# Success Probability Analysis of C-V2X Communications on Irregular Manhattan Grids

Authors: Bin Pan; Hao Wu
Journal: Wireless Communications and Mobile Computing
 (2020)

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## Abstract

To overcome the shortcomings of Dedicated Short Range Communications (DSRC), cellular vehicle-to-everything (C-V2X) communications have been proposed recently, which has a variety of advantages over traditional DSRC, including longer communication range, broader coverage, greater reliability, and smooth evolution path towards 5G. In this paper, we consider an LTE-based C-V2X communications network in irregular Manhattan grids. We model the macrobase stations (MBSs) as a 2D Poisson point process (PPP) and model the roads as a Manhattan Poisson line process (MPLP), with the roadside units (RSUs) modeled as a 1D PPP on each road. As an enhancement architecture to DSRC, C-V2X communications include vehicle-to-vehicle (V2V) communication, vehicle-to-infrastructure (V2I) communication, vehicle-to-pedestrian (V2P) communication, and vehicle-to-network (V2N) communication. Since the spectrum for PC5 interface in 5.9 GHz is quite limited, cellular networks could share some channels to V2I links to improve spectral efficiency. Thus, according to Maximum Power-based Scheme, we adopt the stochastic geometry approach to compute the signal-to-interference ratio- (SIR-) based success probability of a typical vehicle that connects to an RSU or an MBS and the area spectral efficiency of the whole network over shared V2I and V2N downlink channels. In addition, we study the asymptotic characteristics of success probability and provide some design insights according to the impact of several key parameters on success probability.


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## Body

## 1. Introduction

Vehicular communication networks are significant components of intelligent transportation system (ITS), which can provide many benefits, such as enhancing road safety, reducing traffic jam, and providing entertainment services. An important solution of this evolution is cellular vehicle-to-everything (C-V2X) communications, which include vehicle-to-vehicle (V2V) communication, vehicle-to-infrastructure (V2I) communication, vehicle-to-pedestrian (V2P) communication, and vehicle-to-network (V2N) communication. V2V communication can promote information sharing among vehicles without network assistance. V2I and V2N communications can enable vehicles to connect with the core network, which offer many services from basic safety messages and entertainment applications to automatic driving. V2P communication enhances the relationship between running vehicles and walking pedestrians. C-V2X enabled by long-term evolution (LTE) communication has been defined by the third generation partnership project (3GPP) as part of Release 14 [1].Existing vehicular communications are limited by short range, connection interruption, channel congestion, and nonsupporting high-density communication, which is supported by Dedicated Short Range Communications (DSRC) technology [2]. In DSRC-based system, infrastructures consist of roadside units (RSUs), which are sparsely distributed especially in remote areas due to the high cost of deployment. What’s more, single DSRC communication is unable to support everincreasing novel on-board applications like automatic driving. Thus, LTE-based C-V2X communication was proposed to overcome the shortcomings of DSRC, which has a variety of advantages over traditional DSRC, including longer communication range, broader coverage, greater reliability, and smooth evolution path towards 5G.The dissemination of messages in C-V2X system has two kinds of patterns, namely, direct mode and infrastructure mode. In direct mode, vehicles can connect to other vehicles, RSUs, and pedestrians directly over PC5 interface, which are referred to V2V/V2I/V2P links, respectively. As for infrastructure mode, vehicles communicate with cellular networks over Uu interface, which is for V2N link. As there may be large number of vehicles at urban roads and the spectrum for PC5 interface is quite limited, cellular networks can share some channels to V2I links to improve spectral efficiency. In order to analyze the communication quality in C-V2X network, we adopt stochastic geometry approach, which is a powerful tool that has been applied to wireless networks to analyze the mutual interference.


### 1.1. Related Works

The initial studies using stochastic geometry on vehicular networks are limited to a single road or an intersection [3–7]. In [3], the authors proposed two models based on signal-to-interference plus noise ratio (SINR) and Shannon’s law to calculate the optimum transmission probability and transmission range of vehicles on a single road. In [4], the authors proposed a tractable model to study the performance of broadcast protocols with vehicles’ locations modeled as a Poisson point process (PPP) in a single road. In [5], the authors proposed a tractable model to study the multihop transmissions for intervehicle communication in a multilane highway scenario, according to stochastic geometry approach. In [6], the authors proposed a tractable and accurate model to calculate the packet reception probability of a specific link in the intersection scenario of vehicular networks. In [7], the authors also presented a model to evaluate the reliability of packet transmissions near intersections of vehicular networks. They provided the closed form expression of the packet reception probability and can be extended to multilane cases.There is also a little literature focusing on complex road spatial distribution recently. In [8], the authors derived the coverage probability of a typical receiver in vehicular network, which roadways are modeled by Poisson line process (PLP) and the vehicles are modeled by 1D PPP. In [9], the authors proposed a tractable framework of performance analysis of CV2X network over shared cellular and V2V uplink channels and obtained the closed-form expressions for success probability in direct mode and network mode. In [10], the authors analyzed the characteristics of Poisson line Cox point process, which is helpful for network modeling in Internet of Vehicles (IoV). They derived some fundamental properties of the point process, which is useful to quantify the performance of novel vehicular architectures. In particular, a number of researchers pay attention to the orthogonal street system, which is the most common urban road topologies in practice. In [11], the authors focused on the reliability analysis of V2V communications in the regular street system, in which the horizontal and vertical streets have the same fixed intervals. In [12], the authors presented a model for success probability in an urban environment, where the horizontal streets have the same fixed intervals which are different from the intervals of the vertical streets. As far as we know, this paper is the first to calculate the success probability for Manhattan C-V2X network where the intervals of horizontal and vertical streets follow the Poisson distribution. More details of our contributions are followed below.


### 1.2. Contributions

In this paper, we propose a framework for success analysis of an irregular Manhattan C-V2X network. We model the spatial layout of MBSs as a 2D PPP and model the spatial layout of roads as a Manhattan PLP (MPLP), with the RSUs modeled as a 1D PPP on each road. We choose the general power-law path-loss model and Rayleigh fading to describe the channel effects. For an arbitrarily chosen typical vehicle, we compute the signal-to-interference ratio- (SIR-) based success probability, according to Maximum Power-based Scheme. We also provide useful design insights by studying the impact of key parameters on the success probability. More technical details are as follows.


#### 1.2.1. Success Probability

We derive an accurate expression for SIR-based success probability of a typical vehicle in an irregular Manhattan C-V2X network. We first calculate null probability and fundamental distance distributions, which are useful to calculate the success probability. Next, we derive the Laplace transform of interference from three independent sources. Then, we calculate the association probability and success probability of a typical vehicle according to Maximum Power-based Scheme. We also give the expression for area spectral efficiency and asymptotic characteristics of success probability.


#### 1.2.2. Design Insights

According to our theoretical analysis, we explore the impact of three key network parameters on the success probability. It is observed that the success probability increases with the increase of RSU intensity. However, the success probability decreases with the increase of MBS intensity and road intensity. Therefore, in terms of success probability, we should reduce the use of MBSs while other application requirements for MBSs are met and, meanwhile, increase the deployment intensity of RSUs. As for area spectral efficiency, we also require dense deployment of RSUs to improve it.


## 1.1. Related Works

The initial studies using stochastic geometry on vehicular networks are limited to a single road or an intersection [3–7]. In [3], the authors proposed two models based on signal-to-interference plus noise ratio (SINR) and Shannon’s law to calculate the optimum transmission probability and transmission range of vehicles on a single road. In [4], the authors proposed a tractable model to study the performance of broadcast protocols with vehicles’ locations modeled as a Poisson point process (PPP) in a single road. In [5], the authors proposed a tractable model to study the multihop transmissions for intervehicle communication in a multilane highway scenario, according to stochastic geometry approach. In [6], the authors proposed a tractable and accurate model to calculate the packet reception probability of a specific link in the intersection scenario of vehicular networks. In [7], the authors also presented a model to evaluate the reliability of packet transmissions near intersections of vehicular networks. They provided the closed form expression of the packet reception probability and can be extended to multilane cases.There is also a little literature focusing on complex road spatial distribution recently. In [8], the authors derived the coverage probability of a typical receiver in vehicular network, which roadways are modeled by Poisson line process (PLP) and the vehicles are modeled by 1D PPP. In [9], the authors proposed a tractable framework of performance analysis of CV2X network over shared cellular and V2V uplink channels and obtained the closed-form expressions for success probability in direct mode and network mode. In [10], the authors analyzed the characteristics of Poisson line Cox point process, which is helpful for network modeling in Internet of Vehicles (IoV). They derived some fundamental properties of the point process, which is useful to quantify the performance of novel vehicular architectures. In particular, a number of researchers pay attention to the orthogonal street system, which is the most common urban road topologies in practice. In [11], the authors focused on the reliability analysis of V2V communications in the regular street system, in which the horizontal and vertical streets have the same fixed intervals. In [12], the authors presented a model for success probability in an urban environment, where the horizontal streets have the same fixed intervals which are different from the intervals of the vertical streets. As far as we know, this paper is the first to calculate the success probability for Manhattan C-V2X network where the intervals of horizontal and vertical streets follow the Poisson distribution. More details of our contributions are followed below.


## 1.2. Contributions

In this paper, we propose a framework for success analysis of an irregular Manhattan C-V2X network. We model the spatial layout of MBSs as a 2D PPP and model the spatial layout of roads as a Manhattan PLP (MPLP), with the RSUs modeled as a 1D PPP on each road. We choose the general power-law path-loss model and Rayleigh fading to describe the channel effects. For an arbitrarily chosen typical vehicle, we compute the signal-to-interference ratio- (SIR-) based success probability, according to Maximum Power-based Scheme. We also provide useful design insights by studying the impact of key parameters on the success probability. More technical details are as follows.


### 1.2.1. Success Probability

We derive an accurate expression for SIR-based success probability of a typical vehicle in an irregular Manhattan C-V2X network. We first calculate null probability and fundamental distance distributions, which are useful to calculate the success probability. Next, we derive the Laplace transform of interference from three independent sources. Then, we calculate the association probability and success probability of a typical vehicle according to Maximum Power-based Scheme. We also give the expression for area spectral efficiency and asymptotic characteristics of success probability.


### 1.2.2. Design Insights

According to our theoretical analysis, we explore the impact of three key network parameters on the success probability. It is observed that the success probability increases with the increase of RSU intensity. However, the success probability decreases with the increase of MBS intensity and road intensity. Therefore, in terms of success probability, we should reduce the use of MBSs while other application requirements for MBSs are met and, meanwhile, increase the deployment intensity of RSUs. As for area spectral efficiency, we also require dense deployment of RSUs to improve it.


## 1.2.1. Success Probability

We derive an accurate expression for SIR-based success probability of a typical vehicle in an irregular Manhattan C-V2X network. We first calculate null probability and fundamental distance distributions, which are useful to calculate the success probability. Next, we derive the Laplace transform of interference from three independent sources. Then, we calculate the association probability and success probability of a typical vehicle according to Maximum Power-based Scheme. We also give the expression for area spectral efficiency and asymptotic characteristics of success probability.


## 1.2.2. Design Insights

According to our theoretical analysis, we explore the impact of three key network parameters on the success probability. It is observed that the success probability increases with the increase of RSU intensity. However, the success probability decreases with the increase of MBS intensity and road intensity. Therefore, in terms of success probability, we should reduce the use of MBSs while other application requirements for MBSs are met and, meanwhile, increase the deployment intensity of RSUs. As for area spectral efficiency, we also require dense deployment of RSUs to improve it.


## 2. System Model

### 2.1. MPLP

We give a brief introduction to MPLP first. As shown in Figure1, MPLP consists of two types of undirected lines which are orthogonal, namely, horizontal lines and vertical lines. Thus, the spatial layout of these lines in R2 displays the grid-like structure. Instead of the same fixed values, the intervals between adjacent horizontal or vertical lines are generated by a 1D PPP, respectively. Thereby, to construct a MPLP in R2, we can first fill points on the x- and y-axes according to two independent 1D PPPs Ξx and Ξy and then draw vertical and horizontal lines on the basis of these points. According to [13], a MPLP Φl is stationary if the 1D PPPs Ξx and Ξy are stationary, which means the distribution of lines is invariant to any translation Tt,β.Figure 1
Illustration of MPLP in 2D planeR2.


### 2.2. Spatial Model

As shown in Figure2, the cellular macrobase stations (MBSs) are randomly distributed in R2 according to a 2D PPP with intensity λc [14, 15]. We model the road network as a stationary MPLP Φl≡Lh1,Lh2,⋯,Lv1,Lv2⋯ in R2. We denote the set of horizontal and vertical lines by Φlh=Lh1,Lh2,⋯ and Φlv=Lv1,Lv2,⋯, which are generated by two independent homogeneous 1D PPPs Ξy and Ξx with the same intensity λl for conciseness of the final expressions. The locations of vehicles are randomly distributed on each road as 1D PPP with intensity λv. Since RSUs are deployed along the roadside, the locations of RSUs are also randomly distributed on each road as 1D PPP with the same intensity λu. Therefore, driven by the same MPLP Φl, the locations of vehicular nodes and RSU nodes can form two Cox processes Φv and Φu, which are stationary [10, 16]. We name the arbitrary chosen vehicle node in Φv as typical vehicle and the road where the typical vehicle located as typical road. There are altogether two types of typical vehicles: (i) typical general vehicle located at arbitrary position in the road and (ii) typical intersection vehicle located at the intersection of a horizontal and vertical road.Figure 2
System model.Owing to the stationary ofΦv, we pay attention to the analysis of a typical vehicle at the origin O. Thus, without loss of generality, we take the assumption that the typical general vehicle is located on a horizontal road. Therefore, according to Slivnyak’s theorem [17, 18], the translated line process is Φl0,gen=Φl∪Lx. And the resulting point process Φv0,gen is interpreted as the superposition of point process Φv, an 1D PPP with intensity λv on the x-axis Lx and a point at the origin O. Similarly, as the point process Φu is also driven by the same MPLP Φl, the resulting point process Φu0,gen is the superposition of the point process Φu and a 1D PPP with intensity λu on Lx.In the special case, typical intersection vehicle is located at the intersection of a horizontal lineLx and a vertical line Ly, which correspond to x-axis and y-axis, respectively. Using the same method, under this conditioning, the resulting line process is Φl0,int=Φl∪Lx,Ly. Therefore, the resulting point process Φv0,int is interpreted as the superposition of point process Φv, two independent 1D PPPs each with intensity λv on Lx and Ly, and a point at the origin o. In the same way, the resulting point process Φu0,int can be translated as the superposition of Φu and two independent 1D PPPs each with intensity λu along Lx and Ly, respectively. For these two types of typical vehicles, we can apply the similar approach to calculate their success probabilities. Thus, we only give the details for typical general vehicle. In the rest of this paper, typical vehicle refers to typical general vehicle for simplicity of exposition.The number of horizontal and vertical lines that intersect a regionA⊂R2 is denoted by NlhA and NlvA, respectively. And the number of points in set A is denoted by NpA.


### 2.3. Transmitter Association Scheme

The vehicles in the network select to connect to either an MBS or an RSU according to Maximum Power-based Scheme. In this scheme, the vehicle connects to the transmitter (an MBS or an RSU) from which it receives the highest average power.The vehicle chooses the closest RSU and MBS first and then selects to receive data from the one that can supply a higher time-average power.To find the received power, we should introduce the channel model next. We use a general power-law path-loss model with exponentα>2 to describe the long time-scale channel effects. To describe the short time-scale channel effects, we use the Rayleigh fading in which the channel gain is exponentially distributed with unit mean. Thus, the instantaneous power received can be expressed by Prec=Phr−α, where P is the transmit power, h∼exp1 models Rayleigh fading, and r is the distance between transmitter and receiver. We use time-average value of received power as the metric to choose transmitter, because it is undesirable to switch from one transmitter to another frequently due to the rapidly varying instantaneous power. If the average time interval is much larger than the coherence time of channel, the average received power is considered to be independent of h, which can be given by Pav=Pr−α. We assume all MBSs have the same transmit power Pc, and all RSUs have the same transmit power Pu.


## 2.1. MPLP

We give a brief introduction to MPLP first. As shown in Figure1, MPLP consists of two types of undirected lines which are orthogonal, namely, horizontal lines and vertical lines. Thus, the spatial layout of these lines in R2 displays the grid-like structure. Instead of the same fixed values, the intervals between adjacent horizontal or vertical lines are generated by a 1D PPP, respectively. Thereby, to construct a MPLP in R2, we can first fill points on the x- and y-axes according to two independent 1D PPPs Ξx and Ξy and then draw vertical and horizontal lines on the basis of these points. According to [13], a MPLP Φl is stationary if the 1D PPPs Ξx and Ξy are stationary, which means the distribution of lines is invariant to any translation Tt,β.Figure 1
Illustration of MPLP in 2D planeR2.


## 2.2. Spatial Model

As shown in Figure2, the cellular macrobase stations (MBSs) are randomly distributed in R2 according to a 2D PPP with intensity λc [14, 15]. We model the road network as a stationary MPLP Φl≡Lh1,Lh2,⋯,Lv1,Lv2⋯ in R2. We denote the set of horizontal and vertical lines by Φlh=Lh1,Lh2,⋯ and Φlv=Lv1,Lv2,⋯, which are generated by two independent homogeneous 1D PPPs Ξy and Ξx with the same intensity λl for conciseness of the final expressions. The locations of vehicles are randomly distributed on each road as 1D PPP with intensity λv. Since RSUs are deployed along the roadside, the locations of RSUs are also randomly distributed on each road as 1D PPP with the same intensity λu. Therefore, driven by the same MPLP Φl, the locations of vehicular nodes and RSU nodes can form two Cox processes Φv and Φu, which are stationary [10, 16]. We name the arbitrary chosen vehicle node in Φv as typical vehicle and the road where the typical vehicle located as typical road. There are altogether two types of typical vehicles: (i) typical general vehicle located at arbitrary position in the road and (ii) typical intersection vehicle located at the intersection of a horizontal and vertical road.Figure 2
System model.Owing to the stationary ofΦv, we pay attention to the analysis of a typical vehicle at the origin O. Thus, without loss of generality, we take the assumption that the typical general vehicle is located on a horizontal road. Therefore, according to Slivnyak’s theorem [17, 18], the translated line process is Φl0,gen=Φl∪Lx. And the resulting point process Φv0,gen is interpreted as the superposition of point process Φv, an 1D PPP with intensity λv on the x-axis Lx and a point at the origin O. Similarly, as the point process Φu is also driven by the same MPLP Φl, the resulting point process Φu0,gen is the superposition of the point process Φu and a 1D PPP with intensity λu on Lx.In the special case, typical intersection vehicle is located at the intersection of a horizontal lineLx and a vertical line Ly, which correspond to x-axis and y-axis, respectively. Using the same method, under this conditioning, the resulting line process is Φl0,int=Φl∪Lx,Ly. Therefore, the resulting point process Φv0,int is interpreted as the superposition of point process Φv, two independent 1D PPPs each with intensity λv on Lx and Ly, and a point at the origin o. In the same way, the resulting point process Φu0,int can be translated as the superposition of Φu and two independent 1D PPPs each with intensity λu along Lx and Ly, respectively. For these two types of typical vehicles, we can apply the similar approach to calculate their success probabilities. Thus, we only give the details for typical general vehicle. In the rest of this paper, typical vehicle refers to typical general vehicle for simplicity of exposition.The number of horizontal and vertical lines that intersect a regionA⊂R2 is denoted by NlhA and NlvA, respectively. And the number of points in set A is denoted by NpA.


## 2.3. Transmitter Association Scheme

The vehicles in the network select to connect to either an MBS or an RSU according to Maximum Power-based Scheme. In this scheme, the vehicle connects to the transmitter (an MBS or an RSU) from which it receives the highest average power.The vehicle chooses the closest RSU and MBS first and then selects to receive data from the one that can supply a higher time-average power.To find the received power, we should introduce the channel model next. We use a general power-law path-loss model with exponentα>2 to describe the long time-scale channel effects. To describe the short time-scale channel effects, we use the Rayleigh fading in which the channel gain is exponentially distributed with unit mean. Thus, the instantaneous power received can be expressed by Prec=Phr−α, where P is the transmit power, h∼exp1 models Rayleigh fading, and r is the distance between transmitter and receiver. We use time-average value of received power as the metric to choose transmitter, because it is undesirable to switch from one transmitter to another frequently due to the rapidly varying instantaneous power. If the average time interval is much larger than the coherence time of channel, the average received power is considered to be independent of h, which can be given by Pav=Pr−α. We assume all MBSs have the same transmit power Pc, and all RSUs have the same transmit power Pu.


## 3. Success Probability

### 3.1. Null Probability

Lemma 1.
Null probability of RSU: for a MPLPΦl, which are generated by two independent homogeneous 1D PPPs with the same intensity λl and RSUs lie on each road followed by 1D PPP with intensity λu, the probability that there is no RSUs inside a disk bo,R is(1)ℙNubo,R=0=exp−4λl∫0R1−e−2λuR2−r2dr.Proof.
The null probability can be calculated as(2)ℙNubo,R=0=aℙNuΦlh∩bo,R=0×ℙNuΦlv∩bo,R=0=b∑n=0∞ℙNlhbo,R=n×∏i=1nℙNuLhi∩bo,R=0×∑n=0∞ℙNlvbo,R=n×∏i=1nℙNuLvi∩bo,R=0=∑n=0∞e−2λlR2λlRnn!∫−RRe−2λuR2−r212Rdrn2=e−2λlR∑n=0∞λl∫−RRe−2λuR2−r2drnn!2=e−2λlRexp2λl∫0Re−2λuR2−r2dr2=exp−4λl∫0R1−e−2λuR2−r2dr,where (a) follows from that there is no RSU on both horizontal and vertical roads and the distribution of horizontal and vertical roads are independent, and (b) follows from conditioning over the number of horizontal and vertical roads inside the disk.


### 3.2. Distance Distribution

We derive the distance distributions to the closest RSU and MBS, respectively, which will be used in calculating the success probability.Lemma 2.
The cumulative distribution function (CDF) and probability distribution function (PDF) of the distancedu between the typical vehicle and its closest RSU are CDF:(3)CDF:Fduρ=1−exp−4λl∫0ρ1−e−2λuρ2−r2dre−2λuρ,(4)PDF:fduρ=exp−4λl∫0ρ1−e−2λuρ2−r2dr×2λue−2λuρ+8λuλlρe−2λuρ∫0ρe−2λuρ2−r2ρ2−r2dr.Proof.
The CDF ofdu can be computed as Fduρ=ℙdu≤ρ=1−ℙdu>ρ(5)Fduρ=ℙdu≤ρ=1−ℙdu>ρ=ℙdu≤ρ=1−ℙdu>ρ=a1−ℙNubo,ρ=0ℙNuLx∩bo,ρ=0=1−exp−4λl∫0ρ1−e−2λuρ2−r2dre−2λuρ,where (a) follows from that the event du>ρ occurs when both of the following independent events occur simultaneously: (i) there is no RSU on Φl inside the disk bo,ρ and (ii) there is no RSU on Lx inside the disk bo,ρ. The PDF is derived by differentiating the CDF as follows:(6)fduρ=ddρ1−exp−4λl∫0ρ1−e−2λuρ2−r2dre−2λuρ=−ddρexp−4λl∫0ρ1−e−2λuρ2−r2dre−2λuρ=aexp−4λl∫0ρ1−e−2λuρ2−r2dr×2λue−2λuρ+8λuλlρe−2λuρ∫0ρe−2λuρ2−r2ρ2−r2dr,where (a) follows from the Leibniz rule and the differentiation of a definite integral with respect to a parameter.Lemma 3.
The cumulative distribution function (CDF) and probability distribution function (PDF) of the distancedc between the typical vehicle and its closest MBS are(7)CDF:Fdcρ=1−e−πλcρ2,(8)PDF:fdcρ=2πλcρe−πλcρ2.Proof.
The CDF ofdc can be computed as(9)Fdcρ=ℙdc≤ρ=1−ℙdc>ρ=1−ℙNcbo,ρ=0=1−e−πλcρ2.
The PDF can be obtained by differentiating the CDF.


### 3.3. Laplace Transform of Interference

We first give the expression of Laplace transform of interference under Rayleigh fading, since the success probability can be reduced to the Laplace transform of interferenceI, which is a well-known approach [14, 18]. For a wireless network with interfering nodes generated by a PPP Φ, the Laplace transform can be calculated as(10)LIs=Eexp−∑x∈ΦsPhxx−α=E∏x∈ΦEhxexp−sPhxx−α=E∏x∈Φ11+sPx−α,where P is the transmit power of each node and hx∼exp1.Note that the interference at a typical vehicle are from three independent sources: (i) RSU nodes on the other roads (denoted byΦu), (ii) RSU nodes on typical road Lx (denoted by Φux), and (iii) MBS nodes (denoted by Φc). Thus, the interference from these three sources can be denoted by Iu, Iux, and Ic, respectively. We will calculate the Laplace transform of these interferences below.Since the typical vehicle always connects to the closest RSU or MBS, we define some distancesru, rux, and rc, which are the minimum distance between typical vehicle and the closest RSU on the other roads, the closest RSU on typical road Lx, and the closest MBS.Lemma 4.
The Laplace transform of interference distribution from the RSUs on a single road (which is ata distance r) is(11)LIus,ru,r=g1r=exp−2λu∫0+∞1−11+sPur2+t2−α/2dt,r≥ru,g2r=exp−2λu∫ru2−r2+∞1−11+sPur2+t2−α/2dt,r<ru.Proof.
As shown in Figure3, there are two roads inside a disk. One is at a distance r>ru (denoted by D1), which means the interfering RSUs can be located anywhere on D1. The other one is at a distance r<ru (denoted by D2), which means the interfering RSUs can be only located in the segment t=−ru2−r2,−R2−ru2 and t=ru2−r2,R2−ru2. Next, we will analyze these two roads, respectively.
The Laplace transform of interference distribution fromD1 is derived by conditioning on the number of points on this line. Given a line at a perpendicular distance r from the center o, the length of the chord inside a disk (with radius R) can be expressed by 2R2−r2. Thus, the probability of that there are m points on this chord is e−2λuR2−r22λuR2−r2m/m!, according to the properties of PPP. Therefore, the Laplace transform of total interference in the case of r≥rv can be calculated by(12)g1s,ru,r=E∏x∈Dr11+sPux−α=∑m=0∞ℙN=m∫−R2−r2R2−r211+sPur2+t2−α/2ftdtm=∑m=0∞e−2λuR2−r22λuR2−r2mm!×∫−R2−r2R2−r211+sPur2+t2−α/212R2−r2dtm=e−2λuR2−r2exp2λu∫0R2−r211+sPur2+t2−α/2dt=exp−2λu∫0R2−r21−11+sPur2+t2−α/2dt,where Dr refers to a road at a perpendicular distance r from the center o. Similarly, the Laplace transform of total interference in the case of r<rv can be calculated by(13)g2s,ru,r=exp−2λu∫ru2−r2R2−r21−11+sPur2+t2−α/2dt.
The final expressions can be obtained as the radius approaches positive infinity (R⟶+∞) in (12) and (13).Lemma 5.
The Laplace transform of total interference from all other roads can be calculated as follows:(14)LIus,ru=exp−4λl∫0ru1−g2rdr+∫ru+∞1−g1rdr.Proof.
The Laplace transform of total interference from all other roads can be calculated as(15)LIus,ru=Ee−sIu=Ee−sE∑x∈ΦuIx=E∏x∈Φue−sEIx=aE∏x∈Φuhe−sEIx×E∏x∈Φuve−sEIx=∑j=0∞e−2ruλl2ruλljj!∫−rurug2r2rudrj×∑k=0∞e−2R−ruλl2R−ruλlkk!∫ruRg1rR−rudrk2=exp−4λl∫0ru1−g2rdr+∫ruR1−g1rdr,where (a) refers to the fact that the total interference are from the RSUs on horizontal and vertical roads, which are independent and identically distributed (i.i.d.). The final expression can be obtained as the radius approaches positive infinity (R⟶+∞) in (15).Lemma 6.
The Laplace transform of total interference from the typical roadLx can be calculated as follows:(16)LIuxs,rux=exp−2λu∫rux+∞1−11+sPut−αdt.Proof.
We have obtained the Laplace transform of interference from the RSUs on a single node in Lemma4. Therefore, the Laplace transform of interference from all the RSUs on Lx can be derived by substituting r=0 and ru=rux, respectively, in g2r.Figure 3
Distribution of interference RSUs on roads.The expression of Laplace transformLIc from the interference MBSs with a 2D PPP can be derived by [19]:(17)LIcs,rc=exp−2πλc∫rc+∞1−11+sPcr−αrdr.The total interference at origino is the sum of three types of interference sources, i.e.,(18)I=Iu+Iux+Ic.As these three types of interference sources are independent with each other, the Laplace transform of the total interference can be given by:(19)LIs,ru,rux,rc=LIus,ruLIuxs,ruxLIcs,rc.


### 3.4. Association Probability

In Maximum Power-based Scheme, the typical vehicle aims to connect to the transmitter from which it could receive the strongest signal (i.e., average received power). In other words, the typical vehicle chooses to receive signal from the closest RSU or the closest MBS, which can supply a higher time-average power.To derive the overall success probability, we first should calculate the probability of the typical vehicle connecting to RSU and MBS, respectively.Lemma 7.
The probability of an RSU being the transmitter at a distance ofρ from the typical vehicle can be calculated by(20)ℙuρ=exp−λcπduc2,where duc=Pc/Pu1/αρ.Proof.
A typical vehicle connects to the closest RSU when the average received power from it is larger than that from the closest MBS. Thus, given the distance between the vehicle and its closest RSU and MBS (du and dc), the probability of this event occurring can be expressed by(21)ℙPudu−α>Pcdc−α=ℙdc>PcPu1/αdu.
Letduc=Pc/Pu1/αdu for simplicity of notation. Therefore, the typical vehicle at origin o connects to the closest RSU when there is no MBS existing in the disk bo,duc, which happens with the probability(22)ℙudu=exp−λcπduc2.
As the typical vehicle connects to the closest RSU, substitutedu=ρ. Thus, we can obtain the result.Lemma 8.
The probability of an MBS being the transmitter at a distance ofρ from the typical vehicle can be calculated by(23)ℙcρ=exp−4λl∫0dcu1−e−2λudcu2−r2dre−2λudcu,where dcu=Pu/Pc1/αρ.Proof.
Following the same method as in Lemma7, the probability of this event occurring is(24)ℙPcdc−α>Pudu−α=ℙdu>PuPc1/αdc.
Letdcu=Pu/Pc1/αdc for simplicity of notation. The typical vehicle at origin o connects to the closest MBS when there is no RSU existing on Φl and Lx in the disk bo,dcu, which happens with the probability(25)ℙcdc=ℙNubo,dcu=0ℙNuLx∩bo,dcu=0=exp−4λl∫0dcu1−e−2λudcu2−r2dre−2λudcu.
As the typical vehicle connects to the closest MBS, substitutedc=ρ. Thus, we can obtain the result.Thus, given the association probabilities conditioned on the value ofρ in Lemmas 7 and 8, the overall probabilities of associating with the closest RSU and closest MBS can be calculated by(26)ℙu=∫0+∞ℙuρfduρdρ,(27)ℙc=∫0+∞ℙcρfdcρdρ.


### 3.5. Success Probability

In this paper, we use SIR as the metric to judge whether to receive message successfully. If the SIR measured at typical vehicle is larger than a predefined thresholdβ, the message can be detected successfully. Thus, the success probability in a Rayleigh fading network can be derived by(28)ℙsuc=ℙSIR>β=ℙPhρ−αI>β=ℙh>βραIP=Eexp−βραIP=LIβραP.Therefore, the success probability can be expressed by the Laplace transform of interference (LIs, where s=βρα/P).When the typical vehicle connects to the closest RSU at a distanceρ, there is no interfering RSU on Φl in the disk bo,ρ and on typical road Lx in segment −ρ,ρ, and no MBS in the disk bo,duc, i.e., ru=ρ,rux=ρ,rc=duc.Lemma 9.
The success probability that the typical vehicle connects to the closest RSU, which is at a distance ofρ, can be expressed by:(29)ℙsuc∣uρ=LIu1Puβρα,ρLIux1Puβρα,ρ×LIc1Puβρα,duc.
When the typical vehicle connects to the closest MBS at a distanceρ, there is no interfering RSU on Φl in the disk bo,dcu and on typical road Lx in segment −dcu,dcu, and no MBS in the disk bo,ρ, i.e., ru=dcu,rux=dcu,rc=ρ.Lemma 10.
The success probability that the typical vehicle connects to the closest MBS, which is at a distance ofρ, can be expressed by(30)ℙsuc∣cρ=LIu1Pcβρα,dcuLIux1Pcβρα,dcu×LIc1Pcβρα,ρ.
Note that the expressions forLIu, LIux, and LIc can be obtained from the results in Equations (14), (16), and (17).Theorem 11.
The total success probability at a typical vehicle in C-V2X networks is(31)ℙsuc=∫0+∞ℙuρℙsuc∣uρfduρdρ+∫0+∞ℙcρℙsuc∣cρfdcρdρ,where the first term and second term can be named as Psuc,u and Psuc,c, which mean the probabilities of the typical vehicle connecting to an RSU and MBS successfully, respectively.Proof.
The total success probability can be calculated as(32)ℙsuc=Eρℙuρℙsuc∣uρ+ℙcρℙsuc∣cρ=a∫0+∞ℙuρℙsuc∣uρfduρdρ+∫0+∞ℙcρℙsuc∣cρfdcρdρ,where (a) follows from the fact that success probability for RSU or MBS can be derived by taking an expectation over ρ.


### 3.6. Area Spectral Efficiency

Area spectral efficiency can be defined by the average bits transmitted per unit time per unit bandwidth per unit area, which can be calculated by(33)ASE=λalog1+βℙsuc,u+λclog1+βℙsuc,c,where λa is the intensity of vehicle nodes in R2 given by λa=μlλv, and μl is the line density which can be calculated by μl=2λl for a stationary MPLP [13].


### 3.7. Asymptotic Characteristic

Lemma 12.
As the MBS intensity approaches zeroλc⟶0, the association probability with the closest ℙuλc→0 is 1, the association probability with the closest MBS ℙcλc→0 is 0, and the success probability is(34)ℙsucλc→0=∫0+∞LIu1Puβρα,ρLIux1Puβρα,ρfduρdρ.Proof.
By applying the limitλc⟶0 on Pu, we can obtain(35)ℙuλc→0=limλc→0ℙu=limλc→0∫0Rℙuρfduρdρ=a∫0+∞limλc→0ℙuρfduρdρ=∫0+∞1·fduρdρ=1,where (a) follows from applying Dominated Convergence Theorem (DCT). Using the similar approach, we can obtain ℙcλc→0 and ℙsucλc→0.Remark 13.
As the MBS intensity approaches zeroλc⟶0, it can be reduced to a traditional vehicular network without MBS deployment. Thus, the association probability with the closest RSU is equal to 1, and the total interference are all from the other RSU nodes.Lemma 14.
As the road intensity approaches zeroλl⟶0, the association probability with the closest RSU is(36)ℙuλl→0=∫0+∞ℙuρ2λue−2λuρdρ,the association probability with the closest MBS is(37)ℙcλl→0=∫0+∞e−2λudcufdcρdρ,and the success probability is(38)ℙsucλl→0=∫0+∞ℙuρLIux1Puβρα,ρ×LIc1Puβρα,duc2λue−2λuρdρ+∫0+∞e−2λudcuLIux1Pcβρα,dcu×LIc1Pcβρα,ρfdcρdρ.Proof.
The details is omitted, as the method is the same as that in Lemma12.Remark 15.
As the road intensity approaches zeroλl⟶0, it can be reduced to a highway scenario with only one road. Thus, the association node can only be selected between the RSUs on the typical road and surrounding MBSs.Lemma 16.
As the road intensity approaches positive infinityλl⟶+∞ and the RSU intensity approaches to 0 λu⟶0 while the overall intensity of RSUs is constant 2λlλu=λa, the association probability with the closest RSU is(39)ℙuλl→+∞=∫0+∞ℙuρfduλl→+∞ρdρ,the association probability with the closest MBS is(40)ℙcλl→+∞=∫0+∞ℙcλl→+∞ρfdcρdρ,and the success probability is(41)Psuc=∫0+∞PuρLIuλl→+∞1Puβρα,ρ×LIc1Puβρα,ducfduλl→+∞ρdρ+∫0+∞Pcλl→+∞ρLIuλl→+∞1Pcβρα,dcu×LIc1Pcβρα,ρfdcρdρ.Proof.
By applying the limitsλl⟶+∞ and λu⟶0 with λa=2λlλu unchanged, the asymptotic null probability in Lemma 1 can be calculated by(42)limλl→+∞ℙNub0,R=0=limλl→+∞exp−4λl∫0R1−e−2λuR2−r2dr=aexplimλl→+∞−4λl∫0R1−e−2λa/2λlR2−r2dr=bexplimλl→+∞4λl∫0R∑k=1+∞−λa/λlR2−r2kk!dr=explimλl→+∞4λl∫0R−λaλlR2−r2dr+limλl→+∞4λl∫0R∑k=2+∞−λa/λlR2−r2kk!dr=cexplimλl→+∞−4λa∫0RR2−r2dr=e−πR2λa,where (a) follows from substituting λu=λa/2λl, (b) follows from the Taylor series expansion of exponential function, and (c) follows from the Dominated Convergence Theorem on the second term, and the limit of the integrand is equal to 0 for k≥2.
Thus, the cox processΦu approaches to a 2D PPP with the intensity λa. The PDF of distance du between the typical vehicle and its closest RSU can be expressed by(43)fduλl→+∞ρ=2πλaρe−λaπρ2.
The Laplace transform of total interference from RSUs can be calculated by(44)LIuλl→+∞s,ru=exp−2πλa∫ru+∞1−11+sPur−αrdr.
The probability of an MBS being the transmitter at a distance ofρ from the typical vehicle can be calculated by(45)ℙcλl→+∞ρ=exp−λaπdcu2.
Therefore, the overall probabilities of associating with the closest RSU and closest MBS can be calculated by(46)ℙuλl→+∞=∫0+∞ℙuρfduλl→+∞ρdρ,(47)ℙcλl→+∞=∫0+∞ℙcλl→+∞ρfdcρdρ,and the success probability is(48)ℙsuc=∫0+∞ℙuρLIuλl→+∞1Puβρα,ρ×LIc1Puβρα,ducfduλl→+∞ρdρ+∫0+∞ℙcλl→+∞ρLIuλl→+∞1Pcβρα,dcu×LIc1Pcβρα,ρfdcρdρ.


## 3.1. Null Probability

Lemma 1.
Null probability of RSU: for a MPLPΦl, which are generated by two independent homogeneous 1D PPPs with the same intensity λl and RSUs lie on each road followed by 1D PPP with intensity λu, the probability that there is no RSUs inside a disk bo,R is(1)ℙNubo,R=0=exp−4λl∫0R1−e−2λuR2−r2dr.Proof.
The null probability can be calculated as(2)ℙNubo,R=0=aℙNuΦlh∩bo,R=0×ℙNuΦlv∩bo,R=0=b∑n=0∞ℙNlhbo,R=n×∏i=1nℙNuLhi∩bo,R=0×∑n=0∞ℙNlvbo,R=n×∏i=1nℙNuLvi∩bo,R=0=∑n=0∞e−2λlR2λlRnn!∫−RRe−2λuR2−r212Rdrn2=e−2λlR∑n=0∞λl∫−RRe−2λuR2−r2drnn!2=e−2λlRexp2λl∫0Re−2λuR2−r2dr2=exp−4λl∫0R1−e−2λuR2−r2dr,where (a) follows from that there is no RSU on both horizontal and vertical roads and the distribution of horizontal and vertical roads are independent, and (b) follows from conditioning over the number of horizontal and vertical roads inside the disk.


## 3.2. Distance Distribution

We derive the distance distributions to the closest RSU and MBS, respectively, which will be used in calculating the success probability.Lemma 2.
The cumulative distribution function (CDF) and probability distribution function (PDF) of the distancedu between the typical vehicle and its closest RSU are CDF:(3)CDF:Fduρ=1−exp−4λl∫0ρ1−e−2λuρ2−r2dre−2λuρ,(4)PDF:fduρ=exp−4λl∫0ρ1−e−2λuρ2−r2dr×2λue−2λuρ+8λuλlρe−2λuρ∫0ρe−2λuρ2−r2ρ2−r2dr.Proof.
The CDF ofdu can be computed as Fduρ=ℙdu≤ρ=1−ℙdu>ρ(5)Fduρ=ℙdu≤ρ=1−ℙdu>ρ=ℙdu≤ρ=1−ℙdu>ρ=a1−ℙNubo,ρ=0ℙNuLx∩bo,ρ=0=1−exp−4λl∫0ρ1−e−2λuρ2−r2dre−2λuρ,where (a) follows from that the event du>ρ occurs when both of the following independent events occur simultaneously: (i) there is no RSU on Φl inside the disk bo,ρ and (ii) there is no RSU on Lx inside the disk bo,ρ. The PDF is derived by differentiating the CDF as follows:(6)fduρ=ddρ1−exp−4λl∫0ρ1−e−2λuρ2−r2dre−2λuρ=−ddρexp−4λl∫0ρ1−e−2λuρ2−r2dre−2λuρ=aexp−4λl∫0ρ1−e−2λuρ2−r2dr×2λue−2λuρ+8λuλlρe−2λuρ∫0ρe−2λuρ2−r2ρ2−r2dr,where (a) follows from the Leibniz rule and the differentiation of a definite integral with respect to a parameter.Lemma 3.
The cumulative distribution function (CDF) and probability distribution function (PDF) of the distancedc between the typical vehicle and its closest MBS are(7)CDF:Fdcρ=1−e−πλcρ2,(8)PDF:fdcρ=2πλcρe−πλcρ2.Proof.
The CDF ofdc can be computed as(9)Fdcρ=ℙdc≤ρ=1−ℙdc>ρ=1−ℙNcbo,ρ=0=1−e−πλcρ2.
The PDF can be obtained by differentiating the CDF.


## 3.3. Laplace Transform of Interference

We first give the expression of Laplace transform of interference under Rayleigh fading, since the success probability can be reduced to the Laplace transform of interferenceI, which is a well-known approach [14, 18]. For a wireless network with interfering nodes generated by a PPP Φ, the Laplace transform can be calculated as(10)LIs=Eexp−∑x∈ΦsPhxx−α=E∏x∈ΦEhxexp−sPhxx−α=E∏x∈Φ11+sPx−α,where P is the transmit power of each node and hx∼exp1.Note that the interference at a typical vehicle are from three independent sources: (i) RSU nodes on the other roads (denoted byΦu), (ii) RSU nodes on typical road Lx (denoted by Φux), and (iii) MBS nodes (denoted by Φc). Thus, the interference from these three sources can be denoted by Iu, Iux, and Ic, respectively. We will calculate the Laplace transform of these interferences below.Since the typical vehicle always connects to the closest RSU or MBS, we define some distancesru, rux, and rc, which are the minimum distance between typical vehicle and the closest RSU on the other roads, the closest RSU on typical road Lx, and the closest MBS.Lemma 4.
The Laplace transform of interference distribution from the RSUs on a single road (which is ata distance r) is(11)LIus,ru,r=g1r=exp−2λu∫0+∞1−11+sPur2+t2−α/2dt,r≥ru,g2r=exp−2λu∫ru2−r2+∞1−11+sPur2+t2−α/2dt,r<ru.Proof.
As shown in Figure3, there are two roads inside a disk. One is at a distance r>ru (denoted by D1), which means the interfering RSUs can be located anywhere on D1. The other one is at a distance r<ru (denoted by D2), which means the interfering RSUs can be only located in the segment t=−ru2−r2,−R2−ru2 and t=ru2−r2,R2−ru2. Next, we will analyze these two roads, respectively.
The Laplace transform of interference distribution fromD1 is derived by conditioning on the number of points on this line. Given a line at a perpendicular distance r from the center o, the length of the chord inside a disk (with radius R) can be expressed by 2R2−r2. Thus, the probability of that there are m points on this chord is e−2λuR2−r22λuR2−r2m/m!, according to the properties of PPP. Therefore, the Laplace transform of total interference in the case of r≥rv can be calculated by(12)g1s,ru,r=E∏x∈Dr11+sPux−α=∑m=0∞ℙN=m∫−R2−r2R2−r211+sPur2+t2−α/2ftdtm=∑m=0∞e−2λuR2−r22λuR2−r2mm!×∫−R2−r2R2−r211+sPur2+t2−α/212R2−r2dtm=e−2λuR2−r2exp2λu∫0R2−r211+sPur2+t2−α/2dt=exp−2λu∫0R2−r21−11+sPur2+t2−α/2dt,where Dr refers to a road at a perpendicular distance r from the center o. Similarly, the Laplace transform of total interference in the case of r<rv can be calculated by(13)g2s,ru,r=exp−2λu∫ru2−r2R2−r21−11+sPur2+t2−α/2dt.
The final expressions can be obtained as the radius approaches positive infinity (R⟶+∞) in (12) and (13).Lemma 5.
The Laplace transform of total interference from all other roads can be calculated as follows:(14)LIus,ru=exp−4λl∫0ru1−g2rdr+∫ru+∞1−g1rdr.Proof.
The Laplace transform of total interference from all other roads can be calculated as(15)LIus,ru=Ee−sIu=Ee−sE∑x∈ΦuIx=E∏x∈Φue−sEIx=aE∏x∈Φuhe−sEIx×E∏x∈Φuve−sEIx=∑j=0∞e−2ruλl2ruλljj!∫−rurug2r2rudrj×∑k=0∞e−2R−ruλl2R−ruλlkk!∫ruRg1rR−rudrk2=exp−4λl∫0ru1−g2rdr+∫ruR1−g1rdr,where (a) refers to the fact that the total interference are from the RSUs on horizontal and vertical roads, which are independent and identically distributed (i.i.d.). The final expression can be obtained as the radius approaches positive infinity (R⟶+∞) in (15).Lemma 6.
The Laplace transform of total interference from the typical roadLx can be calculated as follows:(16)LIuxs,rux=exp−2λu∫rux+∞1−11+sPut−αdt.Proof.
We have obtained the Laplace transform of interference from the RSUs on a single node in Lemma4. Therefore, the Laplace transform of interference from all the RSUs on Lx can be derived by substituting r=0 and ru=rux, respectively, in g2r.Figure 3
Distribution of interference RSUs on roads.The expression of Laplace transformLIc from the interference MBSs with a 2D PPP can be derived by [19]:(17)LIcs,rc=exp−2πλc∫rc+∞1−11+sPcr−αrdr.The total interference at origino is the sum of three types of interference sources, i.e.,(18)I=Iu+Iux+Ic.As these three types of interference sources are independent with each other, the Laplace transform of the total interference can be given by:(19)LIs,ru,rux,rc=LIus,ruLIuxs,ruxLIcs,rc.


## 3.4. Association Probability

In Maximum Power-based Scheme, the typical vehicle aims to connect to the transmitter from which it could receive the strongest signal (i.e., average received power). In other words, the typical vehicle chooses to receive signal from the closest RSU or the closest MBS, which can supply a higher time-average power.To derive the overall success probability, we first should calculate the probability of the typical vehicle connecting to RSU and MBS, respectively.Lemma 7.
The probability of an RSU being the transmitter at a distance ofρ from the typical vehicle can be calculated by(20)ℙuρ=exp−λcπduc2,where duc=Pc/Pu1/αρ.Proof.
A typical vehicle connects to the closest RSU when the average received power from it is larger than that from the closest MBS. Thus, given the distance between the vehicle and its closest RSU and MBS (du and dc), the probability of this event occurring can be expressed by(21)ℙPudu−α>Pcdc−α=ℙdc>PcPu1/αdu.
Letduc=Pc/Pu1/αdu for simplicity of notation. Therefore, the typical vehicle at origin o connects to the closest RSU when there is no MBS existing in the disk bo,duc, which happens with the probability(22)ℙudu=exp−λcπduc2.
As the typical vehicle connects to the closest RSU, substitutedu=ρ. Thus, we can obtain the result.Lemma 8.
The probability of an MBS being the transmitter at a distance ofρ from the typical vehicle can be calculated by(23)ℙcρ=exp−4λl∫0dcu1−e−2λudcu2−r2dre−2λudcu,where dcu=Pu/Pc1/αρ.Proof.
Following the same method as in Lemma7, the probability of this event occurring is(24)ℙPcdc−α>Pudu−α=ℙdu>PuPc1/αdc.
Letdcu=Pu/Pc1/αdc for simplicity of notation. The typical vehicle at origin o connects to the closest MBS when there is no RSU existing on Φl and Lx in the disk bo,dcu, which happens with the probability(25)ℙcdc=ℙNubo,dcu=0ℙNuLx∩bo,dcu=0=exp−4λl∫0dcu1−e−2λudcu2−r2dre−2λudcu.
As the typical vehicle connects to the closest MBS, substitutedc=ρ. Thus, we can obtain the result.Thus, given the association probabilities conditioned on the value ofρ in Lemmas 7 and 8, the overall probabilities of associating with the closest RSU and closest MBS can be calculated by(26)ℙu=∫0+∞ℙuρfduρdρ,(27)ℙc=∫0+∞ℙcρfdcρdρ.


## 3.5. Success Probability

In this paper, we use SIR as the metric to judge whether to receive message successfully. If the SIR measured at typical vehicle is larger than a predefined thresholdβ, the message can be detected successfully. Thus, the success probability in a Rayleigh fading network can be derived by(28)ℙsuc=ℙSIR>β=ℙPhρ−αI>β=ℙh>βραIP=Eexp−βραIP=LIβραP.Therefore, the success probability can be expressed by the Laplace transform of interference (LIs, where s=βρα/P).When the typical vehicle connects to the closest RSU at a distanceρ, there is no interfering RSU on Φl in the disk bo,ρ and on typical road Lx in segment −ρ,ρ, and no MBS in the disk bo,duc, i.e., ru=ρ,rux=ρ,rc=duc.Lemma 9.
The success probability that the typical vehicle connects to the closest RSU, which is at a distance ofρ, can be expressed by:(29)ℙsuc∣uρ=LIu1Puβρα,ρLIux1Puβρα,ρ×LIc1Puβρα,duc.
When the typical vehicle connects to the closest MBS at a distanceρ, there is no interfering RSU on Φl in the disk bo,dcu and on typical road Lx in segment −dcu,dcu, and no MBS in the disk bo,ρ, i.e., ru=dcu,rux=dcu,rc=ρ.Lemma 10.
The success probability that the typical vehicle connects to the closest MBS, which is at a distance ofρ, can be expressed by(30)ℙsuc∣cρ=LIu1Pcβρα,dcuLIux1Pcβρα,dcu×LIc1Pcβρα,ρ.
Note that the expressions forLIu, LIux, and LIc can be obtained from the results in Equations (14), (16), and (17).Theorem 11.
The total success probability at a typical vehicle in C-V2X networks is(31)ℙsuc=∫0+∞ℙuρℙsuc∣uρfduρdρ+∫0+∞ℙcρℙsuc∣cρfdcρdρ,where the first term and second term can be named as Psuc,u and Psuc,c, which mean the probabilities of the typical vehicle connecting to an RSU and MBS successfully, respectively.Proof.
The total success probability can be calculated as(32)ℙsuc=Eρℙuρℙsuc∣uρ+ℙcρℙsuc∣cρ=a∫0+∞ℙuρℙsuc∣uρfduρdρ+∫0+∞ℙcρℙsuc∣cρfdcρdρ,where (a) follows from the fact that success probability for RSU or MBS can be derived by taking an expectation over ρ.


## 3.6. Area Spectral Efficiency

Area spectral efficiency can be defined by the average bits transmitted per unit time per unit bandwidth per unit area, which can be calculated by(33)ASE=λalog1+βℙsuc,u+λclog1+βℙsuc,c,where λa is the intensity of vehicle nodes in R2 given by λa=μlλv, and μl is the line density which can be calculated by μl=2λl for a stationary MPLP [13].


## 3.7. Asymptotic Characteristic

Lemma 12.
As the MBS intensity approaches zeroλc⟶0, the association probability with the closest ℙuλc→0 is 1, the association probability with the closest MBS ℙcλc→0 is 0, and the success probability is(34)ℙsucλc→0=∫0+∞LIu1Puβρα,ρLIux1Puβρα,ρfduρdρ.Proof.
By applying the limitλc⟶0 on Pu, we can obtain(35)ℙuλc→0=limλc→0ℙu=limλc→0∫0Rℙuρfduρdρ=a∫0+∞limλc→0ℙuρfduρdρ=∫0+∞1·fduρdρ=1,where (a) follows from applying Dominated Convergence Theorem (DCT). Using the similar approach, we can obtain ℙcλc→0 and ℙsucλc→0.Remark 13.
As the MBS intensity approaches zeroλc⟶0, it can be reduced to a traditional vehicular network without MBS deployment. Thus, the association probability with the closest RSU is equal to 1, and the total interference are all from the other RSU nodes.Lemma 14.
As the road intensity approaches zeroλl⟶0, the association probability with the closest RSU is(36)ℙuλl→0=∫0+∞ℙuρ2λue−2λuρdρ,the association probability with the closest MBS is(37)ℙcλl→0=∫0+∞e−2λudcufdcρdρ,and the success probability is(38)ℙsucλl→0=∫0+∞ℙuρLIux1Puβρα,ρ×LIc1Puβρα,duc2λue−2λuρdρ+∫0+∞e−2λudcuLIux1Pcβρα,dcu×LIc1Pcβρα,ρfdcρdρ.Proof.
The details is omitted, as the method is the same as that in Lemma12.Remark 15.
As the road intensity approaches zeroλl⟶0, it can be reduced to a highway scenario with only one road. Thus, the association node can only be selected between the RSUs on the typical road and surrounding MBSs.Lemma 16.
As the road intensity approaches positive infinityλl⟶+∞ and the RSU intensity approaches to 0 λu⟶0 while the overall intensity of RSUs is constant 2λlλu=λa, the association probability with the closest RSU is(39)ℙuλl→+∞=∫0+∞ℙuρfduλl→+∞ρdρ,the association probability with the closest MBS is(40)ℙcλl→+∞=∫0+∞ℙcλl→+∞ρfdcρdρ,and the success probability is(41)Psuc=∫0+∞PuρLIuλl→+∞1Puβρα,ρ×LIc1Puβρα,ducfduλl→+∞ρdρ+∫0+∞Pcλl→+∞ρLIuλl→+∞1Pcβρα,dcu×LIc1Pcβρα,ρfdcρdρ.Proof.
By applying the limitsλl⟶+∞ and λu⟶0 with λa=2λlλu unchanged, the asymptotic null probability in Lemma 1 can be calculated by(42)limλl→+∞ℙNub0,R=0=limλl→+∞exp−4λl∫0R1−e−2λuR2−r2dr=aexplimλl→+∞−4λl∫0R1−e−2λa/2λlR2−r2dr=bexplimλl→+∞4λl∫0R∑k=1+∞−λa/λlR2−r2kk!dr=explimλl→+∞4λl∫0R−λaλlR2−r2dr+limλl→+∞4λl∫0R∑k=2+∞−λa/λlR2−r2kk!dr=cexplimλl→+∞−4λa∫0RR2−r2dr=e−πR2λa,where (a) follows from substituting λu=λa/2λl, (b) follows from the Taylor series expansion of exponential function, and (c) follows from the Dominated Convergence Theorem on the second term, and the limit of the integrand is equal to 0 for k≥2.
Thus, the cox processΦu approaches to a 2D PPP with the intensity λa. The PDF of distance du between the typical vehicle and its closest RSU can be expressed by(43)fduλl→+∞ρ=2πλaρe−λaπρ2.
The Laplace transform of total interference from RSUs can be calculated by(44)LIuλl→+∞s,ru=exp−2πλa∫ru+∞1−11+sPur−αrdr.
The probability of an MBS being the transmitter at a distance ofρ from the typical vehicle can be calculated by(45)ℙcλl→+∞ρ=exp−λaπdcu2.
Therefore, the overall probabilities of associating with the closest RSU and closest MBS can be calculated by(46)ℙuλl→+∞=∫0+∞ℙuρfduλl→+∞ρdρ,(47)ℙcλl→+∞=∫0+∞ℙcλl→+∞ρfdcρdρ,and the success probability is(48)ℙsuc=∫0+∞ℙuρLIuλl→+∞1Puβρα,ρ×LIc1Puβρα,ducfduλl→+∞ρdρ+∫0+∞ℙcλl→+∞ρLIuλl→+∞1Pcβρα,dcu×LIc1Pcβρα,ρfdcρdρ.


## 4. Numerical Results and Discussions

We verify the accuracy of our theoretical analysis through Monte Carlo simulations in this section. We will also analyze and discuss the impact of network parameters on success probability, which can give some design insights to network deployment. For all the numerical results in this paper, we assume that the MBSs’ transmit powerPc=10W (40 dBm), the RSUs’ transmit power Pu=0.1W (20 dBm), and the path-loss exponent α=4.


### 4.1. Association Probability

We first simulate the association probabilities of the typical vehicle withλc=0.5×10−6nodes/m2, λl=1×10−3m−1, and β=0dB. From Figure 4, we observe the simulation results are very close to analytical results, which means that our analytical results are accurate. In addition, given a certain λu, the sum of these two probabilities is always equal to 1, which indirectly verifies our analytical model is effective. We can also see that the probability of associating with RSUs increases with the increase of RSU intensity, because the typical vehicle will have a higher probability to connect with the surrounding RSUs. The probability of associating with MBSs has the opposite trend.Figure 4
Association probabilities with RSU and MBS of the typical vehicle (λc=0.5×10−6nodes/m2, λl=1×10−3m−1, and β=0dB).


### 4.2. Success Probability

We simulate the success probability of the typical vehicle withλc=0.5×10−6nodes/m2, λl=1×10−3m−1, and λu=1×10−3nodes/m. Figure 5 shows that the simulation results match exactly with the analytical results, which means our analytical results about success probability is accurate. In addition, as shown in Figure 5, the success probability of the typical vehicle decreases with the increase of SIR threshold, which accords with common sense.Figure 5
Success probability of the typical vehicle (λc=0.5×10−6nodes/m2, λl=1×10−3m−1, and λu=1×10−3nodes/m).


#### 4.2.1. Impact of Intensity of MBS

We calculate the success probability of the typical vehicle for road intensity ofλl=1×10−3m−1, RSU intensity of λu=1×10−3nodes/m, SIR threshold of β=0dB, and different MBS intensities. Figure 6 shows that the success probability decreases as the MBS intensity increases. This is because as the MBS intensity increases, the increment of success probability that typical vehicle connects to MBSs is less than the decrement of success probability that typical vehicle connects to RSUs and then affects the total success probability. Therefore, in terms of success probability, we should reduce the use of MBSs as far as possible, when other application requirements are met.Figure 6
Impact of MBS intensity on success probability (λl=1×10−3m−1, λu=1×10−3nodes/m, and β=0dB).


#### 4.2.2. Impact of Intensity of Road

We calculate the success probability of the typical vehicle for MBS intensity ofλc=0.5×10−6nodes/m2, RSU intensity of λu=1×10−3nodes/m, SIR threshold of β=0dB, and different road intensities. From Figure 7, we can see that the success probability decreases with the increase of road intensity. This trend can be understood that as the road intensity increases, the increment of success probability that typical vehicle connects to RSUs is less than the decrement of success probability that typical vehicle connects to MBSs, as shown in Figure 7. Thus, there is a lower success probability in dense Manhattan block.Figure 7
Impact of road intensity on success probability (λc=0.5×10−6nodes/m2, λu=2×10−3nodes/m, and β=0dB).


#### 4.2.3. Impact of Intensity of RSU

We calculate the success probability of the typical vehicle for MBS intensity ofλc=0.5×10−6nodes/m2, road intensity of λl=1×10−3m−1, SIR threshold of β=0dB, and different RSU intensities. Different from the features of the impact of MBS intensity and road intensity on success probability, it is observed from Figure 8 that the success probability increases with the increase of the RSU intensity. It is because that the increment of success probability that typical vehicle connects to RSUs is larger than the decrement of success probability that typical vehicle connects to MBSs, as shown in Figure 8. Therefore, to improve success probability, the simplest and most effective way is to increase the deployment intensity of RSUs.Figure 8
Impact of RSU intensity on success probability (λc=0.5×10−3nodes/m2, λl=1×10−3m−1, and β=0dB).


#### 4.2.4. Asymptotic Characteristic of Success Probability

In Figure9, we plot the success probability of the typical vehicle for several certain values of road intensities and RSU intensities, which satisfy λa=2λl×λu=2×10−6nodes/m2.Figure 9
Asymptotic characteristic of success probability (λc=0.5×10−3nodes/m2).According to the discussions above, we know that the success probability decreases with the increase of road intensity and the decrease of RSU intensity. When road intensity approaches positive infinity (λl→+∞) and RSU intensity approaches 0 (λu→0), it provides a lower bound for success probability given a certain value of MBS intensity.


### 4.3. Area Spectral Efficiency

We also analyze the impact of road intensity and RSU intensity on area spectral efficiency. As shown in Figure10, we see that the area spectral efficiency increases with the increase of RSU intensity or the increase of road intensity. It is because as the RSU intensity or road intensity increases, the increment of λu or λl is much larger than the changes of Psuc,u and Psuc,c in (33). Thus, the total area spectral efficiency shows an upward trend. Thereby, for the areas with sparse distribution of roads, we can increase the deployment intensity of RSUs to improve area spectral efficiency.Figure 10
Area spectral efficiency (λc=0.5×10−3nodes/m2).


## 4.1. Association Probability

We first simulate the association probabilities of the typical vehicle withλc=0.5×10−6nodes/m2, λl=1×10−3m−1, and β=0dB. From Figure 4, we observe the simulation results are very close to analytical results, which means that our analytical results are accurate. In addition, given a certain λu, the sum of these two probabilities is always equal to 1, which indirectly verifies our analytical model is effective. We can also see that the probability of associating with RSUs increases with the increase of RSU intensity, because the typical vehicle will have a higher probability to connect with the surrounding RSUs. The probability of associating with MBSs has the opposite trend.Figure 4
Association probabilities with RSU and MBS of the typical vehicle (λc=0.5×10−6nodes/m2, λl=1×10−3m−1, and β=0dB).


## 4.2. Success Probability

We simulate the success probability of the typical vehicle withλc=0.5×10−6nodes/m2, λl=1×10−3m−1, and λu=1×10−3nodes/m. Figure 5 shows that the simulation results match exactly with the analytical results, which means our analytical results about success probability is accurate. In addition, as shown in Figure 5, the success probability of the typical vehicle decreases with the increase of SIR threshold, which accords with common sense.Figure 5
Success probability of the typical vehicle (λc=0.5×10−6nodes/m2, λl=1×10−3m−1, and λu=1×10−3nodes/m).


### 4.2.1. Impact of Intensity of MBS

We calculate the success probability of the typical vehicle for road intensity ofλl=1×10−3m−1, RSU intensity of λu=1×10−3nodes/m, SIR threshold of β=0dB, and different MBS intensities. Figure 6 shows that the success probability decreases as the MBS intensity increases. This is because as the MBS intensity increases, the increment of success probability that typical vehicle connects to MBSs is less than the decrement of success probability that typical vehicle connects to RSUs and then affects the total success probability. Therefore, in terms of success probability, we should reduce the use of MBSs as far as possible, when other application requirements are met.Figure 6
Impact of MBS intensity on success probability (λl=1×10−3m−1, λu=1×10−3nodes/m, and β=0dB).


### 4.2.2. Impact of Intensity of Road

We calculate the success probability of the typical vehicle for MBS intensity ofλc=0.5×10−6nodes/m2, RSU intensity of λu=1×10−3nodes/m, SIR threshold of β=0dB, and different road intensities. From Figure 7, we can see that the success probability decreases with the increase of road intensity. This trend can be understood that as the road intensity increases, the increment of success probability that typical vehicle connects to RSUs is less than the decrement of success probability that typical vehicle connects to MBSs, as shown in Figure 7. Thus, there is a lower success probability in dense Manhattan block.Figure 7
Impact of road intensity on success probability (λc=0.5×10−6nodes/m2, λu=2×10−3nodes/m, and β=0dB).


### 4.2.3. Impact of Intensity of RSU

We calculate the success probability of the typical vehicle for MBS intensity ofλc=0.5×10−6nodes/m2, road intensity of λl=1×10−3m−1, SIR threshold of β=0dB, and different RSU intensities. Different from the features of the impact of MBS intensity and road intensity on success probability, it is observed from Figure 8 that the success probability increases with the increase of the RSU intensity. It is because that the increment of success probability that typical vehicle connects to RSUs is larger than the decrement of success probability that typical vehicle connects to MBSs, as shown in Figure 8. Therefore, to improve success probability, the simplest and most effective way is to increase the deployment intensity of RSUs.Figure 8
Impact of RSU intensity on success probability (λc=0.5×10−3nodes/m2, λl=1×10−3m−1, and β=0dB).


### 4.2.4. Asymptotic Characteristic of Success Probability

In Figure9, we plot the success probability of the typical vehicle for several certain values of road intensities and RSU intensities, which satisfy λa=2λl×λu=2×10−6nodes/m2.Figure 9
Asymptotic characteristic of success probability (λc=0.5×10−3nodes/m2).According to the discussions above, we know that the success probability decreases with the increase of road intensity and the decrease of RSU intensity. When road intensity approaches positive infinity (λl→+∞) and RSU intensity approaches 0 (λu→0), it provides a lower bound for success probability given a certain value of MBS intensity.


## 4.2.1. Impact of Intensity of MBS

We calculate the success probability of the typical vehicle for road intensity ofλl=1×10−3m−1, RSU intensity of λu=1×10−3nodes/m, SIR threshold of β=0dB, and different MBS intensities. Figure 6 shows that the success probability decreases as the MBS intensity increases. This is because as the MBS intensity increases, the increment of success probability that typical vehicle connects to MBSs is less than the decrement of success probability that typical vehicle connects to RSUs and then affects the total success probability. Therefore, in terms of success probability, we should reduce the use of MBSs as far as possible, when other application requirements are met.Figure 6
Impact of MBS intensity on success probability (λl=1×10−3m−1, λu=1×10−3nodes/m, and β=0dB).


## 4.2.2. Impact of Intensity of Road

We calculate the success probability of the typical vehicle for MBS intensity ofλc=0.5×10−6nodes/m2, RSU intensity of λu=1×10−3nodes/m, SIR threshold of β=0dB, and different road intensities. From Figure 7, we can see that the success probability decreases with the increase of road intensity. This trend can be understood that as the road intensity increases, the increment of success probability that typical vehicle connects to RSUs is less than the decrement of success probability that typical vehicle connects to MBSs, as shown in Figure 7. Thus, there is a lower success probability in dense Manhattan block.Figure 7
Impact of road intensity on success probability (λc=0.5×10−6nodes/m2, λu=2×10−3nodes/m, and β=0dB).


## 4.2.3. Impact of Intensity of RSU

We calculate the success probability of the typical vehicle for MBS intensity ofλc=0.5×10−6nodes/m2, road intensity of λl=1×10−3m−1, SIR threshold of β=0dB, and different RSU intensities. Different from the features of the impact of MBS intensity and road intensity on success probability, it is observed from Figure 8 that the success probability increases with the increase of the RSU intensity. It is because that the increment of success probability that typical vehicle connects to RSUs is larger than the decrement of success probability that typical vehicle connects to MBSs, as shown in Figure 8. Therefore, to improve success probability, the simplest and most effective way is to increase the deployment intensity of RSUs.Figure 8
Impact of RSU intensity on success probability (λc=0.5×10−3nodes/m2, λl=1×10−3m−1, and β=0dB).


## 4.2.4. Asymptotic Characteristic of Success Probability

In Figure9, we plot the success probability of the typical vehicle for several certain values of road intensities and RSU intensities, which satisfy λa=2λl×λu=2×10−6nodes/m2.Figure 9
Asymptotic characteristic of success probability (λc=0.5×10−3nodes/m2).According to the discussions above, we know that the success probability decreases with the increase of road intensity and the decrease of RSU intensity. When road intensity approaches positive infinity (λl→+∞) and RSU intensity approaches 0 (λu→0), it provides a lower bound for success probability given a certain value of MBS intensity.


## 4.3. Area Spectral Efficiency

We also analyze the impact of road intensity and RSU intensity on area spectral efficiency. As shown in Figure10, we see that the area spectral efficiency increases with the increase of RSU intensity or the increase of road intensity. It is because as the RSU intensity or road intensity increases, the increment of λu or λl is much larger than the changes of Psuc,u and Psuc,c in (33). Thus, the total area spectral efficiency shows an upward trend. Thereby, for the areas with sparse distribution of roads, we can increase the deployment intensity of RSUs to improve area spectral efficiency.Figure 10
Area spectral efficiency (λc=0.5×10−3nodes/m2).


## 5. Conclusion

In this paper, we derived the success probability of a CV2X communication networks in irregular Manhattan grids over shared downlink channels, according to Maximum Power-based Scheme. We modeled the locations of MBSs as a 2D PPP, the layout of roads as MPLP, and the locations of RSUs as a 1D PPP on each road. We also gave the expression for area spectral efficiency of the whole network and asymptotic characteristic of success probability. What’s more, we studied the impact of several key parameters on success probability. It is observed that the success probability increases with the increase of RSU intensity, while it decreases with the increase of MBS and road intensity. Therefore, in terms of success probability, we should reduce the use of MBSs while other application requirements for MBSs are met and, meanwhile, increase the deployment intensity of RSUs. There are several extensions of this work. A more realistic channel model in urban scenario should be considered. Another meaningful issue is to focus on the handover rate of a moving vehicle, including the handover between MBSs, RSUs, and MBS and RSU. In addition, correcting the spatial road model according to the realistic roads in physical environment is worth to be done in the future.



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*Source: 2746038-2020-08-19.xml*