I Introduction
VEHICLEtoanything (V2X) communications based on cellular infrastructure have been defined by the Third Generation Partnership Project (3GPP) group [1] to cover broader spectrum of communication scenarios, such as vehicletovehicle (V2V), vehicletopedestrian (V2P) and vehicletoinfrastructure/networks (V2I/N). It is regarded as a promising technology to support various novel applications such as road safety, infotainment services, traffic management, traffic optimization and online services to car manufacturers. Specifically, this innovation promises to eliminate 80% of the current road accidents and help in fostering automobile and telecommunication industries for a smarter and safer ground transportation system [2].
Existing V2V communication can be supported via the DSRC standard, however, it has certain limitations such as short range, unable to support high density of networks and has unreliable broadcast services [3]. The underlying carrier sensing multiple access (CSMA); medium access control (MAC) protocol also exhibits signal congestion and connectivity disruptions due to rapid changing network topology and adhoc vehicular networks [4]. More importantly, single DSRC technology can not support variety of incoming vehicular oriented applications, and only vehicletovehicle (V2V) and vehicletoinfrastructure(V2I) communication is achievable using DSRC without the provisioning of V2P, infotainment and traffic management services.
To augment DSRC communication, V2X communication was proposed to meet V2X capacity, latency and coverage requirements [1], [3], [5] and [6] via operating in the following three scenarios as shown in Fig. 1. In V2X Scenario I, V2V, V2P or V2I messages are exchanged directly between vehicles without involvement of cellular nodes. In V2X Scenario II, V2V and V2P messages are first transmitted to the cellular node (eNB or EUTRAN denoted as CE) in uplink, and then being forwarded to multiple vehicles at a local area in the downlink by cellular node. For V2I/N communication, the vehicle can communicate with the EUTRAN type RSU or application server. In V2X scenario III, the UE type RSU or vehicle’s V2X message is being transmitted to the cellular node in the uplink, and then the cellular node forwards it to multiple vehicles or UE type V2X message. In all three scenarios, the vehicular communication can coexist with the cellular carriers called shared (S) or using dedicated carriers referred as dedicated channels (D). To model and analyze these V2X scenarios, stochastic geometry has been proposed, considering that it has been utilized as a powerful tool to model and analyze mutual interference between transceivers in the wireless networks, such as conventional cellular networks [7, 8], wireless sensor networks [9], cognitive radio networks [10, 11], and heterogeneous cellular networks [12, 13, 14].
The initial studies on vehicular communication have focused on modeling the V2V communication (i.e. without involvement of the cellular nodes) using stochastic geometry [15, 16, 17, 18, 19, 20, 21, 22], where simple spatial models with a single road, a multilane road, or orthogonal roads were considered. In [15], the optimum transmission probability of the V2V communication on a single road was computed based on signaltointerference plus noise ratio (SINR) model. In [17], the tradeoff between the aggregate packet progress and spatial frequency reuse for multihop transmission between vehicles in a multilane highway setup with multiple parallel lanes in one road was studied in DSRC. With the vehicles modeled as 1D PPP, [19, 20] derived the packet reception probability of the V2V communication at the intersection area of two perpendicular roads. In [21], the success probability of V2V communication in orthogonal street system with many perpendicular roads was studied for vehicle safety communication. Unfortunately, these early attempts fail to account for the randomness of roads distribution, and thus did not accurately capture the irregular structure of roads and their effect on the performance of vehicular networks. In [22], the authors have modeled the streets in an urban environment as orthogonal streets, and the base stations on each road as a 1D PPP, where the downlink coverage probability of mmWave microcells was derived.
The works in [23, 24, 25, 26, 27, 28] accounted for the randomness of roads distributions. In [23]
, the nodes in the WiFi mesh networks were modeled by a Cox process on a PoissonLine tessellation (PLT), and the nodes on each line are modeled by a inhomogeneous 1D PPP, where the probability density function of the shorted Euclidean distance between two internodes was derived. Later on in
[24], the Cox process on a PLT was generalized to a PoissonLine tessellation (PLT), PoissonVoronoi tessellation (PVT), or a PoissonDelaunay tessellation (PDT), and the nodes on each line are modeled by a homogeneous 1D PPP. Their results have shown that PLT often gains preference over PVT and PDT in modeling road systems^{1}^{1}1 It has also been used in other related applications, such as in modeling the effect of blockages in localization networks [29]. due to its analytical tractability. In [25], the uplink coverage probability was derived for a network where the typical receiver is randomly chosen from a PPP, and the locations of transmitter mobile users alongside roads are modeled as a Cox process on a Poisson line process (PLP). In [28], the coverage probability of the V2V communication was first derived, where the transmitters and receivers were modeled using independent Cox processes on the same PLP (i.e. a doublystochastic spatial model), and it captures the irregularity in the spatial layout of roads via the PLP model, and the distribution of vehicles on each road via the 1D PPP model.Note that [23, 24, 25, 26, 27, 28] are limited to the V2V communication or V2I communication, the first performance characterization of the V2X downlink communication was studied in the master thesis in [30], where the association probabilities and the coverage probabilities for the V2V and the base station to vehicle downlink communications were derived for the maximum power based association scheme and the threshold distance based association scheme.
Different from the downlink communication studied in [30], we focus on the uplink communication of V2X networks driven by the stringent high reliability requirements for safety, traffic management and infotainment applications， where the locations of vehicles are modeled as a Cox process on a Poison Line Process (PLP), and the cellular BSs are deployed as 2D PPP. A flexible V2X link selection scheme is proposed to control the amount of vehicular interference and the number of vehicles operating on the cellular band. Our contributions can be summarized as follows:

We present a comprehensive and tractable analytical framework for analyzing the uplink communication of cellular V2X networks, where the vehicles decide to transmit to other vehicle or cellular BS via shared carriers depending on their corresponding distances and association bias.

Based on the proposed V2X link selection scheme, we derive the shortest distances and the association probabilities of the vehicles using the V2V link or the V2B link via the shared cellular carriers, respectively.

We derive the analytical expressions for the success probabilities of the V2V communication and the V2B communication, and the overall success probability of typical vehicle (i.e., V2X communication) using the uplink cellular resources, which are validated by Monte Carlo simulation.

In V2X network with low and medium vehicle intensity on the roads, there is almost equal probability transmitting via the V2V and the V2B link. Moreover, cellular based V2V link success probability increases at faster rate with the increase of vehicle nodes, and the success probability of the V2X communication improves with increasing the road intensity

Our results shown that the success probability of the V2X communication is much higher than that of the V2V communication, as both cellular based V2V and V2B links supplement each other to achieve higher success probability of V2X communication.
The rest of the paper is organized as follows. The mathematical preliminary, system model along with assumptions, and the methodology of analysis are described in Section II and III. Section IV presents the analysis of association probabilities of V2X communication. The success probability is analyzed in Section V. Section VI presents and discusses the numerical and simulation results. The paper is concluded in Section VII. A list of the key mathematical notations used in this paper is given in Table I. Throughout this paper, we denote the random variables using upper case letters and their corresponding realizations using the lower case letters. For instance,
denotes a random variable, whereas denotes its realization.Ii Preliminary: Poisson Line Process
The V2X networks exhibit unique spatial characteristics due to the fact that vehicles are only driven on roadways, which are predominantly linear in nature. To model these, we model the roadways as a network of lines that are distributed on the plane according to a Poisson Line Process (PLP). In this section, we provide a brief introduction of PLP, the detailed information of the underlying theory can be found in [31, 30, 28].
A Poisson line process is a random collection of lines in a 2D plane. Any undirected line in can be uniquely characterized by its perpendicular distance from the origin and the angle subtended by the perpendicular dropped onto the line from the origin with respect to the positive xaxis in counter clockwise direction, as shown in Fig. 2. The pair of parameters and can be represented as the coordinates of a point on the cylindrical surface as illustrated in Fig. 2. Clearly, there is a onetoone correspondence between the lines in and points on the cylindrical surface . Thus, a random collection of lines can be constructed from a set of points on . In other words, the set of points generated by a PPP with certain density on correspond to the PLP with the same density for lines on .
For a PLP with the intensity in
, the corresponding points are independent and uniformly distributed in representation space
with a surface area of . Thus, the expected number of points in the PPP that lie in is , and the number of lines intersection a disc of radiusis a Poisson distributed with mean
. In PLP, the values of and of each line follow a uniform distribution over an appropriate range defined by . In this work, we limit ourselves to motioninvariant PLP, where the line process is invariant to the rotation of axes to the origin, for analytical simplicity [28]. The PLP is also considered to be stationary, where translated line process of PLP, has the same distribution of lines as that of for any translation in the plane [28].Iii System Model
In this work, we consider a cellular V2X networks with the coexist of V2V, V2P, or V2I/N communications in V2X scenario I and V2B communication in V2X scenario II using the shared carriers as defined in 3GPP Release 14 and shown in Fig. 3. In this scenario, V2V, V2P and V2I/N messages exchange in uplink and downlink for the applications like vehicular safety. The system model are described in detail in the following subsections.
Iiia V2X Nodes (Vehicles, Pedestrians, and RSUs)
As mentioned in Section II, we model the roads as motioninvariant PLP, with line intensity, as per details given in [28], thus the intensity of equivalent Poisson Point Process (PPP) on the representation space is . The V2X nodes including vehicular nodes, pedestrian near roadside, and Road Sensing Units (RSUs), are randomly distributed on each road as homogeneous 1D PPP with intensity . This is because that vehicular nodes, pedestrian and RSUs are distributed according to independent PPP’s of intensities , and , the resulting distributions is still a PPP with intensity as discussed by [30]. For simplicity, the V2V, V2P, V2I/N communications are collectively called the V2V communication throughout the rest of the paper.
Assuming that each vehicle transmits independently with a probability , the locations of transmitting vehicles on each road is then given by a thinned PPP with intensity, , and we denote the set of locations of the transmitting vehicles on a line by . Correspondingly, the distribution of receiving vehicles on each line is also a thinned PPP with intensity, . In other words, the transmitting and receiving vehicles are modeled as the doubly stochastic processes called Cox processes, and , which are driven by the same PLP, .
Notations  Definition 

Intensity of roads, 2D PLP  
Intensity of V2X nodes, ID PPP  
Intensity of basestations, 2D PPP  
Poisson Line Process (PLP) for roads  
Cox process for transmitting nodes  
Cox process for receiving nodes  
2D PPP for cellular basestations  
Association bias,  
Basestation transmit power  
Vehicle transmit power  
Path loss exponent for V2V link  
Path loss exponent for V2B link  
Perpendicular distance of road from origin  
Angle of road from xaxis  
V2X communication channel bandwidth,  
V2V  Vehicle to vehicle uplink 
V2B  Vehicle to basestation uplink 
B2V  Basestation to vehicle downlink 
V2V distance  
V2B uplink distance  
B2V downlink distance  
Thermal noise 
For analytical simplicity, we can translate the origin to the location of the typical receiver vehicle. The translated point process can be treated as the superposition of the point process , an independent 1D PPP with intensity on a line passing through the origin and a vehicle at the origin . This can be realized by applying the Slivnyak’s theorem [31] according to the following steps. We first add a point at the origin to the PPP in the representation space , thereby obtaining a PLP with a line passing through the origin in and second, we add a point at the origin to the 1DPPP on the line passing through the origin in . The line, passing through the origin is referred as typical line in this paper. Since, both and are driven by the same line process, the translated point process is also the superposition of and an independent PPP with intensity on . Note that the other receiving vehicles in the network do not interfere with the typical receiving vehicle in our setup, therefore, we focus only on the distribution of transmitting vehicles. In this work, the impact of the vehicle mobility and direction is neglected as the vehicles are mostly stationary within one packet transmission duration, which is usually less than as discussed in [32].
IiiB Cellular Base Stations
We consider the cellular BSs are spatially distributed in according to the 2D PPP with intensity, . This model for macro BSs has been validated to be as accurate to the typical hexagonal grid model [33]. We assume that each basestation will always have at least one V2X node connected with the BS in the uplink. Due to the shared spectrum between the V2V communication and the V2B communication, there will exist interference between them.
IiiC V2X Link Selection
In our model, the vehicles transmit with the fixed power , which is always smaller than the maximum transmit power. The link selection between V2V and V2B communication is determined by the corresponding distances and association bias factor to limit interference and traffic on cellular network. In other words, it can tune the tradeoff interference and data offloading [34]. To ensure reliable uplink association and avoid the pingpong effects due to handovers, vehicle transmitter associates to the receiving node (cellular or vehicle) based on their longterm average link quality. In this flexible V2X link selection scheme, the vehicle selects the V2V link if , otherwise the vehicle selects the V2B link, where is the shortest V2V link distance, and is the distance between the V2X node and its closest cellular BS. For the extreme case with , the V2V link will never be selected, whereas for the case with , each vehicle transmitter is always selecting the V2V link. It is worth mentioning that one main advantage of the this link selection criterion is that it brings an inherent interference protection to the cellular uplink, which is necessarily required to enable shared link in cellular networks.
IiiD Channel Model
A general powerlaw pathloss model is considered in which the signal power decays at the rate, with the propagation distance , where is the pathloss exponent. Due to the different propagation environments experienced in the V2B and V2V links, each type of link is given its own pathloss exponent, namely, and
, respectively. The smallscale channel fading is modeled as slowflat Rayleigh fading, where its channel gain is assumed to be exponentially distributed with unit mean. All the channel gains are assumed to be independent of each other, independent of the spatial locations, symmetric, and are identically distributed (i.i.d.). For log normal shadowing, we have included the shadowing in a transparent way by using the displacement theorem given by
[35] and method described by [36].IiiE Packet Success Probability
The transmitting vehicle connects to a receiver (either BS or vehicle) depending on the corresponding distance and association bias, and thus the transmitting vehicle can operate in modes , where and modes denote the shared link communication between vehicle and cellular BS and that between vehicle and vehicle, respectively. The success probability (reliability) of the typical receiver (basestation or vehicle) conditioned on minimum distance, between transmitter and receiver and mode can be defined as the probability that the SINR of receiver is greater than a SINR Threshold, , which is given as
(1) 
where is transmit power, is channel gain and is the minimum distance between transmitter and the receiver. Note that the transmitting vehicle always connects to closet typical receiver (BS or vehicle) at distance , thus there will be no interfering vehicles present in the disk with radius as shown in Fig. 2. The size of affects the distribution of interfering nodes, as well as the interference to typical receiver. We need to characterize the distribution of interferers, as per their locations (interferers located on road passing through origin or on all other roads) to determine the Laplace transform of the distribution of interference power conditioned on the serving distance and mode .
Remind that is the minimum possible distance between the receiver and transmitter, and the interfering vehicles are located outside the disk . Therefore, the interferers can be broadly divided into two categories, (a) the interfering vehicles that are located outside the disk on the road passing through the origin and (b) the interfering vehicles located outside the disk on all other roads. The distribution of interferes located on all other roads can be denoted as and the disk does not contain any interfering vehicles. Similarly, the distribution of interferes located on road passing through the origin can be denoted as and there are no vehicles on the road segment from to for road passing through origin. As such, the success probability can be rewritten as
(2) 
where is the thermal noise power, and are the aggregate interference due to the vehicles located on all other roads, and on the road passing through the origin that operate in mode or , respectively.
Iv Association Probabilities
To facilitate the reliability analysis of proposed V2X networks, we first derive the distance distributions of the V2V link and the V2B link, and their corresponding association probability in the following.
Lemma 1 (V2V Distance)
The CDF of the shortest distance between the vehicular transmitter and a typical vehicular receiver in the V2V link is given in [30], we still present results below for completeness
(3) 
Proof 1
See Appendix A.
From Eq. (3), it is obvious that the CDF of the V2V link depends upon the perpendicular distance of roads from origin, intensity of roads and intensity of vehicles. With the increase of road intensities, the perpendicular distance decreases, resulting in the decrease of the V2V link distance. Similarly, with the increase of the intensities of vehicles, the V2V link distance also decreases as vehicular nodes come close to each other. Fig. 4 plots the analytical results for the CDF of the V2V link in Eq. (A), which was validated by the simulation results.
Corollary 1 (The PDF of V2V Link)
The Probability Density Function (PDF), of the shortest distance between the vehicular transmitter and a typical vehicular receiver, in the V2V link is derived as
(4) 
Proof 2
The Probability Density Function (PDF) of can be found by taking derivative of CDF given in Eq. (3), and final result for PDF of is given as Eq. (4). The closed form solution of Eq. (4) is derived as
(5) 
where () denotes the modified Bessel functions of the first kind and () denotes the modified Struve functions. This completes the proof.
Lemma 2 (V2B Distance)
The PDF of the distance between a vehicular transmitter and the nearest BS, for the shared link is given in [7, Eq. (2)] as
(6) 
According to the flexible V2X link selection scheme prosed in section III, the vehicle selects the V2V link if , otherwise the vehicle selects the V2B link, where is the shortest V2V link distance, and is the distance between the transmitting vehicle and its closest cellular BS. Therefore, the vehicular transmitter can connect with the vehicular receiver or BS in the uplink with their corresponding association probabilities. In Lemma 3 and Lemma 4, we derive the association probabilities of the V2V link and the V2B link, respectively.
Lemma 3 (Association probability of the V2V Link)
The probability of the vehicular transmitter selecting the V2V link is derived as
(7) 
where is the cellular basestation intensity, is the V2X nodes intensity, is the road intensities, is the association bias, is the modified Bessel functions of the first kind, and is the modified Struve functions.
Proof 3
See Appendix B.
Lemma 4 (Association probability of the V2B Link)
The probability of the vehicular transmitter selecting V2B link is derived as
(8) 
where is the cellular basestation intensity, is the V2X nodes intensity, is the road intensities, is the association bias.
Proof 4
See Appendix C.
V V2X Success Probability
In this section, we derive the success probability of the V2X communication. To do so, we first need to characterize the interference from each type of interferer category (). Thus, we calculate the general form of Laplace Transform, and we derive the expressions of Laplace Transform of interference from and in this section as concepts given in [30].
Va Laplace Transform of Interference Under Rayleigh Fading
As we know that Laplace Transform of interference, is . The interfering set of vehicles for each category of interfering vehicles based on their location can be represented as
(9) 
where represent various interference sources and is the transmit power of the vehicle. For a vehicle , we denote its distance to the typical receiver as . Although the random variables , are identically distributed, they are not independent in general [7]. However, authors in [7] have shown that this dependence is weak and we will henceforth, assume each to be i.i.d. The expression for is given as
(10) 
By taking expectation over , we get
(11) 
By assuming all as independent, we obtain
(12) 
Based on the fact that , we obtain
(13) 
Now, we calculate the Laplace transform of interference for each category of road ( and ) in the following. Let us denote the outer circular region in which all roads exist as , and inner circular region with radius having minimum distance between transmitter and receiver as shown in Fig. 2. Let us denote two types of road as and , where are the roads intersecting the circular region , and are the roads lie outside the circular region and within circular region . Thus, road is located at distance, and in case of , it is located at distance . In this case, the interferers can be located anywhere on road as it is located outside . However, in case of , the interferers will be located in regions between and . In the following, we calculate the Laplace Transform for the interferences from the vehicular transmitters located in these two types of roads (i.e., and ).
Corollary 2 (Laplace Transform of Interference for Single Road Located Outside Inner Circular Region, )
The conditional Laplace transform of interference at typical receiver, originating from a single road located at a distance (), outside inner circular region, with radius is expressed as
(14) 
For , the closed form solution can be simplified as
(15) 
Proof 5
See Appendix D.
Corollary 3 (Laplace Transform of Interference for Single Road Intersecting the Inner Circular Region, )
The conditional Laplace transform of interference at typical receiver, originating from a single road located at distance (), intersecting the inner circular region, with radius is derived as
(16) 
Based on the results in Corollary 2 and 3, we can derive the Laplace transform of the interference from the vehicles located on the road passing through the origin and that located on all other roads in Corollary 4 and 5, respectively.
Corollary 4 (Laplace Transform of Interference from Vehicles Located on Road Passing Through Origin)
The conditional Laplace transform of interference at typical receiver, originating from road located at a distance of is derived as
(17) 
where is the Hypergeometric function and is the SINR threshold. For , the closed form expression for the above equation is simplified as
(18) 
Proof 7
Corollary 5 (Laplace Transform of Interference from All Roads Excluding Road Passing Through Origin)
The conditional Laplace transform of the total interference at the typical receiver, originating from vehicular transmitters located on all roads except the road passing through the origin is derived as
Proof 8
See Appendix E.
With the help of Corollary 4 and 5, we can derive the success probabilities for the V2V link and that for the V2B link for a given distance in the following theorems.
Theorem 1 (Success Probability of the V2V Shared Link)
Proof 9
The success probability of the V2V shared link for given minimum distance, between transmitter and receiver and operating mode is
(22) 
Using Eq. (2), the above equation can be written as
(23) 
With mathematical simplification, the final equation for the success probability of the V2V link for given minimum distance, between transmitter and receiver is proved in Eq. (1).
Theorem 2 (Uplink Success Probability for the V2B Link)
The uplink success probability for the V2B shared link for given minimum distance, between transmitter and receiver is derived as
(24) 
where and are given in Eqs. (17) and (VA) by substituting .
Proof 10
The success probability for the V2B link for given minimum distance, between transmitter and receiver is presented as
(25) 
By using Eq. (2), the above equation can be written as
(26) 
With mathematical simplification, the final equation for the success probability of the V2B link for given minimum distance, between transmitter and receiver is proved in Eq. (2).
Corollary 6 (Success Probability of Cellular V2X Network)
The success probability of cellular V2X network is derived as
(27) 
where and are given in Eqs. (17) and (VA) by substituting and or , is given in Eq. (35), and is given in Eq. (C).
Proof 11
By using total probability law, the success probability of V2X network is
(28) 
where is given in Eq. (1), is given in Eq. (2), is given in Eq. (35) and is given in Eq. (C). By removing the distance ( and ) condition on Eq. (11), the V2X overall success probability is proved in Eq. (6), which completes the proof.
For the purpose of comparison, we derive the success probability of vehicular communication without involvement of cellular network in the following:
Vi Numerical Results
In this section, the association probability of V2V link, association probability of V2B link, and the success probability are plotted using (3), (4), and (6), respectively. We also plot the success probability of the cellular based V2V and V2B link along with its association probability using the first part of Eq. (6), and the second part of Eq. (6), respectively. The analytical results are validated by Monte Carlo simulations as shown in each figure. In all the figures, we set the path loss at and the thermal noise spectral density, 174 for 10 MHz bandwidth. The transmit power of vehicles are set to be 30 dBm. For comparison purposes, only V2V communication without cellular networks has also been plotted using (7) to exhibit advantages of cellular V2X communication over V2V communication. In the figures, “analyt.” represents analytical plot, “sim” represents simulation plot, “CV2V” represents cellular based V2V communication, “CV2B” represents cellular based V2B communication, “V2X” represents cellular V2X communication, and “V2V” represents V2V communication without cellular.
Via Impact of the SINR threshold
In this subsection, we examine the effect of SINR threshold, on the success probability of the proposed model. In Fig. 5, we set = 0.001 Km/Km, = 0.1 nodes/Km, = 0.00002 BS/Km and . Fig. 5 plots the success probability of the V2X communication and V2V communication at the typical receiver versus the SINR Threshold. Following insights are observed: 1) The cellular V2X network performs better than V2V communication in the range of dB to dB SINR threshold because both V2V and V2B links are contributing in achieving the better reliability of V2X communication. However, after that, the success probability of V2X communication completely matches with V2V communication as there is no contribution by V2B link. 2) We see that the reliability advantage is more significant in V2X communication over V2V communication once vehicle intensities are low or medium. 3) We observe that in a highly dense V2X network, most of the vehicles connect via V2V link instead of V2B link.
ViB Impact of the vehicle intensity
In this subsection, we examine the effect of vehicle intensity, at success probability and association probabilities of V2X network. In Fig. 6, 7, we set = 0.005 Km/Km, = 0.00002 BS/Km, dB and .
Fig. 6 plots the success probability at the typical receiver versus the intensity of vehicular nodes. The following insights can be observed: 1) The success probability of V2X communication decreases as the intensity of vehicles on the roads increases due to decrease in V2B success probability. 2) In low and medium intensity networks, there is almost equal probability of connecting to V2V or V2B link. However, in highly dense networks, this advantage of cellular V2X communication over V2V communication reduces to certain extent. The success probability of V2X communication are still better than V2V communication. 3) We see that the cellular based V2V link success probability increases at faster rate with the increase of vehicle nodes. This is because of reduction in distance between the typical receiver and vehicle located on the lines that are closer to the origin. However, the distance between typical receiver and vehicles located on the lines that are farther away from the origin does not decrease at same rate due to effect of perpendicular distance of roads. This increases the desired signal power at a faster rate than the interference power, thus improving the SINR and hence the success probability at the typical receiver.
In Fig. 7, we plot the association probabilities of V2V and V2B links versus the intensity of vehicles. From the figure, we observed that with the increase of vehicular node intensities, more vehicles start to use the V2V link as the distance between the vehicles reduces in highly dense network. On the other hand, in a low and medium intensity V2X networks, both V2V and V2B links are used by the vehicles.
ViC Impact of the road intensity
In this subsection, we examine the effect of road intensity, at packet success probability of V2X network and V2V communication. In Fig. 8, we set = 0.005 nodes/Km, = 0.00002 BS/Km, and dB.
Fig. 8 plots the success probability at the typical receiver versus the intensity of roads. The following insights can be observed: 1) The success probability of V2X communication in low and medium intensities are much better than V2V communication. 2) In densely populated area with roads, the gain of V2X communication over V2V communication reduces. However, the success probability of the cellular V2X networks is still better than that of the V2V communication. From detailed analysis of results, we see that with the increase of road intensity, the V2V serving distance decreases by bring nodes closer due to which success probability increases and vehicles start using the V2V link instead of the V2B link.
ViD Impact of the basestation intensity
In this subsection, we examine the effect of BS intensity, at the success probability of the V2X communication and sorely V2V communication. In Fig. 9, we set = 0.005 nodes/Km, = 0.005 Km/Km, and dB.
Fig. 9 plots the success probability at the typical receiver versus the intensity of BSs. The following insights can be observed: 1) The success probability of cellular V2X network continuously increases at steady rate due to continuous rise in success probability of V2B link. 2) We observed that with the BS densification, the probability of having basestation in near vicinity to vehicular transmitter is higher than the PLP based V2V link distance, because the BSs are uniformly distributed instead of nonuniform PLP distribution of roads. 3) The success probability of the V2V communication remains constant as variation of BS intensities has no impact on V2V communication. 4) It is expected that in current heterogeneous networks where the intensity of BS is mostly high and sufficient, the number of cellular BSs are available to provide highly reliable coverage to vehicular networks. From this analysis, we can conclude that V2X communication is going provide better reliability performance than V2V communication in current deployment scenarios of cellular networks.
ViE Impact of the association bias
In this subsection, we examine the effect of association bias, at packet success probability of V2X network and V2V communication. In Fig. 10, we set = 0.005 nodes/Km, = 0.005 Km/Km, and dB.
Fig. 10 plots the success probability at the typical receiver versus the association bias. The following insights can be observed: 1) The success probability of V2X communication is much higher than V2V communication as both cellular based V2V and V2B links supplement each other for success probability of V2X communication and there are no coverage gaps for the network. 2) We observed that there is an increase in the success probability of the V2X communication with increase in values of bias, because the success probability of V2V link initially increases at faster rate. 3) We see that at lower bias values, there is an equal opportunity for both links of being selected. At , there is equal probability of association with cellular based V2V or V2B link. 4) The success probability of V2V communication remains constant as association bias has no effect on this type of communication. 5) We can summarize that traffic loading and interference to cellular networks can be controlled through association bias.
Vii Conclusion
In this paper, we presented a comprehensive and tractable analytical framework for the reliability performance of cellular based Vehicletoanything (V2X) communication in which vehicular communication can be established through cellular network or directly between vehicles on shared links. A flexible V2X link selection scheme has been proposed for the vehicular transmitter to decide between the vehicle to vehicle (V2V) and the vehicle to base station (V2B), with a bias factor controlling the amount of vehicular interference and traffic on the cellular network. By modeling the vehicles on roads as doubly stochastic Cox process, and the BSs as 2D PPP, we derived the expressions for the success probabilities of the V2V link, the V2B link, as well as the V2X link, which are validated by the simulations. By comparing the proposed V2X communication with the solely V2V communication, we shown the reliability enhancement brought by the shared communication via cellular networks. Future works can be extended to interference mitigation techniques for cellular V2X network.
Appendix A Proof of Lemma 1
The Complementary Cumulative Distribution Function (CCDF) of
can be calculated by conditioning over the number of roads, in the disk with radius and calculating the probability that no vehicle exists on any of these road having variable lengths, inside the disk. This can be calculated by two independent events i.e. (a) There are no vehicles on any road within circle of radius excluding the typical line passing through the center and (b) There are no vehicles on line passing through the origin.(30) 
Let us define as the number of roads in the disk centered at origin. Similarly, let us denote as number of vehicles on a particular road. Therefore, the CCDF of can be calculated as follows
(31) 
where step (b) is calculated by conditioning over number of roads in the circle with radius and deconditioning over all possible values that there are no vehicles on any of the road. In step (c), we use the fact that the number of roads in disk is a Poisson random variable with mean, . The step (e) has been obtained by using Taylor series expansion . The step (f) has been obtained by using the fact that . By using step (f), the CCDF of is as follow
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