Achieving Congestion Mitigation Using Distributed Power Control for Spectrum Sensor Nodes in Sensor Network-Aided Cognitive Radio Ad Hoc Networks

The data sequence of spectrum sensing results injected from dedicated spectrum sensor nodes (SSNs) and the data traffic from upstream secondary users (SUs) lead to unpredictable data loads in a sensor network-aided cognitive radio ad hoc network (SN-CRN). As a result, network congestion may occur at a SU acting as fusion center when the offered data load exceeds its available capacity, which degrades network performance. In this paper, we present an effective approach to mitigate congestion of bottlenecked SUs via a proposed distributed power control framework for SSNs over a rectangular grid based SN-CRN, aiming to balance resource load and avoid excessive congestion. To achieve this goal, a distributed power control framework for SSNs from interior tier (IT) and middle tier (MT) is proposed to achieve the tradeoff between channel capacity and energy consumption. In particular, we firstly devise two pricing factors by considering stability of local spectrum sensing and spectrum sensing quality for SSNs. By the aid of pricing factors, the utility function of this power control problem is formulated by jointly taking into account the revenue of power reduction and the cost of energy consumption for IT or MT SSN. By bearing in mind the utility function maximization and linear differential equation constraint of energy consumption, we further formulate the power control problem as a differential game model under a cooperation or noncooperation scenario, and rigorously obtain the optimal solutions to this game model by employing dynamic programming. Then the congestion mitigation for bottlenecked SUs is derived by alleviating the buffer load over their internal buffers. Simulation results are presented to show the effectiveness of the proposed approach under the rectangular grid based SN-CRN scenario.


Introduction
Cognitive radio (CR) [1] has newly emerged as a promising solution to improve the spectrum utilization by allowing unlicensed secondary users (SUs) to access the idle licensed spectrum. In a CR network (CRN), SUs can periodically sense the licensed spectrum and opportunistically access the spectrum holes or spectrum opportunities (SOPs) unoccupied by primary users (PUs). Most of the existing research efforts in CRNs mainly focus on the issues of the physical and MAC layers for an infrastructure-based single hop scenario, such as spectrum sensing, spectrum access and sharing techniques [2][3][4]. In addition, SUs can also form a multi-hop ad hoc network without the support of infrastructure. In a cognitive radio ad hoc network (CRANET) [5], SUs can only access the SOPs information about the congestion state of congested SUs. A cross-layer framework to jointly achieve both congestion and power control through a non-convex optimization method was proposed in [19]. In [20], an optimization framework achieving tradeoff between energy efficiency and network utility maximization was devised, which can jointly balance interference, collision, and congestion among SUs by adjusting transmit power, persistence probability, together with data rate simultaneously via interaction between MAC and other layers. However, the proposed frameworks in [19,20] are just suited to mitigate the congestion caused by the data traffic from upstream SUs in multi-hop CRANETs, ignoring the impact of the data sequence of the SSR injected from SSNs on the congestion of SUs.
To the best of our knowledge, aside from some studies on congestion control for CRANETs as mentioned before, there is no related work reported in the literature related to congestion control over SN-CRNs. As a result, there is a strong motivation to explore congestion mitigation approach in SN-CRNs. Under this scenario, it is certainly not a surprise that the channel capacity between any SSN and FC is a concave function of the transmit power of this SSN and channel conditions [21]. In principle, effective transmit power control strategies have been widely used to maximize the total system capacity in conventional celluar wireless networks while adapting to the changing channel and interference conditions. Recent research efforts have achieved the capacity and energy efficiency maximization by devising the optimal power allocation on subchannels in two-tier femtocell networks based on orthogonal frequency division multiple access (OFDMA) [22], together with the optimal power control allocation and sensing time optimization in OFDMA cognitive small cell networks [23]. In addition, the transmission rate of this SSN always depends on channel capacity and is also a function of the transmit power according to the Shannon channel theorem. Thus, the congestion at FC can be controlled and mitigated through the transmission rate adjustment with the help of an optimal power allocation policy for this SSN in the physical layer. During a time interval, the amount of bits of the data sequence of SSR transmitted from this SSN to FC also approximately depends on the channel capacity [24]. For this observation it turns out that we can fully achieve the congestion mitigation for FC by reducing the amount of bits of the data sequence of the SSR transmitted from this SSN, aiming to release the capacity of the internal buffer for FC. In this paper, we propose a congestion mitigation approach by constructing a distributed power control framework for SSNs over the rectangular grid based SN-CRN. The main contributions of this paper are summarized as follows: 1.
To evaluate the performance of local spectrum sensing, we present the relative divergence between the detection probability and the false alarm probability for each SSN under any uplink channel via the Kullback-Leibler divergence framework. By the aid of mathematical statistics, we obtain the detection probability and false alarm probability distributions for each SSN, and also model the stability metric of local spectrum sensing as the relative divergence by applying the entropy modeling framework.

2.
We propose a distributed power control framework for SSNs from the interior tier (IT) and middle tier (MT) perspective in order to achieve the tradeoff between channel capacity and energy consumption. In particular, the power control problem is formulated as a differential game model by taking into account the utility function maximization together with the linear differential equation constraint with respect to energy consumption. We further present the theoretical results of the optimal solutions to this differential game model in a cooperative or noncooperative manner by using dynamic programming. 3.
With the help of the proposed distributed power control framework, we attain the congestion mitigation for bottleneck SU by alleviating its buffer load over its internal buffer. We also rigorously analyze the impact of noncooperative and cooperative optimal transmit power for IT and MT SSNs on the internal buffer of bottleneck SU, respectively.
The rest of paper is organized as follows: Section 2 describes the system model. In Section 3, we present the spectrum sensing quality analysis method based on local spectrum sensing by SSNs. In Section 4, we formulate the distributed power control for IT and MT SSNs as a differential game Sensors 2017, 17, 2132 4 of 28 model, and derive the noncooperative and cooperative optimal solutions. The congestion mitigation approach for bottleneck SU is analyzed rigorously in Section 5. Section 6 presents the simulation results. Finally, Section 7 concludes the paper.

Primary Network and Cognitive Radio ad hoc Network Model
We consider an underlay SN-CRN coexisting with a cellular primary network involving N P PUs and N S SUs in a torus area Ω S = [0, N S /ρ] 2 (ρ is the spatial density of SUs) sharing the spectrum within the same frequency band simultaneously, as depicted in Figure 1. Particularly, PUs have the full privilege of accessing their allocated frequency band whereas SUs can opportunistically utilize idle channels unoccupied by the PUs. In the cellular primary network, PUs send their data traffic to the primary base station (PBS) via the licensed uplink channels constituting a channel set C = {1, 2, · · · , N c }. We employ the independent and identically distributed alternating ON-OFF process to model the occupation time length of PUs in uplink channels. Specifically, the OFF state indicates the idle state where the unoccupied uplink channels or the SOPs can be freely occupied by SUs.
Sensors 2017, 17, 2132 4 of 28 approach for bottleneck SU is analyzed rigorously in Section 5. Section 6 presents the simulation results. Finally, Section 7 concludes the paper.

Primary Network and Cognitive Radio ad hoc Network Model
We consider an underlay SN-CRN coexisting with a cellular primary network involving P N PUs and S N SUs in a torus area ( ρ is the spatial density of SUs) sharing the spectrum within the same frequency band simultaneously, as depicted in Figure 1. Particularly, PUs have the full privilege of accessing their allocated frequency band whereas SUs can opportunistically utilize idle channels unoccupied by the PUs. In the cellular primary network, PUs send their data traffic to the primary base station (PBS) via the licensed uplink channels constituting a channel set . We employ the independent and identically distributed alternating ON-OFF process to model the occupation time length of PUs in uplink channels. Specifically, the OFF state indicates the idle state where the unoccupied uplink channels or the SOPs can be freely occupied by SUs. is probability that the k-th uplink channel transits from OFF to ON state, and k β is probability that the k-th uplink channel transits from ON to OFF state, for k ∈  . We also assume that the occupancy probability of uplink channels by PUs can be acquired by SUs through a priori knowledge of the local spectrum sensing. By bearing in mind the mutually independent occupancy probability of the k-th uplink channel, the SOP usage probability i δ of SU i during time interval [ ] 0 , t T is formulated as follows: In CRANET, SUs denoted by a set N S = {1, 2, · · · , N S } can only leverage the OFF state to access the SOPs over the idle authorized uplink channels. Due to the randomness of data traffic and the dynamic behavior of PUs, we suppose that the SOPs are available for usage by SU i with a probability of δ i , for i ∈ N S . With the aid of the ON-OFF process to characterize the status of the uplink channels, the occupancy probability of the k-th uplink channel by PUs can be given by α k /(α k + β k ), where α k is probability that the k-th uplink channel transits from OFF to ON state, and β k is probability that the k-th uplink channel transits from ON to OFF state, for k ∈ C. We also assume that the occupancy probability of uplink channels by PUs can be acquired by SUs through a priori knowledge of the local spectrum sensing. By bearing in mind the mutually independent occupancy probability of the k-th uplink channel, the SOP usage probability δ i of SU i during time interval [t 0 , T] is formulated as follows: Sensors 2017, 17, 2132 5 of 28 Let ϑ i (t) and B i denote the amount of data traffic in the buffer of SU i at time t ∈ [t 0 , T] and the buffer size of SU i, respectively. The buffer of SU i generally consists of two buffer segments that can hold the offered data load including data traffic injected from upstream SUs and the data sequence of SSR transmitted from SSNs, respectively. Given a time interval ∆t, the first buffer segment of SU i called as the forward buffer holds the amount of data traffic injected from upstream SUs denoted by ϑ F i (∆t). Meanwhile, the second buffer segment of SU i known as the internal buffer is used to store the amount of the data sequence of SSR transmitted from SSNs denoted by ϑ I i (∆t). Thus, the amount of aggregate data traffic from upstream SUs and SSNs in the buffer of SU i with a time interval ∆t can be formulated as ϕ i (∆t) = ϑ F i (∆t) + ϑ I i (∆t). Then, for a given time interval ∆t, the amount of data traffic ϑ i (t) in buffer of SU i at time t evolves as follows: where Λ i (∆t) and χ i (∆t) stand for the amount of data traffic successfully delivered by SU i and the amount of the data sequence of SSR removed by SU i within a time interval ∆t, respectively. It is worth noting that the received data sequence of SSR will be removed by SU i within a time interval ∆t in order to free storage capacity of the internal buffer.
Remark 1. From Figure 1, there are |N IT | + |N ET | hop-by-hop fluid flows of the data sequence of SSR injected from SSNs to a single SU (i.e., FC), in addition to the amount of data traffic from upstream SUs. Apparently, this single SU, also known as a possible bottlenecked SU, is a little more inclined to be a congested SU node as a consequence of its buffer overflow. For convenience, the terms bottlenecked SU is used in the following to describe a possible congested SU. It is worth noting that our work in this paper mainly concentrates on how to attain effective congestion mitigation for a single possible congested SU due to buffer overflow caused by the data sequence of SSR injected from SSNs by means of the proposed distributed power control framework for SSNs. However, it is conceivable that the amount of data traffic from upstream SUs may also lead to the congestion of bottlenecked SUs. This problem can be resolved by the specific congestion control technique (e.g., [25]) which is out of the scope of this work.

Sensor Network Model
As shown in Figure 1, we consider a rectangular grid based sensor network deployed in Ω S to provide the SSR about real-time spectrum availability information to SUs. Each SSN is equipped with a single omnidirectional antenna, a predefined common control channel (CCC), and an energy detector that continuously senses the entire primary licensed uplink channels through individual local real-time measurement. We suppose that the distributed collaborative spectrum sensing is carried out by multiple collaborating SSNs to enhance the sensing performance. Also, each SU serves as the FC collecting the SSR and then makes a global decision on the availability of the monitored uplink channels via a decision fusion rule, e.g., OR-rule fusion mechanism [7]. All SSNs simultaneously communicate to SUs over a narrowband additive white Gaussian noise (AWGN) multiple-access channel with the channel bandwidth denoted by W. The horizontal or vertical distance between any SSNs in rectangular grid is initialized to be d. According to the location of each SU along with the distance between SSN and the corresponding SU, we define the set of SSNs center around the SU as a tier in rectangular grid. More specifically, the three-tier structure is exploited to organize SSNs into three groups due to the simplicity of implementation as shown in Figure 1, including an interior tier (IT) denoted by a set N IT = {1, · · · , n IT } with n IT = 4, a middle tier (MT) denoted by a set N MT = {1, · · · , n MT } with n MT = 8, and an exterior tier (ET) denoted by a set N ET = {1, · · · , n ET } with n ET = 12.

Remark 2.
Because of multiple SUs sharing the authorized uplink channels with PUs, one SSN may also belong to the different tiers based on the presented division criterion to devise the three-tier structure as stated previously. It is important to emphasize that our work in this paper is mainly aimed to study congestion mitigation approaches for one single possible congested SU under the scenario of the single three-tier structure of SSNs. The underlying scenario of the superimposed three-tier structures to organize SSNs is beyond the scope of this work. However, the results about our proposed distributed power control framework for SSNs are easily extendable to the superimposed three-tier structures.
We devise a time-slotted frame structure for sensor network, as illustrated in Figure 2 should satisfy an average power constraint given as follows to mitigate the interference among IT and MT SSNs [26]: where IT av P and MT av P are the total average power assigned to IT and MT SSNs, respectively. We assume that each SSN knows its distance m b d → from bottleneck SU b via the CCC and the channel During the slot τ rp , the data sequence of the SSR will be transmitted by IT and MT SSNs to bottleneck SU. For analytical simplicity, we assume that the starting time and the terminal time of the slot τ rp are equal to t 0 and t 0 + τ rp for both IT and MT SSNs, respectively. The instantaneous transmit power of the m-th SSN from IT or MT at time t ∈ t 0 , t 0 + τ rp , denoted by p m (t), can be adjusted in a continuous way but is also limited by a maximum value P max m , i.e., 0 ≤ p m (t) ≤ P max m . In the case of the simultaneous communications by IT and MT SSNs over an AWGN multiple-access channel, p m (t) should satisfy an average power constraint given as follows to mitigate the interference among IT and MT SSNs [26]: where P IT av and P MT av are the total average power assigned to IT and MT SSNs, respectively. We assume that each SSN knows its distance d m→b from bottleneck SU b via the CCC and the channel path gain h m→b from the m-th SSN to the bottlenecked SU b can be expressed as h m→b = (d m→b ) −κ , where κ ≥ 2 is the path-loss exponent and b ∈ N S . Thus, the channel capacity between the m-th SSN and bottlenecked SU b can be characterized by a concave function of the transmit power and channel conditions as follows [27]: where K is a constant that depends on the transmission frequency and N 0 is the noise power spectral density. Under this channel capacity formulation, the signal-to-noise ratio (SNR) between the m-th SSN and bottlenecked SU b is given by: Remark 3. It is assumed that the data sequence of the SSR will be forwarded by ET SSNs to neighbor MT SSNs via a single-hop fashion during the slot τ ET f . Let t 0 and t 0 + τ ET f denote the starting time and the terminal time of the slot τ ET f , respectively. The transmit power p n (t) of the n-th SSN from ET at time t ∈ t 0 , t 0 + τ ET f is limited to a maximum transmit power P max n , for 0 ≤ p n (t) ≤ P max n and n ∈ N ET . It is clear that the neighbor MT SSN with the shortest distance will be selected by an ET SSN as the next-hop SSN to save energy during the SSR forwarding period. In other words, the selection metric for the next-hop SSN by an ET SSN is only dependent on the distance between neighbor MT SSN and itself. Owing to the scenario of the rectangular grid based sensor network, it should be admitted that the shortest distance between ET SSN and neighbor MT SSN is equal to d for any ET SSN. Therefore, the n-th SSN from ET is inclined to hold the same transmit power p n (t) at time t. Under an AWGN multiple-access channel, the average power constraint should also be satisfied for all ET SSNs, i.e., ∑ n∈N ET p n (t) ≤ n ET P ET av , where P ET av is the total average power assigned to ET SSNs. Thus, our work in this paper primarily focuses on the distributed power control for IT and MT SSNs.

Remark 4.
The channel capacity C m→b in Equation (4) will be rigorously guaranteed if the channel state information (CSI) including the channel path gain h m→b and constant K is perfectly known at the m-th SSN which transmits the data sequence of the SSR. In practice, the perfect knowledge of the CSI measured at the m-th SSN side cannot be available because of time-varying wireless channel impairments along with hardware limitations [28,29]. How to model the uncertain relation between the channel path gain h m→b or constant K and their estimates by taking into account the effect of imperfect CSI and outage constraint on distributed power control for IT and MT SSNs will be our further work in future.

Local Spectrum Sensing Model
We denote by H 1 and H 0 the binary hypotheses of the presence and absence of the PU on the uplink channel, respectively. Without loss of generality, we choose the m-th SSN from the three-tier structure of sensor network to describe its local spectrum sensing model during the slot τ s . This formulation can be easily extendable to the general case for any SSN from one of the tiers including IT, MT, and ET. The sampled signals that are received at the m-th SSN on the k-th uplink channel during the slot of spectrum sensing τ s are given as: where s(u) denotes the signal from the PU on the k-th uplink channel with a sampling frequency f s , υ m,k (u) is the noise at the m-th SSN on the k-th uplink channel, h m,k is the channel gain between the PU and the m-th SSN on the k-th uplink channel implying Rayleigh fading. Then the number of samples that is collected by the m-th SSN on the k-th uplink channel can be defined as U s = f s τ s . We assume that the PU signal s(u) satisfies an independent identically distributed (i.i.d.) random process with zero mean and variance σ 2 s , and the noise υ m,k (u) is i.i.d. circularly symmetric complex Gaussian with zero mean and variance σ 2 υ [8]. Thus, the received SNR from the PU at the m-th SSN on the k-th Let ε m denote a decision threshold by the m-th SSN to decide whether the channel is occupied by the PU. For the m-th SSN, the probabilities of detection and false alarm on the k-th uplink channel are approximately formulated as follows [8]: where Q(·) denotes the right-tail probability of a normalized Gaussian distribution. Hence, the detection probability set P d and the false alarm probability set P f for the m-th SSN over the entire uplink channels can be further expressed as:

Spectrum Sensing Quality Analysis
In this section, our objective is to analyze the spectrum sensing quality of each SSN via a local spectrum sensing model, aiming to provide the quantification result with the emphasis to evaluate the spectrum sensing performance of each SSN. More importantly, the analysis results will be employed to formulate the distributed power control framework for IT and MT SSNs. By revisiting Equations (9) and (10) in local spectrum sensing model, we can observe that the detection probability p d (m, k) in P d and the false alarm probability p f (m, k) in P f can be referred to the random variables for the m-th SSN under the k-th uplink channel due to the uncertainty of the presence and absence of the PU. It is worth noting that the errors in spectrum sensing for a SSN will be generally considered negligible due to imperfect spectrum sensing [27], e.g., misdetection and false alarm caused by hardware capability of SSN and practical time-varying channel conditions. So the errors in spectrum sensing for a SSN will further incur the fact that different uplink channels will hold different probabilities of detection and false alarm. In particular, the higher the detection probability in P d , the better the PUs are protected; the lower the false alarm probability in P f , the more efficiently the uplink channel can be reutilized by SUs [7]. Based on this observation, the higher the relative divergence between p d (m, k) and p f (m, k), the better the performance of local spectrum sensing. It has been revealed that the Kullback-Leibler divergence is an effective measure of how one probability diverges from a second probability [30]. Hence, the relative divergence between p d (m, k) and p f (m, k) for the m-th SSN under the k-th uplink channel can be defined as follows based on a Kullback-Leibler divergence framework: With respect to the entire set of uplink channels, the relative divergence between P d and P f for the m-th SSN can be denoted as: It is noticeable that the relative divergence between the detection probability and the false alarm probability just reflects the performance of local spectrum sensing by each SSN. Viewed from the SU perspective, we are also interested in the impact of the SOP usage probability on the spectrum sensing Sensors 2017, 17, 2132 9 of 28 quality. To this end, we characterize the spectrum sensing quality factor which can be expressed by a function of two parameters including the relative divergence between P d and P f along with the SOP usage probability δ b for bottleneck SU b. Specifically, the spectrum sensing quality factor F m→b of the m-th SSN with respect to bottleneck SU b can be defined as: By using mathematical statistics theory, next we start by formulating the detection probability distribution Υ d (m) and the false alarm probability distribution Υ f (m), which have been derived from Algorithm 1.
The detection probability distribution generation:

1:
Sort the detection probability p d (m, 1), p d (m, 2), · · · , p d (m, N c ) in ascending order and constitute the sorted Calculate the number of the detection probabilities within subinterval µ d −1 , µ d denoted by n d . 7: Calculate the probability Υ d (m) = n d /N c . 8: end for . The false alarm probability distribution generation:
Calculate the number of the false alarm probabilities within subinterval µ f −1 , µ f denoted by n f .

7:
Calculate the probability Υ f (m) = n f /N c .

8: end for
Owing to the fact that the number of the detection probabilities or the false alarm probabilities is calculated under the constraint of N c , it is clear that the distributions Υ d (m) and Υ f (m) fall into the complete probability distributions, i.e., ∑ Apparently, the entropy paradigm should be used for a measure of the uncertainty associated with a random variable of a distribution in information theory [31], and can be also applied to measure the uncertainty of the distributions Υ d (m) and Υ f (m). As a result, for the m-th SSN over the entire uplink channels, the uncertainty characterization of the distributions Υ d (m) and Υ f (m) can be respectively described as based on the entropy modeling framework: (15) In what follows, we are also interested in gaining a better understanding of how to apply this entropy measurement to evaluate the stability of local spectrum sensing. It should be admitted that the entropy tends to be larger when the change of the random variable values in given distribution is disorder or randomness [32]. That is, a more disordered probability distribution will result in larger entropy. Thus, the better performance of spectrum sensing for the m-th SSN will bring about the more ordered probability distributions Υ d (m) and Υ f (m). In this way, the stability of the distributions Υ d (m) and Υ f (m) will decrease because of more disorder for the values of the detection probability and false alarm probability in the distributions Υ d (m) and Υ f (m). Based on the insight, we model the stability metric of local spectrum sensing by the relative divergence between the entropy of the detection probability distribution and the entropy of the false alarm probability distribution. Thus, the stability metric of local spectrum sensing for the m-th SSN over the entire uplink channels denoted by H m Υ d (m) Υ f (m) can be calculated as follows:

Problem Formulation
It is considered that the channel capacity between the SSNs from IT or MT and a bottlenecked SU is a concave function of the transmit power and channel conditions. Therefore, each SSN is expected to increase the transmit power in the physical layer to provide as much channel capacity that each flow of the data sequence of the SSR requires. However, the higher transmit power will result in the more energy consumption for SSN. Meanwhile, the average power constraint will not be guaranteed for all SSNs if each SSN aims to increase the transmit power. As a consequence, it is necessary to require a tradeoff between channel capacity and energy consumption by achieving an optimal power allocation for all IT and MT SSNs in the physical layer during the slot τ rp . Under the constraint of path-loss of wireless channel, the maximum transmit power of the m-th SSN at a distance d m→b is approximately equal to [33]: where P 0 is the receiving reference power of bottlenecked SU b at a reference distance d 0 . Because of the limitation by a maximum value P max m , the value of power reduction for the m-th SSN at time t ∈ t 0 , t 0 + τ rp can be expressed by P max m − p m (t). Then the power reduction efficiency for the m-th SSN can be written as (P max m − p m (t))/P max m . To formulate the revenue for power reduction by the m-th SSN, we firstly define a pricing factor for power reduction by taking into account both the power reduction efficiency and the stability metric of local spectrum sensing according to the distributions by seeking to underlay, overlay, or interweave their signals with those of the existing PUs without significantly impacting their communications. Spectrum sensing is one of the key enabling technologies for the establishment of CRNs, because it constantly allows for the opportunistic identification and use of the SOPs from a licensed primary network without causing harmful interference to the PUs. In order to improve the sensing performance, collaborative spectrum sensing has been proposed as an effective way to reliably detect the activity of PUs by addressing the issues imposed by the hidden PU terminal problem and the wireless channel impairments, such as the heavy shadowing and fading [6][7][8]. In this way, cooperation is achieved by allowing different SUs to collaborate and share their spectrum sensing results (SSR) through a fusion center (FC), which makes a global decision on the occupancy status of the licensed band. However, this centralized FC is not available in decentralized CRANETs. Clearly, each SU under this scenario must perform the distributed collaborative spectrum sensing, which is preferred to the centralized FC scheme because of its scalability, fault tolerance and flexibility [9].
In order to facilitate the spectrum sensing functionality, high sampling rates, high resolution analog to digital converters with large dynamic range, and high speed signal processors are required to be incorporated into an individual SU transceiver [10], which increases hardware cost and power consumption, especially for the double-radio sensing architecture of SU transceiver. An alternative approach is to adopt the cost-effective and dedicated spectrum sensor nodes (SSNs) that perform distributed collaborative spectrum sensing and report SSR to SUs acting as FCs in CRANETs [11]. Technically, s wireless sensor network can be naturally exploited to assist a CRANET by providing SSR about the current spectrum occupancy of PUs in a cooperation fashion. The concept of sensor network embedded into CRANETs has further called sensor network-aided CRANETs (SN-CRNs), which has been considered as one of the most appealing approaches to perform cost-effective spectrum sensing in CR systems [7,11,12]. Υ Similar to most other traditional wireless networks or wireline Internet, network congestion in SN-CRNs will also occur when offered data load that exceed the available capacity of a SU due to buffer overflow caused by the data sequence of the SSR injected from SSNs together with the data traffic from upstream SUs. This therefore leads to energy consumption of SSNs, aggressive retransmission, queuing delay, and blocking of new flows from upstream SUs. Indubitably, a congestion control technique in the transport layer is essential to balance resource loads and avoid excessive congestion. However, the congestion control mechanism for the traditional Transmission Control Protocol (TCP) via the acknowledgement-triggered or window-based methods was initially designed and optimized to perform in reliable wired links with constrained bit error rates and round trip times (RTTs) [13]. A recent study [14] has reported that the performance of HTTP download deteriorates as much as about 40% under the TCP window control in an IEEE P1900.4 based cognitive wireless system by using User Datagram Protocol (UDP) and TCP transport protocols. On the other hand, some other research efforts about congestion control have also been conducted from the perspective of finding methods to modify the TCP protocol, such as TCP monitoring delayed acknowledgment, segment-based selective acknowledgement, TCP adaptive delayed-acknowledgment window, etc. [15], aiming to accommodate the challenging multi-hop wireless environments. Unfortunately, it has been shown that these methods d (m) and by seeking to underlay, overlay, or interweave their signals with those of the existing PUs without significantly impacting their communications. Spectrum sensing is one of the key enabling technologies for the establishment of CRNs, because it constantly allows for the opportunistic identification and use of the SOPs from a licensed primary network without causing harmful interference to the PUs. In order to improve the sensing performance, collaborative spectrum sensing has been proposed as an effective way to reliably detect the activity of PUs by addressing the issues imposed by the hidden PU terminal problem and the wireless channel impairments, such as the heavy shadowing and fading [6][7][8]. In this way, cooperation is achieved by allowing different SUs to collaborate and share their spectrum sensing results (SSR) through a fusion center (FC), which makes a global decision on the occupancy status of the licensed band. However, this centralized FC is not available in decentralized CRANETs. Clearly, each SU under this scenario must perform the distributed collaborative spectrum sensing, which is preferred to the centralized FC scheme because of its scalability, fault tolerance and flexibility [9].
In order to facilitate the spectrum sensing functionality, high sampling rates, high resolution analog to digital converters with large dynamic range, and high speed signal processors are required to be incorporated into an individual SU transceiver [10], which increases hardware cost and power consumption, especially for the double-radio sensing architecture of SU transceiver. An alternative approach is to adopt the cost-effective and dedicated spectrum sensor nodes (SSNs) that perform distributed collaborative spectrum sensing and report SSR to SUs acting as FCs in CRANETs [11]. Technically, s wireless sensor network can be naturally exploited to assist a CRANET by providing SSR about the current spectrum occupancy of PUs in a cooperation fashion. The concept of sensor network embedded into CRANETs has further called sensor network-aided CRANETs (SN-CRNs), which has been considered as one of the most appealing approaches to perform cost-effective spectrum sensing in CR systems [7,11,12]. Υ Similar to most other traditional wireless networks or wireline Internet, network congestion in SN-CRNs will also occur when offered data load that exceed the available capacity of a SU due to buffer overflow caused by the data sequence of the SSR injected from SSNs together with the data traffic from upstream SUs. This therefore leads to energy consumption of SSNs, aggressive retransmission, queuing delay, and blocking of new flows from upstream SUs. Indubitably, a congestion control technique in the transport layer is essential to balance resource loads and avoid excessive congestion. However, the congestion control mechanism for the traditional Transmission Control Protocol (TCP) via the acknowledgement-triggered or window-based methods was initially designed and optimized to perform in reliable wired links with constrained bit error rates and round trip times (RTTs) [13]. A recent study [14] has reported that the performance of HTTP download deteriorates as much as about 40% under the TCP window control in an IEEE P1900.4 based cognitive wireless system by using User Datagram Protocol (UDP) and TCP transport protocols. On the other hand, some other research efforts about congestion control have also been conducted from the perspective of finding methods to modify the TCP protocol, such as TCP monitoring delayed acknowledgment, segment-based selective acknowledgement, TCP adaptive delayed-acknowledgment window, etc. [15], aiming to accommodate the challenging multi-hop wireless environments. Unfortunately, it has been shown that these methods f (m). More precisely, we define a pricing factor by characterizing an efficiency-to-stability ratio for the m-th SSN at time t ∈ t 0 , t 0 + τ rp , denoted by ℘ R m , to describe the power reduction efficiency under the stability of local spectrum sensing, which can be defined as: Spectrum sensing is one of the key enabling technologies for the establishment of CRNs, b it constantly allows for the opportunistic identification and use of the SOPs from a licensed p network without causing harmful interference to the PUs. In order to improve the sensing perfor collaborative spectrum sensing has been proposed as an effective way to reliably detect the act PUs by addressing the issues imposed by the hidden PU terminal problem and the wireless c impairments, such as the heavy shadowing and fading [6][7][8]. In this way, cooperation is achie allowing different SUs to collaborate and share their spectrum sensing results (SSR) through a center (FC), which makes a global decision on the occupancy status of the licensed band. Ho this centralized FC is not available in decentralized CRANETs. Clearly, each SU under this sc must perform the distributed collaborative spectrum sensing, which is preferred to the centrali scheme because of its scalability, fault tolerance and flexibility [9].
In order to facilitate the spectrum sensing functionality, high sampling rates, high reso analog to digital converters with large dynamic range, and high speed signal processors are re to be incorporated into an individual SU transceiver [10], which increases hardware cost and consumption, especially for the double-radio sensing architecture of SU transceiver. An alte approach is to adopt the cost-effective and dedicated spectrum sensor nodes (SSNs) that p distributed collaborative spectrum sensing and report SSR to SUs acting as FCs in CRANE Technically, s wireless sensor network can be naturally exploited to assist a CRANET by providi about the current spectrum occupancy of PUs in a cooperation fashion. The concept of sensor n embedded into CRANETs has further called sensor network-aided CRANETs (SN-CRNs), wh been considered as one of the most appealing approaches to perform cost-effective spectrum s in CR systems [7,11,12]. Υ Similar to most other traditional wireless networks or wireline Internet, network conges SN-CRNs will also occur when offered data load that exceed the available capacity of a SU due to overflow caused by the data sequence of the SSR injected from SSNs together with the data from upstream SUs. This therefore leads to energy consumption of SSNs, aggressive retransm queuing delay, and blocking of new flows from upstream SUs. Indubitably, a congestion technique in the transport layer is essential to balance resource loads and avoid excessive cong However, the congestion control mechanism for the traditional Transmission Control Protoco via the acknowledgement-triggered or window-based methods was initially designed and opt to perform in reliable wired links with constrained bit error rates and round trip times (RTT A recent study [14] has reported that the performance of HTTP download deteriorates as m about 40% under the TCP window control in an IEEE P1900.4 based cognitive wireless system b User Datagram Protocol (UDP) and TCP transport protocols. On the other hand, some other re efforts about congestion control have also been conducted from the perspective of finding meth d (m) Sensors 2017, 17,2132 by seeking to underlay, overlay, or interweave their signals with those of the existi significantly impacting their communications.
Spectrum sensing is one of the key enabling technologies for the establishment o it constantly allows for the opportunistic identification and use of the SOPs from a l network without causing harmful interference to the PUs. In order to improve the sens collaborative spectrum sensing has been proposed as an effective way to reliably dete PUs by addressing the issues imposed by the hidden PU terminal problem and the w impairments, such as the heavy shadowing and fading [6][7][8]. In this way, cooperatio allowing different SUs to collaborate and share their spectrum sensing results (SSR) t center (FC), which makes a global decision on the occupancy status of the licensed this centralized FC is not available in decentralized CRANETs. Clearly, each SU und must perform the distributed collaborative spectrum sensing, which is preferred to th scheme because of its scalability, fault tolerance and flexibility [9].
In order to facilitate the spectrum sensing functionality, high sampling rates, analog to digital converters with large dynamic range, and high speed signal process to be incorporated into an individual SU transceiver [10], which increases hardware consumption, especially for the double-radio sensing architecture of SU transceiver approach is to adopt the cost-effective and dedicated spectrum sensor nodes (SSN distributed collaborative spectrum sensing and report SSR to SUs acting as FCs in Technically, s wireless sensor network can be naturally exploited to assist a CRANET b about the current spectrum occupancy of PUs in a cooperation fashion. The concept o embedded into CRANETs has further called sensor network-aided CRANETs (SN-C been considered as one of the most appealing approaches to perform cost-effective s in CR systems [7,11,12]. Υ Similar to most other traditional wireless networks or wireline Internet, netwo SN-CRNs will also occur when offered data load that exceed the available capacity of a overflow caused by the data sequence of the SSR injected from SSNs together with from upstream SUs. This therefore leads to energy consumption of SSNs, aggressive queuing delay, and blocking of new flows from upstream SUs. Indubitably, a con technique in the transport layer is essential to balance resource loads and avoid exce However, the congestion control mechanism for the traditional Transmission Contro via the acknowledgement-triggered or window-based methods was initially designe to perform in reliable wired links with constrained bit error rates and round trip ti A recent study [14] has reported that the performance of HTTP download deterior about 40% under the TCP window control in an IEEE P1900.4 based cognitive wireless User Datagram Protocol (UDP) and TCP transport protocols. On the other hand, som efforts about congestion control have also been conducted from the perspective of fin f (m) (18) Thus, the revenue of the power reduction for the m-th SSN at time t ∈ t 0 , t 0 + τ rp by attaining the product of the pricing factor together with the power reduction value, i.e., From Equations (18) and (19), it is worth remarking that the smaller of stability metric of local spectrum sensing will generate more pricing ℘ R m and also yield more revenue U R m of the power reduction for the m-th SSN. On the other hand, it is also a natural idea for the m-th SSN to reduce its transmit power to obtain more revenue U R m according to Equation (19). To depict the cost of energy consumption for the m-th SSN, we further define another pricing factor for energy consumption, denoted by ℘ C m , by considering the spectrum sensing quality factor F m→i as: Recall that the higher spectrum sensing quality factor will result in the better performance of local spectrum sensing by each SSN. As can be seen from Equation (20), it will be far more realistic to reduce the pricing ℘ C m for energy consumption for the m-th SSN with respect to the better performance of its local spectrum sensing, aiming to balance local spectrum sensing and the energy efficiency of SSN [34]. Let E m (t) and E A (t) represent the energy consumption value and the available energy value of the m-th SSN at time t, respectively. It is assumed that the available energy of the m-th SSN is derived from its battery of limited capacity and the harvesting energy from the renewable energy sources by exploiting the energy harvesting technology. Thus, the energy consumption of the m-th SSN evolves according to a linear differential equation given as: Then the cost of energy consumption for the m-th SSN at time t ∈ t 0 , t 0 + τ rp can be given as: Therefore, based on the revenue and the cost formulated in Equations (19) and (22), the utility function of the m-th SSN at time t ∈ t 0 , t 0 + τ rp can be constructed as follows: We denote the discount factor by r, for 0 < r < 1. Our optimization objective is to maximize the utility function U m by choosing optimal transmit power p OP m (t) of the m-th SSN during the slot τ rp according to ℘ R m and ℘ C m , at time t ∈ t 0 , t 0 + τ rp , i.e., Maximize : It is noteworthy that the discount factor r is an exponential factor between 0 and 1 by which the future utility must be multiplied in order to obtain the present value with the underlying structure of differential game theory in mind. Therefore, each SSN is required to maximize its discounted utility U m function by discount factor r, implying that discount factor will have a stronger impact on the utility obtained by each SSN in the future. To this end, the power control problem for all IT and MT SSNs in the physical layer can be formulated as a differential game model defined by is the set of players involving all IT and MT SSNs, p OP m (t) is the strategy of player m, p OP m (t) m∈N is the set of strategies or strategy space related to all players, E OP m (t) is the state variable associated with optimal transmit power p OP m (t), and {U m } m∈N is the set of utility function of all players with their strategies.

Noncooperative Optimal Solution
We formulate a dynamic optimization problem P1 to derive the optimal solution to the differential game model G with the objective of the utility function maximization problem under the linear differential equation constraint of the energy consumption for the m-th SSN. From Equations (21) and (24), the problem P1 can be formulated as: For the noncooperation scenario if all the players play noncooperatively, we aim at deriving an optimal solution to the problem P1 in the distributed noncooperative power control (NoCoPC) problem for all IT and MT SSNs by employing the theory of dynamic programming developed by Bellman. Note that the players can abandon the cooperation due to their selfishness and own interests in the NoCoPC problem, e.g., the selfish behavior in forwarding the data sequence of the SSR to bottleneck SU. Specifically, we employ p NC m (t) to represent the noncooperative optimal solution to the problem P1, and assume that there exists a continuously differentiable function V NC m (p m , E m ) satisfying the following partial differential equation: (26) should be subject to the partial differential equation constraint as follows: Proposition 2. The noncooperative optimal solution p NC m (t) constitutes a Nash equilibrium solution to the problem P1 if and only if the optimal transmit power for the m-th SSN can be expressed as: 17,2132 by seeking to underlay, overlay, or interweave their signals with those significantly impacting their communications.
Spectrum sensing is one of the key enabling technologies for the est it constantly allows for the opportunistic identification and use of the S network without causing harmful interference to the PUs. In order to imp collaborative spectrum sensing has been proposed as an effective way to PUs by addressing the issues imposed by the hidden PU terminal prob impairments, such as the heavy shadowing and fading [6][7][8]. In this wa allowing different SUs to collaborate and share their spectrum sensing r center (FC), which makes a global decision on the occupancy status of this centralized FC is not available in decentralized CRANETs. Clearly must perform the distributed collaborative spectrum sensing, which is p scheme because of its scalability, fault tolerance and flexibility [9].
In order to facilitate the spectrum sensing functionality, high sam analog to digital converters with large dynamic range, and high speed s to be incorporated into an individual SU transceiver [10], which increa consumption, especially for the double-radio sensing architecture of S approach is to adopt the cost-effective and dedicated spectrum senso distributed collaborative spectrum sensing and report SSR to SUs act Technically, s wireless sensor network can be naturally exploited to assist about the current spectrum occupancy of PUs in a cooperation fashion. T embedded into CRANETs has further called sensor network-aided CRA been considered as one of the most appealing approaches to perform co in CR systems [7,11,12]. Υ Similar to most other traditional wireless networks or wireline Int SN-CRNs will also occur when offered data load that exceed the available overflow caused by the data sequence of the SSR injected from SSNs from upstream SUs. This therefore leads to energy consumption of SSN queuing delay, and blocking of new flows from upstream SUs. Indu technique in the transport layer is essential to balance resource loads an However, the congestion control mechanism for the traditional Transm via the acknowledgement-triggered or window-based methods was init to perform in reliable wired links with constrained bit error rates and A recent study [14] has reported that the performance of HTTP down about 40% under the TCP window control in an IEEE P1900.4 based cogn User Datagram Protocol (UDP) and TCP transport protocols. On the oth efforts about congestion control have also been conducted from the pers modify the TCP protocol, such as TCP monitoring delayed acknowledgm acknowledgement, TCP adaptive delayed-acknowledgment window, etc the challenging multi-hop wireless environments. Unfortunately, it has b of TCP modification and extension cannot be directly applied into SN-CR bandwidth fluctuation, periodic interruption caused by spectrum sensin Recently, there have also been previous works on congestion control a cross-layer design perspective. In [17], an end-to-end congestion cont under the constraint of the non-uniform channel availability by taking from the physical layer to the transport layer. In [18], a cross-layer fram of MAC, scheduling, routing and congestion control was presented to a set of multi-hop end-to-end packet flows. However, the end-to-end suited for operation over wireless links characterized by higher RTTs. On Sensors 2017, 17,2132 by seeking to underlay, overlay, or interweave their signals wi significantly impacting their communications.
Spectrum sensing is one of the key enabling technologies fo it constantly allows for the opportunistic identification and use network without causing harmful interference to the PUs. In orde collaborative spectrum sensing has been proposed as an effectiv PUs by addressing the issues imposed by the hidden PU termin impairments, such as the heavy shadowing and fading [6][7][8]. In allowing different SUs to collaborate and share their spectrum s center (FC), which makes a global decision on the occupancy s this centralized FC is not available in decentralized CRANETs. must perform the distributed collaborative spectrum sensing, wh scheme because of its scalability, fault tolerance and flexibility [9 In order to facilitate the spectrum sensing functionality, h analog to digital converters with large dynamic range, and high to be incorporated into an individual SU transceiver [10], which consumption, especially for the double-radio sensing architect approach is to adopt the cost-effective and dedicated spectrum distributed collaborative spectrum sensing and report SSR to Technically, s wireless sensor network can be naturally exploited about the current spectrum occupancy of PUs in a cooperation fa embedded into CRANETs has further called sensor network-aid been considered as one of the most appealing approaches to per in CR systems [7,11,12]. Υ Similar to most other traditional wireless networks or wire SN-CRNs will also occur when offered data load that exceed the a overflow caused by the data sequence of the SSR injected from from upstream SUs. This therefore leads to energy consumptio queuing delay, and blocking of new flows from upstream SU technique in the transport layer is essential to balance resource l However, the congestion control mechanism for the traditional via the acknowledgement-triggered or window-based methods to perform in reliable wired links with constrained bit error ra A recent study [14] has reported that the performance of HTT about 40% under the TCP window control in an IEEE P1900.4 bas User Datagram Protocol (UDP) and TCP transport protocols. On efforts about congestion control have also been conducted from modify the TCP protocol, such as TCP monitoring delayed ackno acknowledgement, TCP adaptive delayed-acknowledgment wind the challenging multi-hop wireless environments. Unfortunately, of TCP modification and extension cannot be directly applied int bandwidth fluctuation, periodic interruption caused by spectrum Recently, there have also been previous works on congestion a cross-layer design perspective. In [17], an end-to-end congesti under the constraint of the non-uniform channel availability by from the physical layer to the transport layer. In [18], a cross-la of MAC, scheduling, routing and congestion control was prese a set of multi-hop end-to-end packet flows. However, the end suited for operation over wireless links characterized by higher R Proof. Recall the following expression of the noncooperative optimal solution in Equation (A1) on the basis of Proposition 1. By substituting the expression of function V NC m (p m , E m ) in Equation (27) into Equation (A1), we can easily obtain the noncooperative optimal transmit power p NC m (t) which constitutes a Nash equilibrium solution to the problem P1 can be formulated by Equation (28).
By observing Proposition 2, it is clear that an increased discount factor r will enhance the noncooperative optimal transmit power p NC m (t) for the m-th SSN. From Equation (28), the noncooperative optimal transmit power vector P NC for all IT and MT SSNs in the NoCoPC problem can be further combined as: for n IT IT SSNs , for n MT MT SSNs Given the noncooperative optimal solution p NC m (t), by substituting V NC m (p m , E m ) and p NC m (t) into Equation (26), the function V NC m p NC m (t), E NC m (t) for the m-th SSN is further given by: analog to digital converters with large dynamic range, and high speed s to be incorporated into an individual SU transceiver [10], which increas consumption, especially for the double-radio sensing architecture of S approach is to adopt the cost-effective and dedicated spectrum senso distributed collaborative spectrum sensing and report SSR to SUs act Technically, s wireless sensor network can be naturally exploited to assist about the current spectrum occupancy of PUs in a cooperation fashion. T embedded into CRANETs has further called sensor network-aided CRA been considered as one of the most appealing approaches to perform co in CR systems [7,11,12].
Υ Similar to most other traditional wireless networks or wireline Int SN-CRNs will also occur when offered data load that exceed the available overflow caused by the data sequence of the SSR injected from SSNs t from upstream SUs. This therefore leads to energy consumption of SSN queuing delay, and blocking of new flows from upstream SUs. Indu technique in the transport layer is essential to balance resource loads an However, the congestion control mechanism for the traditional Transm via the acknowledgement-triggered or window-based methods was init to perform in reliable wired links with constrained bit error rates and A recent study [14] has reported that the performance of HTTP down about 40% under the TCP window control in an IEEE P1900.4 based cogn User Datagram Protocol (UDP) and TCP transport protocols. On the oth efforts about congestion control have also been conducted from the pers modify the TCP protocol, such as TCP monitoring delayed acknowledgm acknowledgement, TCP adaptive delayed-acknowledgment window, etc. the challenging multi-hop wireless environments. Unfortunately, it has be of TCP modification and extension cannot be directly applied into SN-CR bandwidth fluctuation, periodic interruption caused by spectrum sensin Recently, there have also been previous works on congestion control a cross-layer design perspective. In [17], an end-to-end congestion cont under the constraint of the non-uniform channel availability by taking from the physical layer to the transport layer. In [18], a cross-layer fram of MAC, scheduling, routing and congestion control was presented to a set of multi-hop end-to-end packet flows. However, the end-to-end suited for operation over wireless links characterized by higher RTTs. On analog to digital converters with large dynamic range, and high s to be incorporated into an individual SU transceiver [10], which consumption, especially for the double-radio sensing architectur approach is to adopt the cost-effective and dedicated spectrum distributed collaborative spectrum sensing and report SSR to S Technically, s wireless sensor network can be naturally exploited to about the current spectrum occupancy of PUs in a cooperation fash embedded into CRANETs has further called sensor network-aide been considered as one of the most appealing approaches to perfo in CR systems [7,11,12].
Υ Similar to most other traditional wireless networks or wirel SN-CRNs will also occur when offered data load that exceed the av overflow caused by the data sequence of the SSR injected from from upstream SUs. This therefore leads to energy consumption queuing delay, and blocking of new flows from upstream SUs. technique in the transport layer is essential to balance resource lo However, the congestion control mechanism for the traditional T via the acknowledgement-triggered or window-based methods w to perform in reliable wired links with constrained bit error rate A recent study [14] has reported that the performance of HTTP about 40% under the TCP window control in an IEEE P1900.4 based User Datagram Protocol (UDP) and TCP transport protocols. On efforts about congestion control have also been conducted from th modify the TCP protocol, such as TCP monitoring delayed acknow acknowledgement, TCP adaptive delayed-acknowledgment windo the challenging multi-hop wireless environments. Unfortunately, it of TCP modification and extension cannot be directly applied into bandwidth fluctuation, periodic interruption caused by spectrum Recently, there have also been previous works on congestion c a cross-layer design perspective. In [17], an end-to-end congestio under the constraint of the non-uniform channel availability by t from the physical layer to the transport layer. In [18], a cross-laye of MAC, scheduling, routing and congestion control was presen a set of multi-hop end-to-end packet flows. However, the end-t suited for operation over wireless links characterized by higher RT where E NC m (t) is the noncooperative optimal energy consumption value of the m-th SSN at time t.

Cooperative Optimal Solution
In this subsection, we move on to explore the distributed cooperative power control (CoPC) problem for all IT and MT SSNs by building up the cooperation scenario if all the players play cooperatively via the differential game model G. Note that this is a natural idea for the players aiming to achieve an optimal power allocation through full cooperation for their common interests. Under this scenario, our optimization objective is to maximize the sum of the utility functions of all players (i.e., ∑ m∈N U m ) while the linear differential equation constraint of the energy consumption should also be satisfied for the m-th SSN. To be specific, we formulate a dynamic optimization problem P2 as follows to attain the objective of maximizing the sum of the utility functions of all players: Under the cooperation scenario, we use p C m (t) to represent the cooperative optimal solution to the problem P2, and assume that there exists a continuously differentiable function W C m (p m , E m ) satisfying the following partial differential equation: Proposition 3. The function W C m (p m , E m ) for the m-th SSN in Equation (32) should be subject to the partial differential equation constraint as follows: Proposition 4. The cooperative optimal solution p C m (t) to the problem P2 if and only if the optimal transmit power for the m-th SSN can be expressed as: 17,2132 by seeking to underlay, overlay, or interweave their signals with those significantly impacting their communications.
Spectrum sensing is one of the key enabling technologies for the esta it constantly allows for the opportunistic identification and use of the S network without causing harmful interference to the PUs. In order to impr collaborative spectrum sensing has been proposed as an effective way to PUs by addressing the issues imposed by the hidden PU terminal probl impairments, such as the heavy shadowing and fading [6][7][8]. In this wa allowing different SUs to collaborate and share their spectrum sensing re center (FC), which makes a global decision on the occupancy status of this centralized FC is not available in decentralized CRANETs. Clearly, must perform the distributed collaborative spectrum sensing, which is pr scheme because of its scalability, fault tolerance and flexibility [9].
In order to facilitate the spectrum sensing functionality, high sam analog to digital converters with large dynamic range, and high speed s to be incorporated into an individual SU transceiver [10], which increas consumption, especially for the double-radio sensing architecture of SU approach is to adopt the cost-effective and dedicated spectrum sensor distributed collaborative spectrum sensing and report SSR to SUs acti Technically, s wireless sensor network can be naturally exploited to assist about the current spectrum occupancy of PUs in a cooperation fashion. T embedded into CRANETs has further called sensor network-aided CRA been considered as one of the most appealing approaches to perform cos in CR systems [7,11,12]. Υ Similar to most other traditional wireless networks or wireline Inte SN-CRNs will also occur when offered data load that exceed the available overflow caused by the data sequence of the SSR injected from SSNs t from upstream SUs. This therefore leads to energy consumption of SSN queuing delay, and blocking of new flows from upstream SUs. Indub technique in the transport layer is essential to balance resource loads and However, the congestion control mechanism for the traditional Transmi via the acknowledgement-triggered or window-based methods was initi to perform in reliable wired links with constrained bit error rates and A recent study [14] has reported that the performance of HTTP downl about 40% under the TCP window control in an IEEE P1900.4 based cogni User Datagram Protocol (UDP) and TCP transport protocols. On the oth efforts about congestion control have also been conducted from the persp modify the TCP protocol, such as TCP monitoring delayed acknowledgm Sensors 2017, 17, 2132 by seeking to underlay, overlay, or interweave their signals wit significantly impacting their communications.
Spectrum sensing is one of the key enabling technologies for it constantly allows for the opportunistic identification and use network without causing harmful interference to the PUs. In order collaborative spectrum sensing has been proposed as an effective PUs by addressing the issues imposed by the hidden PU termin impairments, such as the heavy shadowing and fading [6][7][8]. In allowing different SUs to collaborate and share their spectrum se center (FC), which makes a global decision on the occupancy st this centralized FC is not available in decentralized CRANETs. must perform the distributed collaborative spectrum sensing, wh scheme because of its scalability, fault tolerance and flexibility [9 In order to facilitate the spectrum sensing functionality, h analog to digital converters with large dynamic range, and high to be incorporated into an individual SU transceiver [10], which consumption, especially for the double-radio sensing architectu approach is to adopt the cost-effective and dedicated spectrum distributed collaborative spectrum sensing and report SSR to S Technically, s wireless sensor network can be naturally exploited t about the current spectrum occupancy of PUs in a cooperation fas embedded into CRANETs has further called sensor network-aid been considered as one of the most appealing approaches to perf in CR systems [7,11,12]. Υ Similar to most other traditional wireless networks or wire SN-CRNs will also occur when offered data load that exceed the av overflow caused by the data sequence of the SSR injected from from upstream SUs. This therefore leads to energy consumption queuing delay, and blocking of new flows from upstream SUs technique in the transport layer is essential to balance resource lo However, the congestion control mechanism for the traditional T via the acknowledgement-triggered or window-based methods w to perform in reliable wired links with constrained bit error rat A recent study [14] has reported that the performance of HTTP about 40% under the TCP window control in an IEEE P1900.4 base User Datagram Protocol (UDP) and TCP transport protocols. On efforts about congestion control have also been conducted from t modify the TCP protocol, such as TCP monitoring delayed ackno Proof. Substituting the expression of function W C m (p m , E m ) in Equation (33) into Equation (A4), and applying the result of the indicated maximization operation in Equation (A4), we derive the cooperative optimal solution p C m (t) to the problem P2 as in Equation (34).
Similar to Proposition 2, it is also revealed that an increased discount factor r will enhance the cooperative optimal transmit power p C m (t) for the m-th SSN. According to Equation (34), the cooperative optimal transmit power vector P C for all IT and MT SSNs in the CoPC problem can be also combined as follows: Define the notation Ψ m = P max m H m allowing different SUs to collaborate and share their spectrum sensing results (SSR center (FC), which makes a global decision on the occupancy status of the license this centralized FC is not available in decentralized CRANETs. Clearly, each SU u must perform the distributed collaborative spectrum sensing, which is preferred to scheme because of its scalability, fault tolerance and flexibility [9]. In order to facilitate the spectrum sensing functionality, high sampling rate analog to digital converters with large dynamic range, and high speed signal proce to be incorporated into an individual SU transceiver [10], which increases hardwa consumption, especially for the double-radio sensing architecture of SU transceiv approach is to adopt the cost-effective and dedicated spectrum sensor nodes (SS distributed collaborative spectrum sensing and report SSR to SUs acting as FCs Technically, s wireless sensor network can be naturally exploited to assist a CRANET about the current spectrum occupancy of PUs in a cooperation fashion. The concept embedded into CRANETs has further called sensor network-aided CRANETs (SNbeen considered as one of the most appealing approaches to perform cost-effective in CR systems [7,11,12]. Υ Similar to most other traditional wireless networks or wireline Internet, netw SN-CRNs will also occur when offered data load that exceed the available capacity of overflow caused by the data sequence of the SSR injected from SSNs together w from upstream SUs. This therefore leads to energy consumption of SSNs, aggressi queuing delay, and blocking of new flows from upstream SUs. Indubitably, a c technique in the transport layer is essential to balance resource loads and avoid exc However, the congestion control mechanism for the traditional Transmission Cont via the acknowledgement-triggered or window-based methods was initially design to perform in reliable wired links with constrained bit error rates and round trip A recent study [14] has reported that the performance of HTTP download deteri about 40% under the TCP window control in an IEEE P1900.4 based cognitive wirele User Datagram Protocol (UDP) and TCP transport protocols. On the other hand, so efforts about congestion control have also been conducted from the perspective of fi modify the TCP protocol, such as TCP monitoring delayed acknowledgment, segme acknowledgement, TCP adaptive delayed-acknowledgment window, etc. [15], aimin the challenging multi-hop wireless environments. Unfortunately, it has been shown t of TCP modification and extension cannot be directly applied into SN-CRNs due to bandwidth fluctuation, periodic interruption caused by spectrum sensing and chan Recently, there have also been previous works on congestion control for multi-h a cross-layer design perspective. In [17], an end-to-end congestion control framew under the constraint of the non-uniform channel availability by taking into accou from the physical layer to the transport layer. In [18], a cross-layer framework for of MAC, scheduling, routing and congestion control was presented to maximize a set of multi-hop end-to-end packet flows. However, the end-to-end control pol suited for operation over wireless links characterized by higher RTTs. On the contra allowing different SUs to collaborate and share their spectrum sensing resu center (FC), which makes a global decision on the occupancy status of the this centralized FC is not available in decentralized CRANETs. Clearly, ea must perform the distributed collaborative spectrum sensing, which is prefe scheme because of its scalability, fault tolerance and flexibility [9].
In order to facilitate the spectrum sensing functionality, high sampl analog to digital converters with large dynamic range, and high speed sign to be incorporated into an individual SU transceiver [10], which increases consumption, especially for the double-radio sensing architecture of SU tr approach is to adopt the cost-effective and dedicated spectrum sensor no distributed collaborative spectrum sensing and report SSR to SUs acting Technically, s wireless sensor network can be naturally exploited to assist a C about the current spectrum occupancy of PUs in a cooperation fashion. The embedded into CRANETs has further called sensor network-aided CRANE been considered as one of the most appealing approaches to perform cost-e in CR systems [7,11,12]. Υ Similar to most other traditional wireless networks or wireline Intern SN-CRNs will also occur when offered data load that exceed the available cap overflow caused by the data sequence of the SSR injected from SSNs toge from upstream SUs. This therefore leads to energy consumption of SSNs, a queuing delay, and blocking of new flows from upstream SUs. Indubita technique in the transport layer is essential to balance resource loads and a However, the congestion control mechanism for the traditional Transmissi via the acknowledgement-triggered or window-based methods was initiall to perform in reliable wired links with constrained bit error rates and rou A recent study [14] has reported that the performance of HTTP download about 40% under the TCP window control in an IEEE P1900.4 based cognitiv User Datagram Protocol (UDP) and TCP transport protocols. On the other h efforts about congestion control have also been conducted from the perspec modify the TCP protocol, such as TCP monitoring delayed acknowledgmen acknowledgement, TCP adaptive delayed-acknowledgment window, etc. [15 the challenging multi-hop wireless environments. Unfortunately, it has been of TCP modification and extension cannot be directly applied into SN-CRNs bandwidth fluctuation, periodic interruption caused by spectrum sensing a Recently, there have also been previous works on congestion control for a cross-layer design perspective. In [17], an end-to-end congestion control under the constraint of the non-uniform channel availability by taking int from the physical layer to the transport layer. In [18], a cross-layer framew of MAC, scheduling, routing and congestion control was presented to ma a set of multi-hop end-to-end packet flows. However, the end-to-end con suited for operation over wireless links characterized by higher RTTs. On th f (m) for simplicity in the following. Given the cooperative optimal solution p C where E C m (t) is the cooperative optimal energy consumption value of the m-th SSN at time t. By using P NC of the NoCoPC problem and P C of the CoPC problem, we then design a distributed optimal transmit power adjustment (OTPA) algorithm as stated in Algorithm 2. Recall that the pricing factor ℘ R m is employed to characterize the efficiency-to-stability ratio for the m-th SSN, for m ∈ N , aiming to describe the power reduction efficiency under the stability of local spectrum sensing. Thus, in order to ensure the convergence of optimal transmit power adjustment, we design the average efficiency-to-stability ratio m as a scaling coefficient in Algorithm 2 as follows: It is worth remarking that the adjusted optimal transmit power will be updated with respect ro the scaling coefficient m . In particular, the adjusted optimal transmit power for all IT and MT SSNs via OTPA algorithm can be made locally while guaranteeing that the condition of the average power constraint as given in Equation (3) is satisfied.

Congestion Mitigation for Bottlenecked Secondary User
We have shown that the bottlenecked SU is a little more inclined to be a congested SU node during the slot τ rp due to its buffer overflow under the scenario of the hop-by-hop fluid flows of the data sequence of the SSR injected from |N IT | + |N ET | IT and MT SSNs to this bottlenecked SU. We use the buffer size B b to denote the buffer saturation value of bottlenecked SU b. Recall that the amount of data traffic ϑ b (t 0 ) in buffer of SU b at time t 0 evolves according to Equation (2) for the given slot τ rp . As depicted in Figure 3a, the buffer of bottlenecked SU b is composed of two buffer segments, i.e., the forward buffer that holds data traffic injected from upstream SUs denoted by ϑ F b τ rp , and the internal buffer that stores the data sequence of SSR transmitted from IT and MT SSNs denoted by ϑ I b τ rp . Normally, there exist idle segments within the forward buffer or the internal buffer of bottlenecked SU b when ϑ b t 0 + τ rp < B b . This phenomenon is referred to the normal status where the offered data load does not exceed available buffer capacity of bottleneck SU b because of its higher data rate to deliver the amount of data traffic to other SUs and remove the amount of the data sequence of SSR during the slot τ rp . However, congestion in bottlenecked SU b will occur when the offered data load exceeds its available buffer capacity due to buffer overflow imposed by the data load involving the data sequence of the SSR injected from IT and MT SSNs together with the data traffic from upstream SUs. We assume that the congestion detection information of bottlenecked SU b can be transferred back to the m-th SSN by means of a hop-by-hop backpressure signal via the CCC form bottlenecked SU b to the m-th SSN. As a result, bottlenecked SU b is said to be a congested SU if and only if the amount of its current data traffic ( ) τ , as shown in Figure 3b. We further assume that the m-th SSN has the data sequence of the SSR with m Ξ bits to transmit at time According to [24], the data sequence of SSR with m Ξ bits to transmit from the m-th SSN to ( ) Clearly, m Ξ depends on channel capacity m b C → and is also a function of the transmit power  However, congestion in bottlenecked SU b will occur when the offered data load exceeds its available buffer capacity due to buffer overflow imposed by the data load involving the data sequence of the SSR injected from IT and MT SSNs together with the data traffic from upstream SUs. We assume that the congestion detection information of bottlenecked SU b can be transferred back to the m-th SSN by means of a hop-by-hop backpressure signal via the CCC form bottlenecked SU b to the m-th SSN. As a result, bottlenecked SU b is said to be a congested SU if and only if the amount of its current data traffic ϑ b t 0 + τ rp ≥ B b during the slot τ rp , as shown in Figure 3b. We further assume that the m-th SSN has the data sequence of the SSR with Ξ m bits to transmit at time t ∈ t 0 , t 0 + τ rp . According to [24], the data sequence of SSR with Ξ m bits to transmit from the m-th SSN to bottlenecked SU b during the slot τ rp under the transmit power p m (t) can be approximately expressed as follows based on channel capacity C m→b given by Equation (4): Clearly, Ξ m depends on channel capacity C m→b and is also a function of the transmit power p m (t). For ease of exposition, we use Ξ m (p m (t)) to denote the amount of bits of the data sequence of SSR transmitted from the m-th SSN to bottlenecked SU b under transmit power p m (t). Therefore, the aggregated amount of bits of the data sequence of SSR from all |N IT | + |N ET | IT and MT SSNs to bottleneck SU b during the slot τ rp under transmit power p m (t) is given as ∑ m∈N Ξ m (p m (t)), for N = N IT ∪ N MT . Recall that the internal buffer of bottlenecked SU b is used to store the amount of the data sequence of SSR transmitted from SSNs denoted by ϑ I b τ rp during the slot τ rp by using Equation (2). Hence, within the time interval τ rp , ϑ I b τ rp of bottlenecked SU b can be further formulated by: where χ b τ rp is the amount of the data sequence of SSR removed by bottlenecked SU b within a time interval τ rp . So far, we have mathematically derived the amount of the data sequence of SSR transmitted from SSNs to bottlenecked SU b under the condition that the m-th SSN employs the transmit power p m (t), for m ∈ N . Next, we are concerned with how to mitigate the congestion of bottlenecked SU b by the aid of the proposed distributed power control framework for IT and MT SSNs. The basic idea of congestion mitigation for bottlenecked SU b is to alleviate its buffer load because of the accumulated amount of the data sequence of the SSR transmitted from IT and MT SSNs. It is uncovered that the transmit power of the m-th SSN may have a bearing on Ξ m from Equation (38). Our objective is to leverage the distributed power control to reduce the amount of bits of the data sequence of the SSR transmitted from the m-th SSN to bottlenecked SU b, in order to lower the amount of the data sequence of the SSR in the internal buffer of bottlenecked SU b. This operation will further ensure that offered data load does not exceed available buffer capacity of bottlenecked SU b, i.e., ϑ b t 0 + τ rp < B b . The key point to achieve congestion mitigation for bottlenecked SU b is established with a block diagram shown in Figure 4. It is worth remarking that the amount of the data sequence of SSR in the internal buffer of bottlenecked SU b can be effectively reduced by the proposed distributed power control framework for IT and MT SSNs under the noncooperation and cooperation scenarios. Conceptually, the reduction of the amount of the data sequence of SSR will release the capacity of the internal buffer for bottlenecked SU b, which naturally further attains the congestion mitigation for bottlenecked SU b. In the following, we analyze the impact of optimal transmit power for the m-th SSN on the reduction of the amount of the data sequence of SSR in the internal buffer of bottleneck SU b rigorously from the noncooperation and cooperation cases. Our objective is to leverage the distributed power control to reduce the amount of bits of the data sequence of the SSR transmitted from the m-th SSN to bottlenecked SU b, in order to lower the amount of the data sequence of the SSR in the internal buffer of bottlenecked SU b. This operation will further ensure that offered data load does not exceed available buffer capacity of bottlenecked SU b, i.e., The key point to achieve congestion mitigation for bottlenecked SU b is established with a block diagram shown in Figure 4. It is worth remarking that the amount of the data sequence of SSR in the internal buffer of bottlenecked SU b can be effectively reduced by the proposed distributed power control framework for IT and MT SSNs under the noncooperation and cooperation scenarios. Conceptually, the reduction of the amount of the data sequence of SSR will release the capacity of the internal buffer for bottlenecked SU b, which naturally further attains the congestion mitigation for bottlenecked SU b. In the following, we analyze the impact of optimal transmit power for the m-th SSN on the reduction of the amount of the data sequence of SSR in the internal buffer of bottleneck SU b rigorously from the noncooperation and cooperation cases.     (1) Noncooperative Optimal Transmit Power Case: In this case, suppose, without loss of generality, that τ rp W 1. By replacing p m (t) in Equation (38) with p NC m (t) in Equation (28), Ξ m p NC m (t) for the m-th SSN within a time interval τ rp can be expressed as: must perform the distributed collaborative spectrum sensing, which is preferred t scheme because of its scalability, fault tolerance and flexibility [9]. In order to facilitate the spectrum sensing functionality, high sampling ra analog to digital converters with large dynamic range, and high speed signal pro to be incorporated into an individual SU transceiver [10], which increases hardw consumption, especially for the double-radio sensing architecture of SU transce approach is to adopt the cost-effective and dedicated spectrum sensor nodes ( distributed collaborative spectrum sensing and report SSR to SUs acting as FC Technically, s wireless sensor network can be naturally exploited to assist a CRANE about the current spectrum occupancy of PUs in a cooperation fashion. The concep embedded into CRANETs has further called sensor network-aided CRANETs (SN been considered as one of the most appealing approaches to perform cost-effectiv in CR systems [7,11,12]. Υ Similar to most other traditional wireless networks or wireline Internet, net SN-CRNs will also occur when offered data load that exceed the available capacity overflow caused by the data sequence of the SSR injected from SSNs together w from upstream SUs. This therefore leads to energy consumption of SSNs, aggres queuing delay, and blocking of new flows from upstream SUs. Indubitably, a technique in the transport layer is essential to balance resource loads and avoid e However, the congestion control mechanism for the traditional Transmission Co via the acknowledgement-triggered or window-based methods was initially desig to perform in reliable wired links with constrained bit error rates and round tri A recent study [14] has reported that the performance of HTTP download dete about 40% under the TCP window control in an IEEE P1900.4 based cognitive wire User Datagram Protocol (UDP) and TCP transport protocols. On the other hand, efforts about congestion control have also been conducted from the perspective of modify the TCP protocol, such as TCP monitoring delayed acknowledgment, segm acknowledgement, TCP adaptive delayed-acknowledgment window, etc. [15], aim the challenging multi-hop wireless environments. Unfortunately, it has been shown of TCP modification and extension cannot be directly applied into SN-CRNs due t bandwidth fluctuation, periodic interruption caused by spectrum sensing and cha Recently, there have also been previous works on congestion control for multia cross-layer design perspective. In [17], an end-to-end congestion control frame under the constraint of the non-uniform channel availability by taking into acco from the physical layer to the transport layer. In [18], a cross-layer framework fo of MAC, scheduling, routing and congestion control was presented to maximiz a set of multi-hop end-to-end packet flows. However, the end-to-end control p suited for operation over wireless links characterized by higher RTTs. On the contr d (m) must perform the distributed collaborative spectrum sensing, which is pre scheme because of its scalability, fault tolerance and flexibility [9].
In order to facilitate the spectrum sensing functionality, high samp analog to digital converters with large dynamic range, and high speed sig to be incorporated into an individual SU transceiver [10], which increase consumption, especially for the double-radio sensing architecture of SU approach is to adopt the cost-effective and dedicated spectrum sensor distributed collaborative spectrum sensing and report SSR to SUs actin Technically, s wireless sensor network can be naturally exploited to assist a about the current spectrum occupancy of PUs in a cooperation fashion. Th embedded into CRANETs has further called sensor network-aided CRAN been considered as one of the most appealing approaches to perform cost in CR systems [7,11,12]. Υ Similar to most other traditional wireless networks or wireline Inter SN-CRNs will also occur when offered data load that exceed the available c overflow caused by the data sequence of the SSR injected from SSNs to from upstream SUs. This therefore leads to energy consumption of SSNs queuing delay, and blocking of new flows from upstream SUs. Indubi technique in the transport layer is essential to balance resource loads and However, the congestion control mechanism for the traditional Transmiss via the acknowledgement-triggered or window-based methods was initia to perform in reliable wired links with constrained bit error rates and ro A recent study [14] has reported that the performance of HTTP downlo about 40% under the TCP window control in an IEEE P1900.4 based cogniti User Datagram Protocol (UDP) and TCP transport protocols. On the other efforts about congestion control have also been conducted from the perspe modify the TCP protocol, such as TCP monitoring delayed acknowledgme acknowledgement, TCP adaptive delayed-acknowledgment window, etc. [1 the challenging multi-hop wireless environments. Unfortunately, it has bee of TCP modification and extension cannot be directly applied into SN-CRN bandwidth fluctuation, periodic interruption caused by spectrum sensing Recently, there have also been previous works on congestion control fo a cross-layer design perspective. In [17], an end-to-end congestion contro under the constraint of the non-uniform channel availability by taking in from the physical layer to the transport layer. In [18], a cross-layer frame of MAC, scheduling, routing and congestion control was presented to m a set of multi-hop end-to-end packet flows. However, the end-to-end co suited for operation over wireless links characterized by higher RTTs. On t By plugging Ξ m p NC m (t) into Equation (39), we have: for n IT IT SSNs Thus, the reduction of the amount of the data sequence of SSR in the internal buffer of bottlenecked SU b, denoted by ∆ϑ I b τ rp , is equivalent to: by seeking to underlay, overlay, or interweave their signals wit significantly impacting their communications.
Spectrum sensing is one of the key enabling technologies for it constantly allows for the opportunistic identification and use o network without causing harmful interference to the PUs. In order collaborative spectrum sensing has been proposed as an effective PUs by addressing the issues imposed by the hidden PU termin impairments, such as the heavy shadowing and fading [6][7][8]. In allowing different SUs to collaborate and share their spectrum se center (FC), which makes a global decision on the occupancy st this centralized FC is not available in decentralized CRANETs. C must perform the distributed collaborative spectrum sensing, wh scheme because of its scalability, fault tolerance and flexibility [9 In order to facilitate the spectrum sensing functionality, h analog to digital converters with large dynamic range, and high to be incorporated into an individual SU transceiver [10], which consumption, especially for the double-radio sensing architectu approach is to adopt the cost-effective and dedicated spectrum distributed collaborative spectrum sensing and report SSR to S Technically, s wireless sensor network can be naturally exploited t about the current spectrum occupancy of PUs in a cooperation fas embedded into CRANETs has further called sensor network-aide been considered as one of the most appealing approaches to perf in CR systems [7,11,12]. Υ Similar to most other traditional wireless networks or wire SN-CRNs will also occur when offered data load that exceed the av overflow caused by the data sequence of the SSR injected from from upstream SUs. This therefore leads to energy consumption queuing delay, and blocking of new flows from upstream SUs technique in the transport layer is essential to balance resource lo However, the congestion control mechanism for the traditional T via the acknowledgement-triggered or window-based methods w to perform in reliable wired links with constrained bit error rat A recent study [14] has reported that the performance of HTTP about 40% under the TCP window control in an IEEE P1900.4 base User Datagram Protocol (UDP) and TCP transport protocols. On efforts about congestion control have also been conducted from th modify the TCP protocol, such as TCP monitoring delayed ackno acknowledgement, TCP adaptive delayed-acknowledgment wind the challenging multi-hop wireless environments. Unfortunately, i of TCP modification and extension cannot be directly applied into bandwidth fluctuation, periodic interruption caused by spectrum Recently, there have also been previous works on congestion a cross-layer design perspective. In [17], an end-to-end congestio under the constraint of the non-uniform channel availability by from the physical layer to the transport layer. In [18], a cross-lay of MAC, scheduling, routing and congestion control was presen a set of multi-hop end-to-end packet flows. However, the endsuited for operation over wireless links characterized by higher R d (m) Sensors 2017, 17,2132 by seeking to underlay, overlay, or interweave their sign significantly impacting their communications.
Spectrum sensing is one of the key enabling technolo it constantly allows for the opportunistic identification a network without causing harmful interference to the PUs. collaborative spectrum sensing has been proposed as an e PUs by addressing the issues imposed by the hidden PU impairments, such as the heavy shadowing and fading [6 allowing different SUs to collaborate and share their spec center (FC), which makes a global decision on the occup this centralized FC is not available in decentralized CRA must perform the distributed collaborative spectrum sens scheme because of its scalability, fault tolerance and flexi In order to facilitate the spectrum sensing function analog to digital converters with large dynamic range, an to be incorporated into an individual SU transceiver [10] consumption, especially for the double-radio sensing ar approach is to adopt the cost-effective and dedicated s distributed collaborative spectrum sensing and report S Technically, s wireless sensor network can be naturally exp about the current spectrum occupancy of PUs in a coopera embedded into CRANETs has further called sensor netw been considered as one of the most appealing approache in CR systems [7,11,12]. Υ Similar to most other traditional wireless networks SN-CRNs will also occur when offered data load that excee overflow caused by the data sequence of the SSR injecte from upstream SUs. This therefore leads to energy consu queuing delay, and blocking of new flows from upstre technique in the transport layer is essential to balance res However, the congestion control mechanism for the trad via the acknowledgement-triggered or window-based me to perform in reliable wired links with constrained bit e A recent study [14] has reported that the performance o about 40% under the TCP window control in an IEEE P190 User Datagram Protocol (UDP) and TCP transport protoc efforts about congestion control have also been conducted modify the TCP protocol, such as TCP monitoring delaye acknowledgement, TCP adaptive delayed-acknowledgme the challenging multi-hop wireless environments. Unfortu of TCP modification and extension cannot be directly app bandwidth fluctuation, periodic interruption caused by s Recently, there have also been previous works on con a cross-layer design perspective. In [17], an end-to-end c under the constraint of the non-uniform channel availab from the physical layer to the transport layer. In [18], a c of MAC, scheduling, routing and congestion control wa a set of multi-hop end-to-end packet flows. However, t suited for operation over wireless links characterized by h Proof. Since p m (t) = P max m . Under this condition, using Equation (28), we have: by seeking to underlay, overlay, or interweave their signals with those of th significantly impacting their communications. Spectrum sensing is one of the key enabling technologies for the establis it constantly allows for the opportunistic identification and use of the SOPs network without causing harmful interference to the PUs. In order to improve collaborative spectrum sensing has been proposed as an effective way to relia PUs by addressing the issues imposed by the hidden PU terminal problem a impairments, such as the heavy shadowing and fading [6][7][8]. In this way, co allowing different SUs to collaborate and share their spectrum sensing result center (FC), which makes a global decision on the occupancy status of the l this centralized FC is not available in decentralized CRANETs. Clearly, each must perform the distributed collaborative spectrum sensing, which is prefer scheme because of its scalability, fault tolerance and flexibility [9].
In order to facilitate the spectrum sensing functionality, high samplin analog to digital converters with large dynamic range, and high speed signa to be incorporated into an individual SU transceiver [10], which increases h consumption, especially for the double-radio sensing architecture of SU tra approach is to adopt the cost-effective and dedicated spectrum sensor nod distributed collaborative spectrum sensing and report SSR to SUs acting a Technically, s wireless sensor network can be naturally exploited to assist a CR about the current spectrum occupancy of PUs in a cooperation fashion. The co embedded into CRANETs has further called sensor network-aided CRANET been considered as one of the most appealing approaches to perform cost-eff in CR systems [7,11,12]. Υ Similar to most other traditional wireless networks or wireline Interne SN-CRNs will also occur when offered data load that exceed the available capa overflow caused by the data sequence of the SSR injected from SSNs toget from upstream SUs. This therefore leads to energy consumption of SSNs, ag queuing delay, and blocking of new flows from upstream SUs. Indubitab technique in the transport layer is essential to balance resource loads and avo However, the congestion control mechanism for the traditional Transmission via the acknowledgement-triggered or window-based methods was initially to perform in reliable wired links with constrained bit error rates and roun A recent study [14] has reported that the performance of HTTP download about 40% under the TCP window control in an IEEE P1900.4 based cognitive User Datagram Protocol (UDP) and TCP transport protocols. On the other ha efforts about congestion control have also been conducted from the perspecti modify the TCP protocol, such as TCP monitoring delayed acknowledgment, acknowledgement, TCP adaptive delayed-acknowledgment window, etc. [15], the challenging multi-hop wireless environments. Unfortunately, it has been sh of TCP modification and extension cannot be directly applied into SN-CRNs d bandwidth fluctuation, periodic interruption caused by spectrum sensing an Recently, there have also been previous works on congestion control for m a cross-layer design perspective. In [17], an end-to-end congestion control fr under the constraint of the non-uniform channel availability by taking into from the physical layer to the transport layer. In [18], a cross-layer framewo of MAC, scheduling, routing and congestion control was presented to max a set of multi-hop end-to-end packet flows. However, the end-to-end contr suited for operation over wireless links characterized by higher RTTs. On the d (m) Sensors 2017, 17,2132 by seeking to underlay, overlay, or interweave their signals with th significantly impacting their communications.
Spectrum sensing is one of the key enabling technologies for the it constantly allows for the opportunistic identification and use of th network without causing harmful interference to the PUs. In order to im collaborative spectrum sensing has been proposed as an effective way PUs by addressing the issues imposed by the hidden PU terminal pr impairments, such as the heavy shadowing and fading [6][7][8]. In this allowing different SUs to collaborate and share their spectrum sensin center (FC), which makes a global decision on the occupancy status this centralized FC is not available in decentralized CRANETs. Clea must perform the distributed collaborative spectrum sensing, which i scheme because of its scalability, fault tolerance and flexibility [9].
In order to facilitate the spectrum sensing functionality, high s analog to digital converters with large dynamic range, and high spee to be incorporated into an individual SU transceiver [10], which incr consumption, especially for the double-radio sensing architecture o approach is to adopt the cost-effective and dedicated spectrum sen distributed collaborative spectrum sensing and report SSR to SUs a Technically, s wireless sensor network can be naturally exploited to ass about the current spectrum occupancy of PUs in a cooperation fashion embedded into CRANETs has further called sensor network-aided C been considered as one of the most appealing approaches to perform in CR systems [7,11,12]. Υ Similar to most other traditional wireless networks or wireline SN-CRNs will also occur when offered data load that exceed the availa overflow caused by the data sequence of the SSR injected from SSN from upstream SUs. This therefore leads to energy consumption of S queuing delay, and blocking of new flows from upstream SUs. In technique in the transport layer is essential to balance resource loads However, the congestion control mechanism for the traditional Trans via the acknowledgement-triggered or window-based methods was i to perform in reliable wired links with constrained bit error rates an A recent study [14] has reported that the performance of HTTP dow about 40% under the TCP window control in an IEEE P1900.4 based co User Datagram Protocol (UDP) and TCP transport protocols. On the efforts about congestion control have also been conducted from the pe modify the TCP protocol, such as TCP monitoring delayed acknowled acknowledgement, TCP adaptive delayed-acknowledgment window, e the challenging multi-hop wireless environments. Unfortunately, it has of TCP modification and extension cannot be directly applied into SNbandwidth fluctuation, periodic interruption caused by spectrum sen Recently, there have also been previous works on congestion contr a cross-layer design perspective. In [17], an end-to-end congestion co under the constraint of the non-uniform channel availability by taki from the physical layer to the transport layer. In [18], a cross-layer fr of MAC, scheduling, routing and congestion control was presented a set of multi-hop end-to-end packet flows. However, the end-to-en suited for operation over wireless links characterized by higher RTTs.
Then we obtain the upper bound in Equation (43), thus completing the proof.
(2) Cooperative Optimal Transmit Power Case: In this case, with the assumption τ rp W 1, by replacing p m (t) in Equation (38) with p C m (t) in Equation (34), Ξ m p C m (t) for the m-th SSN within a time interval τ rp can be given by: 17,2132 by seeking to underlay, overlay, or interweave their signals with those of the ex significantly impacting their communications. Spectrum sensing is one of the key enabling technologies for the establishmen it constantly allows for the opportunistic identification and use of the SOPs from network without causing harmful interference to the PUs. In order to improve the se collaborative spectrum sensing has been proposed as an effective way to reliably d PUs by addressing the issues imposed by the hidden PU terminal problem and th impairments, such as the heavy shadowing and fading [6][7][8]. In this way, coopera allowing different SUs to collaborate and share their spectrum sensing results (SSR center (FC), which makes a global decision on the occupancy status of the licens this centralized FC is not available in decentralized CRANETs. Clearly, each SU u must perform the distributed collaborative spectrum sensing, which is preferred to scheme because of its scalability, fault tolerance and flexibility [9].
In order to facilitate the spectrum sensing functionality, high sampling rat analog to digital converters with large dynamic range, and high speed signal proc to be incorporated into an individual SU transceiver [10], which increases hardw consumption, especially for the double-radio sensing architecture of SU transcei approach is to adopt the cost-effective and dedicated spectrum sensor nodes (S distributed collaborative spectrum sensing and report SSR to SUs acting as FCs Technically, s wireless sensor network can be naturally exploited to assist a CRANE about the current spectrum occupancy of PUs in a cooperation fashion. The concep embedded into CRANETs has further called sensor network-aided CRANETs (SN been considered as one of the most appealing approaches to perform cost-effectiv in CR systems [7,11,12]. Υ Similar to most other traditional wireless networks or wireline Internet, netw SN-CRNs will also occur when offered data load that exceed the available capacity o overflow caused by the data sequence of the SSR injected from SSNs together w from upstream SUs. This therefore leads to energy consumption of SSNs, aggress queuing delay, and blocking of new flows from upstream SUs. Indubitably, a technique in the transport layer is essential to balance resource loads and avoid ex However, the congestion control mechanism for the traditional Transmission Con via the acknowledgement-triggered or window-based methods was initially desig to perform in reliable wired links with constrained bit error rates and round trip d (m) Sensors 2017, 17,2132 by seeking to underlay, overlay, or interweave their signals with those of significantly impacting their communications.
Spectrum sensing is one of the key enabling technologies for the estab it constantly allows for the opportunistic identification and use of the SOP network without causing harmful interference to the PUs. In order to improv collaborative spectrum sensing has been proposed as an effective way to re PUs by addressing the issues imposed by the hidden PU terminal problem impairments, such as the heavy shadowing and fading [6][7][8]. In this way, allowing different SUs to collaborate and share their spectrum sensing resu center (FC), which makes a global decision on the occupancy status of th this centralized FC is not available in decentralized CRANETs. Clearly, ea must perform the distributed collaborative spectrum sensing, which is pref scheme because of its scalability, fault tolerance and flexibility [9].
In order to facilitate the spectrum sensing functionality, high samp analog to digital converters with large dynamic range, and high speed sign to be incorporated into an individual SU transceiver [10], which increases consumption, especially for the double-radio sensing architecture of SU t approach is to adopt the cost-effective and dedicated spectrum sensor n distributed collaborative spectrum sensing and report SSR to SUs acting Technically, s wireless sensor network can be naturally exploited to assist a C about the current spectrum occupancy of PUs in a cooperation fashion. The embedded into CRANETs has further called sensor network-aided CRAN been considered as one of the most appealing approaches to perform costin CR systems [7,11,12]. Υ Similar to most other traditional wireless networks or wireline Intern SN-CRNs will also occur when offered data load that exceed the available ca overflow caused by the data sequence of the SSR injected from SSNs tog from upstream SUs. This therefore leads to energy consumption of SSNs, queuing delay, and blocking of new flows from upstream SUs. Indubit technique in the transport layer is essential to balance resource loads and a However, the congestion control mechanism for the traditional Transmiss via the acknowledgement-triggered or window-based methods was initial to perform in reliable wired links with constrained bit error rates and ro By substituting Ξ m p C m (t) into Equation (39), we obtain: Therefore, similar to Equation (42), ∆ϑ I b τ rp is also approximately formulated as: Spectrum sensing is one of the key enabling technologies for th it constantly allows for the opportunistic identification and use of network without causing harmful interference to the PUs. In order to collaborative spectrum sensing has been proposed as an effective w PUs by addressing the issues imposed by the hidden PU terminal p impairments, such as the heavy shadowing and fading [6][7][8]. In thi allowing different SUs to collaborate and share their spectrum sens center (FC), which makes a global decision on the occupancy statu this centralized FC is not available in decentralized CRANETs. Cle must perform the distributed collaborative spectrum sensing, which scheme because of its scalability, fault tolerance and flexibility [9].
In order to facilitate the spectrum sensing functionality, high analog to digital converters with large dynamic range, and high spe to be incorporated into an individual SU transceiver [10], which in consumption, especially for the double-radio sensing architecture approach is to adopt the cost-effective and dedicated spectrum s distributed collaborative spectrum sensing and report SSR to SUs Technically, s wireless sensor network can be naturally exploited to a about the current spectrum occupancy of PUs in a cooperation fashi embedded into CRANETs has further called sensor network-aided been considered as one of the most appealing approaches to perfor in CR systems [7,11,12]. Υ Similar to most other traditional wireless networks or wirelin SN-CRNs will also occur when offered data load that exceed the avai overflow caused by the data sequence of the SSR injected from SS from upstream SUs. This therefore leads to energy consumption of queuing delay, and blocking of new flows from upstream SUs. I technique in the transport layer is essential to balance resource load However, the congestion control mechanism for the traditional Tra via the acknowledgement-triggered or window-based methods was to perform in reliable wired links with constrained bit error rates A recent study [14] has reported that the performance of HTTP d about 40% under the TCP window control in an IEEE P1900.4 based User Datagram Protocol (UDP) and TCP transport protocols. On th efforts about congestion control have also been conducted from the modify the TCP protocol, such as TCP monitoring delayed acknowl acknowledgement, TCP adaptive delayed-acknowledgment window the challenging multi-hop wireless environments. Unfortunately, it h of TCP modification and extension cannot be directly applied into SN bandwidth fluctuation, periodic interruption caused by spectrum se Recently, there have also been previous works on congestion con a cross-layer design perspective. In [17], an end-to-end congestion under the constraint of the non-uniform channel availability by ta from the physical layer to the transport layer. In [18], a cross-layer of MAC, scheduling, routing and congestion control was presente a set of multi-hop end-to-end packet flows. However, the end-tosuited for operation over wireless links characterized by higher RTT d (m) significantly impacting their communications.
Spectrum sensing is one of the key enabling technologi it constantly allows for the opportunistic identification and network without causing harmful interference to the PUs. In collaborative spectrum sensing has been proposed as an effe PUs by addressing the issues imposed by the hidden PU te impairments, such as the heavy shadowing and fading [6][7][8] allowing different SUs to collaborate and share their spectru center (FC), which makes a global decision on the occupan this centralized FC is not available in decentralized CRAN must perform the distributed collaborative spectrum sensin scheme because of its scalability, fault tolerance and flexibil In order to facilitate the spectrum sensing functional analog to digital converters with large dynamic range, and to be incorporated into an individual SU transceiver [10], w consumption, especially for the double-radio sensing arch approach is to adopt the cost-effective and dedicated spe distributed collaborative spectrum sensing and report SSR Technically, s wireless sensor network can be naturally explo about the current spectrum occupancy of PUs in a cooperatio embedded into CRANETs has further called sensor network been considered as one of the most appealing approaches to in CR systems [7,11,12]. Υ Similar to most other traditional wireless networks or SN-CRNs will also occur when offered data load that exceed overflow caused by the data sequence of the SSR injected from upstream SUs. This therefore leads to energy consum queuing delay, and blocking of new flows from upstream technique in the transport layer is essential to balance resou However, the congestion control mechanism for the traditio via the acknowledgement-triggered or window-based meth to perform in reliable wired links with constrained bit err A recent study [14] has reported that the performance of H about 40% under the TCP window control in an IEEE P1900.4 User Datagram Protocol (UDP) and TCP transport protocol efforts about congestion control have also been conducted fr modify the TCP protocol, such as TCP monitoring delayed a acknowledgement, TCP adaptive delayed-acknowledgment the challenging multi-hop wireless environments. Unfortuna of TCP modification and extension cannot be directly applie bandwidth fluctuation, periodic interruption caused by spe Recently, there have also been previous works on conges a cross-layer design perspective. In [17], an end-to-end con under the constraint of the non-uniform channel availabili from the physical layer to the transport layer. In [18], a cros of MAC, scheduling, routing and congestion control was p a set of multi-hop end-to-end packet flows. However, the suited for operation over wireless links characterized by hig Proof. Similar to the proof of Proposition 5, it is easy to identify the upper bound in Equation (48), thus completing the proof.

Simulation Results
In this section, we provide simulation results to evaluate the performance of the proposed congestion mitigation approach by using the distributed power control framework for IT and MT SSNs in the rectangular grid based SN-CRN, and investigate the impact of key system parameters on the performance. Particularly, our simulations pay more attention to evaluate the effect of the proposed distributed power control framework for IT and MT SSNs on the reduction of the amount of the data sequence of SSR in the internal buffer of bottlenecked SU b. Technically, the reduction of the amount of the data sequence of SSR in the internal buffer will free the buffer capacity of bottlenecked SU b, which can bring about the congestion mitigation for bottlenecked SU b. As illustrated in Figure 5, all the simulations are carried out on a rectangular grid topology within a torus area of 100 m × 100 m where one bottlenecked SU is randomly placed within the area of IT over three-tier structure. It suffices to mention that the horizontal or vertical distance between any SSNs is initialized to be d = 20 m. For convenience, the IT SSN and the MT SSN are marked by IT-m and MT-n in sequence for m ∈ N IT and n ∈ N MT , respectively, as shown in Figure 5. We assume that there are N c = 12 licensed uplink channels allocated to PUs. The SOP usage probability δ b is set to 0.65 for the bottlenecked SU. Also, the total average power constraint assigned to IT and MT SSNs are assumed to P IT av = 18 dBm and P MT av = 8 dBm, respectively, to mitigate the interference among IT and MT SSNs. As for the AWGN multiple-access channel, the path-loss exponent κ has been set to 8, and the channel bandwidth used for SSNs is assumed to be 2 MHz according to IEEE 802.15.4a channel model. The constant K under the channel capacity formulation given in Equation (4) is defined as 0.005. In addition, the noise power spectral density N 0 under this channel model is initialized as −3 dBm. We also adopt the receiving reference power by bottleneck SU b P 0 = 20 dBm under the reference distance d 0 = 20 m.
In all the simulations, the detection and false alarm probabilities for IT and MT SSNs over N c = 12 licensed uplink channels have been initialized as in Figure 6. We also set the number of subintervals in Algorithm 1 to be the same value for all SSNs, i.e., ξ = ς = 8. We also assume that the minimum value X d and the maximum value Y d in derivation of the detection probability distribution are set to 0.6 and 1, respectively. Likewise, the minimum value X f and the maximum value Y f in derivation of the false alarm probability distribution are set at 0 and 0.3, respectively.     For simplicity, we deliberately choose some of the IT and MT SSNs in the simulation including IT-1, MT-2, MT-5, and MT-7, to evaluate the performance of our developed approach. Specifically, the detection probability and false alarm probability distributions of the selected SSNs from IT and MT generated from Algorithm 1 are assumed to comply with the corresponding distributions as given by Figure 7. For simplicity, we deliberately choose some of the IT and MT SSNs in the simulation including IT-1, MT-2, MT-5, and MT-7, to evaluate the performance of our developed approach. Specifically, the detection probability and false alarm probability distributions of the selected SSNs from IT and MT generated from Algorithm 1 are assumed to comply with the corresponding distributions as given by Figure 7  The proposed OTPA algorithm for both the NoCoPC problem and the CoPC problem under the distributed power control framework for IT and MT SSNs is compared with the well-known power balancing (PB) algorithm in [35]. The PB algorithm is also a SNR balancing constrained power control iterative method which iteratively searches for decentralized transmit power level updated from the l-th iteration to the (l + 1)-th iteration. Let tar m γ denote the target SNR for the m-th SSN to maintain a certain QoS requirement, for m ∈  . In this simulation, the target SNR can be set to 7 tar m γ = dB.
Therefore, the PB algorithm iteratively updates the transmit power for the m-th SSN according to [35]: First the optimal transmit power is compared between PB algorithm with 250 iterations and our proposed distributed power control framework via the noncooperation and cooperation scenarios under the slot rp τ = 12 ms with the evolution of discount factor r, as exhibited in Figure 8. It is shown that an increased discount factor from 0.1 to 0.9 will enhance the optimal transmit power of selected SSNs (i.e., IT-1, MT-2, MT-5, and MT-7) under the proposed OTPA algorithm for both the NoCoPC problem and the CoPC problem within the slot rp τ = 12 ms. This is due to the fact that the discount factor r has an affect on the optimal transmit power of selected SSNs via Equations (28) and (34). Specifically, the optimal transmit power of selected SSNs grows in proportion to discount factor r. It is also revealed that the optimal transmit power of selected SSNs under a given discount factor satisfy the average power constraint of IT av P = 18 dBm and MT av P = 8 dBm. Thus, the optimal transmit power of selected SSNs will not be adjusted through OTPA algorithm. Comparing the performance of the NoCoPC problem and CoPC problem with the slot rp τ = 12 ms, we can also observe that the optimal transmit power of selected SSNs by using PB algorithm presents a fixed constant value. The reason for this is that the optimal transmit power of the selected SSN by using PB algorithm converges to an expected equilibrium point after 250 iterations. It is also interesting that Proposition 2 has turned out that the optimal transmit power of selected SSNs derived by NoCoPC problem converges to a fixed The proposed OTPA algorithm for both the NoCoPC problem and the CoPC problem under the distributed power control framework for IT and MT SSNs is compared with the well-known power balancing (PB) algorithm in [35]. The PB algorithm is also a SNR balancing constrained power control iterative method which iteratively searches for decentralized transmit power level updated from the l-th iteration to the (l + 1)-th iteration. Let γ tar m denote the target SNR for the m-th SSN to maintain a certain QoS requirement, for m ∈ N . In this simulation, the target SNR can be set to γ tar m = 7 dB. Therefore, the PB algorithm iteratively updates the transmit power for the m-th SSN according to [35]: First the optimal transmit power is compared between PB algorithm with 250 iterations and our proposed distributed power control framework via the noncooperation and cooperation scenarios under the slot τ rp = 12 ms with the evolution of discount factor r, as exhibited in Figure 8. It is shown that an increased discount factor from 0.1 to 0.9 will enhance the optimal transmit power of selected SSNs (i.e., IT-1, MT-2, MT-5, and MT-7) under the proposed OTPA algorithm for both the NoCoPC problem and the CoPC problem within the slot τ rp = 12 ms. This is due to the fact that the discount factor r has an affect on the optimal transmit power of selected SSNs via Equations (28) and (34). Specifically, the optimal transmit power of selected SSNs grows in proportion to discount factor r. It is also revealed that the optimal transmit power of selected SSNs under a given discount factor satisfy the average power constraint of P IT av = 18 dBm and P MT av = 8 dBm. Thus, the optimal transmit power of selected SSNs will not be adjusted through OTPA algorithm. Comparing the performance of the NoCoPC problem and CoPC problem with the slot τ rp = 12 ms, we can also observe that the optimal transmit power of selected SSNs by using PB algorithm presents a fixed constant value. The reason for this is that the optimal transmit power of the selected SSN by using PB algorithm converges to an expected equilibrium point after 250 iterations. It is also interesting that Proposition 2 has turned out that the optimal transmit power of selected SSNs derived by NoCoPC problem converges to a fixed Nash equilibrium point. Moreover, Proposition 4 has also guaranteed that the optimal transmit power of selected SSNs under the CoPC problem will reach to a fixed value. From Figure 8, it is implicitly revealed that the optimal transmit power of selected SSNs by the OTPA algorithm is obviously lower than that of PB algorithm. This observation is reasonable since PB algorithm generates more power consumption to maintain the target SNR. However, the optimal transmit power of selected SSNs via the proposed OTPA algorithm fully depends on the maximum transmit power of the selected SSNs and the pricing factors in differential game model. Nash equilibrium point. Moreover, Proposition 4 has also guaranteed that the optimal transmit power of selected SSNs under the CoPC problem will reach to a fixed value. From Figure 8, it is implicitly revealed that the optimal transmit power of selected SSNs by the OTPA algorithm is obviously lower than that of PB algorithm. This observation is reasonable since PB algorithm generates more power consumption to maintain the target SNR. However, the optimal transmit power of selected SSNs via the proposed OTPA algorithm fully depends on the maximum transmit power of the selected SSNs and the pricing factors in differential game model.  Figure 9 displays the optimal transmit power comparison between PB algorithm with 250 iterations and our proposed distributed power control framework under discount factor r = 0.5 with the evolution of the slot rp τ from 10 ms to 15 ms. As can be seen, the proposed OTPA algorithm for both the NoCoPC problem and CoPC problem under the condition of discount factor r = 0.5 outperforms PB algorithm with 250 iterations significantly in terms of the optimal transmit power of selected SSNs with the growth of the slot rp τ . This result further validates PB algorithm will result in much more power consumption aiming to maintain the target SNR. However, the optimal transmit power of selected SSNs by using the proposed OTPA algorithm entirely depends upon maximum transmit power of selected SSNs and pricing factors in differential game model. We can also observe that the optimal transmit power of selected SSNs of the proposed OTPA algorithm for the CoPC problem under discount factor r = 0.5 is considerably lower than that of the NoCoPC problem. That is, the proposed OTPA algorithm for the CoPC problem outperforms that of the NoCoPC problem. This can be intuitively explained by the fact that there exists an operation of summing with respect   Figure 9 displays the optimal transmit power comparison between PB algorithm with 250 iterations and our proposed distributed power control framework under discount factor r = 0.5 with the evolution of the slot τ rp from 10 ms to 15 ms. As can be seen, the proposed OTPA algorithm for both the NoCoPC problem and CoPC problem under the condition of discount factor r = 0.5 outperforms PB algorithm with 250 iterations significantly in terms of the optimal transmit power of selected SSNs with the growth of the slot τ rp . This result further validates PB algorithm will result in much more power consumption aiming to maintain the target SNR. However, the optimal transmit power of selected SSNs by using the proposed OTPA algorithm entirely depends upon maximum transmit power of selected SSNs and pricing factors in differential game model. We can also observe that the optimal transmit power of selected SSNs of the proposed OTPA algorithm for the CoPC problem under discount factor r = 0.5 is considerably lower than that of the NoCoPC problem. That is, the proposed OTPA algorithm for the CoPC problem outperforms that of the NoCoPC problem. This can be intuitively explained by the fact that there exists an operation of summing with respect to e δ i ·D m (P d P f ) in the denominator of the analytical cooperative optimal transmit power p C m (t). Moreover, the increase of the slot τ rp will generate lower transmit power for both the NoCoPC problem and CoPC problem with discount factor r = 0.5. This is because based on Equations (28) and (34), the optimal transmit power for both the NoCoPC problem and CoPC problem is inversely proportional to the slot In Figure 10, we look at the performance of the transmitted data sequence of SSR from selected SSNs to bottlenecked SU b during the slot rp τ = 12 ms with the evolution of discount factor r from 0.1 to 0.9. From the results, we can see the transmitted data sequence of SSR for selected SSNs gradually increase with the growth of discount factor r. Meanwhile, the transmitted data sequence of SSR for PB algorithm with 250 iterations and our proposed distributed power control framework under the condition of the slot rp τ = 12 ms tends to be close when discount factor r = 0.9. The reason for this is that the transmitted data sequence of SSR can be approximately expressed as a function of channel capacity according to Equation (38), which is in direct proportion to the optimal transmit power of selected SSNs. It has also been shown that an increased discount factor from 0.1 to 0.9 will increase the optimal transmit power of selected SSNs under the proposed OTPA algorithm for both the NoCoPC problem and CoPC problem with the slot rp τ = 12 ms. Consequently, higher optimal transmit power yields more transmitted data sequence of SSR.
In Figure 11, we examine the impact of discount factor r from 0.1 to 0.9 on the reduction of data sequence of SSR with respect to selected SSNs in the internal buffer of bottleneck SU b. It can be observed from the figure that the reduction of data sequence of SSR by our proposed congestion mitigation approach under the slot rp τ = 12 ms will gradually decrease with the growth of discount factor r, except that the reduction result by PB algorithm with 250 iterations appears to a fixed constant value. This trend is the result of the inverse relationship between the reduction of the data sequence of SSR   In Figure 10, we look at the performance of the transmitted data sequence of SSR from selected SSNs to bottlenecked SU b during the slot τ rp = 12 ms with the evolution of discount factor r from 0.1 to 0.9. From the results, we can see the transmitted data sequence of SSR for selected SSNs gradually increase with the growth of discount factor r. Meanwhile, the transmitted data sequence of SSR for PB algorithm with 250 iterations and our proposed distributed power control framework under the condition of the slot τ rp = 12 ms tends to be close when discount factor r = 0.9. The reason for this is that the transmitted data sequence of SSR can be approximately expressed as a function of channel capacity according to Equation (38), which is in direct proportion to the optimal transmit power of selected SSNs. It has also been shown that an increased discount factor from 0.1 to 0.9 will increase the optimal transmit power of selected SSNs under the proposed OTPA algorithm for both the NoCoPC problem and CoPC problem with the slot τ rp = 12 ms. Consequently, higher optimal transmit power yields more transmitted data sequence of SSR.
In Figure 11, we examine the impact of discount factor r from 0.1 to 0.9 on the reduction of data sequence of SSR with respect to selected SSNs in the internal buffer of bottleneck SU b. It can be observed from the figure that the reduction of data sequence of SSR by our proposed congestion mitigation approach under the slot τ rp = 12 ms will gradually decrease with the growth of discount factor r, except that the reduction result by PB algorithm with 250 iterations appears to a fixed constant value. This trend is the result of the inverse relationship between the reduction of the data sequence of SSR ∆ϑ I b τ rp and the optimal transmit power of selected SSNs p NC m (t) for the NoCoPC problem or p C m (t) for the CoPC problem according to Equations (42) and (47). Additionally, in Figure 11, we can also observe that the proposed congestion mitigation approach by using the OTPA algorithm for the CoPC problem with the slot τ rp = 12 ms achieves the higher reduction of the data sequence of SSR compared with those of the NoCoPC problem and PB algorithm with 250 iterations. In other words, through cooperation among all IT and MT SSNs, the reduction of the data sequence of SSR in the internal buffer of bottlenecked SU b can be further improved. This observation has verified the analytical derivation of the proposed congestion mitigation approach. The explanation is twofold: (i) The proposed OTPA algorithm for the CoPC problem outperforms that of the NoCoPC problem owing to the lower cooperative optimal transmit power p C m (t) obtained by the CoPC problem compared with the noncooperative optimal transmit power p NC m (t) by the NoCoPC problem; (ii) The reduction of the data sequence of SSR is inversely proportional to the optimal transmit power of selected SSNs.    Figure 12 shows the comparison of the reduction of data sequence of SSR with respect to selected SSNs in the internal buffer of bottlenecked SU b, versus the slot τ rp under the condition of discount factor r = 0.3, for the proposed congestion mitigation approach and PB algorithm with 250 iterations. As seen from Figure 11, the reduction of data sequence of SSR through PB algorithm has converged to the same constant value after the 250 iterations. This is the direct influence that the optimal transmit power of selected SSN will converge to an expected equilibrium point by using PB algorithm. As expected, the reduction of data sequence of SSR by the proposed congestion mitigation approach with discount factor r = 0.3 is larger than that of PB algorithm with 250 iterations. This is because our proposed approach can obtain smaller optimal transmit power than PB algorithm. According to Equations (42) and (47), the smaller optimal transmit power will lead to the more reduction of data sequence of SSR with respect to selected SSNs. It is also interesting that with the growth of the slot τ rp the reduction of data sequence of SSR by our proposed approach will gradually increase, and the reduction by the CoPC problem is much higher than that of the NoCoPC problem. This result can be interpreted by the fact that the optimal transmit power of selected SSNs by the CoPC problem under discount factor r = 0.3 is clearly lower than that of the NoCoPC problem with the growth of the slot τ rp . This result further gives rise to the larger reduction of data sequence of SSR.
Sensors 2017, 17, 2132 25 of 28 Figure 12 shows the comparison of the reduction of data sequence of SSR with respect to selected SSNs in the internal buffer of bottlenecked SU b, versus the slot rp τ under the condition of discount factor r = 0.3, for the proposed congestion mitigation approach and PB algorithm with 250 iterations. As seen from Figure 11, the reduction of data sequence of SSR through PB algorithm has converged to the same constant value after the 250 iterations. This is the direct influence that the optimal transmit power of selected SSN will converge to an expected equilibrium point by using PB algorithm. As expected, the reduction of data sequence of SSR by the proposed congestion mitigation approach with discount factor r = 0.3 is larger than that of PB algorithm with 250 iterations. This is because our proposed approach can obtain smaller optimal transmit power than PB algorithm. According to Equations (42) and (47), the smaller optimal transmit power will lead to the more reduction of data sequence of SSR with respect to selected SSNs. It is also interesting that with the growth of the slot rp τ the reduction of data sequence of SSR by our proposed approach will gradually increase, and the reduction by the CoPC problem is much higher than that of the NoCoPC problem. This result can be interpreted by the fact that the optimal transmit power of selected SSNs by the CoPC problem under discount factor r = 0.3 is clearly lower than that of the NoCoPC problem with the growth of the slot rp τ . This result further gives rise to the larger reduction of data sequence of SSR.

Conclusions and Future Work
In this paper, we have developed a congestion mitigation approach by employing the distributed power control framework for SSNs in the rectangular grid based SN-CRN. Particularly, we defined the relative divergence between the detection probability and false alarm probability for a SSN under any uplink channel by adopting a Kullback-Leibler divergence framework. After deriving the detection probability and false alarm probability distributions for SSN according to mathematical statistics, we characterized the stability metric of local spectrum sensing based on entropy modeling framework. Aiming to gain the tradeoff between channel capacity and energy consumption, the distributed power control framework for IT and MT SSNs was proposed, and the power control problem was formulated as differential game model by taking into account the utility function maximization with linear differential equation constraint in regard to energy consumption. Further, we derived the theoretic optimal solutions to this game model under the scenario of cooperation or noncooperation via dynamic programming. Based on the obtained optimal transmit

Conclusions and Future Work
In this paper, we have developed a congestion mitigation approach by employing the distributed power control framework for SSNs in the rectangular grid based SN-CRN. Particularly, we defined the relative divergence between the detection probability and false alarm probability for a SSN under any uplink channel by adopting a Kullback-Leibler divergence framework. After deriving the detection probability and false alarm probability distributions for SSN according to mathematical statistics, we characterized the stability metric of local spectrum sensing based on entropy modeling framework. Aiming to gain the tradeoff between channel capacity and energy consumption, the distributed power control framework for IT and MT SSNs was proposed, and the power control problem was formulated as differential game model by taking into account the utility function maximization with linear differential equation constraint in regard to energy consumption. Further, we derived the theoretic optimal solutions to this game model under the scenario of cooperation or noncooperation via dynamic programming. Based on the obtained optimal transmit power of SSNs, we devised the congestion mitigation approach for bottleneck SU by alleviating buffer load over its internal buffer, and validated its performance with simulations.
What we have discussed in this paper is the portion of the foundation for SN-CRN. Possible directions for future work within this research involve examining the effect of imperfect CSI and outage constraint on distributed power control for SSNs by formulating the uncertain relation between the wireless channel conditions and the corresponding estimates. As another future work, we will try to investigate congestion mitigation approaches in future 5G mobile systems with the novel network architecture and networking technologies [36], e.g., fog computing-based radio access networks and network slicing-based mobile networks in currently practical applications.
Sensors 2017, 17,2132 by seeking to underlay, overlay, or interweave their signals w significantly impacting their communications. Spectrum sensing is one of the key enabling technologies it constantly allows for the opportunistic identification and us network without causing harmful interference to the PUs. In ord collaborative spectrum sensing has been proposed as an effecti PUs by addressing the issues imposed by the hidden PU term impairments, such as the heavy shadowing and fading [6][7][8]. allowing different SUs to collaborate and share their spectrum center (FC), which makes a global decision on the occupancy this centralized FC is not available in decentralized CRANET must perform the distributed collaborative spectrum sensing, w scheme because of its scalability, fault tolerance and flexibility In order to facilitate the spectrum sensing functionality analog to digital converters with large dynamic range, and hig to be incorporated into an individual SU transceiver [10], whi consumption, especially for the double-radio sensing archite approach is to adopt the cost-effective and dedicated spectr distributed collaborative spectrum sensing and report SSR t Technically, s wireless sensor network can be naturally exploite about the current spectrum occupancy of PUs in a cooperation embedded into CRANETs has further called sensor network-a been considered as one of the most appealing approaches to p in CR systems [7,11,12]. Υ Similar to most other traditional wireless networks or w SN-CRNs will also occur when offered data load that exceed the overflow caused by the data sequence of the SSR injected fro from upstream SUs. This therefore leads to energy consumpti queuing delay, and blocking of new flows from upstream S technique in the transport layer is essential to balance resource However, the congestion control mechanism for the traditiona via the acknowledgement-triggered or window-based method to perform in reliable wired links with constrained bit error A recent study [14] has reported that the performance of HT about 40% under the TCP window control in an IEEE P1900.4 b User Datagram Protocol (UDP) and TCP transport protocols. O efforts about congestion control have also been conducted from modify the TCP protocol, such as TCP monitoring delayed ack acknowledgement, TCP adaptive delayed-acknowledgment wi the challenging multi-hop wireless environments. Unfortunatel of TCP modification and extension cannot be directly applied in bandwidth fluctuation, periodic interruption caused by spectru Recently, there have also been previous works on congestio a cross-layer design perspective. In [17], an end-to-end conges under the constraint of the non-uniform channel availability from the physical layer to the transport layer. In [18], a cross-l of MAC, scheduling, routing and congestion control was pre a set of multi-hop end-to-end packet flows. However, the en suited for operation over wireless links characterized by higher d (m) Sensors 2017, 17,2132 by seeking to underlay, overlay, or interweave their s significantly impacting their communications.
Spectrum sensing is one of the key enabling techn it constantly allows for the opportunistic identification network without causing harmful interference to the PU collaborative spectrum sensing has been proposed as a PUs by addressing the issues imposed by the hidden P impairments, such as the heavy shadowing and fading allowing different SUs to collaborate and share their sp center (FC), which makes a global decision on the occ this centralized FC is not available in decentralized C must perform the distributed collaborative spectrum se scheme because of its scalability, fault tolerance and fle In order to facilitate the spectrum sensing funct analog to digital converters with large dynamic range, to be incorporated into an individual SU transceiver [ consumption, especially for the double-radio sensing approach is to adopt the cost-effective and dedicated distributed collaborative spectrum sensing and repo Technically, s wireless sensor network can be naturally e about the current spectrum occupancy of PUs in a coop embedded into CRANETs has further called sensor ne been considered as one of the most appealing approac in CR systems [7,11,12]. Υ Similar to most other traditional wireless networ SN-CRNs will also occur when offered data load that ex overflow caused by the data sequence of the SSR inje from upstream SUs. This therefore leads to energy con queuing delay, and blocking of new flows from ups technique in the transport layer is essential to balance However, the congestion control mechanism for the tr via the acknowledgement-triggered or window-based to perform in reliable wired links with constrained b A recent study [14] has reported that the performanc about 40% under the TCP window control in an IEEE P1 User Datagram Protocol (UDP) and TCP transport pro efforts about congestion control have also been conduc modify the TCP protocol, such as TCP monitoring dela acknowledgement, TCP adaptive delayed-acknowledgm the challenging multi-hop wireless environments. Unfo of TCP modification and extension cannot be directly a bandwidth fluctuation, periodic interruption caused by Recently, there have also been previous works on c a cross-layer design perspective. In [17], an end-to-end under the constraint of the non-uniform channel avai from the physical layer to the transport layer. In [18], of MAC, scheduling, routing and congestion control w a set of multi-hop end-to-end packet flows. However suited for operation over wireless links characterized b By substituting p NC m (t) into Equation (25) Spectrum sensing is one of the key enabling technologies for the establishment of CRNs, because it constantly allows for the opportunistic identification and use of the SOPs from a licensed primary network without causing harmful interference to the PUs. In order to improve the sensing performance, collaborative spectrum sensing has been proposed as an effective way to reliably detect the activity of PUs by addressing the issues imposed by the hidden PU terminal problem and the wireless channel impairments, such as the heavy shadowing and fading [6][7][8]. In this way, cooperation is achieved by allowing different SUs to collaborate and share their spectrum sensing results (SSR) through a fusion center (FC), which makes a global decision on the occupancy status of the licensed band. However, this centralized FC is not available in decentralized CRANETs. Clearly, each SU under this scenario must perform the distributed collaborative spectrum sensing, which is preferred to the centralized FC scheme because of its scalability, fault tolerance and flexibility [9].
In order to facilitate the spectrum sensing functionality, high sampling rates, high resolution analog to digital converters with large dynamic range, and high speed signal processors are required to be incorporated into an individual SU transceiver [10], which increases hardware cost and power consumption, especially for the double-radio sensing architecture of SU transceiver. An alternative approach is to adopt the cost-effective and dedicated spectrum sensor nodes (SSNs) that perform distributed collaborative spectrum sensing and report SSR to SUs acting as FCs in CRANETs [11].
Technically, s wireless sensor network can be naturally exploited to assist a CRANET by providing SSR about the current spectrum occupancy of PUs in a cooperation fashion. The concept of sensor network embedded into CRANETs has further called sensor network-aided CRANETs (SN-CRNs), which has been considered as one of the most appealing approaches to perform cost-effective spectrum sensing in CR systems [7,11,12].
Υ Similar to most other traditional wireless networks or wireline Internet, network congestion in SN-CRNs will also occur when offered data load that exceed the available capacity of a SU due to buffer overflow caused by the data sequence of the SSR injected from SSNs together with the data traffic from upstream SUs. This therefore leads to energy consumption of SSNs, aggressive retransmission, queuing delay, and blocking of new flows from upstream SUs. Indubitably, a congestion control technique in the transport layer is essential to balance resource loads and avoid excessive congestion.
However, the congestion control mechanism for the traditional Transmission Control Protocol (TCP) via the acknowledgement-triggered or window-based methods was initially designed and optimized to perform in reliable wired links with constrained bit error rates and round trip times (RTTs) [13].
A recent study [14] has reported that the performance of HTTP download deteriorates as much as about 40% under the TCP window control in an IEEE P1900.4 based cognitive wireless system by using Recently, there have also been previous works on congestion control for multi-hop CRANETs from a cross-layer design perspective. In [17], an end-to-end congestion control framework was proposed under the constraint of the non-uniform channel availability by taking into account the interactions from the physical layer to the transport layer. In [18], a cross-layer framework for joint optimization of MAC, scheduling, routing and congestion control was presented to maximize the throughput of a set of multi-hop end-to-end packet flows. However, the end-to-end control policy in [17,18]  Spectrum sensing is one of the key enabling technologies for the establishment of CRNs, because it constantly allows for the opportunistic identification and use of the SOPs from a licensed primary network without causing harmful interference to the PUs. In order to improve the sensing performance, collaborative spectrum sensing has been proposed as an effective way to reliably detect the activity of PUs by addressing the issues imposed by the hidden PU terminal problem and the wireless channel impairments, such as the heavy shadowing and fading [6][7][8]. In this way, cooperation is achieved by allowing different SUs to collaborate and share their spectrum sensing results (SSR) through a fusion center (FC), which makes a global decision on the occupancy status of the licensed band. However, this centralized FC is not available in decentralized CRANETs. Clearly, each SU under this scenario must perform the distributed collaborative spectrum sensing, which is preferred to the centralized FC scheme because of its scalability, fault tolerance and flexibility [9].
In order to facilitate the spectrum sensing functionality, high sampling rates, high resolution analog to digital converters with large dynamic range, and high speed signal processors are required to be incorporated into an individual SU transceiver [10], which increases hardware cost and power consumption, especially for the double-radio sensing architecture of SU transceiver. An alternative approach is to adopt the cost-effective and dedicated spectrum sensor nodes (SSNs) that perform distributed collaborative spectrum sensing and report SSR to SUs acting as FCs in CRANETs [11].
Technically, s wireless sensor network can be naturally exploited to assist a CRANET by providing SSR about the current spectrum occupancy of PUs in a cooperation fashion. The concept of sensor network embedded into CRANETs has further called sensor network-aided CRANETs (SN-CRNs), which has been considered as one of the most appealing approaches to perform cost-effective spectrum sensing in CR systems [7,11,12].
Υ Similar to most other traditional wireless networks or wireline Internet, network congestion in SN-CRNs will also occur when offered data load that exceed the available capacity of a SU due to buffer overflow caused by the data sequence of the SSR injected from SSNs together with the data traffic from upstream SUs. This therefore leads to energy consumption of SSNs, aggressive retransmission, queuing delay, and blocking of new flows from upstream SUs. Indubitably, a congestion control technique in the transport layer is essential to balance resource loads and avoid excessive congestion.
However, the congestion control mechanism for the traditional Transmission Control Protocol (TCP) via the acknowledgement-triggered or window-based methods was initially designed and optimized to perform in reliable wired links with constrained bit error rates and round trip times (RTTs) [13].
A recent study [14] has reported that the performance of HTTP download deteriorates as much as about 40% under the TCP window control in an IEEE P1900.4 based cognitive wireless system by using Recently, there have also been previous works on congestion control for multi-hop CRANETs from a cross-layer design perspective. In [17], an end-to-end congestion control framework was proposed under the constraint of the non-uniform channel availability by taking into account the interactions from the physical layer to the transport layer. In [18], a cross-layer framework for joint optimization of MAC, scheduling, routing and congestion control was presented to maximize the throughput of a set of multi-hop end-to-end packet flows. However, the end-to-end control policy in [17,18] is ill suited for operation over wireless links characterized by higher RTTs. On the contrary, the hop-by-hop 17,2132 by seeking to underlay, overlay, or interweave their signals with those of t significantly impacting their communications.
Spectrum sensing is one of the key enabling technologies for the establi it constantly allows for the opportunistic identification and use of the SOPs network without causing harmful interference to the PUs. In order to improve collaborative spectrum sensing has been proposed as an effective way to reli PUs by addressing the issues imposed by the hidden PU terminal problem impairments, such as the heavy shadowing and fading [6][7][8]. In this way, co allowing different SUs to collaborate and share their spectrum sensing resul center (FC), which makes a global decision on the occupancy status of the this centralized FC is not available in decentralized CRANETs. Clearly, eac must perform the distributed collaborative spectrum sensing, which is prefe scheme because of its scalability, fault tolerance and flexibility [9].
In order to facilitate the spectrum sensing functionality, high sampli analog to digital converters with large dynamic range, and high speed signa to be incorporated into an individual SU transceiver [10], which increases h consumption, especially for the double-radio sensing architecture of SU tra approach is to adopt the cost-effective and dedicated spectrum sensor no distributed collaborative spectrum sensing and report SSR to SUs acting Technically, s wireless sensor network can be naturally exploited to assist a CR about the current spectrum occupancy of PUs in a cooperation fashion. The c embedded into CRANETs has further called sensor network-aided CRANE been considered as one of the most appealing approaches to perform cost-ef in CR systems [7,11,12].
Υ Similar to most other traditional wireless networks or wireline Interne SN-CRNs will also occur when offered data load that exceed the available cap overflow caused by the data sequence of the SSR injected from SSNs toge from upstream SUs. This therefore leads to energy consumption of SSNs, a queuing delay, and blocking of new flows from upstream SUs. Indubita technique in the transport layer is essential to balance resource loads and av However, the congestion control mechanism for the traditional Transmissio via the acknowledgement-triggered or window-based methods was initially to perform in reliable wired links with constrained bit error rates and rou A recent study [14]  Spectrum sensing is one of the key enabling technologies for the it constantly allows for the opportunistic identification and use of th network without causing harmful interference to the PUs. In order to i collaborative spectrum sensing has been proposed as an effective wa PUs by addressing the issues imposed by the hidden PU terminal p impairments, such as the heavy shadowing and fading [6][7][8]. In this allowing different SUs to collaborate and share their spectrum sensin center (FC), which makes a global decision on the occupancy status this centralized FC is not available in decentralized CRANETs. Clea must perform the distributed collaborative spectrum sensing, which scheme because of its scalability, fault tolerance and flexibility [9].
In order to facilitate the spectrum sensing functionality, high analog to digital converters with large dynamic range, and high spee to be incorporated into an individual SU transceiver [10], which inc consumption, especially for the double-radio sensing architecture o approach is to adopt the cost-effective and dedicated spectrum se distributed collaborative spectrum sensing and report SSR to SUs Technically, s wireless sensor network can be naturally exploited to as about the current spectrum occupancy of PUs in a cooperation fashio embedded into CRANETs has further called sensor network-aided C been considered as one of the most appealing approaches to perform in CR systems [7,11,12].
Υ Similar to most other traditional wireless networks or wireline SN-CRNs will also occur when offered data load that exceed the availa overflow caused by the data sequence of the SSR injected from SSN from upstream SUs. This therefore leads to energy consumption of queuing delay, and blocking of new flows from upstream SUs. In technique in the transport layer is essential to balance resource loads However, the congestion control mechanism for the traditional Tran via the acknowledgement-triggered or window-based methods was to perform in reliable wired links with constrained bit error rates a A recent study [14]  Recently, there have also been previous works on congestion cont a cross-layer design perspective. In [17], an end-to-end congestion c under the constraint of the non-uniform channel availability by tak from the physical layer to the transport layer. In [18], a cross-layer f of MAC, scheduling, routing and congestion control was presented a set of multi-hop end-to-end packet flows. However, the end-to-e suited for operation over wireless links characterized by higher RTTs.
To proceed, we derive the derivative of V NC m (p m , E m ) with respect to E m (t) in Equation (A2). Upon solving the partial differential equation, after some simplifications, we now can express the function V NC m (p m , E m ) as the partial differential equation constraint given by: This completes the proof.

Appendix B
Proof of Proposition 3. The proof is similar to Proposition 1. The only difference is that the objective function of the cooperative power control problem is to maximize the sum of the utility functions of all players. For convenience of derivation, we also relax the time interval of the game and discuss the infinite-horizon differential game (i.e., τ rp → ∞ ), and we also set t 0 = 0. By performing the maximization operation of the right hand side of Equation (31) with respect to the transmit power of all players p 1 (t), p 2 (t), · · · , p |N | (t), after some simplifications, we have: analog to digital converters with large dynamic range, and high to be incorporated into an individual SU transceiver [10], whic consumption, especially for the double-radio sensing architec approach is to adopt the cost-effective and dedicated spectru distributed collaborative spectrum sensing and report SSR to Technically, s wireless sensor network can be naturally exploited about the current spectrum occupancy of PUs in a cooperation f embedded into CRANETs has further called sensor network-ai been considered as one of the most appealing approaches to pe in CR systems [7,11,12].
Υ Similar to most other traditional wireless networks or wir SN-CRNs will also occur when offered data load that exceed the overflow caused by the data sequence of the SSR injected from from upstream SUs. This therefore leads to energy consumptio queuing delay, and blocking of new flows from upstream SU technique in the transport layer is essential to balance resource However, the congestion control mechanism for the traditional via the acknowledgement-triggered or window-based methods to perform in reliable wired links with constrained bit error r A recent study [14] has reported that the performance of HTT about 40% under the TCP window control in an IEEE P1900.4 ba User Datagram Protocol (UDP) and TCP transport protocols. O efforts about congestion control have also been conducted from modify the TCP protocol, such as TCP monitoring delayed ackn acknowledgement, TCP adaptive delayed-acknowledgment win the challenging multi-hop wireless environments. Unfortunately of TCP modification and extension cannot be directly applied in bandwidth fluctuation, periodic interruption caused by spectru Recently, there have also been previous works on congestion a cross-layer design perspective. In [17], an end-to-end congest under the constraint of the non-uniform channel availability b from the physical layer to the transport layer. In [18], a cross-la of MAC, scheduling, routing and congestion control was pres a set of multi-hop end-to-end packet flows. However, the end suited for operation over wireless links characterized by higher d (m) analog to digital converters with large dynamic range, a to be incorporated into an individual SU transceiver [1 consumption, especially for the double-radio sensing a approach is to adopt the cost-effective and dedicated distributed collaborative spectrum sensing and report Technically, s wireless sensor network can be naturally e about the current spectrum occupancy of PUs in a coope embedded into CRANETs has further called sensor net been considered as one of the most appealing approach in CR systems [7,11,12]. Υ Similar to most other traditional wireless network SN-CRNs will also occur when offered data load that exc overflow caused by the data sequence of the SSR injec from upstream SUs. This therefore leads to energy con queuing delay, and blocking of new flows from upstr technique in the transport layer is essential to balance r However, the congestion control mechanism for the tra via the acknowledgement-triggered or window-based m to perform in reliable wired links with constrained bit A recent study [14] has reported that the performance about 40% under the TCP window control in an IEEE P1 User Datagram Protocol (UDP) and TCP transport prot efforts about congestion control have also been conduct modify the TCP protocol, such as TCP monitoring delay acknowledgement, TCP adaptive delayed-acknowledgm the challenging multi-hop wireless environments. Unfor of TCP modification and extension cannot be directly ap bandwidth fluctuation, periodic interruption caused by Recently, there have also been previous works on co a cross-layer design perspective. In [17], an end-to-end under the constraint of the non-uniform channel availa from the physical layer to the transport layer. In [18], a of MAC, scheduling, routing and congestion control w a set of multi-hop end-to-end packet flows. However, suited for operation over wireless links characterized by f (m) τ rp 2 (A4) By substituting p C m (t) into Equation (31), after some algebraic manipulations, we obtain: PUs by addressing the issues imposed by the hidden PU terminal problem and the wireless chann impairments, such as the heavy shadowing and fading [6][7][8]. In this way, cooperation is achieved allowing different SUs to collaborate and share their spectrum sensing results (SSR) through a fusi center (FC), which makes a global decision on the occupancy status of the licensed band. Howev this centralized FC is not available in decentralized CRANETs. Clearly, each SU under this scena must perform the distributed collaborative spectrum sensing, which is preferred to the centralized scheme because of its scalability, fault tolerance and flexibility [9]. In order to facilitate the spectrum sensing functionality, high sampling rates, high resoluti analog to digital converters with large dynamic range, and high speed signal processors are requir to be incorporated into an individual SU transceiver [10], which increases hardware cost and pow consumption, especially for the double-radio sensing architecture of SU transceiver. An alternati approach is to adopt the cost-effective and dedicated spectrum sensor nodes (SSNs) that perfo distributed collaborative spectrum sensing and report SSR to SUs acting as FCs in CRANETs [1 Technically, s wireless sensor network can be naturally exploited to assist a CRANET by providing S about the current spectrum occupancy of PUs in a cooperation fashion. The concept of sensor netwo embedded into CRANETs has further called sensor network-aided CRANETs (SN-CRNs), which h been considered as one of the most appealing approaches to perform cost-effective spectrum sensi in CR systems [7,11,12]. Υ Similar to most other traditional wireless networks or wireline Internet, network congestion SN-CRNs will also occur when offered data load that exceed the available capacity of a SU due to buf overflow caused by the data sequence of the SSR injected from SSNs together with the data traf from upstream SUs. This therefore leads to energy consumption of SSNs, aggressive retransmissi queuing delay, and blocking of new flows from upstream SUs. Indubitably, a congestion cont technique in the transport layer is essential to balance resource loads and avoid excessive congesti However, the congestion control mechanism for the traditional Transmission Control Protocol (TC via the acknowledgement-triggered or window-based methods was initially designed and optimiz to perform in reliable wired links with constrained bit error rates and round trip times (RTTs) [1 A recent study [14] has reported that the performance of HTTP download deteriorates as much about 40% under the TCP window control in an IEEE P1900.4 based cognitive wireless system by usi User Datagram Protocol (UDP) and TCP transport protocols. On the other hand, some other resear efforts about congestion control have also been conducted from the perspective of finding methods modify the TCP protocol, such as TCP monitoring delayed acknowledgment, segment-based select acknowledgement, TCP adaptive delayed-acknowledgment window, etc. [15], aiming to accommod the challenging multi-hop wireless environments. Unfortunately, it has been shown that these metho of TCP modification and extension cannot be directly applied into SN-CRNs due to sudden large-sc bandwidth fluctuation, periodic interruption caused by spectrum sensing and channel switching [1 Recently, there have also been previous works on congestion control for multi-hop CRANETs fro a cross-layer design perspective. In [17], an end-to-end congestion control framework was propos under the constraint of the non-uniform channel availability by taking into account the interactio from the physical layer to the transport layer. In [18], a cross-layer framework for joint optimizati of MAC, scheduling, routing and congestion control was presented to maximize the throughput a set of multi-hop end-to-end packet flows. However, the end-to-end control policy in [17,18] is suited for operation over wireless links characterized by higher RTTs. On the contrary, the hop-by-h d (m) PUs by addressing the issues imposed by the hidden PU terminal problem and the wirele impairments, such as the heavy shadowing and fading [6][7][8]. In this way, cooperation is ac allowing different SUs to collaborate and share their spectrum sensing results (SSR) throug center (FC), which makes a global decision on the occupancy status of the licensed band. this centralized FC is not available in decentralized CRANETs. Clearly, each SU under thi must perform the distributed collaborative spectrum sensing, which is preferred to the cent scheme because of its scalability, fault tolerance and flexibility [9].
In order to facilitate the spectrum sensing functionality, high sampling rates, high analog to digital converters with large dynamic range, and high speed signal processors ar to be incorporated into an individual SU transceiver [10], which increases hardware cost a consumption, especially for the double-radio sensing architecture of SU transceiver. An a approach is to adopt the cost-effective and dedicated spectrum sensor nodes (SSNs) tha distributed collaborative spectrum sensing and report SSR to SUs acting as FCs in CRA Technically, s wireless sensor network can be naturally exploited to assist a CRANET by prov about the current spectrum occupancy of PUs in a cooperation fashion. The concept of senso embedded into CRANETs has further called sensor network-aided CRANETs (SN-CRNs), been considered as one of the most appealing approaches to perform cost-effective spectru in CR systems [7,11,12]. Υ Similar to most other traditional wireless networks or wireline Internet, network con SN-CRNs will also occur when offered data load that exceed the available capacity of a SU du overflow caused by the data sequence of the SSR injected from SSNs together with the d from upstream SUs. This therefore leads to energy consumption of SSNs, aggressive retra queuing delay, and blocking of new flows from upstream SUs. Indubitably, a congesti technique in the transport layer is essential to balance resource loads and avoid excessive c However, the congestion control mechanism for the traditional Transmission Control Prot via the acknowledgement-triggered or window-based methods was initially designed and to perform in reliable wired links with constrained bit error rates and round trip times ( A recent study [14] has reported that the performance of HTTP download deteriorates a about 40% under the TCP window control in an IEEE P1900.4 based cognitive wireless system User Datagram Protocol (UDP) and TCP transport protocols. On the other hand, some othe efforts about congestion control have also been conducted from the perspective of finding m modify the TCP protocol, such as TCP monitoring delayed acknowledgment, segment-base acknowledgement, TCP adaptive delayed-acknowledgment window, etc. [15], aiming to acco the challenging multi-hop wireless environments. Unfortunately, it has been shown that thes of TCP modification and extension cannot be directly applied into SN-CRNs due to sudden bandwidth fluctuation, periodic interruption caused by spectrum sensing and channel swit Recently, there have also been previous works on congestion control for multi-hop CRA a cross-layer design perspective. In [17], an end-to-end congestion control framework was under the constraint of the non-uniform channel availability by taking into account the in from the physical layer to the transport layer. In [18], a cross-layer framework for joint op of MAC, scheduling, routing and congestion control was presented to maximize the thro a set of multi-hop end-to-end packet flows. However, the end-to-end control policy in [1 suited for operation over wireless links characterized by higher RTTs. On the contrary, the h f (m))τ 2 it constantly allows for the opportunistic identification and use of the SO network without causing harmful interference to the PUs. In order to impro collaborative spectrum sensing has been proposed as an effective way to r PUs by addressing the issues imposed by the hidden PU terminal proble impairments, such as the heavy shadowing and fading [6][7][8]. In this way, allowing different SUs to collaborate and share their spectrum sensing res center (FC), which makes a global decision on the occupancy status of th this centralized FC is not available in decentralized CRANETs. Clearly, e must perform the distributed collaborative spectrum sensing, which is pre scheme because of its scalability, fault tolerance and flexibility [9]. In order to facilitate the spectrum sensing functionality, high samp analog to digital converters with large dynamic range, and high speed sig to be incorporated into an individual SU transceiver [10], which increase consumption, especially for the double-radio sensing architecture of SU approach is to adopt the cost-effective and dedicated spectrum sensor n distributed collaborative spectrum sensing and report SSR to SUs actin Technically, s wireless sensor network can be naturally exploited to assist a about the current spectrum occupancy of PUs in a cooperation fashion. Th embedded into CRANETs has further called sensor network-aided CRAN been considered as one of the most appealing approaches to perform cost in CR systems [7,11,12]. Υ Similar to most other traditional wireless networks or wireline Inter SN-CRNs will also occur when offered data load that exceed the available ca overflow caused by the data sequence of the SSR injected from SSNs to from upstream SUs. This therefore leads to energy consumption of SSNs, queuing delay, and blocking of new flows from upstream SUs. Indubi technique in the transport layer is essential to balance resource loads and However, the congestion control mechanism for the traditional Transmiss via the acknowledgement-triggered or window-based methods was initia to perform in reliable wired links with constrained bit error rates and ro A recent study [14] has reported that the performance of HTTP downlo about 40% under the TCP window control in an IEEE P1900.4 based cogniti User Datagram Protocol (UDP) and TCP transport protocols. On the other efforts about congestion control have also been conducted from the perspe modify the TCP protocol, such as TCP monitoring delayed acknowledgme acknowledgement, TCP adaptive delayed-acknowledgment window, etc. [1 the challenging multi-hop wireless environments. Unfortunately, it has bee of TCP modification and extension cannot be directly applied into SN-CRN bandwidth fluctuation, periodic interruption caused by spectrum sensing Recently, there have also been previous works on congestion control fo a cross-layer design perspective. In [17], an end-to-end congestion contro under the constraint of the non-uniform channel availability by taking in from the physical layer to the transport layer. In [18], a cross-layer framew of MAC, scheduling, routing and congestion control was presented to m a set of multi-hop end-to-end packet flows. However, the end-to-end co suited for operation over wireless links characterized by higher RTTs. On t d (m) it constantly allows for the opportunistic identification and use of network without causing harmful interference to the PUs. In order t collaborative spectrum sensing has been proposed as an effective w PUs by addressing the issues imposed by the hidden PU terminal impairments, such as the heavy shadowing and fading [6][7][8]. In th allowing different SUs to collaborate and share their spectrum sen center (FC), which makes a global decision on the occupancy stat this centralized FC is not available in decentralized CRANETs. C must perform the distributed collaborative spectrum sensing, whic scheme because of its scalability, fault tolerance and flexibility [9].
In order to facilitate the spectrum sensing functionality, hig analog to digital converters with large dynamic range, and high sp to be incorporated into an individual SU transceiver [10], which i consumption, especially for the double-radio sensing architectur approach is to adopt the cost-effective and dedicated spectrum distributed collaborative spectrum sensing and report SSR to SU Technically, s wireless sensor network can be naturally exploited to about the current spectrum occupancy of PUs in a cooperation fash embedded into CRANETs has further called sensor network-aided been considered as one of the most appealing approaches to perfo in CR systems [7,11,12]. Υ Similar to most other traditional wireless networks or wireli SN-CRNs will also occur when offered data load that exceed the ava overflow caused by the data sequence of the SSR injected from S from upstream SUs. This therefore leads to energy consumption o queuing delay, and blocking of new flows from upstream SUs. technique in the transport layer is essential to balance resource loa However, the congestion control mechanism for the traditional Tr via the acknowledgement-triggered or window-based methods wa to perform in reliable wired links with constrained bit error rate A recent study [14] has reported that the performance of HTTP d about 40% under the TCP window control in an IEEE P1900.4 based User Datagram Protocol (UDP) and TCP transport protocols. On th efforts about congestion control have also been conducted from the modify the TCP protocol, such as TCP monitoring delayed acknow acknowledgement, TCP adaptive delayed-acknowledgment window the challenging multi-hop wireless environments. Unfortunately, it of TCP modification and extension cannot be directly applied into S bandwidth fluctuation, periodic interruption caused by spectrum s Recently, there have also been previous works on congestion co a cross-layer design perspective. In [17], an end-to-end congestion under the constraint of the non-uniform channel availability by ta from the physical layer to the transport layer. In [18], a cross-laye of MAC, scheduling, routing and congestion control was present a set of multi-hop end-to-end packet flows. However, the end-to suited for operation over wireless links characterized by higher RTT f (m))τ rp 2 − E m (t) r (A5) Next, we try to derive the derivative of W C m (p m , E m ) with respect to E m (t) in Equation (A5). Upon solving the partial differential equation, after some basic mathematical manipulations, we obtain the final result in Equation (32), thus completing the proof.