TY - JOUR
AU1 - Zarei,, Mani
AB - Abstract Vehicular ad hoc networks (VANETs) have emerged as an appropriate class of information propagation technology promising to link us even while moving at high speeds. In VANETs, a piece of information propagates through consecutive connections. In the most previous vehicular connectivity analysis, the provided probability density function of intervehicle distance throughout the wide variety of steady-state traffic flow conditions is surprisingly invariant. But, using a constant assumption, generates approximate communication results, prevents us from improving the performance of the current solutions and impedes designing the new applications on VANETs. Hence, in this paper, a mesoscopic vehicular mobility model in a multilane highway with a steady-state traffic flow condition is adopted. To model a traffic-centric distribution for the spatial per-hop progress and the expected spatial per-hop progress, different intervehicle distance distributions are utilized. Moreover, the expected number of hops, distribution of the number of successful multihop forwarding, the expected time delay and the expected connectivity distance are mathematically investigated. Finally, to model the distribution of the connectivity distances, a set of simplistic closed-form traffic-centric equations is proposed. The accuracy of the proposed model is confirmed using an event-based network simulator as well as a road traffic simulator. 1. INTRODUCTION Vehicular ad hoc networks (VANETs) are one of the most remarkable and well-adopted applications of ad hoc networks, promising to serve us even when moving at maximum speeds [1]. The recent growth in the variety of applications in VANETs and proliferation of such applications has gained considerable attention during the last decade. By offering a wide range of application possibilities, VANETs have attracted a great deal of interest from researchers, industries, consumers and other communities. They play an important role in the future of wireless networking and the internet of things (IoT) by making many desired services possible. Some of such services include movability, ubiquity, on-demand media streaming, fixed-mobile convergence, flexible connectivity, spontaneous networking and autonomic networking to aid smart and pervasive objects. Applying dedicated short-range communications (DSRCs) facilitate a wide range of driver-assistant applications such as vehicle-to-vehicle (V2V) and vehicle-to-roadside (V2R) services, functioning as traffic condition messaging and accident information allowing timely and intelligent communication to develop road safety [2–5]. In V2V level, vehicles operate not only as a host but also as a router and they can jointly create clusters. In a highly dynamic VANET, vehicles join and disjoin clusters frequently along their travel route, resulting in connectivity instability at microscopic level [6]. Most of the prior literature in VANETs has treated connectivity and cluster as synonymous. Hence they have used the term ‘connectivity distance’ and ‘cluster length’ interchangeably. A connected cluster is defined as a sequence of vehicles such that each vehicle is at least within the radio propagation range of one other vehicle in the cluster [7]. The cluster length is defined as the maximum spatial distance of the connected vehicles [5]. In general, vehicular movability patterns in the literature of VANETs can be organized into three approaches (i.e. macroscopic, mesoscopic and microscopic) in accordance with intervehicle communications and mobility factors [8–11]. In the macroscopic model, the intervehicle distance is modeled based on the average intervehicle distance over a highway. On a mesoscopic level, due to traffic stability, the intervehicle distances of individual vehicles are defined by independent and identically distributed (i.i.d.) random variables [8, 12]. In the microscopic model, dissimilar mobility pattern is a norm because of different vehicle types as well as different velocities. In other words, the vehicle mobility is affected by many issues extracted from road situation, mobility pattern of adjacent vehicles, information of the messaging signs along the highway, traffic lights, road environment conditions and driver’s feedbacks to such factors [11, 13]. This, in turn, results in cluster size variations and the possibility of intra-cluster interactions and message forwarding [14, 15]. Vehicular connectivity has been thoroughly investigated by many researchers at different levels spanning microscopic, macroscopic and mesoscopic VANETs mobility models. But, it has been a challenging task to analyze the connectivity of such models due to the dynamic nature of the mobility pattern of different vehicles. In general, the study of vehicular connectivity in microscopic mobility model in the literature includes the product of a set of probabilistic rules that mathematically investigates the probability density function (pdf) of connectivity distance and defines how neighboring vehicles on a dynamic environment might join together to create a cluster [7, 14]. Mesoscopic model is generally adopted to model static mobility in which the intervehicle distances and communication links are unchangeable and vehicles possibly follow the traffic in a dense or near-capacity situations or move freely at a maximum allowed speed in a sparse situation [5, 16]. In our previous study, we have proposed a comprehensive mathematical model of vehicular connectivity at the microscopic level in a dynamic VANET [7]. The connectivity analysis in the mesoscopic level could be quite challenging in dissimilar traffic scenarios. For example, let us consider the analysis of connectivity for a sparse traffic case of an on-road trip versus a congested urban traffic scenario. In the sparse traffic case of on-road trip, vehicles move with a steady-state velocity. This is when in congested urban traffic scenarios vehicles, usually, follow permanent heavy traffic in stationary speed at the mesoscopic level. In spite of having a constant speed, the analysis of these two scenarios is quite dissimilar, and this difference is analyzed and addressed in this paper. This paper proposes a traffic-aware vehicular analysis and utilizes customized solutions for different traffic flow conditions at the mesoscopic level. Our analysis of vehicular connectivity is motivated by grave potentials that the intelligent transportation systems (ITS) have to offer. In the development of the green ITS and renewable electric vehicles, plug-in vehicles and unmanned aerial vehicles, the accuracy of the proposed traffic-centric model would help us promote the network’s energy efficiency and prolong the network’s lifetime. Note that the movement of ships, planes and trains in most of their travel time is invariant, following the mesoscopic mobility model. Therefore, our proposed model can be generalized and applied for the analysis of the mobility pattern of such industries attempting to improve related future applications. Finally, the proposed model can be used to promote the internet of vehicles (IoV) as a possible infrastructure for the IoT. In this work, the characteristics of the traffic-aware vehicular mobility are studied at the mesoscopic level and the corresponding influences on vehicular connectivity are mathematically investigated. To make this paper self-sufficient, we briefly summarized the distribution of intervehicle distance investigated for sparse, dense and intermediate vehicle densities. The contributions of this paper are as follows: (i) Using the categorized steady-state traffic flow conditions, the distribution of spatial per-hop progress and the expected spatial per-hop progress is proposed. (ii) The expected number of hops in a cluster, the distribution of the number of successful multihop forwarding in a time period of a communication link, the expected time delay and the expected connectivity distance are mathematically investigated at the mesoscopic level. (iii) A set of simplistic closed-form traffic-centric equations for modeling the pdf of the connectivity distance is proposed. This paper is organized as follows: Section 2 summarizes the related works and discusses the vehicular connectivity taxonomies at microscopic and mesoscopic levels and discusses their future requirements. Section 3 describes some of the necessary traffic-centric definitions and presents the system model. Section 4 describes the proposed mesoscopic model of vehicular connectivity mathematically. Section 5 summarizes our performance evaluation and simulation result. Finally, Section 6 concludes the paper and discusses the possible future work. 2. RELATED WORK With the development of intervehicle communications, convened by dedicated standards like 802.11p or combined innovative methods, connectivity analysis and its suitable applications, was previously considered as a conventional and significant subject in VANETs and attracted higher attention from research communities [6, 7, 14, 15, 17–28]. In VANETs literature on the connectivity over intervehicle distances, there are various studies that try to model the communication link lifetime based on a wide variety of factors. For example, these factors include fixed or variable velocity, vehicle density, radio propagation range, node degree, connection duration and dissimilar mobility patterns at mesoscopic and/or deterministic microscopic levels [11, 14, 19–21, 23, 28–31]. In the following two subsections, the vehicular mobility is classified at microscopic and mesoscopic levels. 2.1. Microscopic studies Driver behavior and vehicle mobility constraints in microscopic networks create unique characteristics. Different mobility models have different degrees of spatiotemporal dependences, relative speeds and geographic restrictions, which give rise to different link durations between connected vehicles and therefore accessibility of separate paths for multihop transmissions [18]. In [17], the authors proposed an analytical approach for measuring intervehicle distances using an exponential distribution in a 1-D dynamic VANET under different and invariant vehicle velocity assumptions. They assumed that the total number of vehicles that pass an observer during a determined interval follows a predictable homogeneous Poisson process with the density of |$k$|⁠. The authors proposed an equivalent |$M/D/\infty$| queuing model and studied the occupied period in queuing theory. They analyzed the cluster length and defined the Laplace transform used for pdf of the cluster length as well as the tail probability of connectivity distance. Since the proposed Laplace expression might not clearly be inverted, the authors resorted to numerical inverting could not provide a closed-form equation for the distribution of the investigated cluster length. However, their numerical results displayed that growing the radio transmission range of vehicles and traffic flow correspondingly leads to the increase of the discussed connectivity regarding the cluster length and platoon size. In dynamic mobility models at the microscopic level, vehicles join or quit clusters repeatedly along their travel route, resulting in connectivity instability. Some connectivity instability techniques have been proposed in the VANET literature [7, 14, 18, 23, 30, 32]. Focusing on the parameters of speed, direction and radio propagation range, the communication time between two consecutive nodes and an admission control approach for intervehicle connection was presented in [30]. In [18], new metrics for the evaluation of vehicular connectivity was presented regarding the different vehicular mobility patterns. The connection period distribution was characterized by the average duration of the |$k$|-hop path existing between any two nodes. The authors also presented simulation data that recommends single-hop has better connectivity performance than the multihop paths connectivity in VANETs. In [32], authors considered a microscopic 1-D VANET in which vehicles are Poisson distributed with uniform speed distribution in a determined range. The authors in [15, 33] provided a numerical method to calculate the information propagation speed (IPS) under two special network situations when the vehicle density is either very low or very high [15, 33]. They proved that the information propagation process can be modeled as a renewal reward process wherein information propagation cyclically alternates between forward and catch-up processes at the microscopic level. Authors estimated the average cluster size in the forwarding process while considering dissimilar invariant speed model of vehicle mobility. The authors in [14, 23] proposed a closed-form formula used for the pdf of the connectivity motivated by the effort on the connectivity of random interval graph [34] as well as the theory of coverage processes [35]. They considered an innovative method by which time is equally divided into slots and give the opportunity to each node to change its velocity at the beginning of each interval independent of other intervals. Using microscopic mobility model, the authors investigated the information propagation process and obtained an analytical formula for the IPS in a single propagation cycle. In [1] considering vehicular connectivity, we provided a more realistic innovative microscopic model to investigate the average IPS regarding to the total number of consecutive renewal cycles which a message needs to be delivered over clusters. In [7] considering a time-varying velocity assumption, we classified the microscopic mobility model of the head, tail and intermediate vehicles inside a connectivity. In this work, a new mathematical microscopic solution was proposed to investigate the cluster length and pdf of the connectivity accordingly. In [16], we proposed a spatiotemporal synchronized random walk vehicular mobility analysis in which any desired time interval was equally slotted. The distribution of various type of traveling vehicles in a highway was considered as a homogeneous Poisson process through vehicle intensity |$\rho$| at any time slot. Using the suggested microscopic random walk vehicular mobility model, dependent parameters such as the conditional probability, joint Poisson vehicular distribution, tail probability and conditional expected amount of nodes were mathematically investigated. In [36], we extended our model and provided a comprehensive microscopic analysis for a multi-lane highway. The authors in [37] propose several extensions to the publicly available microscopic level simulation framework Artery which itself is derived from Veins. To show Artery’s capabilities, authors set up a number of simulation scenarios with heterogeneous vehicles to reproduce location-based traffic phenomena on an original equipment manufacturer’s backend. In [38], a new clustering method is provided for both dynamic and static vehicular mobility urban scenarios. The proposed model estimates vehicle’s link lifetime and compute relative position as well as moving direction. The authors evaluated the performance of their proposed scheme with related works (i.e. Lowest-ID and VMaSC). In [39], the authors proposed social utility-based dissemination scheme (SUDS), an emergency warning messages in VANETs. They applied social properties of vehicles such as centrality, interests and friendships to moderate inherent wireless environment (i.e. hidden terminal and dissemination storm problems). The authors utilized a hybrid architecture to resolve connectivity and contact duration-related issues in low traffic conditions with high vehicular mobility. The performance evaluation was conducted for a highway scenario under variable vehicular density, vehicular velocity and spatial distance. In [40], the authors used three common routing protocols (i.e. DSR, AODV and DSDV) to find the suitable routing protocol in a high density traffic area in Khartoum. The authors generated MOVE (mobility model generator for vehicular networks) in order to investigate mobility pattern through the simulation with SUMO [41] (microscopic simulator for urban mobility) as a VANET’s traffic simulators. In order to reduce the further deployment of road side unit (RSU) and to guarantee QoS of the safety message dissemination, the authors in [42] proposed a hybrid scheme constructed on VANET-cellular architecture. Their proposed method resolved the problem of the competitive channel and unnecessary data in the vehicular environment. It is evident that in unsteady and dynamic scenarios, vehicular mobility can be modeled at the microscopic level. Reasons for frequent changes in the structure of the connectivity can be defined by dynamic mobility as well as predictable changes in vehicular speeds that are defined with probabilistic models at the microscopic level. On the contrary, vehicular mobility in steady-state traffic scenarios can be modeled at the mesoscopic level. 2.2. Mesoscopic studies In [15], authors provided the IPS in a one dimension VANET in which vehicles were Poisson distributed and moved statically with the same velocity in the same and reverse directions. Regarding the periodic intervehicle disconnection and multihop connectivity, the authors mathematically proposed the upper and lower bounds for the IPS, which focused on the impact of sustained traffic flow condition on the IPS. In [24] authors studied a full analysis of the IPS in both directions of a road or a highway. They verified that under a defined traffic flow threshold, on average, a piece of information propagates at the vehicle speeds. However, beyond the suggested threshold, in a condition that vehicles are joined together to form a connected cluster, a piece of information disseminates much faster. A full cluster was made with vehicles of westbound and eastbound directions and they provided the Laplace transform of the connectivity distance. In [25], based on dissimilar constant-speed mobility model, a metric was defined to consider both single-hop connections between vehicles and multihop forwarding connectivity. In [31], the spatial distance distribution was stated by a tuple |$(h,v,q)$|⁠, wherein |$h$| denoted the intervehicle distance, |$v$| indicated the velocity and |$q$| was the market dissemination ratio, respectively. Regarding these factors, a closed-form equation was given for connectivity distance. In [22], the authors showed that, in uncongested traffic condition, the number of connected nodes grows as either the vehicle’s density or the number of lanes increases. The authors proved that the intervehicle distance has a great influence on connectivity when it is spatially within 3–4 times larger than the radio propagation range and beyond this length, the pdf of connectivity declined slowly. In [19], an innovative analysis on the connectivity over intervehicle distance radio communication was geometrically proposed. In steady-state traffic scenarios, driver behavior and vehicular mobility limit to geographic restrictions and road/highway rules. As a result, vehicular connectedness with stable mobility can be modeled at the mesoscopic level. In previous work, using the dynamic and unstable mobility model we studied vehicular connectivity at microscopic level [7]. This work provides a comprehensive study for analysis of connectivity at the mesoscopic level while considering different steady-state traffic scenarios. In the next section, the system model and mesoscopic analysis of intervehicle distance given different vehicular intensity are summarized. 3. SYSTEM MODEL In this section, the network architecture, in terms of different vehicular densities, is described. Subsequently, the corresponding pdf of the intervehicle distances is classified. This work considers mesoscopic vehicular mobility for a single direction of multilane highway in a steady-state traffic flow condition defined by a time-invariant vehicle density. We denote the vehicle density on the lane under consideration by |$\rho$| in vehicles per meter |$( veh/m)$|⁠. Let time be equally divided to a constant interval size |$\tau$| [7, 14], and |${W}_i=\{{W}_i(m),m=0,1,2,\dots \}$| be a discrete-time slot stochastic process of the |$i$|th intervehicle distance, between vehicle |$i$| and vehicle |$i+1$|⁠, where |${W}_i(m)$| is a random variable representing the intervehicle distance of vehicle |$i$| at the |$m$|th time slot, |$i=0,1,2,\dots, m{\,=\,}0,1,2,\dots$| [11]. For the sake of simplicity as showed in [11], the index |$i$| is omitted when referring to an arbitrary intervehicle distance. At any interval, |${W}_i(m)\in [{W}_{min},{W}_{max}]$| for all |$i,m{\,\ge\,} 0,$| where |${W}_{min}$| and |${W}_{max}$| are the minimum and maximum intervehicle distances, respectively. Furthermore, assume that |${W}_i$|’s are independent with identical statistical behaviors for all |$i{\,\ge\,} 0.$| Assume each vehicle is equipped with DSRC transceiver and capable of communicating and broadcasting information to other vehicles through the fixed radio propagation range, referred to by |$r$|⁠. The proposed model is done for a single direction of a multilane highway (e.g. westbound direction of Fig. 1). FIGURE 1. Open in new tabDownload slide Vehicular mobility example in steady-state traffic flow condition. FIGURE 1. Open in new tabDownload slide Vehicular mobility example in steady-state traffic flow condition. In this paper, three different kinds of steady-state traffic flow conditions are considered. The investigated conditions are sparse, dense and intermediate traffic flow. The classification of vehicular densities to sparse, dense and intermediate conditions is intended for uncongested, congested and near-capacity flow conditions, respectively. A wide variety of traffic flow conditions correspond to a range of vehicle densities is shown in Table 1 [8]. In order to study traffic-centric mesoscopic analysis, the following subsections present the pdf of the intervehicle distance for the three traffic flow conditions of interest. TABLE 1. Traffic flow condition for different vehicle densities [8]. Density (veh/m) Traffic flow condition 0–0.007 Free-flow operations Uncongested flow conditions (sparse) 0.007–0.012 Reasonable free-flow operations 0.012–0.019 Stable operations 0.019–0.026 Borders on unstable operations 0.026–0.042 Extremely unstable flow operations Near-capacity flow conditions (intermediate) 0.042–0.062 Forced on breakdown operations Congested flow conditions (dense) >0.062 Incident situation operations Density (veh/m) Traffic flow condition 0–0.007 Free-flow operations Uncongested flow conditions (sparse) 0.007–0.012 Reasonable free-flow operations 0.012–0.019 Stable operations 0.019–0.026 Borders on unstable operations 0.026–0.042 Extremely unstable flow operations Near-capacity flow conditions (intermediate) 0.042–0.062 Forced on breakdown operations Congested flow conditions (dense) >0.062 Incident situation operations Open in new tab TABLE 1. Traffic flow condition for different vehicle densities [8]. Density (veh/m) Traffic flow condition 0–0.007 Free-flow operations Uncongested flow conditions (sparse) 0.007–0.012 Reasonable free-flow operations 0.012–0.019 Stable operations 0.019–0.026 Borders on unstable operations 0.026–0.042 Extremely unstable flow operations Near-capacity flow conditions (intermediate) 0.042–0.062 Forced on breakdown operations Congested flow conditions (dense) >0.062 Incident situation operations Density (veh/m) Traffic flow condition 0–0.007 Free-flow operations Uncongested flow conditions (sparse) 0.007–0.012 Reasonable free-flow operations 0.012–0.019 Stable operations 0.019–0.026 Borders on unstable operations 0.026–0.042 Extremely unstable flow operations Near-capacity flow conditions (intermediate) 0.042–0.062 Forced on breakdown operations Congested flow conditions (dense) >0.062 Incident situation operations Open in new tab 3.1. Sparse In a VANET, the communication link between neighboring vehicles will be established when their Euclidean distance is smaller than their radio propagation range. For sparse VANET interactions between vehicles are very low and almost negligible. In sparse traffic network environments, it is sensible to accept that, over a significant time interval, vehicles move at unchangeable speed. Whereby, the intervehicle distances between adjacent nodes remain invariant and usually they are larger than the radio propagation range. In such an environment, vehicles move individually at maximum velocity and do not interact with each other [8, 43, 44]. Let us use |${W}_i$| to denote the discrete-time stochastic process of the |$i$|th intervehicle distance. If |${W}_i$| happens to be in a sparse steady-state traffic stream, then, additional label |$S$| (e.g. |${W}_i^L$|⁠) is used to indicate that the traffic flow condition is in the uncongested flow stream. As shown in [11], in the sparse vehicle density, it is reasonable to accept that the intervehicle distances |${W_i^L}^{,\kern0.5em }s$| at any time slot are i.i.d. with an exponential pdf, i.e. $$\begin{equation} {\textrm{f}}_{{\textrm{W}}_{\textrm{i}}^{\textrm{L}}}\left(\textrm{w}\right)={\rho \textrm{e}}^{-\rho \textrm{w}} ,\textrm{w}\ge 0 \end{equation}$$ (1) where |$\rho$| denotes the vehicular density of the spatial Poisson distribution. 3.2. Dense In the literature of VANET, the normal distribution is used to model the time headway for a dense environment [8]. Inspired by the study on the time headway with normal distribution [8], the authors in [11] proposed that for a congested-traffic-flow condition the intervehicle distances |${W_i^D}^{,\kern0.5em }s$| at any time slot are i.i.d. with a normal pdf, i.e. $$\begin{equation} {\textrm{f}}_{{\textrm{W}}_{\textrm{i}}^{\textrm{D}}}\left(\textrm{w}\right)=\frac{1}{\sigma \sqrt{2\pi}}{\textrm{e}}^{-\frac{{\left(\textrm{w}-\mu \right)}^2}{2{\sigma}^2}} ,\textrm{w}\ge 0 \end{equation}$$ (2) where the additional label |$D$| is used to indicate the congested traffic flow condition. Let |$\sigma$| and |$\mu$| be the standard deviation and the mean of the intervehicle spatial distance in meters, respectively, where |$\mu =1/\rho$| and σ are network parameters and take distinct values according to the vehicular intensity. The standard deviation |$\sigma$| for dense vehicle traffic network environments is given by |$\sigma =((\mu -\alpha)/2)$| [8, 11]. 3.3. Intermediate For an intermediate vehicle density condition, neither an exponential nor a normal distribution is an appropriate choice for modeling of intervehicle distances [45]. The authors in [11] showed that the intervehicle distances in near-capacity flow conditions follow Pearson type III, a general distribution that was first proposed for time headways [8]. Hence, for an intermediate traffic-flow condition the intervehicle distances |${W_i^I}^{,\kern0.5em }s$| at any time slot are i.i.d. with a Pearson type III pdf, i.e. $$\begin{equation} {f}_{W_i^I}(w)=\frac{\lambda^z}{\Gamma (z)}{\left(w-\alpha \right)}^{z-1}{e}^{-\lambda \left(w-\alpha \right)},w\ge 0 \end{equation}$$ (3) where the additional label |$I$| is used to indicate the near-capacity flow condition. Parameters |$z$| and |$\lambda$| are the shape and scale factors, respectively, used for the general distribution, Pearson type III, and |$\Gamma (z)={\int}_0^{\infty }{u}^{z-1}{e}^{-u} du$| is the gamma function. The parameters |$\lambda$| and |$z$| are associated to |$\mu$| and |$\sigma$| according to |$\lambda =(\mu -\alpha)/{\sigma}^2$| and |$z={(\mu -\alpha)}^2/{\sigma}^2$| [8, 11]; moreover, for notation simplicity, let |$\mu =\sigma =1/\rho$|⁠. 4. ANALYSIS This section proposes the mesoscopic analysis of connectivity in a VANET customized for the wide variety of steady-state traffic flow conditions. This section is organized as follows: first, to make this section self-sufficient, the distribution of per-hop progress is briefly reviewed. Then, the distribution of expected spatial per-hop progress for each of sparse, dense and intermediate vehicular density is proposed. Next, the expected number of the hops, distribution of the number of successful multihop progress, the expected time delay and the expected connectivity distance for different traffic flows (in accordance with Table 1) is mathematically formulated. This section ends with investigating the pdf of connectivity distance at the mesoscopic level. FIGURE 2. Open in new tabDownload slide Illustration of the spatial per-hop progress in steady-state traffic flow, denoted by |$H$|⁠, which is the distance between reference vehicle (e.g. |$R$|⁠) and the furthest vehicle in the radio propagation range |$r$| of the |$R$| vehicle (i.e. |$F$|⁠). FIGURE 2. Open in new tabDownload slide Illustration of the spatial per-hop progress in steady-state traffic flow, denoted by |$H$|⁠, which is the distance between reference vehicle (e.g. |$R$|⁠) and the furthest vehicle in the radio propagation range |$r$| of the |$R$| vehicle (i.e. |$F$|⁠). 4.1. Expected spatial per-hop progress The probability distribution of the spatial per-hop progress using mesoscopic vehicular mobility is denoted in [11, 23]. Figure 2 shows the spatial per-hop progress that is the spatial distance from the reference vehicle to the farthest vehicle within the radio propagation range of the reference vehicle. The spatial per-hop progress, denoted by H, is upper bounded by the radio propagation range |$r$|⁠. For |$j=L,H,I$|⁠, regardless of traffic flow conditions Section 3 indicated that |${W_i^j}^{,\kern0.5em }s$| are independent with identical statistical behaviors for all |$i\ge 0.$| Therefore, for the sake of simplicity let omit index |$j$| when referring to an arbitrary intervehicle distance. Given a mesoscopic model, the intervehicle distances |${W_i}^{,}$|s are i.i.d. with pdf |${f}_W(w)$| and cumulative distribution function (cdf) |${F}_W(w)$|⁠. Then, the cdf of H is given by [11, 46], $$\begin{equation} {\textrm{F}}_{\textrm{H}}\left(\textrm{h}\right)=\frac{\textrm{P}\left({\textrm{Q}}^{\textrm{c}}\left(\textrm{r}-\textrm{h}\right),\textrm{Q}\left(\textrm{h}\right)\right)}{\textrm{P}\left(\textrm{Q}\left(\textrm{r}\right)\right)} \end{equation}$$ (4) where |$Q(l)$| is the occurrence that there exists at least one vehicle within distance l from a reference vehicle. The event |$Q(l)$| occurs with probability |${F}_W(l)$|⁠. Furthermore, |${Q}^c(l)$| is the complement of event |$Q(l)$| (i.e. the event that there are no vehicles within distance |$l$| from a reference vehicle) [46]. Hence, pdf of the spatial per-hop progress can be expressed as |${f}_H(h)=(d/ dhr){F}_H(h)$|⁠. Expected spatial per-hop progress intended for sparse, dense and intermediate vehicular densities are mathematically investigated as follows. For a VANET with sparse vehicle density, the intervehicle distance is exponentially distributed with pdf given in (1). The pdf of the corresponding spatial per-hop progress is given by [47] $$\begin{align} {\textrm{f}}_{\textrm{H}}\left(\textrm{h}\right)=\frac{{\rho \textrm{e}}^{-\rho \left(\textrm{r}-\textrm{h}\right)}}{1-{\textrm{e}}^{-\rho \textrm{r}}},\kern1em 0<\textrm{h}<\textrm{r}.\end{align}$$ (5) Then, the expected spatial per-hop progress of sparse vehicle traffic network environments is $$\begin{align} {\omega}_{\textrm{L}}&=\textrm{E}\left[{\textrm{H}}_{\textrm{L}}\right]=\underset{0}{\overset{\textrm{r}}{\int }}\left(1-{\textrm{F}}_{{\textrm{H}}_{\textrm{L}}}\left(\textrm{h}\right)\right)\textrm{dh}\nonumber\\ &\qquad\qquad=\underset{0}{\overset{\textrm{r}}{\int }}\left(1-\frac{{\textrm{e}}^{-\rho \left(\textrm{r}-\textrm{h}\right)}.\left(1-{\textrm{e}}^{-\rho \textrm{h}}\right)}{1-{\textrm{e}}^{-\rho \textrm{r}}}\right)\textrm{dh}\nonumber\\&\qquad\qquad=\textrm{r}-\frac{1}{\rho}+\frac{{\textrm{r}\textrm{e}}^{-\rho \textrm{r}}}{1-{\textrm{e}}^{-\rho \textrm{r}}}.\end{align}$$ (6) In a dense VANET, as shown in Section 3.2, the intervehicle distances are i.i.d random variables, each following the normal pdf. Using the cdf normal distribution, i.e. |${F}_{W_m}(w)=(1/2)(1+\operatorname{erf}((w-\mu )/\sqrt{2}\sigma))),$| the cdf for the spatial per-hop progress can be drived from (4) and the expected spatial per-hop progress is given by $$\begin{align} {\omega}_{\textrm{H}}&=\textrm{E}\left[{\textrm{H}}_{\textrm{H}}\right]=\underset{0}{\overset{\textrm{r}}{\int }}\left(1-{\textrm{F}}_{{\textrm{H}}_{\textrm{H}}}\left(\textrm{h}\right)\right)\textrm{dh}\nonumber\\ &=\underset{0}{\overset{\textrm{r}}{\int }}\left(\vphantom{\times{\left(1+\operatorname{erf}\left(\left(\textrm{r}-\mu \right)/\sqrt{2}\sigma \right)\right)}^{-1}}1-\left[\vphantom{\times{\left(1+\operatorname{erf}\left(\left(\textrm{r}-\mu \right)/\sqrt{2}\sigma \right)\right)}^{-1}}\left(\left(1/2\right)\left(1-\operatorname{erf}\left(\left(\textrm{r}-\textrm{h}-\mu \right)/\sqrt{2}\sigma \right)\right)\right)\right.\right.\nonumber\\ &\qquad\qquad\qquad\left.\left.\times \left(1+\operatorname{erf}\left(\left(\textrm{h}-\mu \right)/\sqrt{2}\sigma \right)\right)\right.\right.\nonumber\\ &\qquad\qquad\qquad\left.\left.\times{\left(1+\operatorname{erf}\left(\left(\textrm{r}-\mu \right)/\sqrt{2}\sigma \right)\right)}^{-1}\right]\right)\textrm{dh}\end{align}$$ (7) where erf(.) is the error function, i.e. a special function given by |$\operatorname{erf}(x)=(2/\sqrt{\pi}){\int}_0^x{e}^{-{t}^2} dt.$| For an intermediate vehicular density, as shown in Section 3.3, the intervehicle distances are i.i.d random variables, each following the Pearson type III pfd. The cdf for the spatial per-hop progress can be derived from (4). Therefore, the corresponding expected spatial per-hop progress could be expressed as: $$\begin{align} &{\omega}_{\textrm{I}}=\textrm{E}\left[{\textrm{H}}_{\textrm{I}}\right]=\underset{0}{\overset{\textrm{r}}{\int }}\left(1-{\textrm{F}}_{{\textrm{H}}_{\textrm{I}}}\left(\textrm{h}\right)\right) dh\nonumber\\ &\quad=\underset{0}{\overset{r}{\int }}\left(\vphantom{\times{\left[\frac{\gamma \left(z,\lambda \left(r-\alpha \right)\right)}{\Gamma (z)}\right]}^{-1}}\!1\!-\!\left(\vphantom{\times{\left[\frac{\gamma \left(z,\lambda \left(r-\alpha \right)\right)}{\Gamma (z)}\right]}^{-1}}\left[\!1\!-\!\frac{\gamma \left(z,\lambda \left(r-h-\alpha \right)\right)}{\Gamma (z)}\!\right]\times \left[\!\frac{\gamma \left(z,\lambda \left(h-\alpha \right)\right)}{\Gamma (z)}\!\right]\right.\right.\nonumber\\&\qquad\qquad \left.\left.\times{\left[\frac{\gamma \left(z,\lambda \left(r-\alpha \right)\right)}{\Gamma (z)}\right]}^{-1}\right)\right) dh \nonumber\\ & \quad=\underset{0}{\overset{r}{\int }}\left(1-\left[1-\frac{\gamma \left(z,\lambda \left(r-h-\alpha \right)\right)}{\Gamma (z)}\right]\gamma \left(z,\lambda \left(h-\alpha \right)\right)\right.\nonumber\\&\qquad\qquad\left.\times{\left[\gamma \left(z,\lambda \left(r-\alpha \right)\right)\right]}^{-1}\vphantom{1-\left[1-\frac{\gamma \left(z,\lambda \left(r-h-\alpha \right)\right)}{\Gamma (z)}\right]\gamma \left(z,\lambda \left(h-\alpha \right)\right)}\right) dh \end{align}$$ (8) where |$\Gamma(z,w)={\int}_w^{\infty }{t}^{z-1}{e}^{-t} dt$| and |$\gamma(z,w)={\int}_0^w{t}^{z-1}{e}^{-t} dt$| are the upper and lower incomplete gamma functions, respectively, and |${f}_W(.)$| is given by (3). 4.2. Expected number of hops Let us consider mesoscopic vehicular mobility intended for a multilane highway in a time-variant traffic-flow but a steady-state condition. In the vehicular connectivity, consecutive vehicles are within radio propagation range of one another. The expected number of hops from the reference vehicle to the furthest vehicle in a cluster with length |$y$| is given in following subsections for sparse, dense and intermediate vehicle traffic network environments. For a sparse vehicular density, the expected spatial per-hop progress can be derived from (6) and the corresponding expected number of hops is expressed as: $$\begin{equation} \textrm{E}\left[{\textrm{k}}_{\textrm{L}}|\textrm{y}\right]\approx \frac{\textrm{y}}{\textrm{E}\left[{\textrm{H}}_{\textrm{L}}\right]}=\textrm{y}\times{\left[\textrm{r}-\frac{1}{\rho}+\frac{\textrm{r}{\textrm{e}}^{-\rho \textrm{r}}}{1-{\textrm{e}}^{-\rho \textrm{r}}}\right]}^{-1} \end{equation}$$ (9) where |${k}_L$| denotes the number of hops in uncongested flow condition. For a dense VANET, given the mesoscopic vehicular mobility, the expected spatial per-hop progress can be derived from (7). Let |${k}_H$| denotes the number of progress in congested flow condition. Hence, the corresponding expected number of hops is $$\begin{align} &\textrm{E}\left[{\textrm{k}}_{\textrm{H}}|\textrm{y}\right]\approx \frac{\textrm{y}}{\textrm{E}\left[{\textrm{H}}_{\textrm{H}}\right]}=\nonumber\\ &\quad\textrm{y}\times \left[\underset{0}{\overset{\textrm{r}}{\int }}\left (\vphantom{\times{\left(1+\operatorname{erf}((\textrm{r}-\mu )/\sqrt{2}\sigma )\right)}^{-1}}1-\left[\vphantom{\times{\left(1+\operatorname{erf}((\textrm{r}-\mu )/\sqrt{2}\sigma )\right)}^{-1}}\left((1/2)\left(1-\operatorname{erf}\big((\textrm{r}-\textrm{h}-\mu )/\sqrt{2}\sigma \big)\right)\right)\right.\right.\right.\nonumber\\ &\qquad\qquad\times \left(1+\operatorname{erf}\left((\textrm{h}-\mu )/\sqrt{2}\sigma \right)\right)\nonumber\\&\qquad\qquad\left.\left.\left.\times{\left(1+\operatorname{erf}((\textrm{r}-\mu )/\sqrt{2}\sigma )\right)}^{-1}\right]\right)\textrm{dh}\vphantom{\underset{0}{\overset{\textrm{r}}{\int }}}\right]^{-1}.\end{align}$$ (10) In a near-capacity VANET with intermediate vehicle density traffic flow condition, the expected spatial per-hop progress is given by (8). Thus the expected number of hops is given by: $$\begin{align} &\textrm{E}\left[{\textrm{k}}_{\textrm{I}}|\textrm{y}\right]\approx \frac{\textrm{y}}{\textrm{E}\left[{\textrm{H}}_{\textrm{I}}\right]}=\textrm{y}\nonumber\\& \qquad\times \left[\underset{0}{\overset{r}{\int }}\left(1-\left[1-\frac{\gamma \left(z,\lambda \left(r-h-\alpha \right)\right)}{\Gamma (z)}\right]\gamma \left(z,\lambda \left(h-\alpha \right)\right)\right.\right.\nonumber\\&\qquad\qquad\left.\left.\times{\left[\gamma \left(z,\lambda \left(r-\alpha \right)\right)\right]}^{-1}\vphantom{1-\left[1-\frac{\gamma \left(z,\lambda \left(r-h-\alpha \right)\right)}{\Gamma (z)}\right]}\right) \textrm{dh}\vphantom{\underset{0}{\overset{r}{\int }}}\right]^{-1} \end{align}$$ (11) where |${k}_I$| denotes the number of hops in intermediate flow conditions. 4.3. Distribution of number of successful multihop progress In Section 4.1, the expected spatial per-hop progress, denoted by |${\omega}_j$||$j=L,H,I$| were mathematically investigated for sparse, dense and intermediate vehicular density, respectively, (i.e. a performance-determining factor, which can be used to characterize the multihop forwarding inside a cluster). Using the assumption of mesoscopic mobility, regardless of traffic intensity the number of successful progress, denoted by |${k}_j$||$j=L,H,I$|⁠, follows a geometric distribution with the following pdf [23]: $$\begin{equation} {\textrm{f}}_{{\textrm{K}}_{\textrm{j}}}\left(\textrm{k}\right)=(1-{\textrm{p}}_{\textrm{j}}){\textrm{p}}_{\textrm{j}}^{\textrm{k}-1}\kern2.25em \textrm{j}=\textrm{L},\textrm{H},\textrm{I} \end{equation}$$ (12) where |${p}_j$| is the probability that there is at least one node within a distance |${\omega}_j$|⁠. In other words, for a VANET with sparse vehicle density, a multihop progress through the consecutive connected vehicles can continue as long as at any traffic flow rate there is at least one vehicle within distance |${\omega}_L$| to an arbitrary vehicle, which happens with probability |${p}_L$|⁠. Therefore, there holds $$\begin{equation} {\textrm{p}}_{\textrm{L}}=\textrm{F}\left({\omega}_{\textrm{L}}\right)=1-{\textrm{e}}^{-\rho{\omega}_{\textrm{L}}.}\end{equation}$$ (13) In a dense traffic flow with a mesoscopic vehicular movability, a multihop message forwarding can be continued if only there is at least one vehicle within distance upper bounded with |${\omega}_H$| to an arbitrary vehicle, which happens with probability |${p}_H$|⁠, hence $$\begin{equation} {\textrm{p}}_{\textrm{H}}=\textrm{F}\left({\omega}_{\textrm{H}}\right)=\left(1/2\right)\left(1+\operatorname{erf}\left(\left({\omega}_{\textrm{H}}-\mu \right)/\sqrt{2}\sigma \right)\right). \end{equation}$$ (14) For an intermediate traffic flow in a near-capacity condition as shown in Table. 1, vehicle density is arranged from|$\rho =0.026$| veh/m to |$\rho =0.042$| veh/m. Regarding to (8), a multihop message forwarding progress can be continued as long as there is at least one vehicle within a distance |${\omega}_I$| to an arbitrary vehicle, with probability |${p}_I$|⁠. Therefore, the distribution of the number of successful spatial progress at the mesoscopic level is expresses as: $$\begin{equation} {p}_I=F\left({\omega}_I\right)=\frac{\gamma \left(z,\lambda \left({\omega}_I-\alpha \right)\right)}{\Gamma (z)}. \end{equation}$$ (15) 4.4. Expected time delay In this subsection, the expected time delay of a multihop forwarding in a VANET with mesoscopic mobility for any types of vehicular traffic intensity (in accordance with Table 1) is investigated. For |$j=L,H,I$|⁠, |${k}_j^{\prime }s$| are independent with identical statistical behaviors|$.$| Hence, for notation simplicity, the index |$j$| is omitted when referring to an arbitrary number of hops denoted by |$k$|⁠. The expected time delay in a sparse traffic density is expressed as: $$\begin{equation} \textrm{E}\left[{\textrm{t}}_{\textrm{L}}\right]\approx \beta \textrm{E}\left[\textrm{k}\right]=\frac{\beta}{1-{\textrm{p}}_{\textrm{L}}}=\frac{\beta}{{\textrm{e}}^{-\rho{\omega}_{\textrm{L}}}} \end{equation}$$ (16) where |$\beta$| indicates per-hop delay, i.e. the amount of time required for a vehicle to receive and process a piece of information before it is available for further relaying [48]. The corresponding expected time delay for a dense and intermediate vehicle densities are given by (17) and (18), respectively, i.e. $$\begin{align} &\textrm{E}\left[{\textrm{t}}_{\textrm{H}}\right]\approx \beta \textrm{E}\left[\textrm{k}\right]=\frac{\beta}{1-{\textrm{p}}_{\textrm{H}}}\nonumber\\&\qquad\qquad\qquad=\frac{\beta}{\left(1/2\right)\left(1-\operatorname{erf}\left(\left({\omega}_{\textrm{H}}-\mu \right)/\sqrt{2}\sigma \right)\right)}\nonumber\\&\qquad\qquad\qquad=\frac{2\beta}{1-\operatorname{erf}\left(\left({\omega}_{\textrm{H}}-\mu \right)/\sqrt{2}\sigma \right)} \end{align}$$ (17) $$\begin{align} &E\left[{t}_I\right]\approx \beta E\left[k\right]=\frac{\beta }{1-{p}_I}=\frac{\beta }{1-\frac{\gamma \left(z,\lambda \left({\omega}_I-\alpha \right)\right)}{\Gamma (z)}}\nonumber\\&\qquad\qquad\qquad=\frac{\beta \Gamma (z)}{\Gamma (z)-\gamma \left(z,\lambda \left({\omega}_I-\alpha \right)\right)}\end{align}$$ (18) 4.5. Expected connectivity distance The expected connectivity distance in a sparse VANET is given by (19). Sparse environment corresponds to uncongested conditions in which the intensity of the spatial Poisson distribution of the vehicles is |$\rho <0.026$| veh/m. Hence, $$\begin{equation} \textrm{E}\left[{\textrm{X}}_{\textrm{L}}\right]\approx{\omega}_{\textrm{L}}\textrm{E}\left[\textrm{k}\right]=\frac{\omega_{\textrm{L}}}{1-{\textrm{p}}_{\textrm{L}}}=\frac{\textrm{r}-\frac{1}{\rho}+\frac{\textrm{r}{\textrm{e}}^{-\rho \textrm{r}}}{1-{\textrm{e}}^{-\rho \textrm{r}}}}{{\textrm{e}}^{-\rho{\omega}_{\textrm{L}}}}. \end{equation}$$ (19) For a dense VANET, given the mesoscopic vehicular mobility, the expected spatial per-hop progress denoted by |${\omega}_H$|⁠. As a result, the expected connectivity distance is $$\begin{align} &\textrm{E}\left[{\textrm{X}}_{\textrm{H}}\right]\approx{\omega}_{\textrm{H}}\textrm{E}\left[\textrm{k}\right]=\frac{\omega_{\textrm{H}}}{1-{\textrm{p}}_{\textrm{H}}}\nonumber\\ &\quad=\underset{0}{\overset{\textrm{r}}{\int }}\left(\vphantom{\times{\left(1+\operatorname{erf}\left(\left(\textrm{r}-\mu \right)/\sqrt{2}\sigma \right)\right)}^{-1}}1-\left[\vphantom{\times{\left(1+\operatorname{erf}\left(\left(\textrm{r}-\mu \right)/\sqrt{2}\sigma \right)\right)}^{-1}}\left(\left(1/2\right)\left(1-\operatorname{erf}\left(\left(\textrm{r}-\textrm{h}-\mu \right)/\sqrt{2}\sigma \right)\right)\right)\right.\right.\nonumber\\&\qquad\times \left(1+\operatorname{erf}\left(\left(\textrm{h}-\mu \right)/\sqrt{2}\sigma \right)\right)\nonumber\\&\qquad\left.\left.\times{\left(1+\operatorname{erf}\left(\left(\textrm{r}-\mu \right)/\sqrt{2}\sigma \right)\right)}^{-1}\right]\right)\textrm{dh}\nonumber\\&\qquad\times{\left[\left(1/2\right)\left(1-\operatorname{erf}\left(\left({\omega}_{\textrm{H}}-\mu \right)/\sqrt{2}\sigma\right)\right)\right]}^{-1} \end{align}$$ (20) For an intermediate steady-state traffic flow, a multihop connection can continue as long as there is at least one vehicle within a distance |${\omega}_I$| to an arbitrary vehicle. Therefore, the corresponding expected connectivity distance of near-capacity condition at a mesoscopic level is expressed as: $$\begin{align} E\left[{X}_I\right]\!\approx{\omega}_IE\left[k\right]&=\frac{\omega_I}{1-{p}_I}\nonumber\\ &=\underset{0}{\overset{r}{\int }}\left(1-\left[1-\frac{\gamma \left(z,\lambda \left(r-h-\alpha \right)\right)}{\Gamma (z)}\right]\right.\nonumber\\ &\quad\left.\times \gamma \left(z,\lambda \left(h-\!\alpha \right)\right)\!\times\! {\left[\gamma \left(z,\lambda \left(r-\!\alpha \right)\right)\right]}^{-1}\vphantom{1-\left[1-\frac{\gamma \left(z,\lambda \left(r-h-\alpha \right)\right)}{\Gamma (z)}\right]}\right) dh\nonumber\\ &\quad\times{\left[1-\frac{\gamma \left(z,\lambda \left({\omega}_I-\alpha \right)\right)}{\Gamma (z)}\right]}^{-1}. \end{align}$$ (21) FIGURE 3. Open in new tabDownload slide Snapshot of south side Azadi Sports Complex at Tehran-Karaj highway in OpenStreetMap. FIGURE 3. Open in new tabDownload slide Snapshot of south side Azadi Sports Complex at Tehran-Karaj highway in OpenStreetMap. FIGURE 4. Open in new tabDownload slide Snapshot simulation of Tehran-Karaj highway at south side Azadi Sports Complex in SUMO. FIGURE 4. Open in new tabDownload slide Snapshot simulation of Tehran-Karaj highway at south side Azadi Sports Complex in SUMO. 4.6. Pdf of the connectivity distance This subsection presents the pdf of connectivity distance for different vehicular densities in a stable traffic flow condition. Note that the number of successful multihop progress in connectivity, denoted by |$k$|⁠, is a discrete random variable, whereas the connectivity distance |${x}_m\kern0.75em m=l,h,i$|⁠, is a continuous random variable. In order to simplify the analysis, inspired by [23], let |${k}_c$| be the continuous analog of |$k$|⁠, where |${k}_c$| follows an exponential distribution with parameter |${p}_c$|⁠. Therefore, the probability of having at least |$k$| consecutive successful progress in a connectivity can be given by |${p}_s^k={e}^{-{p}_ck}$|⁠, and there holds |${p}_c=-\ln ({p}_s),$| where |${p}_s\kern0.75em s=L,H,I$|⁠. Hence, the corresponding pdf of connectivity distance for sparse vehicle traffic network environments can be expressed by $$\begin{align} \mathit{\Pr}\left({X}_L={x}_l\right)\approx& \frac{p_c{e}^{-{p}_c\frac{x_l}{\omega_L}}}{\int_{0}^{\infty} {p}_c{e}^{-{p}_c\frac{x_0}{\omega_L}}d{x}_0}=\frac{p_c{e}^{-{p}_c\frac{x_l}{\omega_L}}}{\omega_L}\nonumber\\ &\quad=-\frac{\ln \left(1-{e}^{-\rho{\omega}_L}\right)}{\omega_L}.{e}^{\left(\frac{x_l\ln \left(1-{e}^{-\rho{\omega}_L}\right)}{\omega_L}\right)}\kern1em \end{align}$$ (22) where |${\omega}_L$|is given by (6) and the spatial Poisson distribution of the vehicles for a low density is determined by |$\rho <0.026$| veh/m. Furthermore, the corresponding cdf is $$\begin{align} \mathit{\Pr}\left({X}_L\le{x}_l\right)&\approx \frac{1-{e}^{-{p}_c\frac{x_l}{\omega_L}}}{\omega_L}\nonumber\\ &= \left[\ 1-{e}^{\left(-\ln \left(1-{e}^{-\rho{\omega}_L}\right)\frac{x_l}{\omega_L}\right)}\right]\times{\omega_L}^{-1}.\end{align}$$ (23) The pdf of connectivity distance for dense vehicle traffic network environments can be approximated by: $$\begin{align}\mathit{\Pr}\left({X}_H={x}_h\right)&\approx -\frac{\ln \left(\left(1/2\right)\left(1+\mathit{\operatorname{erf}}\left(\frac{\left({\omega}_H-\mu \right)}{\sqrt{2}\sigma}\right)\right)\right)}{\omega_H}\nonumber\\ &\quad\times{e}^{\left({x}_h\ln \left(\left(1/2\right)\left(1+\mathit{\operatorname{erf}}\left(\frac{\left({\omega}_H-\mu \right)}{\sqrt{2}\sigma}\right)\right)\right)\times{\omega_H}^{-1}\right)} \end{align}$$ (24) where |${\omega}_H$| is given by (7) and as shown in Table 1, the spatial Poisson distribution of the vehicles for a high density can be expressed by |$\rho >0.042$| veh/m. Finally, the corresponding pdf of connectivity distance for an intermediate vehicle density at the mesoscopic level can be approximated by: $$\begin{equation} \mathit{\Pr}\left({X}_I={x}_i\right)\approx -\frac{\ln \left(\frac{\gamma \left(z,\lambda \left({\omega}_I-\alpha \right)\right)}{\Gamma (z)}\right)}{\omega_I}\times{e}^{\left(\left[{x}_i\ln \left(\frac{\gamma \left(z,\lambda \left({\omega}_I-\alpha \right)\right)}{\Gamma (z)}\right)\right]\times{\omega}_I^{-1}\right)}\kern4.25em \end{equation}$$ (25) where |${\omega}_I$| is given by (8) and as depicted in Table 1, the spatial Poisson distribution of the vehicles for an intermediate vehicle density can be considered as |$0.026<\rho <0.042$| veh/m. FIGURE 5. Open in new tabDownload slide Snapshot vehicle mobility in Tehran-Karaj highway at south side Azadi Sports Complex. FIGURE 5. Open in new tabDownload slide Snapshot vehicle mobility in Tehran-Karaj highway at south side Azadi Sports Complex. FIGURE 6. Open in new tabDownload slide Spatial per-hop progress in a low traffic flow condition. FIGURE 6. Open in new tabDownload slide Spatial per-hop progress in a low traffic flow condition. 5. PERFORMANCE EVALUATION The inherent dynamicity of the vehicular mobility makes the VANETs a loosely coupled communication system in which the consecutive vehicles use the chance of periodic interactions. As shown in Section 4, in such a delay tolerant networks DTNs, a set of real-time mechanisms in the shape of the traffic-aware communication model is needed to tolerate the predictable disconnections. Therefore, the vehicular connectivity was mathematically investigated in the previous section at the mesoscopic level with a set of traffic-centric mobility patterns. This section presents the performance evaluation of the proposed mesoscopic model in different traffic-centric scenarios through extensive simulation experiments. To assess the evaluation metrics of discussed analysis VEINS framework is exploited [49] provided by OMNeT++ as a discrete-event network simulator [50]. In order to carry out vehicular mobility simulations over realistic highway scenarios in the city of Tehran, different tools are utilized. The regions of interest are exported from the OpenStreetMap tool [51]. For evaluation of the proposed model in realistic road topology using OpenStreetMap tool, as shown in Fig. 3, an optional part of Tehran-Karaj highway located at south side of the Azadi Sports Complex in the west of Tehran city was selected. With the make use of SUMO tool [41] as a road traffic simulator, vehicular mobility and wide variety types of traffic patterns corresponding to Table 1 were generated accordingly. Figure 4 shows a snapshot simulation of Tehran-Karaj highway at south side Azadi Sports Complex in SUMO. Figure 5 shows a vehicular mobility snapshot extracted from Fig. 4 on a larger scale in SUMO. FIGURE 7. Open in new tabDownload slide Spatial per-hop progress in a congested scenario. FIGURE 7. Open in new tabDownload slide Spatial per-hop progress in a congested scenario. 5.1. Simulation results The mesoscopic analysis performed in this section is based on a stable vehicular mobility assumption. Each of the three steady-state traffic flow conditions (corresponding to low, high and medium vehicular density scenarios) was extensively simulated. Figure 6 captures the analysis results for the expected per-hop progress versus simulation results collected in sparse scenario scattered at the end of multiple simulations for a given variety of steady-state vehicle density. As shown in Table 1, vehicular traffic intended for low traffic situations is defined from |$\rho =0.00$||$veh/m$| to|$\rho =0.026$||$veh/m$|⁠. As the vehicle’s density increases, the per-hop progress increases accordingly. Comparison of the data generated by the proposed model with the simulation results reveals that the exponential distribution tends to be more accurate than the normal distribution and Pearson type III. In the provided analysis and simulation area, the radio propagation range was considered to be |$r=300\ m$|⁠. Therefore, as shown in Fig. 6, for the lowest traffic flow (e.g. |$\rho =0.002\ veh/m$|⁠) the spatial per-hop progress was estimated from |$130\ m$| to |$170\ m$|⁠. However, at the highest level of vehicular density in sparse scenario (i.e. |$\rho =0.026$||$veh/m$|⁠), the per-hop progress improved to |$250$||$m$|⁠. FIGURE 8. Open in new tabDownload slide Spatial per-hop progress in a near-capacity scenario. FIGURE 8. Open in new tabDownload slide Spatial per-hop progress in a near-capacity scenario. FIGURE 9. Open in new tabDownload slide Expected number of hops in a sparse traffic flow condition. FIGURE 9. Open in new tabDownload slide Expected number of hops in a sparse traffic flow condition. In high density condition, the results indicate that when a vehicle’s density is greater than |$\rho=0.042$||$veh/m$|⁠, the usage of normal distribution is better suited for computing the expected per-hop progress compared with the other two methods. As depicted in Fig. 7, in the case of |$r=300$||$m$|⁠, when the traffic changes from |$\rho =0.04$||$veh/m$| to |$\rho =0.18$||$veh/m$|⁠, the per-hop progress increases from about |$275\ m$| to |$295\ m$|⁠. The extracted results confirm that the normal distribution is the best choice for computing the expected per-hop progress in congested traffic flow conditions. With respect to Table 1, in the case of a near-capacity condition as shown in Fig. 8, when the traffic flow changes from |$\rho =0.026\ veh/m$| to |$\rho =0.042\ veh/m$|⁠, the Pearson type III generates a better results than the analytical models when they are compared with the actual simulation results. The comparison of the uncongested, near-capacity and the congested flow condition of each of the provided analytical and simulation results reveals that as the vehicle density increase, the corresponding spatial per-hop progress significantly increases. Figure 9 shows the expected number of the hops relative to the growth rate of the connectivity distance when the vehicle density is |$\rho =0.01\ veh/m$|⁠. Regardless of the type of analysis, the number of per-hop progress increases as a function of the cluster length. Section 4.2 showed that the average number of steps for information progress can be implemented using exponential distribution for reasonable free-flow traffic in sparse condition. As shown in Fig. 9, the exponential distribution is the most appropriate analytical model in uncongested scenarios. FIGURE 10. Open in new tabDownload slide Expected number of hops in a dense traffic flow condition. FIGURE 10. Open in new tabDownload slide Expected number of hops in a dense traffic flow condition. Figure 10 captures the expected number of hops in a dense (congested) traffic flow condition, where |$\rho=0.18$||$veh/m$|⁠. This is when the radio range, similar to previous scenarios, is fixed at |$r=300\ m$|⁠. As illustrated in this figure, the analysis results generated from (10) better match the simulation results compared to that suggested by (9) and (11) at the mesoscopic level. Figure 11 captures the expected number of hops in a near-capacity traffic flow condition. The near-capacity condition is simulated using |$\rho=0.03$||$veh/m$|⁠. As suggested in Section 4.2, in such traffic density the Pearson type III would generate the best match against the utilized discrete-event simulator. Comparison of Figs 9–11 reveals that increasing the traffic volume reduces the average number of steps. This is because with the growth of vehicle density, it is more probable for a piece of information to travel as far as the radio propagation boundaries. Thus, the expected number of hops reduces significantly. Figure 12 shows the distribution of successful multihop progress with respect to the growth rate of the number of hops in low-density conditions with a constant value of |$\rho=0.01$||$veh/m$|⁠. As illustrated in this figure, regardless of the type of analysis, increasing the number of hops would reduce the probability of successful multihop progress. In other words, since the vehicle’s density is low, the possibility of creating large-scale connectivity is very low. In addition, this figure suggests that the behavior predicted by the exponential distribution would best match the result of the simulator. FIGURE 11. Open in new tabDownload slide Expected number of hops in a near-capacity traffic flow condition. FIGURE 11. Open in new tabDownload slide Expected number of hops in a near-capacity traffic flow condition. FIGURE 12. Open in new tabDownload slide Distribution of number successful per of hop progress in a sparse scenario. FIGURE 12. Open in new tabDownload slide Distribution of number successful per of hop progress in a sparse scenario. FIGURE 13. Open in new tabDownload slide Distribution of number successful per of hop progress in a dense scenario. FIGURE 13. Open in new tabDownload slide Distribution of number successful per of hop progress in a dense scenario. Figure 13 shows the distribution of successful progress in a high-density scenario with a value of |$\rho =0.044$||$veh/m$|⁠. Using the analytical model in (14), it is clear that the model generated by normal distribution at the mesoscopic level is the closest match to the actual simulation results. In the case of near-capacity flow conditions with |$\rho =0.03\ veh/m$|⁠, as depicted in Fig. 14, the distribution of Pearson type III generates the best prediction when compared to the simulation results generated by OMNeT ++. FIGURE 14. Open in new tabDownload slide Distribution of number successful per of hop progress in an intermediate scenario. FIGURE 14. Open in new tabDownload slide Distribution of number successful per of hop progress in an intermediate scenario. Comparison of Figs 12–14 reveals that for |$r=300\ m$| and with increasing vehicular density, the probability of successful propagation significantly reduces. However, as traffic density increases, the distribution of successful multihop relay increases, reducing the total number of hops. This is because with the growth of vehicle density, it is more probable that the information travel as far as possible to the radio propagation boundary, resulting in a significant reduction in the average number of hops. The expected time delay at the steady-state traffic flow condition as a function of vehicle density is presented in Fig. 15. In this figure, similar to [48], the per-hop delay is considered to be |$\beta =4\ ms$|⁠. This figure illustrates that by increasing the vehicle density from |$\rho =0.01$||$veh/m$| to |$\rho=0.024$||$veh/m$|⁠, the expected time delay in the information dissemination process increases. In order to clarify the obtained result, a special case of traffic flow is explained next. FIGURE 15. Open in new tabDownload slide The expected time delay in a sparse scenario. FIGURE 15. Open in new tabDownload slide The expected time delay in a sparse scenario. FIGURE 16. Open in new tabDownload slide Expected connectivity distance in a sparse scenario. FIGURE 16. Open in new tabDownload slide Expected connectivity distance in a sparse scenario. Let us assume that in a sparse highway, a single vehicle, with no other vehicle at its radio propagation boundary, forms connectivity. In this situation, the delay of the propagation time is as much as one step, although no vehicle can receive this message. By increasing the vehicle density, the number of adjacent vehicles increases, resulting in an increase in the created connectivity distances. Therefore, with increasing the connectivity distance, the expected time delay for the dissemination of information increases. Based on the discussion in Section 4.4, it is expected that the proposed analytical model, which is generated based on exponential distribution, to best match the simulation result and to generate a better fit compared to the normal and Pearson type III distributions. FIGURE 17. Open in new tabDownload slide Expected connectivity distance in a low traffic flow condition. FIGURE 17. Open in new tabDownload slide Expected connectivity distance in a low traffic flow condition. FIGURE 18. Open in new tabDownload slide Pdf of connectivity distance in a sparse scenario. FIGURE 18. Open in new tabDownload slide Pdf of connectivity distance in a sparse scenario. In order to verify the proposed analytical model, the result of simulating a comparative scenario is captured in Fig. 16. In this simulation, a sparse traffic flow with vehicle density of |$\rho =0.02\ veh/m$| at the mesoscopic level is considered. Figure 16 shows the expected connectivity distance as the radio frequency grows from range |$r=50\ m$| to |$r=300\ m.$| By increasing the radio propagation range, the possibility of connection with adjacent vehicles increases. This in turn increases the probability of connectivity as well as the expected connectivity distance. Figure 17 shows the expected connectivity distance as a function of vehicle density. In this scenario, unlike Fig. 16, the radio range is fixed at |$r=300\ m.$| Note that with an increase in vehicle density, the chances of interacting with adjacent vehicles increases. This in turn directly impacts the growth of the expected connectivity distances. Figure 18 shows the pdf of the connectivity distance density in a steady-state and sparse traffic condition as a function of the size of the connectivity distance. Since traffic intensity is |$\rho =0.02\ veh/m$|⁠, for a sparse and uncongested steady-state traffic network, the chances of forming a large-scale connectivity distance is very low. The analytical model very closely matches the simulation results at the mesoscopic level. FIGURE 19. Open in new tabDownload slide Pdf of connectivity distance in uncongested flow condition. FIGURE 19. Open in new tabDownload slide Pdf of connectivity distance in uncongested flow condition. FIGURE 20. Open in new tabDownload slide Expected number of hops in different traffic scenarios. FIGURE 20. Open in new tabDownload slide Expected number of hops in different traffic scenarios. Figure 19 shows the pdf of the cluster length as a function of the vehicle density in a sparse traffic flow condition. In this scenario, the pdf of connectivity with the length of |$5000\ m$| at the mesoscopic level is simulated. As depicted in Fig. 19, due to the growth of vehicle density (from about |$\rho =0.008\ veh/m$| to |$\rho =0.013\ veh/m$|⁠), the pdf of the cluster length has also increased. By increasing the traffic flow from |$\rho =0.013\ veh/m$| to |$\rho =0.025\ veh/m$|⁠, the pdf of the connectivity distance was significantly reduced. The result is expected because, with increasing density, over time, the resulting cluster size grows beyond |$5000\ m$|⁠. This reduces the pdf of the connectivity distance significantly. 5.2. Discussion Figure 20 summarizes the results collected from Figs 9–11 and verifies that each traffic flows is best modeled using a customized analytical model. Figs 21 and 22 verify that the exponential and the normal distributions are the best choices for the connectivity analysis in sparse and dense traffic flow conditions at the mesoscopic level, respectively. FIGURE 21. Open in new tabDownload slide Pdf of connectivity distance in three sparse scenarios. FIGURE 21. Open in new tabDownload slide Pdf of connectivity distance in three sparse scenarios. FIGURE 22. Open in new tabDownload slide Pdf of connectivity distance in three congested scenarios. FIGURE 22. Open in new tabDownload slide Pdf of connectivity distance in three congested scenarios. VANETs play a great role in the growth of ITS, future technologies and the IoT. Hence, the accuracy of the extracted results from this study will be a useful choice to develop the IoV as a practical and attractive part of the IoT. Therefore, improving the accuracy of the intervehicle communications results in customizing the location-based services such as vehicular fog computing as well as mobile edge computing. 6. Conclusion In this paper, a new mathematical steady-state traffic-centric model for investigating the connectivity at the mesoscopic level is proposed. Using the proposed model, the expected spatial per-hop progress, the distribution of intervehicle distances and the distribution of the number of successful multihop progress were investigated. Furthermore, the expected time delay and the expected connectivity distance for a variety of stable traffic flows were studied, and the pdf of connectivity distance at the mesoscopic level was mathematically formulated. The accuracy of the proposed model was validated through discrete-event network and traffic simulations. Using the proposed model, it was shown that in a vehicular network, although the mobility model was limited to the roadmap, the variety of traffic flow conditions provided different mobility patterns as well as different communicating results. The future research includes the extension of the proposed model to investigate vehicular connectivity at multi-level highways, as well as vehicles traveling in opposite directions in steady-state traffic flow conditions. 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TI - Traffic-Centric Mesoscopic Analysis of Connectivity in VANETs
JF - The Computer Journal
DO - 10.1093/comjnl/bxz094
DA - 2020-02-19
UR - https://www.deepdyve.com/lp/oxford-university-press/traffic-centric-mesoscopic-analysis-of-connectivity-in-vanets-KcUS9YJLTi
SP - 203
VL - 63
IS - 2
DP - DeepDyve
ER -