) signifies the exact measure of the optimal transmit power on subcarrier m of all the CRV-SU n. Therefore, if ∼ P * mn < ∼ P * mn , then the CRV-SU n requires more transmit power ∼ P * mn compared to CRV-SU n , so as to be able to transmit the same amount of message on subcarrier n. Hence, the CRV-SU that will be selected is the one that requires the minimum transmit power. To achieve this, a linear search is performed amongst the dynamically available N C subcarriers of the optimum CRV-SU n * , which is given by n * = arg min ∼ P * mn . Proof of Theorem 4 is completed.
Proof. Furthermore, based on Equation (37), the binary representation of Equation (35) gives the proposed novel SNO-CRAVNET optimal subcarrier scheduling strategy C * N C ×R O N C ×R = [C * mn ]; for instance, C * mn = 0 when n n * , and C * mn = 1 when n = n * . Proof of Theorem 5 is completed.