Interference Characterization in Cellular-Assisted Vehicular Communications With Jamming

The automobile industry has been evolving with rapid growth over the last two decades. Conventionally, the narrow band dedicated short-range communications is used for the wireless connectivity of vehicular networks in the automobile industry. However, due to its shorter wireless range, higher latency, and lower capacity; cellular-assisted vehicular communications can be adopted to improve networks performance by disrupting the nearby infrastructure. The performance of cellular-assisted vehicular communications is typically compromised if jammers are present in their vicinity. Thus, it is important to characterize the interference in cellular-assisted vehicular communications with jammers. In this paper, we investigate jamming interference due to clustered-jammers for vehicular networks by considering that vehicles, road-side units, and pedestrians are modeled using an independent one-dimensional Poisson line process while cellular base stations are modeled using an independent two-dimensional homogeneous Poisson point process. The results show that jamming disrupts the performance of the vehicular networks and that this performance is severely compromised by increasing the density of jamming clusters and their transmission power. Therefore, the prime focus of the system designers should be to introduce anti-jamming scenario, when the transmission power and the number of jamming clusters in a network increase.


I. INTRODUCTION
There has been rapid growth in the automobile industry over the past two decades. The automobile industry facilitates the wireless connectivity of modern vehicular networks to solve issues such as road accidents, traffic congestion, etc. The automobiles can communicate with pedestrians, infrastructures, vehicles, road-side units (RSUs), cloud platforms, etc., and thus, evolve the concept of vehicle-to-everything (V2X) communications [1], [2]. The third-generation partnership project (3GPP) defines cellular V2X (C-V2X) communications as vehicle-to-vehicle (V2V), vehicle-to-infrastructure (V2I), and vehicle-to-pedestrian (V2P) [3], [4] (see, Fig.1). Traditionally, a dedicated PC5 sidelink interface in C-V2X communications can support robust and enhanced network capacity in vehicular communications without requiring the need for cellular infrastructure. This allows vehicular The associate editor coordinating the review of this manuscript and approving it for publication was Ángel F. García-Fernández . networks to autonomously select their radio resources and communicate in C-V2X communications.
Existing vehicular communications consider the protocols of a narrow band dedicated short-range communications (DSRC) as a standard for V2V communications [5], [6]. However, the DSRC protocol has several limitations in terms of its performance. For instance, in the U.S., the wireless access vehicular environments (WAVE) exploit the DSRC protocol along with the carrier sense multiple access (CSMA); however, the performance of DSRC protocol with CSMA decreases for congested roads due to rapidly changing topology. Moreover, DSRC protocol mainly depends upon the RSUs which are unable to support a variety of vehicularbased applications. Furthermore, due to its short-range, the communication between entities with DSRC protocol is typically restricted to 300 meters [4]. Nevertheless, deploying and managing the infrastructure for RSUs is expensive.
To address the above-mentioned limitations of DSRC protocol in vehicular communications and to provide lower latency with higher reliability and capacity; cellular-assisted vehicular communications have been proposed in several researches [7]- [10], where networks' infrastructure such as cellular BSs is used for vehicular communications. Particularly, C-V2X has been defined in 3GPP and illustrates the key features like direct and cellular-assisted vehicular transmissions. Furthermore, in C-V2X communications, RSUs can be replaced by entities of long-term evolution (LTE) and fifth generation (5G); to reduce management and cost issues. The performance of the communication technologies in C-V2X communications is often compromised by intentional-jamming attacks [11]- [13]. For instance, communication between a platoon of tanks or armored ground vehicles can be disrupted by jamming attacks, if jammers are intentionally placed in the vicinity. Similarly, the performance of the aerial vehicles can be disrupted by jammers, if jammers are located in the region-of-interest [14], [15]. The most common types of jamming attacks in vehicular communications include denial of service (DoS), Sybil attack, malware, spam, man in the middle attack (MiM), etc. [11], [16]. The jamming attacks in C-V2X communications can pose threats to wireless interface, software, and hardware. Moreover, jammers can also damage the sensors in vehicles and the infrastructure behind the wireless access. Such jamming attacks are very critical and can disrupt the performance of C-V2X communications. Therefore, it is crucial and important to investigate jamming attacks in C-V2X communications for better network performance.
In C-V2X communications, networks' performance is investigated in [17]- [19] without considering the randomness of the vehicular locations and roads which is an important aspect for signal-to-interference ratio (SIR) distribution and coverage-performance. The randomness of vehicles and roads is stochastically modeled in [20], [21] with Matern hardcore point processes (MHPPs). MHPPs consider that vehicles located within a certain radius are not allowed to transmit and that vehicles are distributed in the region-ofinterest using a homogeneous 1-dimensional (1D) point process. However, the studies are limited to single lane scenarios and do not consider multi-lane and orthogonal road scenarios. In [22]- [24], authors modeled vehicular networks by considering road segments as a Poisson line process (PLP), vehicles along those roads as an independent 1D homogeneous Poisson point process (PPP), and cellular base-stations (BSs) as an independent 2D homogeneous PPP; and improved the performance of a network. However, the work lacks the performance analysis of C-V2X communications under jamming attacks; which can extensively disrupt networks' performance in terms of its SIR distribution and success in coverage.
In this paper, we consider vehicular communications under intentional-jamming attacks. Previously, intentionaljamming is analyzed for V2V communications in multiple studies [25]- [28]. However, the studies are limited to direct communications. To the best of the authors' knowledge, this is the first research effort to investigate intentional-jamming for cellular-assisted vehicular communications. Particularly, we consider intentional-jamming due to clusters by modeling jammers as a clustering process [14], [15] and investigate networks' performance by modeling cellular BSs as a homogeneous PPP; while vehicles, pedestrians, and RSUs as a PLP [4], [5]. The novel contributions of this paper include i) Investigating the effect of jamming interference due to multiple clustered-jammers in cellular-assisted vehicular communications. ii) Deriving the analytical expression of success probability for V2V, V2I, I2V, and overall success probability of cellular-assisted vehicular communications in the presence of jammers. iii) Evaluating probabilities of association for direct and cellular-assisted vehicular transmission in the presence of jamming interference. iv) Prioritizing focus on anti-jamming when the transmission power of jammers and number of jamming clusters are increased in the network.
The rest of the paper is organized as follows. Section II provides details for the system model of vehicular communications, Section III investigates the success probability of a network with V2V communications, V2I communications, I2V communications, and overall success probability of cellular-assisted V2X transmissions. Section IV presents results and their discussion. Finally, Section V concludes this paper.

II. SYSTEM MODEL A. INFRASTRUCTURE DEPLOYMENT
Consider a vehicular network comprising of vehicles, RSUs, pedestrians, and cellular macro BSs (MBSs) (see, Fig.2). The road system in vehicular communications is modeled by motion-invariant PLP L = {L 1 , L 2 , . . . , } with line (road) intensity λ L and is referred in the representation space as C ≡ [0, 2π)×[0, ∞) with corresponding line process C and average length per unit space as µ L = λ L /π [5], [23], [29]. The vehicles are modeled on each line using an independent and 1D homogeneous PPP v with vehicle intensity µ v . RSUs and pedestrians are typically located near roads, thus they are also modeled on each line using an independent and 1D homogeneous PPP u and p with intensities µ u and µ p , respectively. Remind that v , u , and p are stationary point processes and the resulting intensity of these point processes is given by µ V = µ v + µ u + µ p [4]. Furthermore, MBSs are modeled using an independent and 2D homogeneous PPP M with intensity λ M and it is assumed that at least one vehicle is connected to an MBS in uplink.
Assume that nodes (e.g., vehicles, RSUs, or pedestrians) transmit with probability p independently at a particular frequency such that nodes at each road transmit with intensity µ t = pµ V . Similarly, nodes receive transmissions independently at each road with intensity µ r = (1 − p)µ V . Thus, the transmitting and receiving node locations can be modeled with doubly stochastic processes, referred as t and r , respectively [4], [23].
By Slivnyak's theorem [29], the distribution of a point process remains the same if a point is translated to origin. Thus, for simple tractability, origin o ≡ (0, 0) is translated to the location of a typical receiver. This can be achieved by first placing a point at the origin and passing a line L o through it using a thinned PLP L o ≡ L ∪ L o and then, adding a point or a receiving node (e.g., o) at the origin using 1D homogeneous PPP on a line L o that passes through the origin. The Line L o is then referred as a typical line. Thus, the thinned point process of transmitting and receiving nodes (i.e., t o ≡ t ∪ L o ∪ o and r o ≡ r ∪ L o ∪ o) with density µ t o and µ r o , respectively is the superposition of original transmitting and receiving point processes t and r with intensities µ t and µ r , respectively.

B. JAMMERS DEPLOYMENT
The performance of vehicular communications decreases if jammers are present in the vicinity of jamming site [14], [30]. Typically, low-power ground-jammers are intentionally placed around a jamming site or receiving node in clusters. Thus, the legitimate communication in vehicular networks is disrupted by intentional-jamming created by groundjammers. Similar to [14], [15], we consider ground-jammers that can jam signals at multiple frequencies (e.g., they can jam F1 and F2 in Fig. 1) and can be modeled using a Matern cluster process (MCP) J which consists of parents (or number of jamming clusters) with density λ j and children (or number of ground-jammers in each cluster) with cluster-radius R j such that the density of ground-jammers in a network is given as λ J = λ jc , wherec is the average number of children (or ground-jammers) per cluster.

C. SIGNAL PROPAGATION
The vehicles, RSUs, and pedestrians transmit with the same power P V , while the MBSs transmit with the power P M . The distance-dependent pathloss model is used between transmitting and receiving nodes and the received power decays as r −α e with propagation distance r, where α e > 2 is the pathloss exponent (i.e., e ∈ {V , M }). The channel gain for nodes and MBSs is given as h V and h M , respectively that is independent and identically distributed (i.i.d) and is assumed to be slow-flat Rayleigh fading such that the fading gains are exponentially distributed with mean power one [4], [5], [31], [32]. The received signal power from vehicles, RSUs, or pedestrians at a distance r is given as while the received signal power from MBSs at a distance r is given as S M = P M h M r −α M . Furthermore, jammers transmit with power P J . The notations given in this paper are summarized in Table 1.

D. ASSOCIATION POLICY
We consider that a typical receiver associates itself with either an MBS or a vehicular node (such as vehicle, RSU, or pedestrian) based on maximum average received power. This can be obtained by comparing maximum received power from MBSs and vehicular nodes at the typical receiver. For instance, a typical receiver associates with an MBS if , otherwise, it associates itself with the vehicular node, where r M and r V are the distance between MBS and origin and vehicular node and origin, respectively.

E. SIR OF DIRECT AND CELLULAR-ASSISTED VEHICULAR TRANSMISSIONS
We assume that vehicular networks adopt either direct transmissions (like vehicles transmitting to vehicular nodes) or cellular-assisted vehicular transmissions (i.e., V2I and then I2V) by considering link-distances, environmental impacts, and transmission power. Additionally, we consider the scheme for channel access as a slotted-ALOHA; which approximates the performance of carrier sense multiple access with low-and high-node intensities [21], [33]. Furthermore, similar to [4], we assume that the transmission link in between vehicular nodes and between vehicular nodes and BSs share their resources. Note that we do not exploit PC5-based dedicated sidelink for C-V2X communications addressed in 3GPP. Moreover, we consider interferencelimited vehicular communications by ignoring thermal noise power and assuming τ as the pre-defined SIR threshold for success in coverage event.
The SIR in the presence of jammers at a typical receiving entity 1 with transmissions T ∈ {D, C}, where D represents direct transmission and C represents cellular-assisted vehicular transmission and is expressed as where P e is the transmission power of the entities (i.e., vehicles, RSUs, pedestrians, or MBSs), h e is the fading gain of entities, I e is the interference power of entities, and I J is the jamming interference at the typical node. Particularly, for direct transmission, the SIR at a vehicular node in the 1 In the rest of the article, entity refers to MBSs, RSUs, pedestrians, and vehicles. While, vehicular nodes refer to RSU, pedestrians, and vehicles.
presence of jammers is expressed as where I V represents the interference power of vehicular nodes. The SIR for cellular-assisted vehicular transmissions from a vehicular node to an MBS in the presence of jammers is given as where r V 2M is the distance between a transmitting vehicular node and an MBS. Similarly, the SIR for cellular-assisted vehicular transmissions from an MBS to a vehicular node in the presence of jammers is given as where r M 2V is the distance between an MBS and a receiving vehicular node and I M represents the interference power of the MBSs. In our system model, we translate origin to the location of a serving (typical) vehicle. Then, we find the serving transmitting (test) vehicle by calculating the maximum received power from all the vehicular nodes at a typical vehicle. Remind that the test vehicle can be located on the same road (i.e., typical line) or on a different road. The test vehicle then finds whether it should transmit to a typical vehicle directly or it should transmit via cellular MBS by calculating maximum received power from a test vehicle at MBS and vehicular node. If the maximum received power at the MBS is smaller then the maximum received power at the vehicular node; the test vehicle considers the direct transmission. While, if the maximum received power at the MBS is larger then the maximum received power at the vehicular node; the test vehicle considers the cellular-assisted vehicular transmission. Generally, there can be interfering transmission entities (like RSUs, vehicles, etc.) inside the disc of B(o, r V ) and B(o, r M ) due to transmitter receiver pairs. However, they can be ignored as our focus is the receiver that is present at the origin. After selecting direct transmission, we then calculate SIR and the corresponding success probability using (2) and (27). While, after selecting cellular-assisted vehicular transmission, we then calculate SIR and the corresponding success probabilities using (3) and (29) for transmitting to an MBS and (4) and (31) for transmitting from an MBS.

III. PRELIMINARIES
In this section, we derive distance between vehicular transmitting and receiving nodes and vehicular transmitting and cellular nodes. We also derive association probabilities of the transmitter (test) vehicle with the typical receiving vehicle and cellular BS. Furthermore, we also derive interference of vehicular nodes, cellular nodes, and jammers.

A. DISTANCE DISTRIBUTION
Here, we derive distance distribution to vehicular nodes and MBSs which will be used in deriving the association probabilities of vehicular transmitting node with vehicular receiving node and MBS.

1) DISTANCE DISTRIBUTION OF VEHICULAR NODES
The probability density function (PDF) of the distance distribution between receiving and transmitting vehicular node is derived in Appendix VI and is expressed as

2) DISTANCE BETWEEN MBS AND VEHICULAR NODE
The distance between transmitting vehicular node and receiving MBS in terms of its CDF is given as The PDF of the distance between transmitting vehicular node and MBS is expressed as Note that the distance distribution between the vehicular transmitting node and receiving MBS is the same as vehicular receiving node and transmitting MBS.

B. ASSOCIATION PROBABILITIES
Here, we derive the probability of association of the serving (transmitter) test vehicular node with the typical receiving vehicular node or the cellular BS.

1) DIRECT TRANSMISSION PROBABILITY
Direct transmission considers the direct communication of vehicular nodes. The probability that a typical vehicular node located on a typical line will associate with the transmitting vehicular node is referred as direct transmission probability or association probability of a typical receiving vehicular node with the transmitting vehicular node. The direct transmission is selected by the transmitter (test) vehicular node if the average received signal strength at the typical receiving vehicular node is equal or larger than the average received signal strength at the cellular BS. The probability that a typical transmitting vehicular node selects direct transmission by considering distance to vehicular receiving node as r V is given as Eq. (8) represents that there is not even a single cellular BS within the radius of r and that vehicular transmitter associates to vehicular receiver located at origin. Thus, the CCDF of (8) is expressed as By removing condition on distance r V and considering network's radius R, we get association probability of a typical receiving vehicular node with the transmitting vehicular node in direct transmission as 2) CELLULAR-ASSISTED VEHICULAR TRANSMISSION PROBABILITY Cellular-assisted vehicular transmission considers communication of vehicular nodes by considering cellular BSs of the given infrastructure. The probability that a vehicular transmitting node selects cellular BS for communicating to other vehicular nodes is referred as cellular-assisted vehicular transmission probability. The cellular-assisted vehicular transmission is selected by the transmitting vehicular node if the average received signal strength at the MBS from transmitting vehicular node is larger than the average received signal strength at the typical receiving vehicular node from the transmitting vehicular node. The probability that the transmitting vehicular node selects cellular-assisted vehicular transmission by considering distance to cellular BS as r M is derived as Eq. (11) represents that no vehicular node within a radius of r α M /α V M exists and that vehicular transmitter associates to cellular BS. The association probability of the cellular-assisted vehicular transmission can be obtained from CCDF of r V using (38) (i.e., 1 − F R V (r V )) and by removing condition on distance r M . It is then expressed as

C. INTERFERENCE CHARACTERIZATION
Here, interference of vehicular nodes located on a typical line (road), interference of vehicular nodes located on roads excluding a typical line, and interference of cellular BSs is derived in terms of Laplace transform of interference. Moreover, interference of jammers in terms of jamming interference is also derived. Conventionally, for vehicular networks, interference at a typical node originates from entities such as vehicular nodes VOLUME 10, 2022

1) INTERFERENCE OF VEHICULAR NODES LOCATED ON ROADS (EXCLUDING A TYPICAL ROAD)
Here, we derive interference of vehicular nodes located on all other roads excluding a typical line (road). This can be obtained by first deriving interference from a single road (excluding a typical road) and then obtain the results for all the other roads (excluding a typical road).
The interference originating from a single road excluding a typical road is referred as I V L and is derived by considering that the perpendicular distance y from origin to that road is either greater than r V or less than r V . If y r V , all the vehicular nodes located on the considered line [− R 2 − y 2 , R 2 − y 2 ] (referred as * ) will take part in the interference process (i.e., 1 (y)). Whereas, if y < r V , then only those vehicular nodes on the considered line that are located outside B(o, r V ) but inside B(o, R) (i.e., [− R 2 − y 2 , − r 2 V − y 2 ] and [ r 2 V − y 2 , R 2 − y 2 ] referred as * * ) will take part in the interference process (i.e., 2 (y)). we first derive interference from a single road in terms of 1 (y) and then 2 (y). Later on, we obtain the interference originating from all such roads excluding a typical road.
The interference of a single road lying in a disc B(o, R) has a road length of 2 R 2 − y 2 between vehicles and horizontal distance of x. Conventionally, interference from vehicular nodes at a typical receiver is expressed in terms of a Laplace transform of interference L I (s) = E[e −Is ], where s = P e y α e . Thus, interference originating from k vehicles at a typical receiver residing on the road length 2 R 2 − y 2 under Rayleigh fading environment is given as The vehicular nodes are distributed using independent and homogeneous PPP. Thus, mean number of points (vehicular nodes) lying on a line segment 2 R 2 − y 2 is given as 2µ V R 2 − y 2 and PDF of the vehicular nodes in the considered length of line segment is given as f (x) = 1/2µ V R 2 − y 2 . The probability that there are k vehicular nodes lying on a line segment 2 R 2 − y 2 is given (13) can be expressed as where x is uniformly distributed between − R 2 − y 2 and R 2 − y 2 in * . Substituting f (x) and after simple mathematical manipulations, we get Using even function property for Taylor series (i.e., ∞ k=0 e X = X k /k!, we get 1 (y) as The final expression for the Laplace transform of interference from single road excluding a typical road and from roads located at y r V is obtained by simple mathematical manipulations and is expressed as Similarly, follow the same procedure to obtained the final expression for the Laplace transform of interference from single road excluding a typical road and from roads located at y < r V and is expressed as VOLUME 10, 2022 Note that for y < r V , vehicular nodes will be located on line segments with lengths [− R 2 − y 2 , − r 2 V − y 2 ] and [ r 2 V − y 2 , R 2 − y 2 ]. The total interference of all the vehicular nodes and roads located in a disc B(o, R) (excluding a typical road) is computed by integrating 1 (y) and 2 (y) over the entire given region. Consider the number of roads t 1 intersecting B(o, r) is a Poisson process with mean 2πλ L r and the interfering vehicular nodes located at these roads are located in [r, R]. Similarly, consider the number of roads t 2 located between disc B (o, R) and disc B(o, r) intersecting disc B(o, r) is also a Poisson process with mean 2πλ L (R − r) and the interfering vehicular nodes are located in (0, r]. Then, the Laplace transform of interference originating from vehicular nodes located on all the roads (excluding typical road) can be expressed as (a) is obtained by simplifying mathematical expression and considering r 0 dr = 2 r. (b) is obtained by considering even function property for Taylor series expansion and by further simple mathematical manipulations.

2) INTERFERENCE OF VEHICULAR NODES LOCATED ON A TYPICAL ROAD
The interference of vehicular nodes located at a typical road with y = 0 can be derived by following the same procedure of 2 (r) and is expressed by Laplace transform of interference as where r V o is the horizontal distance of the serving typical transmitter to the typical receiver.

3) INTERFERENCE OF MBS
The interference of cellular BSs at a typical receiver is a well-known expression given in [34], [35] and is expressed by Laplace transform of interference as where

4) INTERFERENCE OF JAMMERS
Here, interference of clustered-jammers at a receiving entity is derived in terms of Laplace transform of jamming interference L I J . The jamming interference from various clusters located in the vicinity of a typical entity can be derived as [36] where β(z, y) the PGF of clustered-jammers. The PGF is simplified in [14], [15] for clustered-jammers as where (a) follows by changing variables, (b) follows by mathematical simplification, and (c) follows by considering ω = τ −2/α e r −2 y 2 . Furthermore, b is simplified using [37] and is given as b = R 2 exp{−cβ(z, y)}f (y)dy = exp{−J r 2 τ 2/α e 2π 2 α e csc(2π/α e )}, where J = λ jc (πR 2 j ) −1 . The final expression for the Laplace transform of clustered VOLUME 10, 2022 jamming interference is obtained by substituting the values of a and b in (22) and is expressed as

IV. SUCCESS PROBABILITY ANALYSIS
The probability that SIR of a typical vehicular node without selecting any transmission mode when exceeds a pre-set threshold value is defined as the success probability of V2V communication and is given as whereas, the probability that SIR of a typical receiving entity with a specific transmission mode T exceeds a pre-set threshold value is defined as the success probability under specific selected transmission mode and is given as The success probability directly relates to the reliability of the link connection between entities (e.g., vehicles and MBSs). Thus, the success probability of a typical vehicle with direct transmission is given as the the probability that the SIR of that vehicle exceeds a pre-set threshold value and is expressed as The analytical expression for the success probability of direct transmission in V2V and in the presence of clustered-jammers can be derived using CCDF of SIR and is given as The final expression for the success probability of direct transmission is obtained in (24), as shown at the bottom of the page, by substituting the values of (19), (20), (23), and (5), respectively in (28) by considering e = V and r = r V . For cellular-assisted vehicular transmission, the success probability of a typical vehicular node in V2I is given as the the probability that the SIR at an MBS exceeds a pre-set threshold value and is expressed as The analytical expression for the success probability of cellular-assisted vehicular transmission in the presence of clustered-jammers at an MBS can be derived as Substituting the values of L I V L (r), L I V Lo (r), L I J (r), and f R V (r), from (19), (20), (23), and (5), respectively in (30) by considering e = V , R M = R V 2M , and r = r M = r V 2M , the final expression for the success probability of cellular-assisted vehicular transmission is obtained in (34), as shown at the bottom of the next page. Similarly, the success probability of a typical test vehicular node with cellular-assisted vehicular transmission in I2V is given as the the probability that the SIR at a typical vehicular node exceeds a pre-set threshold value and is expressed as The analytical expression for the success probability of cellular-assisted vehicular transmission in the presence of clustered-jammers at a typical vehicular node can be derived as Substituting the values of L I M (r), L I J (r), and f R V (r), from (21), (23), and (5), respectively in (32) by considering e = M , R M = R M 2V , and r = r M = r M 2V , the final expression for the success probability of cellular-assisted vehicular transmission is obtained in (35), as shown at the bottom of the next page.
The overall success probability of cellular-assisted vehicular transmissions is expressed as Substituting the values of P S D (τ, r V ), A D , P S C (τ, r V 2M ), P S C (τ, r M 2V ), and A C from (28), (10), (30), (32), and (12), respectively in (33), we obtain the final expression for the success probability of cellular-assisted vehicular transmission.

V. RESULTS AND DISCUSSION
In this section, we present overall success probability results of cellular-assisted vehicular communications. The analytical results are validated using simulations with 100,000 Monte  Fig. 4 shows the effect of increasing vehicular node density on the success probability in the presence of jamming nodes. It is shown that the success probability increases by increasing the density of vehicular nodes; however, the success probability of the vehicular networks in the presence of jamming nodes is lower than the success probability of vehicular networks without jamming nodes. Moreover, the success probability further decreases with the increasing power of jamming nodes. Thus, it is suggested to introduce and focus on anti-jamming techniques in cellular-assisted vehicular communications, when the transmission power of the jamming nodes is increased in the network. Note that the success probability increases by increasing the density of vehicular nodes because the distance-dependent pathloss between transmitting test vehicular node and receiving vehicular node decreases which increases the received SIR and the success probability. Fig. 5 shows the effect of increasing vehicular node density on the association probability of transmitting vehicular node with the cellular BS and typical receiving vehicular node. It is shown that the association probability of the direct transmission (i.e., vehicular node association) increases by VOLUME 10, 2022  increasing the density of vehicular nodes in comparison with the association probability of the cellular-assisted vehicular transmission (i.e., MBS association). This is because the distance between the transmitting and receiving vehicular node decreases by increasing the density of vehicular nodes which increases the received power at the typical vehicular node in comparison with the received power at the MBS. Moreover, it is also shown that at a lower density of vehicular nodes both MBS and vehicular node association are used by vehicular networks. Fig. 6 shows the effect of increasing road density on the success probability in the presence of jamming nodes. It is shown that the success probability increases by increasing the density of roads. However, the success probability of the vehicular networks in the presence of jammers is lower than the success probability of vehicular networks without jammers. Moreover, the success probability is further decreased by increasing the power of jamming nodes. The success probability of vehicular networks increases with road density because the distance-dependent pathloss between transmitting and receiving entities decreases by increasing the density of roads in the considered region which increases the SIR and the probability of success in coverage. Fig. 7 shows the effect of increasing the density of MBSs on the success probability in the presence of jammers. It is shown that the success probability decreases by increasing the MBS density. This is because, with the increase in the density of cellular BSs, interference power also increases which decreases the SIR and success probabilities. Moreover, the success probability is further decreased by increasing the power of jamming nodes. Fig. 8 shows the effect of the SIR threshold value on the success probability in the presence of jamming clusters. It is shown that the success probability decreases by increasing the SIR threshold value. It is also shown that the success probability further decreases by increasing the density of jamming clusters because with the increasing number of jamming clusters in the region of interest, the cumulative jamming power will increase which will increase the network interference and as a result, success probability will decrease. Thus, it is suggested to introduce and focus on anti-jamming techniques in cellular-assisted vehicular communications, when the number of jamming clusters is increased in the network.

VI. CONCLUSION
In this paper, jamming interference is analyzed for cellular-assisted vehicular communications. MBSs are modeled using independent and 2D homogeneous PPP while, vehicles, RSUs, and pedestrians are modeled using 1D homogeneous PLP. The jammers are placed in clusters using MCP and the impact of jamming interference on vehicular networks is obtained in terms of the success probability. It is shown that jammers severely disrupt the success probability of vehicular networks. It is also shown that the network's performance decreases by increasing the density of jamming clusters and their transmission power. Therefore, the main focus of the network operators should be to introduce antijamming, when the transmission power of the jammers and the number of jamming clusters in the network increase.

APPENDIX A DERIVATION OF (5)
The distance between typical vehicular transmitter and vehicular receiver considers that there is no vehicular node in a disc of radius r V centered at origin (i.e., B(o, r V )) that can create interference. The test vehicle can be located either on the same typical line (road) where typical receiving vehicle is present or it can be present on any other line (road). The shortest distance of the vehicular transmitting (test) and receiving (typical) node is given as cumulative distance distribution (CDF) F R V (r V ) in [31] and is expressed as where Pr{N L (B(o, r V )) = 0} represents that the number of vehicular nodes on lines (roads) in a point process L centered at origin with radius r V is zero and Pr{N L o (−r V , r V )} represents that the number of vehicular nodes located on a typical line L o with distance [−r V , r V ]. Pr{N L (B(o, r V )) = 0} can be derived by finding the probability that there are k lines (roads) present in a disc B(o, r V ) and is given as Pr{N roads (B(o, r V )) = k}. Then, for each of the k-th line, find the probability that the number of vehicular nodes on that road is zero (i.e., Pr{N V (road i ) = 0}). Pr{N L (B(o, r V )) = 0} is derived as where (a) represents that the number of roads intersecting a disc of radius r V centered at the origin is Poisson distributed with average roads given as 2π λ L r V and the number of vehicular nodes that are uniformly distributed between [−r V , r V ] and located on a road is also Poisson distributed with average vehicular nodes per road given as 2µ V r 2 V − y 2 . (b) is obtained by simple mathematical manipulations, (c) is obtained using even function property for Taylor series (i.e., ∞ k=0 e X = X k /k!), and (d) is obtained by simple mathematics and considering r V −r V = 2r V . Thus, the CDF of distance between a typical receiving and transmitting vehicular node is obtained by substituting 2 = exp(−2µ V r V ) and (37) in (36) and is expressed as