# Joint V2V-Assisted Clustering, Caching, and Multicast Beamforming in Vehicular Edge Networks

Authors: Kan Wang; Ruijie Wang; Junhuai Li; Meng Li
Journal: Wireless Communications and Mobile Computing
 (2020)

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## Abstract

As an emerging type of Internet of Things (IoT), Internet of Vehicles (IoV) denotes the vehicle network capable of supporting diverse types of intelligent services and has attracted great attention in the 5G era. In this study, we consider the multimedia content caching with multicast beamforming in IoV-based vehicular edge networks. First, we formulate a joint vehicle-to-vehicle- (V2V-) assisted clustering, caching, and multicasting optimization problem, to minimize the weighted sum of flow cost and power cost, subject to the quality-of-service (QoS) constraints for each multicast group. Then, with the two-timescale setup, the intractable and stochastic original problem is decoupled at separate timescales. More precisely, at the large timescale, we leverage the sample average approximation (SAA) technique to solve the joint V2V-assisted clustering and caching problem and then demonstrate the equivalence of optimal solutions between the original problem and its relaxed linear programming (LP) counterpart; and at the small timescale, we leverage the successive convex approximation (SCA) method to solve the nonconvex multicast beamforming problem, whereby a series of convex subproblems can be acquired, with the convergence also assured. Finally, simulations are conducted with different system parameters to show the effectiveness of the proposed algorithm, revealing that the network performance can benefit from not only the power saving from wireless multicast beamforming in vehicular networks but also the content caching among vehicles.


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## Body

## 1. Introduction

Smart devices equipped with the capability to interact with the physical environment and the function to communicate with each other are prompting the Internet towards the so-called Internet of Things (IoT) [1]. IoT holds the promise to improve our lives and the way we interact with devices, such as actuators, sensors, cell phones, and home automation. With the ever-increasing proliferation of IoT devices, the worldwide multimedia traffic is anticipated to experience a rapid growth in the 5G era [2]. Different from the legacy IoT with information exchange at the byte level, image, audio, video, and other traffic in the 5G era are typically with a large volume of information, thereby bringing forth the new terminology, namely, multimedia IoT (MIoT). As an emerging type of IoT, MIoT denotes the IoT with multimedia traffic as outputs and inputs and has been extensively used in healthcare, smart homes, communication-based train control system (CBTC), and Internet of Vehicles (IoV) [3–6].The introduction of IoV-based vehicular edge networks and the increasing demand for IoV services have imposed more stringent constraints on the quality-of-service (QoS) requirements of image and video, particularly for high-quality real-time multimedia services in fast-moving vehicles. These constraints indeed pose challenges in exploiting the full potential of vehicle-to-vehicle (V2V) communications, e.g., resource allocation, multiple-input-multiple-output- (MIMO-) based beamforming, network architecture, multicast routing, and dynamic controls [7–9].To better cope with these constrains imposed on multimedia content retrieval in IoV-based vehicular edge networks, wireless content caching begins to emerge as promising solutions [10]. By bringing multimedia contents closer to vehicles via prefetching them from the Internet or core networks, the traffic loads and network costs can be significantly reduced. In the meantime, the QoS for IoV applications could also be improved, since lower access delay and higher data rate are enabled by caching in edge nodes (e.g., base stations (BSs) and roadside units (RSUs)). Therefore, wireless content caching contributes to the agile multimedia content distribution with a higher data rate and enhanced broadband connectivity. On the other hand, as the extension of unicast transmission and by exploiting the broadcast nature of wireless channels, the point-to-multipoint multicast transmission provides a more efficient capacity-offloading approach, to deliver the identical content to multiple vehicles on the same frequency band. Thus, the integration of caching with multicasting becomes the key enabler in IoV-based edge networks, which could not only reduce the flow (traffic) cost in core or transport networks but also improve the spectrum efficiency in vehicular edge networks [11].Besides, there have been some works related to caching-based multicast beamforming design [12–15]. Nevertheless, the integration of caching with multicasting in vehicular edge networks is not smooth and still faces several challenges, due to the following observations. First, the caching decision needs to operate at the large timescale (e.g., several minutes), and the multicast beamforming has to be tailored at the small timescale (e.g., several seconds) to exploit the mobility. That is, it is a prerequisite to decouple the joint caching and multicast beamforming to different timescales. Second, when the edge network is confronted with IoV in the 5G era, numerous vehicles emerge to assist the V2V communications between each other. Thus, although with smaller storage size compared to the BS, caching profits in vehicles are nonnegligible, which yet has not been studied thoroughly, especially in the caching-based multicast beamforming. Third, the cooperative multicasting necessitates the clustering of vehicles (namely, deciding on which vehicle should be picked into the cluster to serve the same receiver), which needs to be optimized at the large timescale as well.In this paper, we study the joint V2V-assisted clustering, caching, and wireless multicast beamforming in vehicular edge networks. The distinctive features of this work are as follows:The distinctive features of this work are as follows:(1)
We present a two timescale-based V2V-assisted clustering, caching, and multicast beamforming problem, with the objective of minimizing the total network cost (involving both flow and power costs), subject to the QoS constraint for each multicast group, and the caching memory limitation for each vehicle. With the two-timescale setup, different types of variables are decoupled at separate timescales(2)
At the large timescale, we leverage the sample average approximation (SAA) technique to resolve the joint clustering and caching problem. We first reformulate and relax the stochastic original problem as a deterministic integer linear programming (ILP) one and then exhibit the equivalence of optimal solutions between the original one and its relaxed linear programming (LP) counterpart, thereby simplifying the computation substantially(3)
We leverage the successive convex approximation (SCA) method to solve the multicast beamforming problem at the small timescale. As such, the original nonconvex problem can be transformed into a series of convex subproblems, with the convergence assured(4)
Simulations are executed with different system parameters to show the effectiveness of the proposed algorithm, as well as the convergence of the proposed two-timescale setup. Simulation results reveal that the system performance could benefit from both the power saving from multicasting and the content caching among vehiclesThe remainder of this work is organized as follows. In Section2, we list works related to caching-based multicasting and V2V-assisted caching, respectively. In Section 3, we present the system model and formulate a joint clustering, caching, and multicast beamforming problem, to minimize the total network cost subject to QoS constraints. In Section 4, we leverage the SAA and SCA methods to solve the large- and small-timescale problem, respectively. In Section 5, we present and discuss the simulation results. We conclude this paper in Section 6.


## 2. Related Work

Some recent studies related to caching-based multicast beamforming and V2V-assisted caching in edge networks are presented. We believe these works can motivate more achievements in the academia and industrial domain.


### 2.1. Caching-Based Multicasting

In the literature, multicast beamforming has emerged as an effective approach to mitigate the interference in cellular systems, e.g., in smart-grid powered cellular networks [16] or in full-duplex cellular networks [17]. Furthermore, the integration of caching and multicasting could multicast the identical content to multiple receivers in the same frequency band, and thus attracts significant interests.In particular, in the cache-enabled cloud radio access network (RAN), Tao et al. [12] investigated the joint design of content-centric BS clustering and multicast beamforming for wireless content delivery. In particular, the problem is formulated as one mixed-integer nonlinear programming (MINLP) problem, and a sparse multicast beamforming algorithm is proposed based on the L0 norm approximation. In the two-tier heterogeneous networks, Cui et al. [13] considered a random caching and multicasting mechanism with caching distributions as design parameters, to enable the efficient content dissemination. First, the successful transmission probabilities in the general and high signal-to-noise ratio region are derived via stochastic geometry; then, a nonconvex joint caching and multicasting problem is formulated to maximize the successful transmission probability in the asymptotic region.Also, in cache-enabled ultradense cellular networks, Nguyen et al. [14] proposed a cooperative multicast beamforming approach to improve the cost-efficiency. Besides, Zhou et al. [15] minimized the energy cost in a multicell multigroup setup, involving caching, computing, and communication resources.


### 2.2. V2V-Assisted Caching

As an analogous terminology to V2V, device-to-device (D2D) caching refers to the caching of popular contents in mobile devices. D2D could directly deliver the content to adjacent devices, thereby offloading the BS’s traffic. Malak et al. [18] leveraged the stochastic geometry theory to derive the outage probability in the presence of interference and noise, designed the distributed caching strategy, and maximized the density of successful receptions. Jiang et al. [19] modelled the local caching as a backpack problem and characterized the D2D pairing as the maximum weight matching problem in the bipartite graph, to maximize the BS traffic offloading. In addition, Cacciapuoti et al. [20] claimed that in D2D networks, caching performance is closely related to devices’ interests and preferences in multimedia contents. In particular, content popularity focuses on community interests, while device preference reflects the difference of request probability between different individuals. Therefore, Zhang et al. [21] designed a unified D2D-caching utility function, not only considering the D2D communication distance but also incorporating the similarity of preferences among adjacent devices.With recent advents in the IoV paradigm, Su et al. [7] developed an edge caching mechanism in RSUs. Specifically, involving the content access pattern, vehicle velocity, and traffic density, the features of content requests are analysed, followed by a model to decide whether and where to cache the content. By proposing an analytical model, Tan et al. [22] quantified the effects of velocity, traffic density, and service rate on the content hit ratio. Also focusing on the edge caching in RSUs (termed as edge nodes therein), Zhao et al. [23] proposed one cost-efficient method to minimize the time-averaged service response delay, by jointly studied caching, request routing, and wireless resource allocation over the long run in an online manner. To solve the dynamics in IoV, Lyapunov optimization is leveraged to make near-optimal decisions, with a stable system assured. In addition, Xiao et al. [24] proposed one adaptive and dynamic user-centric virtual cell scheme to facilitate the multicasting of vehicular-to-everything (V2X) message, followed by a max-min fair problem formulation. Furthermore, focusing on 802.11p protocols rather than 3GPP standards, Wu et al. [25] proposed one semi-Markov decision process- (SMDP-) based formulation to maximize the long-term reward in vehicular edge networks, by jointly considering the transmission delay, computing delay, and task diversity.


### 2.3. Discussions

Edge computing and edge caching have attracted lots of attention [26–28]. The aforementioned works typically solve the clustering and multicast beamforming optimization at the same timescale. Nevertheless, it is often the case that the clustering and caching execute at a much larger timescale, while the multicast beamforming is tailored per small timescale to exploit the fast fading of wireless channels. Although Qiao et al. [28] decoupled content placement and content delivery at different timescales in vehicular edge networks, the multicast transmission was ignored, which possibly incurs low-spectrum efficiency. That is, it is a prerequisite to propose a joint clustering, caching, and multicast beamforming algorithm in vehicular edge networks on the basis of a two-timescale setup.


## 2.1. Caching-Based Multicasting

In the literature, multicast beamforming has emerged as an effective approach to mitigate the interference in cellular systems, e.g., in smart-grid powered cellular networks [16] or in full-duplex cellular networks [17]. Furthermore, the integration of caching and multicasting could multicast the identical content to multiple receivers in the same frequency band, and thus attracts significant interests.In particular, in the cache-enabled cloud radio access network (RAN), Tao et al. [12] investigated the joint design of content-centric BS clustering and multicast beamforming for wireless content delivery. In particular, the problem is formulated as one mixed-integer nonlinear programming (MINLP) problem, and a sparse multicast beamforming algorithm is proposed based on the L0 norm approximation. In the two-tier heterogeneous networks, Cui et al. [13] considered a random caching and multicasting mechanism with caching distributions as design parameters, to enable the efficient content dissemination. First, the successful transmission probabilities in the general and high signal-to-noise ratio region are derived via stochastic geometry; then, a nonconvex joint caching and multicasting problem is formulated to maximize the successful transmission probability in the asymptotic region.Also, in cache-enabled ultradense cellular networks, Nguyen et al. [14] proposed a cooperative multicast beamforming approach to improve the cost-efficiency. Besides, Zhou et al. [15] minimized the energy cost in a multicell multigroup setup, involving caching, computing, and communication resources.


## 2.2. V2V-Assisted Caching

As an analogous terminology to V2V, device-to-device (D2D) caching refers to the caching of popular contents in mobile devices. D2D could directly deliver the content to adjacent devices, thereby offloading the BS’s traffic. Malak et al. [18] leveraged the stochastic geometry theory to derive the outage probability in the presence of interference and noise, designed the distributed caching strategy, and maximized the density of successful receptions. Jiang et al. [19] modelled the local caching as a backpack problem and characterized the D2D pairing as the maximum weight matching problem in the bipartite graph, to maximize the BS traffic offloading. In addition, Cacciapuoti et al. [20] claimed that in D2D networks, caching performance is closely related to devices’ interests and preferences in multimedia contents. In particular, content popularity focuses on community interests, while device preference reflects the difference of request probability between different individuals. Therefore, Zhang et al. [21] designed a unified D2D-caching utility function, not only considering the D2D communication distance but also incorporating the similarity of preferences among adjacent devices.With recent advents in the IoV paradigm, Su et al. [7] developed an edge caching mechanism in RSUs. Specifically, involving the content access pattern, vehicle velocity, and traffic density, the features of content requests are analysed, followed by a model to decide whether and where to cache the content. By proposing an analytical model, Tan et al. [22] quantified the effects of velocity, traffic density, and service rate on the content hit ratio. Also focusing on the edge caching in RSUs (termed as edge nodes therein), Zhao et al. [23] proposed one cost-efficient method to minimize the time-averaged service response delay, by jointly studied caching, request routing, and wireless resource allocation over the long run in an online manner. To solve the dynamics in IoV, Lyapunov optimization is leveraged to make near-optimal decisions, with a stable system assured. In addition, Xiao et al. [24] proposed one adaptive and dynamic user-centric virtual cell scheme to facilitate the multicasting of vehicular-to-everything (V2X) message, followed by a max-min fair problem formulation. Furthermore, focusing on 802.11p protocols rather than 3GPP standards, Wu et al. [25] proposed one semi-Markov decision process- (SMDP-) based formulation to maximize the long-term reward in vehicular edge networks, by jointly considering the transmission delay, computing delay, and task diversity.


## 2.3. Discussions

Edge computing and edge caching have attracted lots of attention [26–28]. The aforementioned works typically solve the clustering and multicast beamforming optimization at the same timescale. Nevertheless, it is often the case that the clustering and caching execute at a much larger timescale, while the multicast beamforming is tailored per small timescale to exploit the fast fading of wireless channels. Although Qiao et al. [28] decoupled content placement and content delivery at different timescales in vehicular edge networks, the multicast transmission was ignored, which possibly incurs low-spectrum efficiency. That is, it is a prerequisite to propose a joint clustering, caching, and multicast beamforming algorithm in vehicular edge networks on the basis of a two-timescale setup.


## 3. System Model and Problem Formulation

Vehicular edge network with one BS and multiple IoV devices is considered. The BS is equipped withL antennas, and each vehicle has one single antenna. As shown in Figure 1, in the context of V2V communications, the content requesting vehicle (CRV) could not only attach to the BS via cellular links but also associate with content caching vehicles (CCVs) directly. The BS is indexed by j=0, while CCVs are denoted by J=1,⋯,j,⋯,J. Furthermore, J0=J∪0 denotes as the set of all transmitters. Meanwhile, the CRVs with the identical content request are categorized into one multicast group, and all multicast groups are denoted by S=1,⋯,s,⋯,S. For each multicast group s∈S, its all associated CRVs are denoted by Is, and ∪s∈SIs=I holds, where I is the set of all CRVs. In addition, we follow the equal content size for all contents [12], namely, each of which is chunked and normalized to the size of one.Figure 1
System model of vehicular edge networks.


### 3.1. Two-Timescale Setup

Mixed-timescale setup in [29, 30] is incorporated in this study. Each large timescale accommodates multiple successive small timescale slots, e.g., a total of T slots, denoted by T=1,⋯,T. In addition, in each slot, the beamformer is designed for each multicast group, to exploit the fast fading and mobility experienced by CRVs. As revealed in Figure 2, at the end of each large timescale, the network operator has to decide the CCV clustering (i.e., the CCVs serving for the same multicast group) and the content caching for transmitters. Then, following this decision unchanged throughout the next large timescale, the picked content is cached in the local memory or storage in both BS and clustered CCVs.Figure 2
Two-timescale setup.In this work, we assume that all multicast groups can be admitted and satisfied with their content requests. The case also occurs that insufficient resources cannot accommodate all multicast groups (e.g., in [31]), which is yet beyond the scope of this work.Afterward, following the two-timescale setup, two types of variables can be specified as:(i)
Large-timescale variables:xs,j∈0,1 and zs,j∈0,1, indicating whether or not that transmitter j belongs to the serving cluster of group s, and whether or not that transmitter j caches the content requested by group s, respectively. That is, xs,j∈0,1 indicates a clustering variable: if xs,j=1, then transmitter j is within the cluster of s; and xs,j=0 vice versa(ii)
Small-timescale variables:ws,0t∈ℂL×1 (or ws,jt∈ℂ,j∈J), denoting the per-slot multicast beamforming vector of the BS (or CCV j) for group sThe two-timescale setup along with its two types of variables necessitates a network cost involving both the flow and power costs, which will be introduced in the following, respectively.


### 3.2. Caching Model

At the end of large timescale, both the clustering variablexs,j and caching variable zs,j have to be designed for the next large timescale. It is straightforward that if the content is not cached in the network, then it must be retrieved via the backhaul link from the Internet, as shown in Figure 1, thus incurring flow cost in the backhaul link or transport networks. The data rate of fetching content from core networks should be as least as the transmission rate of its associated multicast group. Following the fixed transmission rate Rs in [12] for each multicast group s, the total flow passing through the backhaul link on the BS can be represented as ∑s∈S1−zs,0Rs. In particular, when zs,0=1, the immediate transmission from the BS to CRV i does not incur any flow cost on the backhaul link, and zs,0=0 vice versa. Besides, different from the BS, CCVs could not directly retrieve the content through the backhaul link from the Internet or core networks. Without caching, CCVs have to resort to the BS for the content retrieval, which would claim a dedicated frequency band to avert the interference to CRVs, thereby contributing to the flow cost as well. As such, the flow cost should involve both the backhaul cost and frequency bandwidth cost, and the latter one could be equivalently translated to the former one in terms of fixed transmission rate Rs. Till now, the total flow cost (on the backhaul link and dedicated frequency band) can be represented as
(1)CF=∑s∈S∑j∈J01−zs,jRs.Then, for the clustering variablexs,j, it is irrational to cache the requested content in transmitter j when it is not picked to join the cluster of multicast group s. That is, zs,j could only take value zero when xs,j=0 holds, which can be formulated as
(2)zs,j≤xs,j,∀s,j.Next, also revealed in Equation (2), zs,j tends to take value one to save the flow cost, given xs,j=1. Yet, finite storage size in CCVs constrains the cached content number, namely, only parts of contents can be accommodated in each transmitter j, written as
(3)∑s∈Szs,j≤Cj,∀j,which indicates the caching capacity constraint Cj with a normalized content size of one.Finally, although the caching function does also incur costs for both BS and CCVs, we only focus on the trade-off between backhaul cost and power cost, and the study on caching cost is beyond the scope of this work.


### 3.3. Multicast Model

The multicast beamforming vector is tailored per slot. For simplicity, letwst=ws,0Ht,ws,1Ht,⋯,ws,JHtH be the network-wide beamforming vector for group s from all transmitters [32]. Furthermore, for each CRV i∈Is, hit∈ℂL+J×1 denotes as the aggregate channel vector from all transmitters to CRV i. As such, the received signal-to-noise-ratio (SINR) for CRV i at slot t can be written as
(4)SINRit=hiHtwst2∑s′≠shiHtws′t2+σi2,∀i∈Is,s,where σi2 is the additive white Gaussian noise power per CRV and ∑s′≠shiHtws′t2 is the intergroup interference from any other multicast group s′≠s. For each group s, it is ensured that the achievable data rate of any CRV i is no smaller than its group’s fixed transmission rate Rs, namely,
(5)Blog21+SINRit≥Rs,∀i∈Is,which can be further recast as
(6)SINRit≥2Rs/B−1,∀i∈Is,s,where B denotes the frequency bandwidth.Furthermore, considering any pair of transmitterj and multicast group s, if j does not belong to the serving clustering of s, then it is irrational to design a nonzero beamforming vector for s. That is to say, if xs,j=0 holds, then we have ws,0t=0 (or ws,jt=0) at any time slot t, which can be represented as
(7)1−xs,0ws,0t=0,∀s,t1−xs,jws,jt=0,∀s,j∈J,t.Moreover, since each transmitter is with the maximum transmit power threshold, the peak power budgetEj per transmitter should be imposed as
(8)∑s∈Sws,0t22≤E0,∀t,∑s∈Sws,jt22≤Ej,∀j∈J,t.In addition, summing over the transmit power of all network-wide beamforming vectors, the total power cost can be computed as(9)CP=1T∑t∈T∑s∈Swst22,which is averaged over all time slots in one large timescale. In particular, CP is exactly defined as the time average of small-timescale transmit power cost ∑s∈Swst22.


### 3.4. Problem Formulation

Finally, involving both the flow cost and power cost, the overall cost minimization problem can be formulated as(10)P0:minCF+ηCPs.t.2,3,6,7,8varxs,j,zs,j∈0,1,∀s,j,varwst,∀s,t,where η is a coefficient to balance the trade-off between flow and power costs and could be regulated artificially in line with the price of backhaul link and transmit power.


## 3.1. Two-Timescale Setup

Mixed-timescale setup in [29, 30] is incorporated in this study. Each large timescale accommodates multiple successive small timescale slots, e.g., a total of T slots, denoted by T=1,⋯,T. In addition, in each slot, the beamformer is designed for each multicast group, to exploit the fast fading and mobility experienced by CRVs. As revealed in Figure 2, at the end of each large timescale, the network operator has to decide the CCV clustering (i.e., the CCVs serving for the same multicast group) and the content caching for transmitters. Then, following this decision unchanged throughout the next large timescale, the picked content is cached in the local memory or storage in both BS and clustered CCVs.Figure 2
Two-timescale setup.In this work, we assume that all multicast groups can be admitted and satisfied with their content requests. The case also occurs that insufficient resources cannot accommodate all multicast groups (e.g., in [31]), which is yet beyond the scope of this work.Afterward, following the two-timescale setup, two types of variables can be specified as:(i)
Large-timescale variables:xs,j∈0,1 and zs,j∈0,1, indicating whether or not that transmitter j belongs to the serving cluster of group s, and whether or not that transmitter j caches the content requested by group s, respectively. That is, xs,j∈0,1 indicates a clustering variable: if xs,j=1, then transmitter j is within the cluster of s; and xs,j=0 vice versa(ii)
Small-timescale variables:ws,0t∈ℂL×1 (or ws,jt∈ℂ,j∈J), denoting the per-slot multicast beamforming vector of the BS (or CCV j) for group sThe two-timescale setup along with its two types of variables necessitates a network cost involving both the flow and power costs, which will be introduced in the following, respectively.


## 3.2. Caching Model

At the end of large timescale, both the clustering variablexs,j and caching variable zs,j have to be designed for the next large timescale. It is straightforward that if the content is not cached in the network, then it must be retrieved via the backhaul link from the Internet, as shown in Figure 1, thus incurring flow cost in the backhaul link or transport networks. The data rate of fetching content from core networks should be as least as the transmission rate of its associated multicast group. Following the fixed transmission rate Rs in [12] for each multicast group s, the total flow passing through the backhaul link on the BS can be represented as ∑s∈S1−zs,0Rs. In particular, when zs,0=1, the immediate transmission from the BS to CRV i does not incur any flow cost on the backhaul link, and zs,0=0 vice versa. Besides, different from the BS, CCVs could not directly retrieve the content through the backhaul link from the Internet or core networks. Without caching, CCVs have to resort to the BS for the content retrieval, which would claim a dedicated frequency band to avert the interference to CRVs, thereby contributing to the flow cost as well. As such, the flow cost should involve both the backhaul cost and frequency bandwidth cost, and the latter one could be equivalently translated to the former one in terms of fixed transmission rate Rs. Till now, the total flow cost (on the backhaul link and dedicated frequency band) can be represented as
(1)CF=∑s∈S∑j∈J01−zs,jRs.Then, for the clustering variablexs,j, it is irrational to cache the requested content in transmitter j when it is not picked to join the cluster of multicast group s. That is, zs,j could only take value zero when xs,j=0 holds, which can be formulated as
(2)zs,j≤xs,j,∀s,j.Next, also revealed in Equation (2), zs,j tends to take value one to save the flow cost, given xs,j=1. Yet, finite storage size in CCVs constrains the cached content number, namely, only parts of contents can be accommodated in each transmitter j, written as
(3)∑s∈Szs,j≤Cj,∀j,which indicates the caching capacity constraint Cj with a normalized content size of one.Finally, although the caching function does also incur costs for both BS and CCVs, we only focus on the trade-off between backhaul cost and power cost, and the study on caching cost is beyond the scope of this work.


## 3.3. Multicast Model

The multicast beamforming vector is tailored per slot. For simplicity, letwst=ws,0Ht,ws,1Ht,⋯,ws,JHtH be the network-wide beamforming vector for group s from all transmitters [32]. Furthermore, for each CRV i∈Is, hit∈ℂL+J×1 denotes as the aggregate channel vector from all transmitters to CRV i. As such, the received signal-to-noise-ratio (SINR) for CRV i at slot t can be written as
(4)SINRit=hiHtwst2∑s′≠shiHtws′t2+σi2,∀i∈Is,s,where σi2 is the additive white Gaussian noise power per CRV and ∑s′≠shiHtws′t2 is the intergroup interference from any other multicast group s′≠s. For each group s, it is ensured that the achievable data rate of any CRV i is no smaller than its group’s fixed transmission rate Rs, namely,
(5)Blog21+SINRit≥Rs,∀i∈Is,which can be further recast as
(6)SINRit≥2Rs/B−1,∀i∈Is,s,where B denotes the frequency bandwidth.Furthermore, considering any pair of transmitterj and multicast group s, if j does not belong to the serving clustering of s, then it is irrational to design a nonzero beamforming vector for s. That is to say, if xs,j=0 holds, then we have ws,0t=0 (or ws,jt=0) at any time slot t, which can be represented as
(7)1−xs,0ws,0t=0,∀s,t1−xs,jws,jt=0,∀s,j∈J,t.Moreover, since each transmitter is with the maximum transmit power threshold, the peak power budgetEj per transmitter should be imposed as
(8)∑s∈Sws,0t22≤E0,∀t,∑s∈Sws,jt22≤Ej,∀j∈J,t.In addition, summing over the transmit power of all network-wide beamforming vectors, the total power cost can be computed as(9)CP=1T∑t∈T∑s∈Swst22,which is averaged over all time slots in one large timescale. In particular, CP is exactly defined as the time average of small-timescale transmit power cost ∑s∈Swst22.


## 3.4. Problem Formulation

Finally, involving both the flow cost and power cost, the overall cost minimization problem can be formulated as(10)P0:minCF+ηCPs.t.2,3,6,7,8varxs,j,zs,j∈0,1,∀s,j,varwst,∀s,t,where η is a coefficient to balance the trade-off between flow and power costs and could be regulated artificially in line with the price of backhaul link and transmit power.


## 4. Joint Clustering, Caching, and Multicast Beamforming Algorithm

In this section, we first present the challenges to solve problemP0 and then leverage the SAA technique to decouple the two-timescale problem into one large-timescale problem and a series of independent small-timescale subproblems.


### 4.1. Algorithm Design Challenges

We desire to solveP0 at the end of each large timescale for the next one. Nevertheless, it poses challenges to solve P0 due to the following observations:
(i)
The channel vectorshit∀t∈T are unknown for the operator, since all channel vectors are future ones in the next large timescale(ii)
Bothx=xs,j and z=zs,j are binary variables, rending P0 an MINLP problem(iii)
Even thoughx and z are relaxed, both Equations (6) and (7) are still nonconvex constraintsThus, to makeP0 tractable, an approximation approach is a prerequisite to solve it.


### 4.2. SAA-Based Cost Minimization

The channel vectorshit∀t∈T are exactly stochastic at each decision moment, since they are scattered in the next large timescale and thus unpredictable. As in [29, 30], the SAA technique is leveraged to approximate the random variables hit∀t∈T, with the basic principle assuming that hit∀t∈T is drawn from a certain distribution. As such, Equation (9) can be deemed as the time average of T random variables, and its expectation can be approximately computed as
(11)Eh^∑s∈Sw^s22≈1T∑t∈T∑s∈Swst22,where h^ is the stochastic channel vector space, and w^s denotes the h^-based beamforming vector. Then, by leverage the SAA, a series of samples are generated, and the expectation (11) can be approximated by its sample average. To distinguish the sample from real channel vectors per time slot, v is utilized to denote the sample index, and a total of samples are produced. Therefore, by substituting t with v at the decision moment of each large timescale, an approximate problem of P0 can be recast as
(12)P1:minCF+ηV∑v∈V∑s∈Swsv22s.t.2,3,6,7,8varxs,j,zs,j∈0,1,∀s,j,varwsv,∀s,v,where hiv is the v-th sample from a certain distribution, and wsv is the hiv-based beamforming vector.From Equation (12), it follows that P1 turns out to be a deterministic problem rather than a stochastic one. Nevertheless, solving P1 still poses challenges, since both Equations (2) and (6) are nonconvex constraints. Furthermore, small-timescale variable wsv and large-timescale variable xs,j are still tied in Equation (7). For this problem with two types of coupled variables, an intuitive approach is to decouple them firstly and then optimize them separately. Thus, we would propose an iterative algorithm, with the outline as follows:
(i)
First, given any feasiblew, search for the optimal x∗,z∗ in P1(ii)
Then, givenx∗,z∗, search for the optimal w∗ with the sample hiv∀i,v(iii)
Repeat aforementioned procedures until convergence(iv)
With the acquiredx∗,z∗, the optimal wst∀s,t for problem P0 can be obtained by solving the following multicast beamforming problem per slot as follows:(13)P2:min1T∑t∈T∑s∈Swst22s.t.6,7,8varwst,∀s,t.By comparingP1 and P2, it reaches the conclusion that P1 is a large-timescale problem with sample channel vectors, while P2 reduces to a small-timescale one with actual channel vectors. In the next, we would separately introduce approaches to solve them.


### 4.3. Joint Clustering and Caching Algorithm

Given any feasiblew, any group-transmitter pair with nonzero beamforming vector can be determined. Define the set K1=s,j∣ws,0v≠0orws,jv≠0,∃v∈V. Following Equation (7), if ws,0v≠0 or ws,jv≠0,∃v∈V, then xs,j=1 must hold. On the contrary, although ws,0v=0 or ws,jv=0,∀v cannot straightforwardly reach the conclusion with xs,j=0 from Equation (7), we can claim that xs,j=0 holds, namely, j does not need to belong to the serving cluster of s when ws,0v=0 or ws,jv=0,∀v. Likewise, define the set K2=s,j∣ws,0v=0orws,jv=0,∀v∈V.As such, the clustering variablexs,j is fixed given any feasible w, and P1 reduces to the following large-timescale problem as
(14)P1−I:minCFs.t.2,3,xs,j=1,∀s,j∈K1,xs,j=1,∀s,j∈K2,varxs,j,zs,j∈0,1∀s,j,which exactly involves only the caching variable zs,j, since xs,j is determined.It follows thatP1−I is a typical 0-1 ILP problem. On one hand, the celebrated cutting-plane or branch-and-bound methods have been extensively utilized to solve it, yet with the computational complexity scaled with S and J. Thus, solving it is prohibitively complicated in a larger-sized network. On the other hand, there are also extensive works (e.g., in [2]) to leverage heuristic algorithms to solve the ILP problem. Nevertheless, the optimality cannot be ensured in this case, and it poses challenges to analyse the gap between the heuristic and optimality as well.Motivated by our previous works in [33, 34], we resort to the LP counterpart of P1−I. If the equivalence of optimal solutions between P1−I and its LP counterpart can be established, then we claim that the optimal solution to LP counterpart is also integer-valued and optimal to P1−I as well.First, with relaxedzs,j, P1−I can be recast as
(15)P1−R:minCFs.t.2,3,xs,j=1,∀s,j∈K1,xs,j=0,∀s,j∈K2,var0≤zs,j≤1,∀s,j,where the clustering variable xs,j is eliminated on the basis of K1 and K2 for conciseness.It is widely known that the optimal solution of LP must locate on the vertex of the polyhedron for a feasible set. Thus, what we only need to do next is to establish that any vertex of the polyhedron ofP1−R is integer-valued. As such, the optimal solution to P1−R must be integer-valued and is also optimal to P1−I. Henceforth, we further provide a sufficient condition, under which the relaxed P1−R has the integer-valued optimal solution.Theorem 1.
If each transmitter’s storage capacity is the integer multiplier of content size, i.e.,Cj∈ℤ,∀j∈J0, then the optimal solution to P1−R is integer-valued.Proof.
From [35], it follows that if the constraint matrix is totally unimodular, then the optimal solution is integer-valued and locates at the vertex of the polyhedron. Therefore, we prove this theorem by establishing the total unimodularity of the constraint matrix of P1−R. By introducing a caching variable vector Y=y1,0,⋯,yS,0,y1,1,⋯,yS,1,⋯,⋯,y1,J,⋯,yS,JT and a storage capacity vector C=C0,C1,⋯,CjT, Equation (3) can be recast as WY≤C, and W turns out to be as
(16)W=W0W1W2⋯Wj=11⋯100⋯0⋯00⋯000⋯011⋯1⋯00⋯0∗∗⋮∗∗∗⋮∗⋱∗∗⋮∗00⋯000⋯0⋯11⋯1where each Wj is with the dimension of J+1×S. In particular, the (j+1)-th row of Wj is all one and all zero for the other rows. Since each column of Wj contains only one nonzero entry belonging to 0,+1,−1, we claim that the constraint matrix W meets the four subconditions of total unimodularity in [35]. As such, W is totally unimodular, and thus the optimal solution of P1−R is integer-valued.
The proof is completed.With Theorem 1, we can effortlessly solveP1−I via classical simplex method or ellipsoid algorithm, due to the equivalence of optimal solutions between P1−R and P1−I, thus significantly reducing the computational complexity. Although the classical cutting-plane or branch-and-bound algorithm can also work for the ILP, they are only applicable to small- and moderate-sized vehicular edge networks. When it comes to a large-sized network, Theorem 1 exhibits more effectiveness than legacy ones.


### 4.4. SCA-Based Multicast Beamforming

Once the aforementionedx∗,z∗ is given, P1 reduces to the following one as
(17)P1−S:min1V∑v∈V∑s∈Swsv22s.t.6,7,8varwsv,∀s,v,which is exactly the same as P2 with the only difference in substituting the index t in P2 with v in P1−S. Notice that, P2 is the small-timescale multicast beamforming problem with actual channel vectors hit∀t∈T, while P1−S is solved at the large timescale with channel samples hiv∀v∈V, just to proceed with the algorithm iteration in solving P1−S. Thus, in the following, by putting P2 and P1−S together, we take P2 as a goal to find its approximation algorithm, and do not distinguish them strictly.Although the objective as well as constraints 7 and 8 turn into convex ones, the nonconvex constraint 6 rendersP2 a nonconvex one. Thus, by resorting to the SCA technique in [12], we first recast Equation (6) as
(18)2Rs/B−1∑s′≠shiHtws′t2+σi2−hiHtwst2≤0,∀i∈Is,s,t.It is evident that the left-hand side of Equation (18) is the difference of two convex functions. Next, by substituting the second term with its first-order Taylor expansion, at the o+1-th iteration, we have
(19)2Rs/B−1∑s′≠shiHtws′t2+σi2−2RwsotHhithiHtwst−hiHtwsot2≤0,∀i∈Is,s,t,which becomes a convex constraint, with wsot acting as the initial feasible solution obtained from the o-th iteration.Till now, at theo+1-th iteration, P2 can be transformed to the following one as
(20)P2−C:min1T∑t∈T∑s∈Swst22s.t.7,8,19arwst,∀s,t,which is a convex quadratically constrained quadratic programming (QCQP) problem and can be solved via many mature convex algorithms effortlessly, e.g., interior-point method [36]. At each iteration, a convex QCQP problem needs to be solved until convergence. From Equation (19), it follows that the obtained wsot from the o-th iteration is always feasible to the subproblem at the o+1-th iteration. Besides, since the objective of P2−C is the minimization of power cost, wso+1t must be a better solution than wsot, revealing the monotonicity of objective.


### 4.5. Algorithm Outline and Computational Complexity

Algorithm 1: Joint V2V-assisted clustering, caching, and multicasting algorithm.
Initialization:xs,j=1,∀s∈S,∀j∈J0 (Full cooperative multicasting is as the initial solution).RepeatSolve problemP1−S via SCA to acquire w∗;ObtainK1 and K2 from Equation (14);Solve problemP1−I via Theorem 1 to acquire x∗,z∗.Until the error tolerance is met or maximum iterations are reached.fort=1,2⋯,TSolve problemP2 via SCA with the available x∗,z∗.end forAt the end of a large timescale, the operator has to iteratively resolve problemsP1−S and P1−I, until the termination criteria are satisfied, namely, one convergent solution is acquired or the maximum iteration number is reached, as described in Algorithm 1. Then, with the available x∗,z∗ from large timescale, the operator proceeds to tailor the multicast beamforming per slot, to solve P2 via SCA for the optimal wsot.From Algorithm 1, it follows that the SCA is a prerequisite at both timescales. Assume the maximum iteration numbers for joint V2V clustering and caching algorithm, and SCA method reachϱmax and omax, respectively. From [37], it follows that the interior point method to solve each per-iteration subproblem for P1−S or P2 is with the computational complexity of Φ=OJ+L3S3V3 or Φ=OJ+L3S3T3. Thus, the overall computational complexity reaches Oϱmax+1omaxΦ=OϱmaxomaxΦ.


## 4.1. Algorithm Design Challenges

We desire to solveP0 at the end of each large timescale for the next one. Nevertheless, it poses challenges to solve P0 due to the following observations:
(i)
The channel vectorshit∀t∈T are unknown for the operator, since all channel vectors are future ones in the next large timescale(ii)
Bothx=xs,j and z=zs,j are binary variables, rending P0 an MINLP problem(iii)
Even thoughx and z are relaxed, both Equations (6) and (7) are still nonconvex constraintsThus, to makeP0 tractable, an approximation approach is a prerequisite to solve it.


## 4.2. SAA-Based Cost Minimization

The channel vectorshit∀t∈T are exactly stochastic at each decision moment, since they are scattered in the next large timescale and thus unpredictable. As in [29, 30], the SAA technique is leveraged to approximate the random variables hit∀t∈T, with the basic principle assuming that hit∀t∈T is drawn from a certain distribution. As such, Equation (9) can be deemed as the time average of T random variables, and its expectation can be approximately computed as
(11)Eh^∑s∈Sw^s22≈1T∑t∈T∑s∈Swst22,where h^ is the stochastic channel vector space, and w^s denotes the h^-based beamforming vector. Then, by leverage the SAA, a series of samples are generated, and the expectation (11) can be approximated by its sample average. To distinguish the sample from real channel vectors per time slot, v is utilized to denote the sample index, and a total of samples are produced. Therefore, by substituting t with v at the decision moment of each large timescale, an approximate problem of P0 can be recast as
(12)P1:minCF+ηV∑v∈V∑s∈Swsv22s.t.2,3,6,7,8varxs,j,zs,j∈0,1,∀s,j,varwsv,∀s,v,where hiv is the v-th sample from a certain distribution, and wsv is the hiv-based beamforming vector.From Equation (12), it follows that P1 turns out to be a deterministic problem rather than a stochastic one. Nevertheless, solving P1 still poses challenges, since both Equations (2) and (6) are nonconvex constraints. Furthermore, small-timescale variable wsv and large-timescale variable xs,j are still tied in Equation (7). For this problem with two types of coupled variables, an intuitive approach is to decouple them firstly and then optimize them separately. Thus, we would propose an iterative algorithm, with the outline as follows:
(i)
First, given any feasiblew, search for the optimal x∗,z∗ in P1(ii)
Then, givenx∗,z∗, search for the optimal w∗ with the sample hiv∀i,v(iii)
Repeat aforementioned procedures until convergence(iv)
With the acquiredx∗,z∗, the optimal wst∀s,t for problem P0 can be obtained by solving the following multicast beamforming problem per slot as follows:(13)P2:min1T∑t∈T∑s∈Swst22s.t.6,7,8varwst,∀s,t.By comparingP1 and P2, it reaches the conclusion that P1 is a large-timescale problem with sample channel vectors, while P2 reduces to a small-timescale one with actual channel vectors. In the next, we would separately introduce approaches to solve them.


## 4.3. Joint Clustering and Caching Algorithm

Given any feasiblew, any group-transmitter pair with nonzero beamforming vector can be determined. Define the set K1=s,j∣ws,0v≠0orws,jv≠0,∃v∈V. Following Equation (7), if ws,0v≠0 or ws,jv≠0,∃v∈V, then xs,j=1 must hold. On the contrary, although ws,0v=0 or ws,jv=0,∀v cannot straightforwardly reach the conclusion with xs,j=0 from Equation (7), we can claim that xs,j=0 holds, namely, j does not need to belong to the serving cluster of s when ws,0v=0 or ws,jv=0,∀v. Likewise, define the set K2=s,j∣ws,0v=0orws,jv=0,∀v∈V.As such, the clustering variablexs,j is fixed given any feasible w, and P1 reduces to the following large-timescale problem as
(14)P1−I:minCFs.t.2,3,xs,j=1,∀s,j∈K1,xs,j=1,∀s,j∈K2,varxs,j,zs,j∈0,1∀s,j,which exactly involves only the caching variable zs,j, since xs,j is determined.It follows thatP1−I is a typical 0-1 ILP problem. On one hand, the celebrated cutting-plane or branch-and-bound methods have been extensively utilized to solve it, yet with the computational complexity scaled with S and J. Thus, solving it is prohibitively complicated in a larger-sized network. On the other hand, there are also extensive works (e.g., in [2]) to leverage heuristic algorithms to solve the ILP problem. Nevertheless, the optimality cannot be ensured in this case, and it poses challenges to analyse the gap between the heuristic and optimality as well.Motivated by our previous works in [33, 34], we resort to the LP counterpart of P1−I. If the equivalence of optimal solutions between P1−I and its LP counterpart can be established, then we claim that the optimal solution to LP counterpart is also integer-valued and optimal to P1−I as well.First, with relaxedzs,j, P1−I can be recast as
(15)P1−R:minCFs.t.2,3,xs,j=1,∀s,j∈K1,xs,j=0,∀s,j∈K2,var0≤zs,j≤1,∀s,j,where the clustering variable xs,j is eliminated on the basis of K1 and K2 for conciseness.It is widely known that the optimal solution of LP must locate on the vertex of the polyhedron for a feasible set. Thus, what we only need to do next is to establish that any vertex of the polyhedron ofP1−R is integer-valued. As such, the optimal solution to P1−R must be integer-valued and is also optimal to P1−I. Henceforth, we further provide a sufficient condition, under which the relaxed P1−R has the integer-valued optimal solution.Theorem 1.
If each transmitter’s storage capacity is the integer multiplier of content size, i.e.,Cj∈ℤ,∀j∈J0, then the optimal solution to P1−R is integer-valued.Proof.
From [35], it follows that if the constraint matrix is totally unimodular, then the optimal solution is integer-valued and locates at the vertex of the polyhedron. Therefore, we prove this theorem by establishing the total unimodularity of the constraint matrix of P1−R. By introducing a caching variable vector Y=y1,0,⋯,yS,0,y1,1,⋯,yS,1,⋯,⋯,y1,J,⋯,yS,JT and a storage capacity vector C=C0,C1,⋯,CjT, Equation (3) can be recast as WY≤C, and W turns out to be as
(16)W=W0W1W2⋯Wj=11⋯100⋯0⋯00⋯000⋯011⋯1⋯00⋯0∗∗⋮∗∗∗⋮∗⋱∗∗⋮∗00⋯000⋯0⋯11⋯1where each Wj is with the dimension of J+1×S. In particular, the (j+1)-th row of Wj is all one and all zero for the other rows. Since each column of Wj contains only one nonzero entry belonging to 0,+1,−1, we claim that the constraint matrix W meets the four subconditions of total unimodularity in [35]. As such, W is totally unimodular, and thus the optimal solution of P1−R is integer-valued.
The proof is completed.With Theorem 1, we can effortlessly solveP1−I via classical simplex method or ellipsoid algorithm, due to the equivalence of optimal solutions between P1−R and P1−I, thus significantly reducing the computational complexity. Although the classical cutting-plane or branch-and-bound algorithm can also work for the ILP, they are only applicable to small- and moderate-sized vehicular edge networks. When it comes to a large-sized network, Theorem 1 exhibits more effectiveness than legacy ones.


## 4.4. SCA-Based Multicast Beamforming

Once the aforementionedx∗,z∗ is given, P1 reduces to the following one as
(17)P1−S:min1V∑v∈V∑s∈Swsv22s.t.6,7,8varwsv,∀s,v,which is exactly the same as P2 with the only difference in substituting the index t in P2 with v in P1−S. Notice that, P2 is the small-timescale multicast beamforming problem with actual channel vectors hit∀t∈T, while P1−S is solved at the large timescale with channel samples hiv∀v∈V, just to proceed with the algorithm iteration in solving P1−S. Thus, in the following, by putting P2 and P1−S together, we take P2 as a goal to find its approximation algorithm, and do not distinguish them strictly.Although the objective as well as constraints 7 and 8 turn into convex ones, the nonconvex constraint 6 rendersP2 a nonconvex one. Thus, by resorting to the SCA technique in [12], we first recast Equation (6) as
(18)2Rs/B−1∑s′≠shiHtws′t2+σi2−hiHtwst2≤0,∀i∈Is,s,t.It is evident that the left-hand side of Equation (18) is the difference of two convex functions. Next, by substituting the second term with its first-order Taylor expansion, at the o+1-th iteration, we have
(19)2Rs/B−1∑s′≠shiHtws′t2+σi2−2RwsotHhithiHtwst−hiHtwsot2≤0,∀i∈Is,s,t,which becomes a convex constraint, with wsot acting as the initial feasible solution obtained from the o-th iteration.Till now, at theo+1-th iteration, P2 can be transformed to the following one as
(20)P2−C:min1T∑t∈T∑s∈Swst22s.t.7,8,19arwst,∀s,t,which is a convex quadratically constrained quadratic programming (QCQP) problem and can be solved via many mature convex algorithms effortlessly, e.g., interior-point method [36]. At each iteration, a convex QCQP problem needs to be solved until convergence. From Equation (19), it follows that the obtained wsot from the o-th iteration is always feasible to the subproblem at the o+1-th iteration. Besides, since the objective of P2−C is the minimization of power cost, wso+1t must be a better solution than wsot, revealing the monotonicity of objective.


## 4.5. Algorithm Outline and Computational Complexity

Algorithm 1: Joint V2V-assisted clustering, caching, and multicasting algorithm.
Initialization:xs,j=1,∀s∈S,∀j∈J0 (Full cooperative multicasting is as the initial solution).RepeatSolve problemP1−S via SCA to acquire w∗;ObtainK1 and K2 from Equation (14);Solve problemP1−I via Theorem 1 to acquire x∗,z∗.Until the error tolerance is met or maximum iterations are reached.fort=1,2⋯,TSolve problemP2 via SCA with the available x∗,z∗.end forAt the end of a large timescale, the operator has to iteratively resolve problemsP1−S and P1−I, until the termination criteria are satisfied, namely, one convergent solution is acquired or the maximum iteration number is reached, as described in Algorithm 1. Then, with the available x∗,z∗ from large timescale, the operator proceeds to tailor the multicast beamforming per slot, to solve P2 via SCA for the optimal wsot.From Algorithm 1, it follows that the SCA is a prerequisite at both timescales. Assume the maximum iteration numbers for joint V2V clustering and caching algorithm, and SCA method reachϱmax and omax, respectively. From [37], it follows that the interior point method to solve each per-iteration subproblem for P1−S or P2 is with the computational complexity of Φ=OJ+L3S3V3 or Φ=OJ+L3S3T3. Thus, the overall computational complexity reaches Oϱmax+1omaxΦ=OϱmaxomaxΦ.


## 5. Simulation Results and Discussions

In this section, we demonstrate the performance of proposed joint V2V-assisted clustering, caching, and multicast beamforming algorithm in vehicular edge networks via simulation results. In particular, the impact of the following two parameters is studied: (1) the number of CCVs; and (2) the average data rate requirement per CRV. Meanwhile, two costs are leveraged as performance metrics: (1) total flow cost; and (2) total transmit power cost. In addition, for performance comparison, three benchmark schemes are also evaluated, listed as follows:(i)
Multicast without (w.o.) V2V caching refers to the multicast beamforming scheme proposed in [11], which only involves the BS caching but neglecting the V2V caching. For fair comparison, the two-timescale setup is also used in this scheme(ii)
Unicast with (w.) V2V caching refers to the unicast transmission scheme proposed in [23], where both V2V and BS caching are involved. In this scheme, each CRV is assigned an individual beamforming vector, regardless of its requested content(iii)
Unicast w.o. V2V caching. In this scheme, each CRV has to access the BS to fetch its requested content


### 5.1. Simulation Parameters

We investigate a time-slotted wireless network consisting of one BS and 20 CRVs, with a radius of 500 m for the cell coverage. The BS is equipped with 20 antennas, and each CRV and CCV is identically equipped with one antenna. Overall, 20 types of multimedia contents exist in the system. The BS has a storage size of 6 contents, while each CCV could accommodate at most 1 content. For simplicity, we do not employ the well-known Zipf distribution in [2]; instead, each CRV equally requests any content with a probability of 5%.The system bandwidth is 5 MHz, and the additive white Gaussian noise power spectral is -174 dBm/Hz. The transmit power budgets for the BS and CCV are 46 dBm and 24 dBm, respectively. The path loss model is35.3+37.6log2dm, the log-normal shadowing parameter is 8 dB, and the multipath channel model with Rayleigh fading is assumed [20]. Overall, 200 samples are produced to simulate stochastic channels vectors. Besides, the Monte Carlo approach is utilized, both CRVs and CCVs are uniformly distributed and dropped in the cell coverage, and all simulation results are averaged over 1000 droppings. In addition, the vehicle speed follows a truncated normal distribution ranging from 15 m/s and 31 m/s.


### 5.2. Convergence Performance

Figure3 illustrates the convergence performance of joint V2V clustering and caching algorithms. In this setup, the number of CCVs is 20, and the data rate requirement per group is 2 Mbps. To overcome the impact of η with too large or small values on the objective of P0 and for fair comparison, only the system power cost is evaluated. First, it can be observed from Figure 3 that the joint V2V clustering and caching algorithm tends to saturate within 30 iterations. In spite of moderate convergence, the result is acceptable since it is operated at a large timescale and does not need to work per small timescale slot. Second, with the increase of η, the total power cost gradually reduces. This is because the objective of P0 is the minimization of system cost; a larger η will increase the weight of power cost, thus reducing the power proportion in the optimal value.Figure 3
The convergence of the proposed algorithm.Figure4 reveals the convergence of the SCA-based multicast beamforming algorithm. Likewise, the number of CCVs is 20, and the data rate requirement per group is 2 Mbps. We leverage the L2 norm on the difference of successive optimal solutions, i.e., wso+1t−wsot2 as the metric. As shown in Figure 4, all settings under different values of η have good convergence, saturating within 10 steps.Figure 4
The convergence of the SCA method.


### 5.3. Performance Comparison

Figure5 shows the power-flow cost tradeoff curve of the proposed algorithm under different values of η. Similar to the result in Figure 3, when η approaches infinity, the proportion of power cost is getting larger, and thus the power cost decreases in the minimization of the weighted sum. In particular, the flow cost reduces to 21 W while the flow cost reaches 34 Mbps, given η=106. On the contrary, when η approaches zeros, the proportion of power cost gets smaller, and more weights are imposed on the flow cost. In particular, the flow cost increases to 35 W while the flow cost reduces to 18 Mbps, given η=10−6.Figure 5
Flow-power cost tradeoff.Figure6 shows the impact of CCV number on the flow cost. In this scenario, the average data rate requirement is 2 Mbps, η=103, and the CRV number ranges from 15 to 30. As shown in Figure 6, the curve is broken line-shaped, since the flow cost must be the integer multiplier of the data rate requirement. The flow cost reduces with the increase of CCV number. This is because the wireless network incorporating more CCVs would introduce multiuser diversity gain and a more flexible clustering combination. In the unicast w.o. V2V caching scheme, the caching functions are only available in the BS, and each CCV provided a requested content individually, thus resulting in the largest flow cost. In addition, unicast w. V2V caching outperforms multicast w.o. V2V caching in terms of flow cost, indicating that the V2V caching gains overwhelm the multicasting gains in this setting, especially when the CCV number is relatively large.Figure 6
Flow cost versus the CCV number.Figure7 shows the impact of CCV number on the power cost, with identical settings as in Figure 6. Likewise, unicast w.o. V2V caching still gets the largest power cost compared to the other three schemes. Meanwhile, unicast w. V2V caching is also with worse performance compared to multicast w.o. V2V caching. This is because multicast w.o. V2V caching tends to save more transmit power than unicast w. V2V caching, while the latter one must allocate an individual beamformers to each CCV. The performance gap also indicates the multicast gains over unicast. Furthermore, the proposed algorithm gets close performance with multicast w.o. V2V caching, only with a slightly small gap, since in this setting, multicast gains overwhelm V2V caching gains.Figure 7
Power cost versus the CCV number.In Figure8, we compare the performance with different data rate requirements Rs, ranging from 2 Mbps to 3.8 Mbps. In this scenario, there are 20 CCVs, and η=103. Figure 8 shows the flow cost is almost proportional to the data rate requirement. In the meantime, there exists a significant gap between the proposed algorithm and the other three schemes (approximately 30 Mbps on average compared to multicast w.o. V2V caching), which can be explained as follows: on one hand, the unicast transmission would result in more flow costs, since each CRV is equipped with a CCV cluster and a content flow; on the other hand, the multicast w.o. V2V caching neglects the caching functions in CCVs, thus incurring the traffic congestion.Figure 8
Flow cost versus the average data rate requirement.Figure9 shows the impact of data rate requirement on power cost, with identical settings as in Figure 8. Different from the results in Figure 8, both the proposed algorithm and multicast w.o. V2V almost remain unchanged with the increase of data rate requirements. This is because in this setting, sufficient power is available to support a larger data rate, and thus multicast transmissions could save more power than unicast. Meanwhile, the performance gap between the proposed algorithm and multicast w.o. V2V caching reveals that, the lack of V2V caching would incur that all CRVs tend to access the BS, the power resource of CCVs is wasted, and thus a worse feasible solution is produced. In addition, unicast would allocate an individual beamformer to each CRV, thus getting the worst performance.Figure 9
Power cost versus the average data rate requirement.


## 5.1. Simulation Parameters

We investigate a time-slotted wireless network consisting of one BS and 20 CRVs, with a radius of 500 m for the cell coverage. The BS is equipped with 20 antennas, and each CRV and CCV is identically equipped with one antenna. Overall, 20 types of multimedia contents exist in the system. The BS has a storage size of 6 contents, while each CCV could accommodate at most 1 content. For simplicity, we do not employ the well-known Zipf distribution in [2]; instead, each CRV equally requests any content with a probability of 5%.The system bandwidth is 5 MHz, and the additive white Gaussian noise power spectral is -174 dBm/Hz. The transmit power budgets for the BS and CCV are 46 dBm and 24 dBm, respectively. The path loss model is35.3+37.6log2dm, the log-normal shadowing parameter is 8 dB, and the multipath channel model with Rayleigh fading is assumed [20]. Overall, 200 samples are produced to simulate stochastic channels vectors. Besides, the Monte Carlo approach is utilized, both CRVs and CCVs are uniformly distributed and dropped in the cell coverage, and all simulation results are averaged over 1000 droppings. In addition, the vehicle speed follows a truncated normal distribution ranging from 15 m/s and 31 m/s.


## 5.2. Convergence Performance

Figure3 illustrates the convergence performance of joint V2V clustering and caching algorithms. In this setup, the number of CCVs is 20, and the data rate requirement per group is 2 Mbps. To overcome the impact of η with too large or small values on the objective of P0 and for fair comparison, only the system power cost is evaluated. First, it can be observed from Figure 3 that the joint V2V clustering and caching algorithm tends to saturate within 30 iterations. In spite of moderate convergence, the result is acceptable since it is operated at a large timescale and does not need to work per small timescale slot. Second, with the increase of η, the total power cost gradually reduces. This is because the objective of P0 is the minimization of system cost; a larger η will increase the weight of power cost, thus reducing the power proportion in the optimal value.Figure 3
The convergence of the proposed algorithm.Figure4 reveals the convergence of the SCA-based multicast beamforming algorithm. Likewise, the number of CCVs is 20, and the data rate requirement per group is 2 Mbps. We leverage the L2 norm on the difference of successive optimal solutions, i.e., wso+1t−wsot2 as the metric. As shown in Figure 4, all settings under different values of η have good convergence, saturating within 10 steps.Figure 4
The convergence of the SCA method.


## 5.3. Performance Comparison

Figure5 shows the power-flow cost tradeoff curve of the proposed algorithm under different values of η. Similar to the result in Figure 3, when η approaches infinity, the proportion of power cost is getting larger, and thus the power cost decreases in the minimization of the weighted sum. In particular, the flow cost reduces to 21 W while the flow cost reaches 34 Mbps, given η=106. On the contrary, when η approaches zeros, the proportion of power cost gets smaller, and more weights are imposed on the flow cost. In particular, the flow cost increases to 35 W while the flow cost reduces to 18 Mbps, given η=10−6.Figure 5
Flow-power cost tradeoff.Figure6 shows the impact of CCV number on the flow cost. In this scenario, the average data rate requirement is 2 Mbps, η=103, and the CRV number ranges from 15 to 30. As shown in Figure 6, the curve is broken line-shaped, since the flow cost must be the integer multiplier of the data rate requirement. The flow cost reduces with the increase of CCV number. This is because the wireless network incorporating more CCVs would introduce multiuser diversity gain and a more flexible clustering combination. In the unicast w.o. V2V caching scheme, the caching functions are only available in the BS, and each CCV provided a requested content individually, thus resulting in the largest flow cost. In addition, unicast w. V2V caching outperforms multicast w.o. V2V caching in terms of flow cost, indicating that the V2V caching gains overwhelm the multicasting gains in this setting, especially when the CCV number is relatively large.Figure 6
Flow cost versus the CCV number.Figure7 shows the impact of CCV number on the power cost, with identical settings as in Figure 6. Likewise, unicast w.o. V2V caching still gets the largest power cost compared to the other three schemes. Meanwhile, unicast w. V2V caching is also with worse performance compared to multicast w.o. V2V caching. This is because multicast w.o. V2V caching tends to save more transmit power than unicast w. V2V caching, while the latter one must allocate an individual beamformers to each CCV. The performance gap also indicates the multicast gains over unicast. Furthermore, the proposed algorithm gets close performance with multicast w.o. V2V caching, only with a slightly small gap, since in this setting, multicast gains overwhelm V2V caching gains.Figure 7
Power cost versus the CCV number.In Figure8, we compare the performance with different data rate requirements Rs, ranging from 2 Mbps to 3.8 Mbps. In this scenario, there are 20 CCVs, and η=103. Figure 8 shows the flow cost is almost proportional to the data rate requirement. In the meantime, there exists a significant gap between the proposed algorithm and the other three schemes (approximately 30 Mbps on average compared to multicast w.o. V2V caching), which can be explained as follows: on one hand, the unicast transmission would result in more flow costs, since each CRV is equipped with a CCV cluster and a content flow; on the other hand, the multicast w.o. V2V caching neglects the caching functions in CCVs, thus incurring the traffic congestion.Figure 8
Flow cost versus the average data rate requirement.Figure9 shows the impact of data rate requirement on power cost, with identical settings as in Figure 8. Different from the results in Figure 8, both the proposed algorithm and multicast w.o. V2V almost remain unchanged with the increase of data rate requirements. This is because in this setting, sufficient power is available to support a larger data rate, and thus multicast transmissions could save more power than unicast. Meanwhile, the performance gap between the proposed algorithm and multicast w.o. V2V caching reveals that, the lack of V2V caching would incur that all CRVs tend to access the BS, the power resource of CCVs is wasted, and thus a worse feasible solution is produced. In addition, unicast would allocate an individual beamformer to each CRV, thus getting the worst performance.Figure 9
Power cost versus the average data rate requirement.


## 6. Conclusions

In this study, wireless content caching along with multicast beamforming was studied in vehicular edge networks. First, a two-timescale setup was proposed to decouple the joint clustering, caching, and multicast beamforming problem at separate timescales. Next, at the large timescale, with the SAA technique, a theorem was verified to reveal the equivalence of optimal solutions between the original ILP problem and its relaxed LP counterpart. Then, with the SCA method, the multicast beamforming-based power minimization problem was solved per slot. Simulation results revealed that the network performance could benefit from not only the power saving from wireless multicast beamforming but also the content caching and sharing among vehicles.



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*Source: 8837751-2020-11-19.xml*