Intelligent transportation systems (ITSs) are promising technology for improving drivers/pedestrians safety and the efficiency of transportation . In general, vehicle-to-infrastructure (V2I) and vehicle-to-vehicle (V2V) communications play a key role in achieving ITSs. These communications are commonly based on narrow-band dedicated short range protocols (DSRC). For instance, wireless access in vehicular environments (WAVE) is the protocol suite adopted in the U.S. In WAVE, IEEE 802.11p  is standardized for the media access control (MAC) and physical layers, and it uses 5.9-GHz bandwidth and carrier sense multiple access (CSMA) similar to IEEE 802.11a. In addition, IEEE 1609  designs higher-layer functions such as networking and multichannel operations.
Cooperative vehicle safety (CVS) systems  can be considered as one of the key applications of ITSs. CVS systems, working on DSRC protocols, include many applications such as cooperative collision warning, electronic emergency brake lights, and slow/stopped vehicle alerts . In these systems, vehicles periodically broadcast their information e.g., car positions (Global Positioning System; GPS), speed, braking status, traffic status, and emergency information. By exchanging the information, vehicles can track the positions of other vehicles and avoid traffic congestion, collisions, or unknown hazards. CVS systems has been attracting much attention in recent decades because their rich applications will drastically change our lives.
Because of the critical nature of CVS systems, their performance analysis and management are hot research topics. Broadcasting with a high transmit power and high broadcast rate in congested roadways may significantly degrade the wireless communication quality due to high interference. To reduce the interference caused by a large number of vehicles sharing the same channel, several adaptive control schemes of the transmission power or broadcasting rate have recently been proposed [5, 6, 7, 8, 9, 10]. However, most of these schemes are not based on theoretical analysis and are commonly evaluated through simulations. Because the environments in which V2V communications occur may quickly and frequently change, a more general understanding of performance is crucial for the effective control of CVS systems. Furthermore, most studies consider only homogeneous environments, such as multi-lane highways, in which vehicles are running in the same direction with the same traffic density. However, to deploy CVS systems in urban environments in the future, more realistic inhomogeneous situations, such as intersections, must be taken into account. Most recently, a theoretical analysis was studied toward an optimization of transmission power of vehicles at an intersection . However, the obtained analytical results are highly complicated and mathematically intractable, and thus the analysis cannot be applied to real-time control due to its high computational time.
In this paper, we perform theoretical analysis of V2V broadcast communications at an intersection and derive performance metrics in a tractable form. By expressing key performance metrics as closed-form approximate formulae, we can optimize the broadcast rate in a reasonable computational time so that the number of successful receivers per unit time is maximized. We consider an intersection model, in which locations of vehicles are separated into queueing segments and running segments. In the former segment, vehicles are assumed to be queueing, i.e., stopped, at even intervals; and in the latter, vehicles are distributed in accordance with a homogeneous Poisson point process (PPP). By using a stochastic geometry approach, theoretical values of the key performance metrics of V2V broadcast communications are obtained: the probability of successful transmission and the mean number of successful receivers. The former is defined as the probability that the signal-to-interference-ratio (SIR) of a receiver exceeds a certain threshold, and the latter as the expected number of vehicles that can successfully receive information from a transmitter. However, these results from exact analysis are expressed in non-analytical form and need numerical integrals and multi-level summations for computing. Thus, they are too complicated not only to understand the forms of the function but also to optimize their system parameters due to large computational time. To address this problem, we developed a closed-form approximation for the performance metrics of V2V broadcast communications. More precisely, when vehicles broadcast to their neighbors, the probability of successful transmission geometrically decreases as the distance to the receiver increases in our approximation. In addition, the decay rate depends only on a few system parameters, such as the density of vehicles. Using the approximate formulae, we propose a method for optimizing the broadcast rate of vehicles that maximizes the number of successful receivers per unit time. By using the closed-form expression, we can easily compute the optimal broad cast rate without time-consuming numerical computation. Therefore, our optimization method can be applied to real-time broadcast rate control for CVS systems to mitigate the interference problem caused by congestion. Numerical experiments revealed our approximation fitted well to both simulation and exact analysis in a realistic setting. We also found that the proposed optimization could improve the interference problem at an intersection.
We briefly summarize our contributions as follows.
We developed a closed-form approximation for the theoretical values of two key performance metrics of the performance of V2V broadcast communications at an intersection: the probability of successful transmission and the mean number of successful receivers. The approximate formulae depend only on system parameters and thus do not require time-consuming numerical computation.
By using the approximation formulae, we developed a method for optimizing the broadcast rate of vehicles. Balancing the broadcast rate and the mean number of successful receivers, we improved V2V broadcast communications at a congested intersection. In accordance with the closed-form formulae, the optimal broadcast rate can be obtained in a reasonable computational time and thus can be applied to real-time broadcast rate control.
We evaluated our approximation and optimization methods through simulation and numerical experiments. We found that our optimization method had sufficient performance. We also found that our approximation fitted well to the results from simulation and exact analysis in a realistic setting.
The remainder of this paper is organized as follows. Section II summarizes previous studies. In Section III, we explain the proposed model considered in this paper. Section IV presents the main results of this paper, the approximate analysis of the key performance metrics of V2V communications at an intersection. In Section V, we explain a broadcast-rate-optimization method based on the analytical results. Finally, we discuss several numerical experiments in Section VI, and conclude the paper in Section VII.
Ii Related work
Due to the importance of ITSs, there have been a lot of studies in the area of the performance evaluation of V2I/V2V communications in the past decade. Most of the earlier work is simulation-based [9, 12, 13, 14]. However, simulation-based approaches often require much computational time and resources. The previous work [15, 16, 17, 18]
conducted a theoretical analysis of the CSMA behaviors of IEEE 802.11p on the basis of a Markov chain model approach. Fallah et al. studied the impact of the rate and range of broadcasting on network performance in a highway environment considering the hidden terminal problem. Han et al.  and Yao et al.  analyzed the enhanced distributed channel access (EDCA) behavior in IEEE 802.11p, in which different access categories have different contention windows and arbitration inter-frame space. However, these studies did not consider the geographical effects or interference in V2V communications and assumed only simple communication scenarios.
To reduce the interference of V2V broadcast communications, several adaptive control schemes for transmission power [5, 9, 10] or broadcasting rate [5, 6, 7, 8] have recently been proposed. The method proposed by Moreno et al.  adaptively controls the transmission power of vehicles so that their max-min fairness is satisfied. In 
, a segment-based power control method based on a distributed vehicle density estimation algorithm is proposed. Huang et al. developed broadcast rate and power control algorithms, in which the rate is determined by estimating the channel error rate and the power is determined by observing the channel status. Tielert et al.  introduced a rate adaptation algorithm based on the channel busy ratio. Most recently, Fallah et al.  updated the algorithm of  so that the power changes in each iteration can be configurable and stable. None of the adaptive control methods above was based on theoretical interference analysis and were considered in simple environments such as multi-lane highways, in which vehicles are running in the same direction with the same traffic density. However, theoretical guidelines for more realistic situations, such as intersections, are crucial to deploy CVS systems in more complex urban environments.
Stochastic geometry is a powerful mathematical tool for modeling random spatial events  and has recently been applied to the area of wireless communication [20, 21] including vehicular ad-hoc networks (VANET) [22, 23, 24, 25]. By modeling the locations of communication devices, such as vehicles and road side units (RSUs), as a spatial point process, theoretical values of various performance metrics can be calculated. Such mathematical understanding of the ITS system not only frees us from time consuming simulation but also helps in optimization or sensitivity analysis of system parameters. In previous studies [24, 25], the behavior of CSMA used in DSRC was analyzed. More specifically, Nguyen et al.  showed that CSMA behaves like an ALOHA-type transmission pattern in dense networks and derived the theoretical expression of performance metrics in broadcast V2V communications while assuming that vehicles are distributed in accordance with spatially homogeneous PPP. In , the performance of DSRC in both the spatial and time domains was studied by using a Markov chain model approach for CSMA, which is similar to that of Nguyen et al. . However, these studies focused on MAC behaviors of DSRC, considered only simple situations and did not take into account the impacts of heterogeneous urban structures, such as intersections.
Iii Model description
In this section, we explain the system model. Figure 1 shows a conceptual image of our model. We consider an intersection where two streets are crossing. One of the streets runs parallel along the -axis, and the other along the -axis. On the street along the -axis, there are vehicles queueing, i.e., stopped, at the intersection, and on each street, vehicles are running. We call these parts a queueing segment or a running segment . In addition, and denote the running segments on the - or -axis, respectively, i.e., . We assume that vehicles in are distributed in accordance with a homogeneous PPP on each street. Let and [km] denote the intensity of vehicles in and . Let and denote the numbers of vehicles stopped at an intersection in each part, where the subscript represents the positive or negative direction on the -axis. We assume that vehicles have length and the widths of the streets are negligible. Note that there is no queue on the -axis because we consider the case where the traffic signals on the -axis are green. We can easily apply the same discussion in this paper to the case where those on the -axis are green.
We next explain the channel model considered in this paper. Vehicles periodically broadcast radio signals, such as GPS, their braking information, or emergency information to neighbors, i.e., V2V communications. All vehicles are assumed to be equipped with devices necessary for V2V communications. We assume that each vehicle requires [sec.] for each transmission and vehicles in independently transmit with rate [Hz] and those in with . Thus, the probability that each vehicle in (resp. in ) is transmitting is equal to [resp. ]. We also assume that vehicles currently transmitting cannot simultaneously receive information from other vehicles. The transmission power of each vehicle is assumed to be equal to
and constant. Antenna gain is assumed to be equal to 1 throughout this paper. All transmission channels have the effect of Rayleigh fading the random variable of which is denoted asfor vehicle . The path loss model is for distance , where is a path loss exponent. Thus, for instance, the received power from vehicle at distance can be expressed as . Table I summarizes the notations used in this paper.
|length of vehicle|
|set of vehicles running on street|
|set of vehicles running on street along -axis|
|set of vehicles running on street along -axis|
|set of vehicles stopping/queueing at intersection|
|each transmission time|
|broadcast rate of vehicles in and|
|probability that vehicles in and are transmitting|
|intensity of vehicles in|
|number or vehicles stopped at intersection|
|transmission power of vehicles in|
|interference from vehicles in s|
|interference from vehicles in s|
Note that CSMA is designed as the MAC layer protocol in IEEE 802.11p . Since vehicles that are close to each other do not transmit simultaneously in CSMA, hard-core point processes have been used for modeling such CSMA-based protocols [25, 26, 28]; however, they are not mathematically tractable because they are obtained by dependent thinning of a PPP. In addition, Nguyen et al.  claimed that CSMA behaves like an ALOHA-type transmission pattern in dense networks. Therefore, in this paper, we assume that transmitting vehicles use the ALOHA-type MAC protocol and follow a PPP because we focus on the congestion at the intersection and maintain the mathematical tractability.
Let denote a random variable representing the total received power, i.e., interference power, from all the vehicles. If we consider the target channel in which the communication distance is equal to , the SIR can be written as
We then define the probability of successful transmission as the probability that the SIR of the target receiver exceeds a threshold , i.e.,
where is the Laplace transform of . Probability is a key performance metric of V2V broadcast communications, and thus, we mainly focus on the analysis of this value in our paper.
Iii-a Performance Metrics
In this section, we provide theoretical expressions of performance metrics of V2V communications via a stochastic geometry approach. Since equation (1) shows that can be obtained as the Laplace transform of the interference from vehicles, we first consider the interference distribution and then provide the theoretical values of the probability of successful transmission.
Iii-A1 Interference distribution
We first consider the interference from vehicles in . For this purpose, we assume that the target receiver is in the positive part on the -axis and at distance from the intersection. In addition, let denote the distance between the target receiver and the -th vehicle from the intersection. We then have the following total interference power received from .
where if the -th vehicle transmits, and otherwise. Note that is exponential with mean according to the Rayleigh fading assumption. Note also that the vehicles in the are transmitting with the probability . Therefore, the Laplace transform of can be obtained as
We next consider the interference from the vehicles in . Similar to the previous case, we assume that the target receiver is at distance from the intersection in the positive part on the -axis. Let and denote PPPs corresponding to and . The total interference from can be represented as . Recall that vehicles in transmit with probability . By following a well-known computation of the Laplace functional of the Poisson point process (see e.g., Proposition 1.5 and Corollary 2.9 in ), we can compute the Laplace transform of as follows.
Note that the distance from the tagged vehicle to a vehicle at distance from the intersection on the -axis is equal to . Thus, if denotes the total interference from , we have . Therefore, similar to (3), we obtain (see also Section 2 in ),
Iii-A2 Probability of successful transmission
Note that the Laplace transform of the total interference from all the vehicles can be represented as
Suppose that the target transmitter is at distance from an intersection. The probability of successful transmission to a vehicle at distance from the transmitter on the -axis is equal to
where if the receiver is on the right-hand side of the transmitter, and otherwise.
Iii-A3 Mean number of successful receivers
Using Proposition III.1, we can also obtain the mean number of successful receivers, which is defined as the expected number of vehicles to which the target transmitter can transmit. This metric can be considered as a key performance metric of V2V broadcast communications and is also considered in  under a homogeneous PPP environment. Note that there are three types of receivers: vehicles in , in , and in . Recall also that vehicles transmitting radio waves cannot simultaneously receive information from other vehicles. From these facts, we obtain the following result.
If the target transmitter is the -th vehicle from an intersection, the mean number of successful receivers is given by
Iv Approximate Formulae of Performance Metrics
Although theoretical values of the performance metrics can be obtained as in Propositions III.1 and III.2, they are expressed in non-analytical forms [see (2)–(7)]. Therefore, it is difficult not only to see the impacts of various parameters on them but also to optimize their system parameters because numerical integrals and multi-level summations are included. Thus, if we apply any numerical optimization methods to obtain optimal system parameters, a large computational time is required because of these heavy computations in each step of the numerical methods. To solve this problem, we attempt to obtain a closed-form approximation for and that depends only on system parameters. We then propose an optimization method for the broadcast rate of vehicles in (see Section V). In accordance with the closed-form approximation, we can solve the optimization problem in a reasonable computational time, and thus, it can be applied to real-time broadcast rate control for CVS systems.
In general, the characteristics of and depend on the location of the target transmitter. To obtain closed-form formulae, we consider three typical locations of the target transmitter instead of considering arbitrary locations of the transmitter: the target transmitter is in the positive direction on the -axis and (A) at the intersection, (B) at the end of the queue, and (C) in the middle of the queue (see Figure 2). Since a vehicle at (or near) the intersection is affected by interferences from both directions (- and -axis directions) and queues, the vehicle is expected to have the worst performance, and its performance can be approximated by case (A). A vehicle at (or near) the end of the queue is said to be in an intermediate state of vehicles between the queueing and running segments. Thus, we approximate it by considering case (B). The performance in this case can characterize the traffic heterogeneity at the intersection. In case (C), the transmitter is far from both the end of the queue and the street along the -axis. Thus, if the queue is sufficiently long, the performance in this case can be approximated as vehicles stopping at even intervals on a 1-d line with infinite length.
Note that the performance of vehicles at other positions can be estimated by interpolating or extrapolating those in cases (A)–(C) (detailed discussion is given in SectionVI-C). Note also that if a vehicle is in and far from the end of the queue, its performance can be estimated by considering vehicles homogeneously distributed on a 1-d line, i.e., the effect of queues is negligible. Thus, we do not consider this simple case in this paper. In addition, we can consider the case where the transmitter is in by combining case (A) and a model with a single street. Therefore, we analyze cases (A)–(C) because they characterize the effect of the intersection.
As we can see later, we can calculate the analytical values of and in special cases, such as . However, the term , i.e., the interference from , cannot be expressed in an analytical form even in such cases. Therefore, we mainly focus on giving a closed-form approximation for in this paper. The obtained approximate formulae for basically hold under conditions in which and queue lengths and are sufficiently large.
Iv-a Case (A): Transmitter at Intersection
We first consider case (A), where the transmitter is at an intersection. As mentioned in Section III-A3, there are three types of receivers: a receiver in , in , and in .
Iv-A1 Probability of successful transmission
We first provide approximation for the probability of successful transmission when transmitting to a receiver in . Note that if a receiver is in and the -th vehicle from the intersection, the communication distance is equal to . Detailed explanation for the derivation of the formulae below is given in Appendix A-A.
Approximate formulae of in case (A): Suppose that the transmitter is at an intersection. If 111A typical value of the outage threshold is 10–15 dB (for example, dB () in IEEE 802.11p). In addition, the optimal was often less than in our experiments. Thus, this assumption can be considered as valid. In addition, we can also derive an approximate formula for other cases using the results in Appendix A. and , the probability of successful transmission can be approximated as follows. (i) If a receiver is the -th vehicle from an intersection, then
where, for ,
(ii) if a receiver is in at distance from the intersection, then
and (iii) if a receiver is in at distance from the intersection, then
The approximate formulae (8), (12), and (13) suggest that, in our approximation, the probability of successful transmission decreases geometrically with the distance to receivers. For example, if a receiver is in , then the geometric decay rate is equal to
Iv-A2 Mean number of successful receivers
From the results in the previous section, we can approximate the mean number of successful receivers. Since the approximate formulae of are expressed in a geometric form, we can also obtain a closed-form approximation for . Let , , and denote the mean numbers of successful receivers in , , and , respectively. First, applying (8) to Proposition III.2 and considering sufficiently large and , we obtain
Iv-B Case (B): Transmitter at End of Queue
We next consider case (B), where the transmitter is at the end of the queue. In this case, the transmitter is far from the -axis due to the queueing segment. Therefore, the interferences from the vehicles in and the receivers in are both negligible. This case can be divided into three sub-cases: a receiver is in (i) , in (ii) in the negative direction, or (iii) in the positive direction, i.e., the left-hand side of the transmitter or the right-hand side (see Figure 2).
Iv-B1 Probability of successful transmission
As mentioned above, the interference from the vehicles in is relatively much smaller than that from and regardless of the position of a receiver. Thus, the term is negligible, i.e.,
We then have the following results, in which also decreases geometrically as increases; however, the decay rate is different from that in case (A). Detailed explanation for the derivation of the formulae below is given in Appendix A-B.
Approximate formulae of in case (B): Suppose that the transmitter is at the end of the queue and . If , the probability of successful transmission can be approximated as follows. (i) If a receiver is in and the -th vehicle from the end of the queue, then
(ii) if a receiver is in in the negative direction and at distance from the end of the queue, then
and (iii) if a receiver is in in the positive direction and at distance from the end of the queue, then
Iv-B2 Mean number of successful receivers
Similar to case (A) considered in Section IV-A, the approximate formulae presented in Section IV-B1 are in geometric forms. This fact again enables us to obtain the closed-form approximation . Recall here that is negligible in this case due to the distance between the transmitter and the -axis. Thus, by using (18), (20), (21), and Proposition III.2, we can approximate and as
Iv-C Case (C): Transmitter in Middle of Queue
Finally, we consider case (C), where the transmitter is in the middle of the queue.
Iv-C1 Probability of successful transmission
As well as case (B), if the queue is sufficiently long, then we can neglect the interference from and . Thus, we approximate this case by considering vehicles queuing at even intervals on a single street with infinite length, i.e., a single infinite queue. Under this assumption, we can obtain the approximate formulae for this case by simply removing the effect of the interference from vehicles on the -axis in the results in Section IV-A1. Thus, substituting into (8) and (12), we can immediately obtain the following.
Approximate formulae of in case (C): Suppose that the transmitter is in the middle of the queue. If and , the probability of successful transmission can be approximated as follows. (i) If a receiver is the -th vehicle from the transmitter, then
and (ii) if a receiver is in at distance from the intersection, then
Iv-C2 Mean number of successful receivers
As mentioned in the above, the number of the successful receivers in is relatively small in this case. Therefore, it is sufficient to consider receivers in and . In a similar way to the derivation of (14), we also easily obtain an approximation for and by substituting into (14) and (LABEL:eq-M_R-case-A-x), respectively.
V Broadcast Rate Optimization
We next consider the optimization of the broadcast rate of vehicles in the on the basis of the approximate formulae of the performance metrics presented in Section IV. If the vehicles in transmit with a high broadcast rate, then they have higher interference than those in due to the congestion of vehicles at the intersection. Therefore, the probability of successful transmission for the transmitter in becomes lower than that for the transmitter in . However, there are more vehicles near the intersection than in , and thus, the transmitter in has more vehicles to transmit. In addition, if a vehicle transmits with a high broadcast rate, it has more chance to successfully transmit to its neighbors. Thus, to characterize and balance this relationship, we consider the mean number of successful transmissions per unit time, which is equal to
Note here that we directly optimize the value , not , because and is assumed to be constant. By using as a cost function of , we consider the optimization problem
By solving the above problem, we can obtain the optimal broadcast rate that maximizes the mean number of successful transmissions per unit time. In addition, by substituting the approximate formulae of , , and [shown in (14)–(16), (22), and (23)] with (28), we can consider the optimization problem of the broadcast rate in cases (A)–(C). In our experiments, we found that in case (C) can roughly maximize in cases (A) and (B) if the traffic intensity in is not very high. Therefore, by applying this value to the broadcast rate of all the vehicles in , we can optimize the performance of broadcasting in all three cases (for details, see Section VI-B).
Since the values from exact analysis shown in Propositions III.1 and III.2 include numerical integrals and multi-level summations, the calculation of the exact value of requires a large computational cost. Furthermore, its optimization becomes much more time-consuming because such a heavy computation is needed in each step of numerical optimization methods. However, by using the closed-form approximation of , we can compute the optimal in a reasonable computational time. Because we found that and are mostly insensitive to and , we adopted a long queue assumption in our approximation. As a result, if we assume that , , and are given in advance, can be determined by only and . This fact suggests that if we prepare a look-up table in our vehicles that describes the optimal broadcast rate corresponding to each value of and , we can immediately optimize the broadcast rate by just obtaining the current traffic density in the running segments. This fact leads to the possibility of (near-)optimal real-time broadcast rate control. However, considering all possible cases for , , , and is unrealistic for the exact analysis.
In this section, we provide the results from several numerical experiments. We first show the experimental results for the performance metrics and . We evaluate our approximate formulae in cases (A)–(C) considered in Sections IV-A–IV-C with both simulation and exact results. We then discuss our broadcast-rate optimization method presented in Section V. Finally, we investigate the performance of vehicles at other locations by interpolating or extrapolating the results for cases (A)–(C).
Before we move on to the experimental results, we will explain the parameters used in the experiments. The interval of vehicles was fixed to [m] and in all experiments. By considering realistic settings, we chose and [dB]. In addition, [km] and .
Vi-a Evaluation of Performance Metrics
In this subsection, we provide the experimental results for the performance metrics and and evaluate the accuracy of our approximate formulae for them. Figure 3 compares the simulation results and the exact and approximate values of in case (A) i.e., the case where the transmitter is at the intersection (see Section IV-A). The left graph corresponds to the case where the receiver is in the and the -th vehicle from the intersection. In addition, the right graph shows the case where the receiver is in
and the horizontal axis represents the transmission distance. All error-bars in the graphs in this paper represent 95% confidence intervals. We calculated the values from exact analysis using (5) and those from approximate analysis using (8) [left graph] and (13) [right graph]. We can see from the left graph that geometrically decreased, and if increases also decreases due to higher interference from vehicles in . We can also see that our approximate formula fit well to the results from simulation and exact analysis in all cases and the error became larger when was larger. The reason for this can be considered as follows. Since we assume that is sufficiently large in our approximation, if the distance from the receiver to the end of the queue is closer, then the approximation error becomes large. From the right graph, we can find that the approximate formulae took higher values than the theoretical results and the error increased subject to . The reason for this is that we approximate the Euclidean distance from the receiver in to the transmitter at the intersection by the Manhattan distance (see (41) in Appendix A-A). Thus, the interference became smaller and became larger than in the simulation and exact analysis. In addition, the larger suggests that there were greater impacts from the interference from the vehicles in . Therefore, the errors increased subject to . Although the errors in the right graph were larger than those in the left one, we could obtain a rough estimation for . Indeed, we later determined that the errors could be negligible when considering (see Figure 6).
Similarly, Figure 5 shows the results of our approximation and the simulation and exact analysis for in case (B) where the transmitter is at the end of the queue. The horizontal axis represents the distance to the receiver where the positive (resp. negative) part corresponds to the vehicles in the right-hand side, i.e., in (resp. the left-hand side, i.e., in ) of the transmitter. We used (5) for the exact analysis and (18) and (21) for the approximate one. Thus, we can see from the figure that if is smaller, the results on the positive and negative parts become closer because the interference from the decreases. This is because the interference from the decreases as decreases. The figure also shows that our approximate formula achieved quite small errors in all cases. Figure 5 shows the experimental results for case (C), where the transmitter is in the middle of the queue. The values of exact analysis in (5) were approximated by using (24). The figure shows a similar tendency to case (A) (see Figure 3) because the results can be computed very similarly.
We next show the results for . Figure 6 compares the simulation results and the exact and approximate values of . The left, middle, and right graphs correspond to cases (A), (B), and (C), respectively. The values from the exact analysis are calculated by (7) whereas those from the approximate analysis corresponding to cases (A), (B), and (C) are calculated by using (14)–(16), (22) and (23), and (LABEL:eq-M_Q-case-C) and (27), respectively. We can see from the graphs that rapidly decreased as increased. In addition, if was larger, became smaller because the interference at the intersection became higher. We can also see that our approximation performed well except for region , which was larger than in case (A). Furthermore, the errors increased when was small. Similar to the evaluation of , this is because we assume that the queue length is sufficiently large in our approximation. However, the optimal is usually around (see also Section VI-B), and thus, we regard our approximate formulae for as having sufficient accuracy in realistic settings. In addition, we can find that the errors in the graph in Figure 3 did not have much effect on the approximation for .
Vi-B Effectiveness of Optimization Method
In this subsection, we provide the evaluation results for the optimization method presented in Section V. Figure 7 shows the results for the cost function with different and . We also plotted the optimal ’s that maximized the approximate in the same graphs. The left, middle, and right graphs correspond to cases (A), (B), and (C), respectively. All values in the graphs were calculated by using (7) or approximate formulae in Section IV. We first focus on the results from the exact analysis. We can see from the graphs that there exist local maximum values in the domain in all cases. The figure shows that the optimal ’s achieved roughly 10% higher at the maximum. Furthermore, when approached , from the exact analysis slightly increased. This is because we fixed the broadcast rate of the vehicles in . Thus, if increases, the transmitter has more of a chance to transmit to vehicles in although the probability of successful transmission becomes smaller. However, larger than is unrealistic when considering a realistic receiving time and other computational time, e.g., encoding. Thus, we assume optimal is in the domain . Indeed, in our experimental settings, we can see that unique optimal ’s exist in that domain.
We next discuss the accuracy of our approximation. We can see from the graphs that the values of with our approximation fitted well to those from the exact analysis in the domain . We can also see that the value of was not very sensitive to the value of in all cases. This fact suggests that our approximate not depending on is valid. However, the errors between the values from the exact and approximate analysis became larger when was smaller or was larger. The reason for this is that, in our approximation, we assume that is sufficiently large and . Thus, in the region where is small or is close to , the accuracy of the approximate formulae becomes worse. In addition, the errors in the left graph in the domain were larger than those in the middle and right ones. The reason for this is that we approximated the Euclidean distance from the receiver in to the vehicles in by the Manhattan distance, which is larger than the Euclidean one. Thus, the interference from the vehicles in to the receivers in was underestimated. Consequently, the approximate values of were larger than those in the exact analysis. However, the errors are relatively small in the domain where the optimal existed and . Therefore, we can say that the optimization with our approximation provides a good guideline for the optimal .
We next discuss how we can determine the single optimal for the vehicles in . Since our optimization problem depends on cases (A)–(C), we need to choose or derive a single that can be optimal for the vehicles in the from these solutions. Figure 9 compares results of the values of in all three cases with different . We also plotted in cases (A) and (B) as the dashed and dotted lines, respectively, where is equal to in case (C). In addition, we fixed . We can see from the figure that and took similar values when was smaller than . This fact is supported by Fig. 7, in which was insensitive to around . Therefore, by adopting to determine the broadcast rate of all vehicles in , we can roughly maximize in all the cases if the traffic intensity in the running segment is not high. We now evaluate the impact of on in case (C), i.e., . Figure 9 shows the results for of approximate and exact analysis in case (C) when varying . From the figure, we can see that increased subject to . The figure also shows that the approximate values for again fitted well to those from the exact analysis. Recall here that is the key parameter for determining the optimal (see Section V). As a result, the results in Figures 9 and 9 show that we can determine the optimal broadcast rate for all the vehicles in by only observing the traffic intensity .
Vi-C Vehicles at Other Locations
We next consider the case where a transmitter is at other locations, i.e., not at the intersection, at the end of the queue, or the middle of the queue. Since our approximation assumes that is sufficiently large and does not consider the distance from the end of the queue, we cannot obtain closed-form formulae for or in general cases. However, vehicles at other locations can be considered as being in an intermediate state between the vehicle at the intersection and that at the end of the queue. Thus, it is expected that we can roughly estimate their performance by interpolating or extrapolating the values of approximate formulae for cases (A)–(C).
Figure 11 shows the values of when varying the positions of the transmitter and the distance to the receiver in . We fixed and , and the -axis was in log scale. In the graph, the dashed lines represent the interpolation line using the approximation formulae for cases (A)–(C), where cases (A), (B), and (C) correspond to , , and , respectively. In other words, the dashed line shows the log-linear interpolation. From the figure, we can see that the interpolation can give a rough estimation for the values of in all cases. We can also see that if increased, the results from the exact analysis become close to log-linear, whereas they tended to be a constant value when was small. This is because if the receiver is closer to the transmitter, i.e., is smaller, the impacts of the intersection or the end of the queue rapidly disappear as the distance from the transmitter to them increases. Similarly, Figure 11 shows the results for , i.e., the mean number of successful receivers in when varying the positions of the transmitter . Note that the -axis is in linear scale. The dashed line was plotted by interpolating the results of the approximation for the three cases. The broken line was plotted by extrapolating the approximation for cases (A) and (C). We can see that if the positions of the transmitter were close to the middle of the queue, i.e., , the extrapolation became a good estimation for the exact values. However, if the transmitter was close to the intersection or the end of the queue, the error increased. This tendency is similar to that in Figure 11. As a result, we can conclude the following. If the transmitter is close to the middle of the queue, we can use extrapolation on the basis of the approximate formulae for cases (A) and (C); otherwise, the approximate formulae for cases (A) and (B) should be used instead.
In this paper, we conducted a theoretical performance evaluation of vehicle-to-vehicle (V2V) broadcast communications at an intersection. We first derived the theoretical values of the probability of successful transmission and the mean number of successful receivers. We then provided a closed-form approximation for them. By using the approximation formulae, we provided a method for optimizing the broadcast rate of vehicles in the queueing segment. By using the closed-form formulae, we can obtain the optimal broadcast rate in a reasonable time without time-consuming numerical computation, which is required in exact analysis.
To maintain mathematical tractability, we assumed a simple channel model and media access control (MAC) layer in this paper, e.g., Rayleigh fading assumption or ALOHA. Thus, the generalization of the distribution of vehicles and fading are for future work. In addition, power control can be a good solution for the interference problem at an intersection. Therefore, the joint modeling and optimization of the broadcast rate and the transmission power of vehicles are also for future work. In addition, in our optimization method, we assume that the traffic intensities in running segments are given. However, vehicles/road side units need to infer these values in a practical situation, e.g., by measuring the distance to the vehicle running at the front. Such inference schemes and their impacts on our optimization method are also for future work.
Appendix A Approximation Methodology
In this appendix, we give detailed explanations for the derivation of the approximate formulae presented in Section IV. For later use, we first introduce approximate formulae for