Joint Optimization for Uplink/Downlink Intelligent Decoupled Access in Heterogeneous C-V2X Communications

Abstract

The uplink/downlink (UL/DL) decoupled access, which allows users to associate with different base stations (BSs), including small BSs (SBSs) and macro BSs (MBSs), has emerged as a network architecture in heterogeneous cellular vehicle-to-everything (C-V2X) communications. It can be tailored to mitigate the signal interference and attenuation impairments that cell-edge vehicles face, while vehicles closer to a BS can opt for coupled access. Therefore, a UL/DL intelligent decoupled access network that integrates decoupled and coupled access approaches is urgently needed for C-V2X communications. In this paper, we present a novel framework for UL/DL intelligent decoupled access in C-V2X networks in the context of fifth-generation mobile communications (5G) and beyond 5G (B5G). We propose a joint optimization approach for radio resource allocation, power control, and user association to enhance the network throughput of UL and DL while meeting the service quality requirements of vehicle users. Specifically, we formulate the problem as a mixed-integer nonlinear programming (MINLP) problem and transform it into a standard convex optimization problem by introducing various auxiliary variables. An efficient iterative algorithm based on successive convex optimization techniques is introduced to obtain a sub-optimal solution. The proposed framework uniquely integrates decoupled and coupled access modes within a unified optimization formulation, enabling dynamic mode selection based on network load. Extensive simulation results demonstrate a significant performance improvement of the proposed UL/DL intelligent decoupled access in C-V2X networks compared with benchmark schemes.

1. Introduction

With the rapid development of cellular vehicle-to-everything (C-V2X) technology in the fifth-generation mobile communication networks (5G) and beyond 5G (B5G), its potential to enhance road safety [1], improve transportation efficiency through intelligent scheduling [2], and enrich in-vehicle information services via dynamic resource management [3] is being progressively harnessed. With an increase in vehicle density, C-V2X networks face a series of challenges in achieving efficient spectrum reuse through centralized resource allocation [4], reliability of ultra-high-reliability connections [5], accommodation of heterogeneity in network deployment [6], and adaptability to diverse quality-of-service (QoS) requirements [7]. Particularly, vehicles at the edge of a cell face more signal interference and attenuation than those closer to the base station (BS), further exacerbating the asymmetry of uplink (UL) and downlink (DL) connections [8]. Thus, the UL/DL decoupled access approach, as an innovative network solution, allows user devices to connect to different BSs and different types of BSs, such as small BS (SBS) and macro BS (MBS) in the UL and DL, thereby addressing the aforementioned challenges [9,10], especially for C-V2X communications [1,11].

The UL/DL decoupled access can effectively enhance signal quality and communication stability in the cell-edge areas by allowing vehicles to independently select the best BSs for UL and DL [12,13,14]. Vehicle users in the core area of a cell, who are closer to the BS, can opt for the coupled access method [15,16]. Therefore, UL/DL decoupled access can be an effective complement to traditional UL/DL coupled access. To enhance network performance, by integrating UL/DL decoupled access and coupled access, the C-V2X can allocate resources and associate users based on real-time network conditions and user demands. The hybrid network is referred to as the UL/DL intelligent decoupled access C-V2X network.

The main advantage of the intelligent decoupled access network avoids the limitation of traditional coupled technology that UL must be associated with the same BS in DL [17]. Instead, the network dynamically selects the most appropriate BSs for both UL and DL, taking into account the user’s real-time location, service requirements, and prevailing network conditions. Subsequently, the user’s access mode is ascertained based on the specific BSs with which they have established a connection. This user-centric network design not only can significantly improve the communication quality for users at the edge of a cell, but also can fully exploit the overall performance of C-V2X, achieving efficient resource utilization and satisfactory service quality.

In this paper, we explore the potential of UL/DL intelligent decoupled access to address the challenges associated with C-V2X We propose an effective resource allocation and user association strategy to enhance the overall network performance while meeting the QoS requirements of vehicle users both in DL and UL. The key contributions of this work are summarized below:

• We present a novel UL/DL intelligent decoupled access framework for C-V2X in the 5G/B5G era. This framework provides a foundation for potential performance improvement of UL/DL intelligent decoupled access in C-V2X. The framework uniquely supports dynamic switching between decoupled and coupled access modes, enabling vehicles to adaptively select the optimal access strategy based on their location and network conditions.
• We aim to maximize system throughput by jointly optimizing the association matrices, transmission power, and radio spectrum allocation between vehicles and BSs for UL and DL. Thus, we formulate the maximum throughput of all vehicles as a mixed-integer nonlinear programming problem (MINLP), which is non-convex and difficult to solve. By introducing auxiliary variables, we transform the MINLP into a standard convex optimization problem. This joint formulation captures the inherent coupling between user association, power control, and bandwidth allocation across both UL and DL.
• We propose an efficient iterative algorithm for a sub-optimal solution by employing successive convex optimization techniques. Extensive simulations are conducted to demonstrate the effectiveness of the successive convex approximation (SCA) iterative algorithm. Simulation results demonstrate that the proposed framework achieves substantial throughput gains in both UL and DL compared with coupled access and distance-based association benchmarks.

The subsequent sections of this paper are structured as follows. We briefly introduce the existing research works related to UL/DL decoupled access and intelligent decoupled access in Section 2. In Section 3, we propose the UL/DL intelligent decoupled access framework for C-V2X. Section 4 and Section 5 presents an iteration optimization method to maximize the system throughput for both UL and DL. As a comparison, the coupled access optimization problem is formulated in Section 6. The SCA-based optimization iterative algorithm is provided in Section 7. In Section 8, simulation results are presented to evaluate the performance of the proposed UL/DL intelligent decoupled access C-V2X framework. Section 9 presents the concluding remarks of this study.

2. Related Work

The UL/DL decoupled access has emerged as a promising network solution for enabling C-V2X in intelligent transportation systems (ITSs) [18]. Yu et al. first introduced the UL/DL decoupled access into C-V2X and investigated the feasibility of C-V2X [19]. Jiao et al. modeled the C-V2X as a Cox process and analyzed the performance of UL/DL decoupled access in C-V2X [1].

For resource optimization of UL/DL decoupled access in C-V2X, Liu et al. proposed a streamlined algorithm to enhance energy efficiency in the context of 5G/B5G vehicular networks [20]. Yu et al. introduced a two-tier radio access network (RAN) slicing strategy to dynamically distribute radio resources for V2X services [21]. Jiang et al. considered DL/UL decouple access and proposed a resource allocation algorithm in a two-tier heterogeneous network to improve the total system throughput in the UL [22]. Beyond vehicular networks, joint optimization of clustering, transmission, and trajectory has also been studied for UAV-assisted wireless powered communication networks to minimize age-of-information [23], demonstrating the broad applicability of joint resource optimization in heterogeneous wireless systems.

Despite potential advantages of UL/DL decoupled access, users in close proximity to the BS typically benefit from a strong signal from the BS, making coupled access a prevalent choice for these users [24]. As a result, the application of UL/DL decoupled access should be tailored to the peripheral regions of the cell, where the benefits of UL/DL decoupled access can be effectively realized [12]. Thus, a UL/DL intelligent decoupled access framework that integrates both decoupled and coupled access should be studied for C-V2X. In cellular networks, preliminary research on hybrid decoupled and coupled access networks has been conducted. Sekander et al. developed a comprehensive framework to analyze the usefulness of UL/DL decouple access in a full-duplex multi-tier cellular network [15]. Different from most existing studies, in this paper, we focus on the framework analysis and resource allocation of UL/DL intelligent decoupled access in C-V2X, for applications such as real-time vehicular communication, traffic management, and road safety in ITS [25].

3. System Model and Problem Formulation

In this section, we first introduce the system model for UL/DL intelligent decoupled access C-V2X. Then, we present the channel and interference models. Finally, a MINLP is formulated to maximize the throughputs of UL and DL for all vehicle users.

3.1. UL/DL Intelligent Decoupled Access

As depicted in Figure 1, we consider a scenario of UL/DL intelligent decoupled access, where vehicles have the freedom to choose BSs for their UL and DL transmissions, which can be either the same or different. When the selected BSs for UL and DL are not identical, it is referred to as UL/DL decoupled access; otherwise, it is coupled access. A central controller collects the information of the entire network, including signal-to-noise-and-interference ratio (SINR) and transmit power [15]. This centralized coordination assumption is consistent with the C-RAN and NFV-enabled 5G/B5G network architecture, and has been widely adopted in the UL/DL decoupled access literature [15,19,20]. In this heterogeneous network, two types of BSs are deployed along the road, i.e., MBSs with higher transmit power located farther away from the road, and SBSs with lower transmit power situated closer to the road [1]. Here, $\mathcal{B}=\{1,\cdots ,B\}$ and $𝒱=\{1,2,\cdots ,V\}$ denote the index sets of BSs and vehicles, respectively. The MBS set is denoted as $\mathcal{M}=\{1,\cdots ,M\}$ and the SBS set is denoted as $S=\{1,\cdots ,S\}$. Thus, $\mathcal{B}=\mathcal{M}\cup S$ and $B=M+S$. The radio spectrum bandwidth of BS b allocates to the vehicle v in DL and UL is denoted by $W_{v,b}^d$ and $W_{v,b}^u$, respectively. Each BS can serve only a limited number of vehicles [15], with the maximum number for MBS/SBS in DL/UL given by

$$q_b^d=\left\{\begin{array}{cc}{q}_m^d,& b\in \mathcal{M}\\ q_s^d,& b\in S,\end{array}\right.q_b^u=\left\{\begin{array}{cc}{q}_m^u,& b\in \mathcal{M}\\ q_s^u,& b\in S.\end{array}\right.$$

The UL/DL decoupled access allows a vehicle to access different BSs in UL and DL, with the two BSs not being the same, while coupled access requires that the UL and DL must use the same BS [26]. Thus, there are six access cases:

• Coupled Case 1: $U=D=\mathrm{MBS} i$
• Coupled Case 2: $U=D=$ SBS j
• Decoupled Case 3: $U=$ MBS i, $D=$ MBS j, ($i\ne j$)
• Decoupled Case 4: $U=$ SBS i, $D=$ MBS j
• Decoupled Case 5: $U=$ MBS i, $D=$ SBS j
• Decoupled Case 6: $U=$ SBS i, $D=$ SBS j, ($i\ne j$),

where U and D denote the connected BS in UL and DL, respectively.

3.2. Channel Propagation and Interference Models

When a vehicle connects to BS b, the BS is activated with transmit power $P_{v,b}^d$ in DL, while the UL transmit power between BS b and vehicle v is denoted by $P_{v,b}^u$. The maximum transmit power of MBS, SBS, and vehicle is denoted by $P_{max}^m,P_{max}^s$, and $P_{max}^v$, respectively.

The channel gain between BS b and vehicle v, $G_{v,b}^d$ for DL is modeled as $G_{v,b}^d=g_{v,b}^d{l}_{v,b}^d$, and $G_{v,b}^u$ for UL is modeled as $G_{v,b}^u=g_{v,b}^u{l}_{v,b}^u$. Here, $g_{v,b}^{(·)}$ denotes the composite channel power gain between BS b and vehicle v due to Rayleigh fading ${\gamma }_{v,b}^{(·)}$ and log-normal shadowing ${\chi }_{v,b}^{(·)}$ under consideration [27], where the fading component follows an exponential distribution with a mean of $1/\mu$, $\gamma \sim exp(1/\mu )$ [28]. The shadowing component follows a log-normal distribution, given by $10{log}_{10}{\chi }_d\sim \aleph \left({\omega }_d,{\delta }_d^2\right)$, where ${\omega }_d$ represents the mean of ${\chi }_d$ in dB and ${\delta }_d$ represents the standard deviation of ${\chi }_d$ in dB [27]. The path loss, $l_{v,b}^{(·)}$ with $(·)=\{d,u\}$, between BS b and vehicle v, is a function of BS-vehicle distance [29]. We consider an Urban Macro (UMa) scenario path model, with details given on Page 28 of [29] and omitted here.

The received power is $P_{rec}^d\left(v\right)=x_{v,b}^d{P}_{v,b}^d{G}_{v,b}^d$ in the DL and $P_{rec}^u\left(b\right)=x_{v,b}^u{P}_{v,b}^u{G}_{v,b}^u$ in the UL, where $x_{v,b}^d\in \{0,1\}$ is a binary decision variable given by

$$x_{v,b}^d=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}\mathrm{vehicle} v \mathrm{is}\mathrm{associated}\mathrm{to}\mathrm{BS} b \in \mathcal{B} \mathrm{in}\mathrm{DL}\hfill \\ 0,\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

$$x_{v,b}^u=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}\mathrm{vehicle} v \mathrm{is}\mathrm{associated}\mathrm{to}\mathrm{BS} b \in \mathcal{B} \mathrm{in}\mathrm{UL}\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Under the assumption that a vehicle can access only a BS in UL or DL, we have ${\sum }_{b=1}^B{x}_{v,b}^{(·)}=1,$ $(·)=\{d,u\}$.

Different frequency bands are used for the UL and DL, respectively [18]. In the DL, the aggregate interference, $I_d$, at a typical vehicle is composed of the interference from the MBSs and SBSs. In the UL, assuming directional antennas at vehicles, interference experienced by a BS is from nearby vehicles [1]. Thus, when vehicle v is associated with MBS or SBS in the DL, the SINR is

$$SIN{R}_d={\displaystyle \frac{x_{v,b}^d{P}_{v,b}^d{G}_{v,b}^d}{I_d+{\sigma }_d^2}},$$

(1)

where ${\sigma }_d^2$ is the received noise power and $I_d$ is given by

$$I_d={\displaystyle \sum _{b=1}^B}\left(1-x_{v,b}^d\right)P_{v,b}^d{G}_{v,b}^d.$$

(2)

The SINR of UL is

$$SIN{R}_u={\displaystyle \frac{x_{v,b}^u{P}_{v,b}^u{G}_{v,b}^u}{I_u+{\sigma }_u^2}},$$

(3)

where ${\sigma }_u^2$ is the received noise power, interference $I_u$ is

$$I_u={\displaystyle \sum _{v=1}^V}{P}_{v,b}^u{G}_{v,b}^u-x_{v,b}^u{P}_{v,b}^u{G}_{v,b}^u.$$

(4)

In DL, each vehicle can connect to only one BS, so the interference to the BS can be directly represented by $\left(1-x_{v,b}^d\right)P_{v,b}^d{G}_{v,b}^d$. However, in UL, one BS can serve multiple vehicles. Therefore, when determining UL interference, the binary decision variable matrix may have multiple 1s in a column. The UL interference should be determined as in Equation (4).

3.3. Data Transmission Rate

According to the Shannon–Hartley theorem [30], the rate for vehicle v in DL and UL is given by the following:

$${\tau }_{v,b}^d=W_{v,b}^d{log}_2\left(1+SIN{R}_d\right),$$

(5)

$${\tau }_{v,b}^u=W_{v,b}^u{log}_2\left(1+SIN{R}_u\right).$$

(6)

3.4. Problem Formulation

The objective is to maximize the cumulative throughput of all vehicles in both UL and DL. The optimization problem is formulated as follows:

$$\begin{array}{cc}\hfill \underset{W_{v,b}^{(·)},P_{v,b}^{(·)},x_{v,b}^{(·)}}{max}& \hspace{1em}{\displaystyle \sum _{b=1}^B}{\displaystyle \sum _{v=1}^V}({\tau }_{v,b}^d+{\tau }_{v,b}^u)\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.\hspace{1em}& {\tau }_{v,b}^{(·)}>{\tau }_{min}^{(·)}, \forall b\in \mathcal{B},\forall v\in 𝒱\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill \hspace{1em}& {\displaystyle \sum _{v=1}^V}{W}_{v,b}^{(·)}\le W_{b,t}^{(·)}, \forall b\in \mathcal{B},\forall v\in 𝒱\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill \hspace{1em}& 0\le W_{v,b}^{(·)}\le W_{b,max}^{(·)}, \forall b\in \mathcal{B},\forall v\in 𝒱\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill \hspace{1em}& {\displaystyle \sum _{b=1}^B}{x}_{v,b}^{(·)}=1, \forall v\in 𝒱\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill \hspace{1em}& {\displaystyle \sum _{v=1}^V}{x}_{v,b}^{(·)}\le q_b^{(·)}, \forall b\in \mathcal{B},\forall v\in 𝒱\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill \hspace{1em}& x_{v,b}^{(·)}\in \{0,1\}, \forall b\in \mathcal{B},\forall v\in 𝒱\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill \hspace{1em}& 0\le P_{v,b}^{(·)}\le P_{c,max}^{(·)}, \forall b\in \mathcal{B},\forall v\in 𝒱\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill \hspace{1em}& {\displaystyle \sum _{v=1}^V}{P}_{v,b}^{(·)}\le P_{c,t}^{(·)}, \forall b\in \mathcal{B},\forall v\in 𝒱.\hfill \end{array}$$

(15)

In problem (7), $(·)=\{d,u\}$, $c\in \mathcal{B}$ in DL and $c\in 𝒱$ in UL, $W_{b,t}^{(·)}$ is the total radio spectrum bandwidth in DL and UL, $P_{c,t}^{(·)}$ is the total transmit power of BS in DL and vehicle in UL. Constraint (8) ensures QoS satisfaction for both DL and UL communication of each vehicle, respectively. In constraints (11) and (15), to avoid extreme cases where most of the bandwidth and power are allocated to a single vehicle during the multi-vehicle and multi-BS optimization process, we have set a maximum bandwidth limit $W_{b,max}^{(·)}$ and maximum transmit power $P_{c,max}^{(·)}$ allocated for each vehicle from each BS in DL and from each vehicle in UL. This ensures a balanced bandwidth and power distribution, to align with real-world scenarios. Constraint (11) indicates that each vehicle can access only one BS both in UL and DL, and constraint (12) ensures that the number of vehicles accessing the BS does not exceed the maximum.

Problem (7) is a MINLP problem [31,32], and it is challenging to find an optimal solution in general. In the following, we aim to obtain an efficient sub-optimal solution.

In an intelligent UL/DL decoupled access network, there are no restrictions on the allocation of UL and DL connections to the same BS. Furthermore, as the frequency bands used for UL and DL are different, the problem can be simplified to separately optimize the throughput of the UL and DL. The optimization problem is transformed to

$$\begin{array}{cc}\hfill \underset{W_{v,b}^d,P_{v,b}^d,x_{v,b}^d}{max}& \hspace{1em}{\displaystyle \sum _{b=1}^B}{\displaystyle \sum _{v=1}^V}{\tau }_{v,b}^d\hfill \\ \hfill \mathrm{s}.\mathrm{t}.\hspace{1em}& \left(8\right)\sim \left(15\right),\hfill \end{array}$$

(16)

for the DL with $(·)=d,c\in \mathcal{B}$ and

$$\begin{array}{cc}\hfill \underset{W_{v,b}^u,P_{v,b}^u,x_{v,b}^u}{max}& \hspace{1em}{\displaystyle \sum _{b=1}^B}{\displaystyle \sum _{v=1}^V}{\tau }_{v,b}^u\hfill \\ \hfill \mathrm{s}.\mathrm{t}.\hspace{1em}& \left(8\right)\sim \left(15\right),\hfill \end{array}$$

(17)

for the UL with $(·)=u,c\in 𝒱$.

4. DL Joint Optimization

Problem (16) and its constraints do not fit the standard convex optimization problem form; convex optimization (CVX) tools are inapplicable. We use some mathematical methods for a sub-optimal solution. Specifically, we first introduce a penalty term to relax the discrete BS-vehicle association binary variables into continuous variables. Then, by introducing an alternative variable, the objective function is transformed into a problem of maximizing the lower bound. Subsequently, through some mathematical transformations, the radio resource constraints are converted into standard concave constraints.

4.1. Association Variable and Objective Function Optimization for DL

Due to the discrete nature of association variable $x_{v,b}^d$, we first relax $x_{v,b}^d$ to a continuous variable in the range of 0 to 1; thus, $0\le x_{v,b}^d\le 1$ [31]. At the same time, to maintain the binary nature of the association variables, we introduce a penalty, $x_{v,b}^d(1-x_{v,b}^d)$. Thus, ${\tau }_{v,b}^d$ is rewritten as

$$\begin{array}{c}\hfill {\tau }_{v,b}^d+{\lambda }_d{x}_{v,b}^d(1-x_{v,b}^d),\end{array}$$

(18)

where ${\lambda }_d$ is a hyperparameter. Although the penalty term is a concave function, the penalty term is a univariate quadratic function, with its maximum point at 0.5. In order to guide $x_{v,b}^d$ towards convergence to 0 or 1, we perform a first-order Taylor expansion on it as follows [33]:

$$\begin{array}{c}\hfill x_{v,b}^{d,o}(1-x_{v,b}^{d,o})+(x_{v,b}^d-x_{v,b}^{d,o})(1-2x_{v,b}^{d,o}),\end{array}$$

(19)

where $x_{v,b}^{d,o}$ is the expansion point.

Since the function in Equation (5) for rate ${\tau }_{v,b}^d$ is not concave, it cannot be optimized using convex optimization tools. To maximize an objective function, we can find its maximum value approximately by maximizing its lower bound [34]. Thus, we introduce an auxiliary variable, ${{\rm Y}}_{v,b}^d$, such that ${log}_2\left(1+SIN{R}_d\right)>{{\rm Y}}_{v,b}^d.$ Therefore, ${\tau }_{v,b}^d$ is relaxed to $W_{v,b}^d{{\rm Y}}_{v,b}^d$.

In convex optimization, maximizing an objective function requires that the function itself be concave [35]. To meet this requirement, we perform the following mathematical transformation. Using mathematical identity $4xy={(x+y)}^2-{(x-y)}^2$, $W_{v,b}^d{{\rm Y}}_{v,b}^d$ can be equivalently expressed as ${\displaystyle \frac{1}{4}}\left[{(W_{v,b}^d+{{\rm Y}}_{v,b}^d)}^2-{(W_{v,b}^d-{{\rm Y}}_{v,b}^d)}^2\right]$. As the function is still not concave, we proceed to linearize it using a first-order Taylor expansion:

$$\begin{array}{cc}\hfill & W_{v,b}^d{{\rm Y}}_{v,b}^d\approx {\displaystyle \frac{1}{4}}\left[{(W_{v,b}^{d,o}+{{\rm Y}}_{v,b}^{d,o})}^2+2(W_{v,b}^d-W_{v,b}^{d,o})(W_{v,b}^{d,o}+\right.\hfill \\ \hfill & \left.{{\rm Y}}_{v,b}^{d,o})+({{\rm Y}}_{v,b}^d-{{\rm Y}}_{v,b}^{d,o})(W_{v,b}^{d,o}+{{\rm Y}}_{v,b}^{d,o})-{(W_{v,b}^d-{{\rm Y}}_{v,b}^d)}^2\right],\hfill \end{array}$$

(20)

where $W_{v,b}^{d,o}$ and ${{\rm Y}}_{v,b}^{d,o}$ are the expansion points. Thus, $W_{v,b}^d{{\rm Y}}_{v,b}^d$ is transformed into a concave function. Due to the overly complex form of Equation (20), for ease of expression, we denote it as ${\mathcal{J}}_{v,b}^d$.

Thus, ${\tau }_{v,b}^d$ in Equation (16) is rewritten as

$$\begin{array}{cc}\hfill {\mathcal{R}}_{v,b}^d& ={\mathcal{J}}_{v,b}^d+{\lambda }_d\left[x_{v,b}^{d,o}(1-x_{v,b}^{d,o})+\right.\hfill \\ \hfill \hspace{1em}& \left.(x_{v,b}^d-x_{v,b}^{d,o})(1-2x_{v,b}^{d,o})\right].\hfill \end{array}$$

(21)

4.2. Radio Resource Allocation for DL

After executing the above mathematical transformation in Section 4.1, constraint (8) is transformed into the following two constraints:

$$\begin{array}{c}\hfill {\mathcal{J}}_{v,b}^d>{\tau }_{min}^d, \forall b\in \mathcal{B},\forall v\in 𝒱,\end{array}$$

(22)

$$\begin{array}{c}\hfill {log}_2\left(1+SIN{R}_d\right)>{{\rm Y}}_{v,b}^d,\forall b\in \mathcal{B},\forall v\in 𝒱,\end{array}$$

(23)

where constraint (22) is concave, while constraint (23) is not. To handle constraint (23), which involves power $P_{v,b}^d$ and continuous variable $x_{v,b}^d$, we transform it to the following relaxed form:

$$\begin{array}{cc}\hfill \hspace{1em}& {\displaystyle \frac{1}{ln2}}\left[ln\left({\displaystyle \sum _{b=1}^B}{P}_{v,b}^d{G}_{v,b}^d+{\sigma }_d^2\right)-\right.ln\left({\displaystyle \sum _{b=1}^B}{\mathcal{F}}_{v,b}^{d,o}{G}_{v,b}^d+{\sigma }_d^2\right)-\hfill \\ \hfill \hspace{1em}& \left.{\displaystyle \frac{{\displaystyle \sum _{b=1}^B}\left({\mathcal{F}}_{v,b}^d-{\mathcal{F}}_{v,b}^{d,o}\right)G_{v,b}^d}{{\displaystyle \sum _{b=1}^B}{\mathcal{F}}_{v,b}^{d,o}{G}_{v,b}^d+{\sigma }_d^2}}\right]>{{\rm Y}}_{v,b}^d,\forall b\in \mathcal{B},\forall v\in 𝒱,\hfill \end{array}$$

(24)

where ${\mathcal{F}}_{v,b}^d$ is an alternative variable, and ${\mathcal{F}}_{v,b}^d\prec \left(1-x_{v,b}^d\right)P_{v,b}^d$, ${\mathcal{F}}_{v,b}^{d,o}$ is the first-order Taylor expansion point. Constraint (24) is a concave constraint with respect to $P_{v,b}^d$ and ${\mathcal{F}}_{v,b}^d$. The detailed derivation of Equation (24) is given as follows:

Proof. 

We first expand the $SIN{R}_d$ term and perform the following operation.

$$\begin{array}{cc}\hfill \hspace{1em}& {log}_2\left(1+{\displaystyle \frac{x_{v,b}^d{P}_{v,b}^d{G}_{v,b}^d}{I_d+{\sigma }_d^2}}\right)\hfill \\ \hfill \hspace{1em}& ={log}_2\left({\displaystyle \frac{I_d+x_{v,b}^d{P}_{v,b}^d{G}_{v,b}^d+{\sigma }_d^2}{I_d+{\sigma }_d^2}}\right)\hfill \\ \hfill \hspace{1em}& \stackrel{\left(a\right)}{=}lo{g}_2\left({\displaystyle \frac{{\displaystyle \sum _{b=1}^B}{P}_{v,b}^d{G}_{v,b}^d+{\sigma }_d^2}{{\displaystyle \sum _{b=1}^B}\left(1-x_{v,b}^d\right)P_{v,b}^d{G}_{v,b}^d+{\sigma }_d^2}}\right)\hfill \\ \hfill \hspace{1em}& ={log}_2\left({\displaystyle \sum _{b=1}^B}{P}_{v,b}^d{G}_{v,b}^d+{\sigma }_d^2\right)-{log}_2\left({\displaystyle \sum _{b=1}^B}\left(1-x_{v,b}^d\right)P_{v,b}^d{G}_{v,b}^d+{\sigma }_d^2\right),\hfill \end{array}$$

(25)

where $I_d={\sum }_{b=1}^B\left(1-x_{v,b}^d\right)P_{v,b}^d{G}_{v,b}^d$ in (a).

The convexity of (25) is uncertain. To address this, we introduce alternative variables, such that ${\mathcal{F}}_{v,b}^d\prec \left(1-x_{v,b}^d\right)P_{v,b}^d$. By substituting ${\mathcal{F}}_{v,b}^d$ into (25), the (25) is directly converted to the difference of two concave functions with regard to $P_{v,b}^d$ and ${\mathcal{F}}_{v,b}^d$ and the DC programming method can be used [36]. We perform the first-order Taylor expansion on the DC form in (25) and obtain its lower bound as

$$\begin{array}{cc}\hfill & {log}_2\left({\displaystyle \sum _{b=1}^B}{P}_{v,b}^d{G}_{v,b}^d+{\sigma }_d^2\right)-{log}_2\left({\displaystyle \sum _{b=1}^B}\left(1-x_{v,b}^d\right)P_{v,b}^d{G}_{v,b}^d+{\sigma }_d^2\right)\hfill \\ \hfill \stackrel{\left(a\right)}{=}& {\displaystyle \frac{1}{ln2}}\left[ln\left({\displaystyle \sum _{b=1}^B}{P}_{v,b}^d{G}_{v,b}^d+{\sigma }_d^2\right)-\right.\left.ln\left({\displaystyle \sum _{b=1}^B}{\mathcal{F}}_{v,b}^d{G}_{v,b}^d+{\sigma }_d^2\right)\right]\hfill \\ \hfill \stackrel{\left(b\right)}{=}& {\displaystyle \frac{1}{ln2}}\left[ln\left({\displaystyle \sum _{b=1}^B}{P}_{v,b}^d{G}_{v,b}^d+{\sigma }_d^2\right)-\right.ln\left({\displaystyle \sum _{b=1}^B}{\mathcal{F}}_{v,b}^{d,o}{G}_{v,b}^d+{\sigma }_d^2\right)-\left.{\displaystyle \frac{{\displaystyle \sum _{b=1}^B}\left({\mathcal{F}}_{v,b}^d-{\mathcal{F}}_{v,b}^{d,o}\right)G_{v,b}^d}{{\displaystyle \sum _{b=1}^B}{\mathcal{F}}_{v,b}^{d,o}{G}_{v,b}^d+{\sigma }_d^2}}\right].\hfill \end{array}$$

(26)

In (a), the first half is in the form of $ln\left(x\right)$, which is concave. For the second half, we perform a first-order Taylor expansion around the point ${\mathcal{F}}_{v,b}^{d,o}$ to obtain a lower bound in (b).

   □

Note that the introduced variable ${\mathcal{F}}_{v,b}^d$ has

$$\begin{array}{c}\hfill {\mathcal{F}}_{v,b}^d\ge \left(1-x_{v,b}^d\right)P_{v,b}^d,\forall b\in \mathcal{B},\forall v\in 𝒱,\end{array}$$

(27)

which is not concave; we can approximately linearize it using a first-order Taylor expansion as in Equation (20), and obtain

$$\begin{array}{cc}\hfill \hspace{1em}& P_{v,b}^d-{\displaystyle \frac{1}{4}}\left[2\left(x_{v,b}^d-x_{v,b}^{d,o}+P_{v,b}^d-P_{v,b}^{d,o}\right)\left(x_{v,b}^{d,o}+P_{v,b}^{d,o}\right)\right.\hfill \\ \hfill \hspace{1em}& +{(x_{v,b}^{d,o}+P_{v,b}^{d,o})}^2\left.-{(x_{v,b}^d-P_{v,b}^d)}^2\right]\le {\mathcal{F}}_{v,b}^d,\forall b\in \mathcal{B},\forall v\in 𝒱.\hfill \end{array}$$

(28)

Through the mathematical processing above, constraint (27) has been transformed into a concave constraint, allowing for further calculations using SCA.

By substituting the obtained upper or lower bounds derived above into problem (16), we can obtain a convex optimization problem for the t-th iteration as follows:

$$\begin{array}{cc}\hfill \underset{{\mathbf{T}}^d}{max}& \hspace{1em}{\displaystyle \sum _{b=1}^B}{\displaystyle \sum _{v=1}^V}{\mathcal{R}}_{v,b}^d\hfill \\ \hfill \mathrm{s}.\mathrm{t}.\hspace{1em}& \left(9\right)\sim \left(15\right),(·)=d,c\in \mathcal{B}\hfill \\ \hfill \hspace{1em}& \left(22\right),\left(24\right) \mathrm{and} \left(28\right)\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill \hspace{1em}& W_{v,b}^d/W_{b,max}^d\le x_{v,b}^d,\forall b\in \mathcal{B},\forall v\in 𝒱\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill \hspace{1em}& P_{v,b}^d/P_{b,max}^d\le x_{v,b}^d,\forall b\in \mathcal{B},\forall v\in 𝒱,\hfill \end{array}$$

(31)

where matrix ${\mathbf{T}}^d=\{W_{v,b}^d,P_{v,b}^d,x_{v,b}^d,{\mathcal{F}}_{v,b}^d,{{\rm Y}}_{v,b}^d\}$ is the set of optimization variables. The complete expression of problem (29) is at the top of the next page. Specifically, the convex optimization problem for the $(t+1)$-th iteration is as shown in Equations (29)–(31). Constraints (30) and (31) ensure that radio spectral bandwidth and power are allocated as much as possible to the connected vehicles, thereby enhancing the optimization effect.

Proof. 

The detailed expressions of problem (29) are at the top of the next page.

$$\begin{array}{ccc}\hfill \underset{W_{v,b}^d,P_{v,b}^d,x_{v,b}^d,{\mathcal{F}}_{v,b}^d,{{\rm Y}}_{v,b}^d}{\max }& & {\displaystyle \sum _{b=1}^B}{\displaystyle \sum _{v=1}^V}{\displaystyle \frac{1}{4}}[{(W_{v,b}^{d,o}+{{\rm Y}}_{v,b}^{d,o})}^2+2(W_{v,b}^d-W_{v,b}^{d,o})(W_{v,b}^{d,o}+{{\rm Y}}_{v,b}^{d,o})\hfill \\ & & \hspace{1em}+({{\rm Y}}_{v,b}^d-{{\rm Y}}_{v,b}^{d,o})(W_{v,b}^{d,o}+{{\rm Y}}_{v,b}^{d,o})-{(W_{v,b}^d-{{\rm Y}}_{v,b}^d)}^2]\hfill \\ & & \hspace{1em}+{\lambda }_d{x}_{v,b}^{d,o}(1-x_{v,b}^{d,o})+(x_{v,b}^d-x_{v,b}^{d,o})(1-2x_{v,b}^{d,o})\hfill \end{array}$$

(32)

$$\begin{array}{ccc}\hfill \mathrm{s}.\mathrm{t}.& & \left(9\right)\sim \left(15\right),(·)=d,c\in \mathcal{B}\hfill \\ & & {\displaystyle \frac{1}{4}}[{(W_{v,b}^{d,o}+{{\rm Y}}_{v,b}^{d,o})}^2+2(W_{v,b}^d-W_{v,b}^{d,o})(W_{v,b}^{d,o}+{{\rm Y}}_{v,b}^{d,o})\hfill \\ & & \hspace{1em}+({{\rm Y}}_{v,b}^d-{{\rm Y}}_{v,b}^{d,o})(W_{v,b}^{d,o}+{{\rm Y}}_{v,b}^{d,o})\hfill \\ & & \hspace{1em}-{(W_{v,b}^d-{{\rm Y}}_{v,b}^d)}^2]>{\tau }_{min}^d,\forall b\in \mathcal{B},\forall v\in 𝒱\hfill \end{array}$$

(33)

$$\begin{array}{ccc}& & {\displaystyle \frac{1}{ln2}}\left[ln\left({\displaystyle \sum _{b=1}^B}{P}_{v,b}^d{G}_{v,b}^d+{\sigma }_d^2\right)-\right.ln\left({\displaystyle \sum _{b=1}^B}{\mathcal{F}}_{v,b}^{d,o}{G}_{v,b}^d+{\sigma }_d^2\right)-\hfill \\ & & \left.{\displaystyle \frac{{\displaystyle \sum _{b=1}^B}\left({\mathcal{F}}_{v,b}^d-{\mathcal{F}}_{v,b}^{d,o}\right)G_{v,b}^d}{{\displaystyle \sum _{b=1}^B}{\mathcal{F}}_{v,b}^{d,o}{G}_{v,b}^d+{\sigma }_d^2}}\right]>{{\rm Y}}_{v,b}^d,\forall b\in \mathcal{B},\forall v\in 𝒱,\hfill \end{array}$$

(34)

$$\begin{array}{ccc}& & {\mathcal{F}}_{v,b}^d\ge P_{v,b}^d-{\displaystyle \frac{1}{4}}\left[2\left(x_{v,b}^d-x_{v,b}^{d,o}+P_{v,b}^d-P_{v,b}^{d,o}\right)\left(x_{v,b}^{d,o}+P_{v,b}^{d,o}\right)+\right.\hfill \\ & & {(x_{v,b}^{d,o}+P_{v,b}^{d,o})}^2\left.-{(x_{v,b}^d-P_{v,b}^d)}^2\right],\forall b\in \mathcal{B},\forall v\in 𝒱,\hfill \end{array}$$

(35)

$$\begin{array}{ccc}& & W_{v,b}^d/W_{b,max}^d\le x_{v,b}^d,\forall b\in \mathcal{B},\forall v\in 𝒱,\hfill \end{array}$$

(36)

$$\begin{array}{ccc}& & P_{v,b}^d/P_{b,max}^d\le x_{v,b}^d,\forall b\in \mathcal{B},\forall v\in 𝒱.\hfill \end{array}$$

(37)

   □

5. UL Joint Optimization

The UL throughput maximization problem has been formulated in problem (17). In this section, we aim to maximize the throughput of UL by following a similar method as described in Section 4.

5.1. Association Variable and Objective Function Optimization for UL

Following the similar steps of Section 4.1, ${\tau }_{v,b}^u$ in Equation (17) is rewritten as

$$\begin{array}{cc}\hfill {\mathcal{R}}_{v,b}^u=& {\mathcal{J}}_{v,b}^u+{\lambda }_u\left[x_{v,b}^{u,o}(1-x_{v,b}^{u,o})+\right.\left.(x_{v,b}^u-x_{v,b}^{u,o})(1-2x_{v,b}^{u,o})\right],\hfill \end{array}$$

(38)

where ${\lambda }_u$ is a hyperparameter, ${\mathcal{J}}_{v,b}^u$ is the UL formula expression symbol corresponding to ${\mathcal{J}}_{v,b}^d$ in DL, and $x_{v,b}^{u,o}$ is the first-order Taylor expansion point.

5.2. Wireless Resource Allocation Optimization for UL

After executing the above mathematical transformation in Section 5.1, constraint (8) in UL is transformed into the following two constraints:

$$\begin{array}{c}\hfill {\mathcal{J}}_{v,b}^u\ge {\tau }_{min}^u, \forall b\in \mathcal{B},\forall v\in 𝒱,\end{array}$$

(39)

$$\begin{array}{c}\hfill {log}_2\left(1+SIN{R}_u\right)>{{\rm Y}}_{v,b}^u,\forall b\in \mathcal{B},\forall v\in 𝒱,\end{array}$$

(40)

where the constraint (40) is a non-concave constraint; we first expand it and then perform the following mathematical transformation:

$$\begin{array}{cc}\hfill \hspace{1em}& {log}_2\left(1+{\displaystyle \frac{x_{v,b}^u{P}_{v,b}^u{G}_{v,b}^u}{I_u+{\sigma }_u^2}}\right)\hfill \\ \hfill \hspace{1em}& ={log}_2\left({\displaystyle \frac{I_u+x_{v,b}^u{P}_{v,b}^u{G}_{v,b}^u+{\sigma }_u^2}{I_u+{\sigma }_u^2}}\right)\hfill \\ \hfill \hspace{1em}& \stackrel{\left(a\right)}{=}{log}_2\left({\displaystyle \sum _{v=1}^V}{P}_{v,b}^u{G}_{v,b}^u+{\sigma }_u^2\right)-{log}_2\left({\displaystyle \sum _{1=1}^V}{P}_{v,b}^u{G}_{v,b}^u-x_{v,b}^u{P}_{v,b}^u{G}_{v,b}^u+{\sigma }_u^2\right),\hfill \end{array}$$

(41)

where $I_u={\sum }_{1=1}^V{P}_{v,b}^u{G}_{v,b}^u-x_{v,b}^u{P}_{v,b}^u{G}_{v,b}^u$. In (a), the  first part is a concave function, while the second part is of indeterminate concavity or convexity due to the two variables $P_{v,b}^u$, $x_{v,b}^u$. Therefore, we introduce an auxiliary variable ${\mathcal{F}}_{v,b}^u\prec {\sum }_{1=1}^V{P}_{v,b}^u{G}_{v,b}^u-x_{v,b}^u{P}_{v,b}^u{G}_{v,b}^u$. Constraint (41) is transformed to a concave function as follows:

$$\begin{array}{cc}\hfill \hspace{1em}& {\displaystyle \frac{1}{ln2}}\left[ln\left({\displaystyle \sum _{v=1}^V}{P}_{v,b}^u{G}_{v,b}^u+{\sigma }_u^2\right)-\right.ln\left({\displaystyle \sum _{v=1}^V}{\mathcal{F}}_{v,b}^{u,o}{G}_{v,b}^u+{\sigma }_u^2\right)-\left.{\displaystyle \frac{{\displaystyle \sum _{v=1}^V}\left({\mathcal{F}}_{v,b}^u-{\mathcal{F}}_{v,b}^{u,o}\right)G_{v,b}^u}{{\displaystyle \sum _{v=1}^V}{\mathcal{F}}_{v,b}^{u,o}{G}_{v,b}^u+{\sigma }_u^2}}\right],\hfill \end{array}$$

(42)

The detailed derivation of Equation (42) is shown as follows:

Proof. 

The detailed derivation of (42) is formulated as

$$\begin{array}{cc}\hfill & {log}_2\left({\displaystyle \sum _{v=1}^V}{P}_{v,b}^u{G}_{v,b}^u+{\sigma }_u^2\right)-lo{g}_2\left({\displaystyle \sum _{1=1}^V}{P}_{v,b}^u{G}_{v,b}^u-x_{v,b}^u{P}_{v,b}^u{G}_{v,b}^u+{\sigma }_u^2\right)\hfill \\ \hfill =& {\displaystyle \frac{1}{ln2}}\left[ln\left({\displaystyle \sum _{v=1}^V}{P}_{v,b}^u{G}_{v,b}^u+{\sigma }_u^2\right)-\right.\left.ln\left({\displaystyle \sum _{v=1}^V}{\mathcal{F}}_{v,b}^u{G}_{v,b}^u+{\sigma }_u^2\right)\right]\hfill \\ \hfill \stackrel{\left(a\right)}{=}& {\displaystyle \frac{1}{ln2}}\left[ln\left({\displaystyle \sum _{v=1}^V}{P}_{v,b}^u{G}_{v,b}^u+{\sigma }_u^2\right)-\right.ln\left({\displaystyle \sum _{v=1}^V}{\mathcal{F}}_{v,b}^{u,o}{G}_{v,b}^u+{\sigma }_u^2\right)-\left.{\displaystyle \frac{{\displaystyle \sum _{v=1}^V}\left({\mathcal{F}}_{v,b}^u-{\mathcal{F}}_{v,b}^{u,o}\right)G_{v,b}^u}{{\displaystyle \sum _{v=1}^V}{\mathcal{F}}_{v,b}^{u,o}{G}_{v,b}^u+{\sigma }_u^2}}\right],\hfill \end{array}$$

(43)

where ${\mathcal{F}}_{v,b}^u\prec {\sum }_{1=1}^V{P}_{v,b}^u{G}_{v,b}^u-x_{v,b}^u{P}_{v,b}^u{G}_{v,b}^u$, (a) is Taylor expanded at point ${\mathcal{F}}_{v,b}^{u,o}$, transforming the function into an affine function.    □

Constraint (40) turns into a concave constraint as follows:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{ln2}}\left[ln\left({\displaystyle \sum _{v=1}^V}{P}_{v,b}^u{G}_{v,b}^u+{\sigma }_u^2\right)-\right.ln\left({\displaystyle \sum _{v=1}^V}{\mathcal{F}}_{v,b}^{u,o}{G}_{v,b}^u+{\sigma }_u^2\right)-\hfill \\ \hfill \hspace{1em}& \left.{\displaystyle \frac{{\displaystyle \sum _{v=1}^V}\left({\mathcal{F}}_{v,b}^u-{\mathcal{F}}_{v,b}^{u,o}\right)G_{v,b}^u}{{\displaystyle \sum _{v=1}^V}{\mathcal{F}}_{v,b}^{u,o}{G}_{v,b}^u+{\sigma }_u^2}}\right]>{{\rm Y}}_{v,b}^u,\forall b\in \mathcal{B},\forall v\in 𝒱.\hfill \end{array}$$

(44)

Since constraint (44) is taking the lower bound of ${log}_2\left(1+SIN{R}_u\right)$, an additional constraint needs to be added for the auxiliary variable ${\mathcal{F}}_{v,b}^u$ to require

$$\begin{array}{c}\hfill {\mathcal{F}}_{v,b}^u\ge {\displaystyle \sum _{1=1}^V}{P}_{v,b}^u{G}_{v,b}^u-x_{v,b}^u{P}_{v,b}^u{G}_{v,b}^u,\forall b\in \mathcal{B},\forall v\in 𝒱.\end{array}$$

(45)

The introduced constraint (45) is not a standard convex constraint. Therefore, we introduce auxiliary variable $A_{v,b}^u\prec (-x_{v,b}^u{P}_{v,b}^u{G}_{v,b}^u)$ and constraint (45) is rewritten as

$$\begin{array}{c}\hfill {\mathcal{F}}_{v,b}^u\ge {\displaystyle \sum _{1=1}^V}{P}_{v,b}^u{G}_{v,b}^u+A_{v,b}^u,\forall b\in \mathcal{B},\forall v\in 𝒱,\end{array}$$

(46)

where constraint (46) is a affine constraint and can be solved by the CVX tool. We perform the first-order Taylor expansion on $(-x_{v,b}^u{P}_{v,b}^u{G}_{v,b}^u)$ and obtain its upper bound as

$$\begin{array}{cc}\hfill A_{v,b}^u& \ge {\displaystyle \frac{1}{4}}{G}_{v,b}^u\left[{\left(x_{v,b}^u-P_{v,b}^u\right)}^2-{\left(x_{v,b}^{u,o}+P_{v,b}^{u,o}\right)}^2-\right.\hfill \\ \hfill \hspace{1em}& \left.2\left(x_{v,b}^u-x_{v,b}^{u,o}+P_{v,b}^u-P_{v,b}^{u,o}\right)\left(x_{v,b}^{u,o}+P_{v,b}^{u,o}\right)\right]\hfill \\ \hfill \hspace{1em}& \ge \left(-x_{v,b}^u{P}_{v,b}^u{G}_{v,b}^u\right).\hfill \end{array}$$

(47)

Thus, constraint (48) is added as follows to constrain the auxiliary variable $A_{v,b}^u$.

$$\begin{array}{cc}\hfill A_{v,b}^u& \ge {\displaystyle \frac{1}{4}}{G}_{v,b}^u\left[{\left(x_{v,b}^u-P_{v,b}^u\right)}^2-{\left(x_{v,b}^{u,o}+P_{v,b}^{u,o}\right)}^2-\right.\hfill \\ \hfill \hspace{1em}& \left.2\left(x_{v,b}^u-x_{v,b}^{u,o}+P_{v,b}^u-P_{v,b}^{u,o}\right)\left(x_{v,b}^{u,o}+P_{v,b}^{u,o}\right)\right],\hfill \\ \hfill \hspace{1em}& \forall b\in \mathcal{B},\forall v\in 𝒱.\hfill \end{array}$$

(48)

By substituting these bounds into the related terms of problem (17), the optimization problem can be rewritten as in problem (49).

$$\begin{array}{cc}\hfill \underset{{\mathbf{T}}^u}{max}& \hspace{1em}{\displaystyle \sum _{b=1}^B}{\displaystyle \sum _{v=1}^V}{\mathcal{R}}_{v,b}^u\hfill \end{array}$$

(49)

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.\hspace{1em}& \left(9\right)\sim \left(15\right),(·)=u,c\in 𝒱\hfill \\ \hfill \hspace{1em}& \left(39\right),\left(44\right),\left(46\right) and \left(48\right)\hfill \\ \hfill \hspace{1em}& W_{v,b}^u/W_{b,max}^u<=x_{v,b}^u,\forall b\in \mathcal{B},\forall v\in 𝒱\hfill \\ \hfill \hspace{1em}& P_{v,b}^u/P_{b,max}^u<=x_{v,b}^u,\forall b\in \mathcal{B},\forall v\in 𝒱.\hfill \end{array}$$

(50)

In problem (49), ${\mathbf{T}}^u=\{W_{v,b}^u,P_{v,b}^u,x_{v,b}^u,{\mathcal{F}}_{v,b}^u,A_{v,b}^u,{{\rm Y}}_{v,b}^u\}$ is the set of variables. The complete expressions of problem (49) are at the top of the next page. Constraint (50) ensures that vehicles only transmit power to the connected BS in UL, thereby reducing UL interference among vehicles. Both problems (29) and (49) are standard convex optimization problems and can be solved by the CVX toolbox.

Proof. 

The detailed expressions of problem (49) are at the top of the next page.

$$\begin{array}{ccc}\hfill \underset{W_{v,b}^u,P_{v,b}^u,x_{v,b}^u,{\mathcal{F}}_{v,b}^u,A_{v,b}^u,{{\rm Y}}_{v,b}^u}{\max }& & {\displaystyle \sum _{b=1}^B}{\displaystyle \sum _{v=1}^V}{\displaystyle \frac{1}{4}}[{(W_{v,b}^{u,o}+{{\rm Y}}_{v,b}^{u,o})}^2+2(W_{v,b}^u-W_{v,b}^{u,o})(W_{v,b}^{u,o}+{{\rm Y}}_{v,b}^{u,o})\hfill \\ & & \hspace{1em}+({{\rm Y}}_{v,b}^u-{{\rm Y}}_{v,b}^{u,o})(W_{v,b}^{u,o}+{{\rm Y}}_{v,b}^{u,o})-{(W_{v,b}^u-{{\rm Y}}_{v,b}^u)}^2]\hfill \\ & & \hspace{1em}+{\lambda }_u{x}_{v,b}^{u,o}(1-x_{v,b}^{u,o})+(x_{v,b}^u-x_{v,b}^{u,o})(1-2x_{v,b}^{u,o})\hfill \end{array}$$

(51)

$$\begin{array}{ccc}\hfill \mathrm{s}.\mathrm{t}.& & \left(9\right)\sim \left(15\right),(·)=u,c\in 𝒱\hfill \\ & & {\displaystyle \frac{1}{4}}[{(W_{v,b}^{u,o}+{{\rm Y}}_{v,b}^{u,o})}^2+2(W_{v,b}^u-W_{v,b}^{u,o})(W_{v,b}^{u,o}+{{\rm Y}}_{v,b}^{u,o})\hfill \\ & & \hspace{1em}+({{\rm Y}}_{v,b}^u-{{\rm Y}}_{v,b}^{u,o})(W_{v,b}^{u,o}+{{\rm Y}}_{v,b}^{u,o})\hfill \\ & & \hspace{1em}-{(W_{v,b}^u-{{\rm Y}}_{v,b}^u)}^2]>{\tau }_{min}^u,\forall b\in \mathcal{B},\forall v\in 𝒱\hfill \end{array}$$

(52)

$$\begin{array}{ccc}& & {\displaystyle \frac{1}{ln2}}\left[ln\left({\displaystyle \sum _{u=1}^V}{P}_{v,b}^u{G}_{v,b}^u+{\sigma }_u^2\right)-\right.ln\left({\displaystyle \sum _{v=1}^V}{\mathcal{F}}_{v,b}^{u,o}{G}_{v,b}^u+{\sigma }_u^2\right)-\hfill \\ & & \left.{\displaystyle \frac{{\displaystyle \sum _{v=1}^V}\left({\mathcal{F}}_{v,b}^u-{\mathcal{F}}_{v,b}^{u,o}\right)G_{v,b}^u}{{\displaystyle \sum _{v=1}^V}{\mathcal{F}}_{v,b}^{u,o}{G}_{v,b}^u+{\sigma }_u^2}}\right]>{{\rm Y}}_{v,b}^u,\forall b\in \mathcal{B},\forall v\in 𝒱,\hfill \end{array}$$

(53)

$$\begin{array}{ccc}& & {\mathcal{F}}_{v,b}^u\ge {\displaystyle \sum _{1=1}^V}{P}_{v,b}^u{G}_{v,b}^u+A_{v,b}^u,\forall b\in \mathcal{B},\forall v\in 𝒱,\hfill \end{array}$$

(54)

$$\begin{array}{ccc}& & A_{v,b}^u\ge {\displaystyle \frac{1}{4}}{G}_{v,b}^u\left[{\left(x_{v,b}^u-P_{v,b}^u\right)}^2-{\left(x_{v,b}^{u,o}+P_{v,b}^{u,o}\right)}^2-\right.\hfill \\ & & \left.2\left(x_{v,b}^u-x_{v,b}^{u,o}+P_{v,b}^u-P_{v,b}^{u,o}\right)\left(x_{v,b}^{u,o}+P_{v,b}^{u,o}\right)\right],\forall b\in \mathcal{B},\forall v\in 𝒱.\hfill \end{array}$$

(55)

$$\begin{array}{ccc}& & W_{v,b}^u/W_{b,max}^u\le x_{v,b}^u,\forall b\in \mathcal{B},\forall v\in 𝒱.\hfill \end{array}$$

(56)

$$\begin{array}{ccc}& & P_{v,b}^u/P_{b,max}^u\le x_{v,b}^u,\forall b\in \mathcal{B},\forall v\in 𝒱.\hfill \end{array}$$

(57)

   □

6. Coupled Access Optimization

For the coupled access optimization problem, unlike intelligent decoupled access, which imposes no restrictions on the UL and DL access BSs, coupled access requires that the UL and DL BSs must be the same one. Therefore, the optimization problem for coupled access is as follows:

$$\begin{array}{ccc}\hfill ccl\underset{T_c^u,T_c^d}{\max }& & {\displaystyle \sum _{b=1}^B}{\displaystyle \sum _{v=1}^V}{\mathcal{J}}_{v,b}^d+{\mathcal{J}}_{v,b}^u+\hfill \\ & & {\lambda }_c\left[x_{v,b}^{u,o}(1-x_{v,b}^{u,o})+(x_{v,b}^u-x_{v,b}^{u,o})(1-2x_{v,b}^{u,o})\right]\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \mathrm{s}.\mathrm{t}.& & \left(9\right)\sim \left(15\right),(·)=d,c\in \mathcal{B},\hfill \\ & & \left(9\right)\sim \left(15\right),(·)=u,c\in 𝒱,\hfill \\ & & \left(33\right)\sim \left(36\right),\hfill \\ & & \left(52\right)\sim \left(56\right),\hfill \end{array}$$

(58)

$$\begin{array}{ccc}& & x_{v,b}^u=x_{v,b}^d,\hfill \end{array}$$

(59)

where ${\mathbf{T}}_c^u$ and ${\mathbf{T}}_c^d$ are the variable sets, and ${\mathbf{T}}_c^u=\{W_{v,b}^u,P_{v,b}^u,x_{v,b}^u,F_{v,b}^u,A_{v,b}^u,{{\rm Y}}_{v,b}^u\},{\mathbf{T}}_c^d=\{W_{v,b}^d,P_{v,b}^d,x_{v,b}^d,F_{v,b}^d,{{\rm Y}}_{v,b}^d\}$. In Equations (10)–(15), $(·)=d,c\in \mathcal{B}$ and (33)–(36) are DL constraints. In Equations (10)–(15), $(·)=u,c\in 𝒱$ and (52)–(56) are UL constraints. Constraint (59) ensures that the BS accessed for UL and DL is the same BS. Therefore, the penalty term involving $x_{v,b}^{(·)}$ can represent both $x_{v,b}^u$ and $x_{v,b}^d$. Here, we choose to use $x_{v,b}^u$.

After the mathematical transformations in the previous two sections, problem (58) is a standard convex optimization problem and can be solved by using the CVX toolbox, as in Algorithm 1 in the following section. The detailed steps are omitted here.

Algorithm 1: SCA for maximizing system throughput

7. Optimization Algorithm for Maximizing System Throughput

Based on the mathematical transformations in Section 4 and Section 5, we convert the non-standard convex optimization problems of UL and DL into standard convex optimization problems. Using the CVX toolbox, we propose Algorithm 1 to solve the optimization variables in problems (29) and (49). Firstly, we obtain the channel information for UL and DL, with initial values ${\mathbf{T}}_o^d=\{W_{v,b}^{d,o},P_{v,b}^{d,o},x_{v,b}^{d,o},{\mathcal{F}}_{v,b}^{d,o},{{\rm Y}}_{v,b}^{d,o}\},{\mathbf{T}}_o^u=\{W_{v,b}^{u,o},P_{v,b}^{u,o},x_{v,b}^{u,o},{\mathcal{F}}_{v,b}^{u,o},A_{v,b}^{u,o},{{\rm Y}}_{v,b}^{u,o}\}$ that satisfy all constraints. Then, we use the CVX toolbox to solve problems (29) and (49) and obtain the optimal values ${\mathbf{T}}^d=\{W_{v,b}^d,P_{v,b}^d,x_{v,b}^d,{\mathcal{F}}_{v,b}^d,{{\rm Y}}_{v,b}^d\},{\mathbf{T}}^u=\{W_{v,b}^u,P_{v,b}^u,x_{v,b}^u,{\mathcal{F}}_{v,b}^u,A_{v,b}^u,{{\rm Y}}_{v,b}^u\}$ for the current iteration. Subsequently, we assign optimal values ${\mathbf{T}}^d,{\mathbf{T}}^u$ to the initial values ${\mathbf{T}}_o^d,{\mathbf{T}}_o^u$ and continue the iterative optimization process. This process continues until the difference between the maximum values of the objective function before and after optimization in two consecutive iterations is less than a predefined threshold. When this condition is met or the maximum number of iterations is reached, we consider the problem solved and terminate the solution process.

The computational complexity of Algorithm 1 is analyzed as follows. At each iteration, the DL and UL convex subproblems are solved using an interior-point method. Each subproblem involves $O\left(VB\right)$ optimization variables, and the interior-point method has a per-iteration complexity of $\mathcal{O}\left({\left(VB\right)}^{3.5}\right)$. Therefore, the total complexity of Algorithm 1 is $\mathcal{O}(N_{iter}·{\left(VB\right)}^{3.5})$, where $N_{iter}$ is the number of SCA iterations. As shown in Figure 3, the algorithm converges within 4∼5 iterations in practice, demonstrating its computational efficiency.

To determine the vehicle access mode, we propose Algorithm 2. By comparing the types and identifiers of BSs accessed by each vehicle in UL and DL, the BS access mode is determined, thereby achieving a intelligent decoupled BS access selection strategy.

Algorithm 2: Access Mode Classification for Intelligent Decoupled Access

8. Simulation Results

In this section, we consider a scenario of UL/DL intelligent decoupled access heterogeneous C-V2X with 1200 m two bidirectional single lanes, where two MBSs and four SBSs are evenly deployed along the road with distances of 30 m and 10 m from the road center, respectively, as in Figure 2. Other system parameters are listed in

Table 1. Main parameters.

Channel Parameters	Value
MBS transmit power $P_{max}^m=P_{m,t}^d$ (dBm)	46
SBS transmit power $P_{max}^s=P_{s,t}^d$ (dBm)	30
Vehicle transmit power $P_{max}^v=P_{v,t}^u$ (dBm)	29
Mean of shadowing gain ${\omega }_d$ for LOS, NLOS (dB)	$-2,-6$
STD of shadowing gain ${\omega }_d$ for LOS, NLOS (dB)	$4,6$
Mean of Rayleigh fading $1/\mu$ (dBi)	1
The antenna heights of MBS, SBS, vehicle (m)	25, 10, 1.8
The DL communication frequency (GHz)	5
The UL communication frequency (GHz)	5.9
Simulation Parameters	Value
The minimum DL rate (Mbps)	0.5
The minimum UL rate (Mbps)	0.1
The length of road (m)	1200
The total bandwidth $W_{b,t}^d$ of MBS, SBS (MHz)	15, 6
The total bandwidth $W_{b,t}^u$ of MBS, SBS (MHz)	15, 6
The maximum bandwidth $W_{b,max}^d$ of MBS, SBS (MHz)	3, 3
The maximum bandwidth $W_{b,max}^u$ of MBS, SBS (MHz)	3, 3
The maximum $q_b^d$ for MBS/SBS in DL	5, 2
The maximum $q_b^u$ for MBS/SBS in UL	5, 2
The proportion of vehicles in LOS	80%
The number V of vehicles	6∼18
The number M of MBS	2
The number S of SBS	4

[29].

To demonstrate the performance of Algorithm 1, we have chosen to present the iterative results in DL as an example for illustration, as shown in Figure 3. In Figure 3, we depict the convergence process for various numbers of vehicles, with each iteration process repeated 10 times to ensure the reliability of the results. It can be observed that Algorithm 1 requires only 4∼5 iterations to convergence. Moreover, the number of vehicles has a negligible impact on the iteration speed.

Figure 4 presents the cumulative distribution function (CDF) of the DL/UL rates. In Figure 4, we have also distinguished the rates under MBS and SBS. In Figure 4a, it can be observed that the DL CDF curves for the intelligent decoupled access are all positioned to the right of those for the coupled access. This indicates that a higher proportion of vehicles with intelligent access have higher DL rates. The CDF curve for coupled access to the MBS consistently lies further to the right compared to that for coupled access to the SBS. Meanwhile, there is an intersection between the MBS and SBS segments in the case of intelligent decoupled access. This is primarily because intelligent decoupled access breaks the restriction that UL and DL must be accessed through the same BS, fully exploiting the latent performance potential of SBS due to its lower transmit power.

Figure 4b presents a comparison of the CDF for UL. The enhancement in UL rates due to intelligent decoupled access is not significant. However, the UL rate for vehicles connected to an SBS is higher than that for those connected to an MBS. This is primarily because SBSs are typically closer to the vehicles compared to MBSs, resulting in overall better channel conditions. Considering Figure 4a,b together, the UL gain remains limited while the DL gain is significant, primarily due to C-V2X’s inherent characteristics. In UL, vehicles inherently prefer SBS over distant MBS even in coupled access, yet UL performance is dominated by vehicle-to-vehicle interference—not BS selection—as vehicle density increases. This is confirmed by Figure 7b, where SBS’s UL proportion decreases under high density, indicating that SBS capacity saturation and escalating vehicle-to-vehicle interference negate proximity advantages. Conversely, DL gain thrives via decoupling, avoiding MBS interference. UL gains emerge under two conditions: (1) sparse SBS deployment forcing MBS reliance in coupled access, or (2) vehicles in MBS coverage but far from SBS, leveraging SBS proximity advantage.

Figure 5 presents the average rate of DL/UL. In Figure 5a, regardless of whether it is MBS or SBS access, the average DL rate for intelligent decoupled access is higher than that for coupled access. Additionally, the impact of the number of vehicles on the average rate of intelligent decoupled access is relatively small. In Figure 5b, the UL rate exhibits a trend similar to the DL. When the number of vehicles is large, the average UL rate for access to an SBS is greater than that for an MBS. This is because the UL rate is more significantly affected by the channel conditions, especially path loss. Therefore, although the number of vehicles increases and interference grows, the SBS is closer to the vehicles compared to the MBS. As a result, the SINR for SBS access is overall higher, leading to better performance under high vehicle load conditions.

Figure 6 depicts the proportion of access modes versus the number of vehicles in UL/DL intelligent access C-V2X. Figure 6a compares the proportion of vehicles choosing decoupled and coupled access in intelligent decoupled access C-V2X. With the increase in the number of vehicles, the proportion of decoupled access first decreases and then rises, because the UL/DL decoupled access significantly improves the communication quality for edge users. Therefore, when the load is low, most vehicles still opt for coupled access. As the vehicle load increases to a certain level and coupled access becomes fully loaded, the decoupled mode begins to gradually increase. Figure 6b more intricately displays the proportion of the six access modes. The proportions for Case 1 and Case 2 have consistently been higher. During the optimization process of intelligent decoupled access, the restriction that UL and DL must be accessed through the same BS in a purely coupled access is broken, leading to a situation where there is more access to SBSs for UL and DL. As the number of vehicles increases, the distance between SBSs and vehicles effectively decreases, and UL interference grows, resulting in a rise in the proportion of Case 6.

Figure 7 illustrates the proportion of UL/DL access to MBS and SBS by vehicle users in the intelligent decoupled access network, choosing between coupled access, Case 1 and Case 2, and decoupled access, Case 3∼6. Figure 7a illustrates the UL/DL access ratios in Case 1 and Case 2. With the increase in vehicle number, the proportion of UL access to the MBS has been continuously rising. This is because, as the number of vehicles increases, the SBSs are in close proximity, resulting in less interference when accessing the MBS compared to the SBS; consequently, the SINR for accessing the MBS is relatively higher. In DL, the proportion of access to MBS first increases and then decreases, because the power of MBS is higher, leading to more access. As the number of vehicles grow, the load on an MBS increases. However, the proximity to SBSs compensates for the power disadvantage. In Figure 7b, it can be observed that the trend for both MBS and SBS is similar to that in Figure 7a. However, in the decoupled scenario, the proportion of DL access to SBSs is overall higher than in the coupled access scenario. This is because vehicles that are decoupled are often users at the edge of the cell, which are farther away from the MBS.

Figure 8 provides a comparison of the average rates of UL and DL under different association policies. We present a comparison between the UL/DL intelligent decoupled access, which is the optimization method proposed in this paper, and two other association policies based on the nearest distance and the best channel gain, as well as the coupled access method. It can be observed that regardless of whether it is UL or DL, UL/DL intelligent decoupled access achieves the highest average rate. As the number of vehicles increases and the network load becomes heavier, the average rates for both UL and DL decrease.

9. Conclusions

We have proposed a UL/DL intelligent decoupled access framework for C-V2X, which significantly enhances network performance. By modeling the maximizing system throughput for both UL and DL as an MINLP, we have optimized the radio resource bandwidth, transmit power, and association variables of UL and DL, and have solved this NP-hard problem by using an efficient iterative algorithm while meeting the QoS requirements. The simulation results validate the effectiveness of the proposed framework and resource allocation strategy, showcasing the potential of UL/DL intelligent decoupled access in addressing the challenges faced by C-V2X. The findings provide important insight for the practical deployment of C-V2X in ITS, demonstrating that intelligent decoupled access achieves substantial throughput gains in both UL and DL while satisfying QoS requirements. The current work focuses on steady-state resource allocation optimization under given channel conditions. Incorporating vehicle mobility, dynamic channel variability, and control plane signaling overhead into the optimization framework represents an important direction for future work.