Outage in Motorway Multi-Lane VANETs with Hardcore Headway Distance Using Synthetic Traces

Outage in Motorway Multi-Lane VANETs with Hardcore Headway Distance Using Synthetic Traces

Konstantinos Koufos and Carl P. Dettmann
K. Koufos and C.P. Dettmann are with the School of Mathematics, University of Bristol, BS8 1TW, Bristol, UK. {K.Koufos, Carl.Dettmann}@bristol.ac.ukThis work was supported by the EPSRC grant number EP/N002458/1 for the project Spatially Embedded Networks. All underlying data are provided in full within this paper.
Abstract

In this paper we analyze synthetic mobility traces generated for three-lane unidirectional motorway traffic to find that the locations of vehicles along a lane are better modeled by a hardcore point process instead of the widely-accepted Poisson point process (PPP). In order to capture the repulsion between successive vehicles while maintaining a level of analytical tractability, we make a simple extension to PPP: We model the inter-vehicle distance along a lane equal to the sum of a constant hardcore distance and an exponentially distributed random variable. We calculate the J-function and the Ripley’s K-function for this point process. We fit the parameters of the point process to the available traces, and we illustrate that the higher the average speed along a lane, the more prominent the hardcore component becomes. In addition, we consider a transmitter-receiver link on the same lane, and we generate simple formulae for the moments of interference under reduced Palm measure for that lane, and without conditioning for other lanes. Finally, we illustrate that under Rayleigh fading a shifted-gamma approximation for the distribution of interference per lane provides a very good fit to the simulated outage probability using synthetic traces while the PPP fails.

Headway distance models, probability generating functional, reduced Palm measure, synthetic mobility traces.

I Introduction

Applications and protocols for vehicle-to-vehicle communication have been extensively investigated during the past two decades [1, 2]. The cost of deploying large scale testbeds is high, and the proposed solutions had been mostly assessed using computer simulations [3]. Modern simulators for vehicular networks can include street maps, and realistic micro-mobility behavior, e.g., lane changing, acceleration/deceleration and car-following patterns [4]. In addition, they may be calibrated with real measurement data for macroscopic features like intensity and average speed of vehicles, giving rise to synthetic mobility traces. The traces available in [5, 6] are valuable, because they can be used to validate the performance obtained with simplified deployment models, as we will do in this paper.

With the recent advent of wireless networks with irregular structure, e.g., small cells, sensors and wireless ad hoc networks, point processes have been employed to model the locations of network elements and investigate their performance [7]. In vehicular networks, the spatial model can be divided into two components: the road infrastructure and the deployment of vehicles along a road. Instead of running time-consuming simulations, the analysis with point processes can give a quick insight into the impact of various parameters on the properties of the network. The analytical results should be trusted only if the adopted processes are realistic.

Modeling the road infrastructure, and the headway distance and time (measured from the tip of a vehicle to the tip of its successor as they pass a point on the roadway) has long been a subject studied in transportation research. The adopted models are often complicated: Random iterated tessellations have been fitted to real infrastructure data, e.g., inter-city main roads and side streets, minimizing some distance metric [8]. Empirical studies revealed that the distribution of time headway depends on traffic status; it is well-approximated by the log-normal distribution under free flow [9] and by the log-logistic distribution under congestion [10]. The distribution of vehicles may directly be modeled with a two-dimensional point process instead of modeling first the road network. The Thomas, Matèrn cluster and log-Gaussian Cox processes were recently found to fit well real snapshots of taxis, independently of the regularity of urban street layouts [11]. Despite their impressive accuracy, all the models above seem quite complex to incorporate into coverage and capacity performance evaluation of Vehicular ad hoc networks. In order to balance between model accuracy and analytical tractability, the random orientation of roads has been modeled by Poisson line process, coupled with one-dimensional (1D) Poisson Point Processs for the locations of vehicles along each road [12, 13]. Under these assumptions, the distribution of vehicles becomes a Cox process in the plane, and the coverage probability of a typical vehicle is calculated in [12, Theorem 1]. The conflicting effect of road intensity (higher intensity increases the interference level) and vehicle intensity (higher intensity increases the average link gain) has also been investigated. Regular street layouts have been modeled by the Manhattan Poisson line process with the empty space filled-in with buildings to resemble urban districts [14]. Near intersections, the packet reception probability in VANETs decreases because there is dominant interference both from horizontal and vertical streets [15].

Inter-city motorway traffic does not require a complex model for the road network. A superposition of 1D point processes should be sufficient to model the locations of vehicles along multi-lane motorways. Not surprisingly, the PPP has been widely adopted in these scenarios. Due to the independence assumption, it has been used in performance evaluation of complex communication protocols like multi-hop networks with interference [16] and collision-avoidance type of channel access schemes [17]. Unfortunately, the PPP will be accurate only under certain conditions. For instance, the independence assumption may not hold near traffic lights due to clustering [18]. In high-speed motorways, the drivers maintain a safety distance from the vehicle ahead, depending on their speed and reaction time. When there are few lanes, the PPP does not capture correctly the distribution of inter-vehicle distances because it does not constrain the minimum spacing between successive vehicles [23]. In [19], it is shown that for a Poisson flow of vehicles entering a road, the headway distance is exponentially distributed in the steady state, under the assumption that each vehicle selects in the entrance its speed from a common Probability Distribution Function (PDF). The suitability of 1D PPP has been established in [20], under the assumption of low transmission probability per vehicle. This assumption might be true with the underlying automotive radar application, where each vehicle sends a short pulse and waits for the response during the duty cycle. Intuitively, under strong thinning, the interference field due to a lattice converges to that due to a PPP of equal intensity, and the PPP becomes a valid model for target detection. Diverting from the PPP’s independence assumption adds very high complexity in the performance evaluation. The lifetime for a link with log-normally distributed headway distances is studied in [21], but the impact of interference is neglected. The bit error probability and channel capacity are studied in [22] with a non-uniform intensity measure modeling clustering of vehicles, but interference is neglected there too.

In [23, 24], we have taken a step away from the 1D PPP, modeling the headway distance along motorways equal to the sum of a constant hardcore distance and an exponentially distributed Random Variable (RV). The shifted-exponential distribution of inter-arrivals is known in transportation research as the Cowan M2 model [25]. The hardcore distance models the safety distance between successive vehicles along a lane, and it makes the locations of vehicles correlated. The correlation properties have been studied since the early 1950’s in statistical mechanics, under the name of radial distribution function for hard spheres, where the spheres are the particles of 1D fluids [26]. In [23, 24] we simplified the Pair Correlation Function (PCF) keeping only short-range correlations, and we generated simple expressions for the variance of interference, the skewness and the probability of outage at the origin. In the current paper, we will make an extension to multi-lane networks, superimposing independent Cowan M2 models for each lane. Most importantly, we will validate whether the probability of outage predicted using Cowan M2 agrees with the probability of outage using synthetic traces [5, 6].

Empirical data has already been used in performance evaluation of wireless cellular networks and VANETs. In [27], real snapshots of macro base stations are fitted by pseudo-likelihood maximization and minimum contrast to PPP, Strauss and Poisson hardcore processes. The unsuitability of PPP is demonstrated through spatial statistics, and the Strauss process gives the best fit to the coverage probability. In vehicular networks, the study in [11] first points out the dependency in taxis’ locations through sampling. Then, it identifies the point process minimizing the contrast to the Ripley’s K-function and to the connection probability of the snapshot. In [5, 6], the authors processed measurement data about the intensity and speed of vehicles per lane as they pass a point of a motorway. They generated synthetic traces to investigate the topology of motorway traffic, and highlight the impact of communication range on full connectivity, however, without interference. Instead, we would like to see whether a simple enhancement to the PPP model, i.e., the Cowan M2, can capture better than PPP the outage probability of VANETs using these traces. The contributions of our work are:

  • We analyze the synthetic traces through spatial statistics, and we illustrate that the envelopes of the J- and the L- functions, see Section III, for small distances indicate repulsion for all lanes. The leftmost lane which is characterized by the highest average speed of vehicles experiences the highest degree of repulsion. This suggests that Cowan M2 might be an adequate model because the repulsion can be captured through the hardcore distance.

  • We calculate the J-function of the hardcore process in closed-form and the Ripleys’s K-function as a finite sum. These functions can be used for fitting the parameters of the hardcore process to the snapshot. Even if the fitting is done with some other method, it is important to see the comparison between the summary statistics of the snapshot and those of the fitted point process.

  • We generate simple but accurate approximations for the mean, the variance and the skewness of interference under reduced Palm for the hardcore point process. Then, we approximate the Laplace Transform (LT) of interference from the lane containing the transmitter-receiver link. Note that the moments of interference derived in [23, 24] are without conditioning and thus, they will be used to approximate the LT of interference from other lanes.

  • We illustrate that the J-function for Cowan M2 fits well the empirical J-function for small distances and all lanes, capturing the repulsion between successive vehicles. We use a shifted-gamma approximation for the distribution of interference per lane, which fits well the outage probability calculated using synthetic traces, while PPP fails. We assess the accuracy by visual inspection of the outage probability and also by goodness-of-fit metrics.

The remainder of this paper is organized as follows: In Section II, we introduce the system model for the lane containing the transmitter-receiver link. In Section III, we calculate summary statistics for the hardcore point process. In Section IV we generate the statistics’ envelope for the synthetic traces, and we show how to fit the parameters of the point process to the snapshot. In Section V, we approximate the first three moments of interference under reduced Palm, and we fit a shifted-gamma approximation for the distribution of interference. In Section VI, we extend the interference model to multiple lanes, and in Section VII we validate our approximation for the outage probability with synthetic traces. Finally, in Section VIII, we summarize the main results of this study and highlight few directions for future work.

Ii System model

We consider 1D point process of vehicles , where the inter-vehicle distance follows the shifted-exponential PDF 111The extension to multi-lane unidirectional vehicular deployments will be given in Section VI.. The shift is denoted by , and the rate of the exponential by . The intensity of vehicles is calculated from , or equivalently . The joint probability there are two vehicles at and , is , where

(1)

and  [26, equation (32)].

Fig. 1: Normalized PCF with respect to the normalized distance . The dashed lines correspond to , or,  [23, Fig.2].

The PCF, , is depicted in Fig. 1. For small hardcore distance as compared to the mean inter-vehicle distance , the PCF converges quickly to , which is the PCF of a PPP of intensity . The locations of vehicles become uncorrelated at few multiples of , when the hardcore distance does not dominate over the random part of the deployment.

The higher-order correlations are naturally more complicated than the PCF. For ordered points, , the th order correlation is  [26, equation (27)]. For instance, the third-order correlation describing the probability to find a triple of distinct vehicles at and , is

(2)

We condition on the location of a transmitter at the origin. The receiver associated to it is at distance away, see Fig. 2. The locations of vehicles behind the transmitter and in front of the receiver follow the point process . The distance follows the shifted-exponential distribution too, with shift and rate . We assume that only the vehicles behind the transmitter generate interference. Other vehicles may also interfere due to antenna backlobes radiation, but this would not dominate the overall interference level, and it is currently neglected. Given , the distance-based useful signal level is denoted by .

Fig. 2: The vehicles are modeled as identical impenetrable disks of diameter . Their antenna is placed at the right side of the disk. A transmitter (black disk) is conditioned at the origin and paired with the receiver (hollow disk) at . The vehicles behind the transmitter (red disks) generate interference at the receiver, while the rest (blue disks) do not. All vehicles move rightwards but the interferers are assumed in the positive half-axis to simplify the expressions.

The transmit power level is normalized to unity for all the vehicles. The propagation pathloss exponent is denoted by . The distance-based pathloss is . The fading power level over the interfering links, , and over the transmitter-receiver link, , is exponential (Rayleigh distribution for the fading amplitudes) with mean unity. The fading is independent and identically distributed (i.i.d.) over different links. The interferers are active with probability .

Iii Spatial statistics

Perhaps the most classical measure characterizing the behavior for a set of points (repulsion, clustering or complete randomness) is the J-function at distance ,  [28, Chapter 2.8]. It is defined as the ratio of two complementary Cumulative Distribution Functions: the nearest-neighbor distance CDF divided by the contact CDF . The nearest-neighbor distance is the distance between a point and its nearest neighbor . The CDF describes the distance between a reference location and the nearest point , i.e., . For a PPP , e.g., in 1D both and are exponential CDFs with rate twice the intensity. For a repulsive point process with hardcore distance , , while , resulting to . On the other hand, is associated with clustering.

The function for the point process introduced in Section II is shifted-exponential, . The function for the point process has been derived in [24, Section III], and it has a piecewise form with breakpoint at . After substituting , into J, we end up with

(3)
(a)
(b)
Fig. 3: Spatial statistics for the hardcore process. Intensity , hardcore distance m, simulations within a line segment of km. (a) In each run, we measure the nearest-neighbor distance for a point of the process, and the contact distance at the origin. The calculation is given in (3). (b) In each run we select a point of the process close to the origin and we count the points of the process within distance from that point. Then we normalize by . The calculation is given in (4) after scaling by half.

It is straightforward to see that since . The J-function can indicate the repulsion, however, it uses empty-space distributions and thus, it does not give any info about the long-range behavior of the process. A more appropriate measure directly targeting the second-order properties is the Ripley’s K-function [29, Chapter 2.5]. The K-function counts the mean number of points falling within some distance from a point of the process without counting this point, and normalizes the outcome with the intensity . In 1D, , where is a ball with radius centered at . For a PPP of intensity , the mean number of points within and excluding is . Therefore normalizing the K-function by two can be used to distinguish between repulsion and clustering in 1D. For instance, means that the point process contains fewer points as compared to the PPP of equal intensity within indicating repulsion. The normalized Ripley’s K-function is usually referred to as the L-function. Informally speaking, the K-function is the integral of within the area centered at and extending up to distance from  [29, Chapter 2.5]. In our 1D system and notational set-up we get

where , is the indicator function, the factor two accounts for the negative half-axis, and also note that in (1), the PCF is defined as the joint probability to find a pair of vehicles at , thereby we have to divide by since the K-function conditions on the location of a vehicle at .

The above expression can be simplified by exchanging the orders of summation (over ) and integration, and adding up terms of the same -th order.

After carrying out the integration in terms of we get

(4)

where is the incomplete Gamma function.

Example illustrations for the and are depicted in Fig. 3. Both functions capture the repulsive property of the point process and quantify the hardcore distance by visual inspection. For a realization within a finite domain, the evaluation of is constrained by the half of maximum inter-point distance, denoted by , and the evaluation of by the maximum nearest-neighbor distance , both increasing logarithmically with the total system size. For most realizations , and the function will take very high values close to . Because of that, we see in Fig. (a)a that the simulated average starts to rise for . In Fig. (b)b, the slope of the L-function for m becomes practically unity, indicating that the correlation for distance separations larger than approximately is not strong for . This remark is in accordance with the ’blue curve’ in Fig. 1, and it cannot be deduced from the graph of J-function. Next, we shall generate the summary statistics of synthetic traces, and study whether the point process can capture them better than the PPP.

Iv Validating the model with synthetic traces

The studies in [5, 6] use measurement data from a three-lane motorway, M40, outside Madrid, Spain from 8:30 a.m. to 9:00 a.m. (busy hour), and from 11:30 a.m. to 12:00 p.m. (off-peak) on May 7 2010. The traffic direction for all the lanes is the same. Sensors buried under the concrete layer of the roadway collected per-lane measurements every second about the number of passing vehicles and their speed. The measurements have been fed to calibrate a microscopic simulator, whose output is the location of each vehicle, i.e., lane and horizontal position over a road segment of km with one second granularity. The corresponding simulation time is half an hour, which means that snapshots of vehicles per time span are available. The traces have been calibrated to represent quasi-stationary road traffic for each lane [5]. In the validation of our models, we will drop snapshots from the initial ten minutes of the simulation, in order to allow the first vehicles entering the roadway reach at the exit. This is to have enough samples of inter-vehicle distances for each snapshot while constructing its empirical CDF. The locations of a vehicle and subsequently, the distribution of inter-vehicle distances between the snapshots are correlated. We will fit the PPP and the hardcore point process for each snapshot and lane, independently of other snapshots and lanes. The interference and outage models in the following sections are for a single snapshot too. Modeling the temporal aspects of interference, see for instance [30, 31] is left for future work.

(a) Left lane
(b) Middle lane
(c) Right lane
Fig. 4: Envelope of J-function over 1200 snapshots for the busy hour (gray-shaded areas). The solid line is an example illustration of for the 1000-th snapshot. The CDFs are estimated within a window of km to avoid boundary effects. The PPP corresponds to , dashed-line.
(a) Left lane
(b) Middle lane
(c) Right lane
Fig. 5: Envelope of J-function over 1200 snapshots during off-peak. See the caption of previous figure for explanation of the lines.

In order to estimate the distributions , we first take an inner segment (or window) of the -th lane for that snapshot to avoid boundary effects. The endpoints of the window should be at least away from the endpoints of the roadway, where is the maximum considered in the evaluation of the J-function [29, Chapter 1.10]. In order to estimate the distribution of for a snapshot, we calculate the nearest-neighbor distance for each point in the window. In order to estimate the distribution of , we randomly select locations within the window, and for each one of them we calculate its minimum distance over the snapshot’s points. In Fig. 4, it is evident that all lanes exhibit repulsion for small distances, for all snapshots. The vehicles do not risk approaching each other very closely. The repulsion is stronger at the left lane, because this lane experiences the highest average speeds [6, Fig. 5b]: The envelope stays larger than unity up to m and there are only few instances of clustering for the considered range of distances. The example illustration for the 1000-th snapshot and the right lane shows that the traces may also exhibit clustering for distance m. Nevertheless, for small distances, all snapshots behave similarly to the J-function of the hardcore process, see Fig. (a)a. This is due to the physical dimension of vehicles and the safety distance maintaining from the vehicle ahead. During off-peak, there is still clear repulsion at small distances for all lanes and snapshots, but the differences between the lanes are less prominent. Off-peak is characterized by lower intensity of vehicles. Because of that, the drivers select their lane more freely as compared to the busy hour, making the lanes to look more alike to each other. In Fig. 6 we reconfirm, through the envelope of L-function, that the left lane has the highest repulsion. In order to generate the L-function for a snapshot, we take each point in the window and count the number of points within a distance from that point. Then, we average over all points and normalize by . The envelope already quantifies the range of hardcore distance for each lane. As expected, the left lane gives the highest values.

(a) Left lane
(b) Middle lane
(c) Right lane
Fig. 6: Envelope of over 1200 snapshots for the busy hour. Example illustration of for the 1000-th snapshot (solid line). For each snapshot the intensity per lane required in the calculation of the K-function is estimated as being the inverse of the mean inter-vehicle distance. The L-function is evaluated within an inner window of km to avoid boundary effects, m. The dashed-line corresponds to PPP.

The analysis through summary statistics has given enough evidence that a hardcore type of process is more suitable to model the available traces than PPP. We will use Cowan M2 for the locations of vehicles along each lane. Apart from being used in transportation research [25], this model will allow us to construct simple approximations for the moments of interference. Before starting with interference modeling, we still need to estimate the model parameters, i.e, per-lane intensity and hardcore distance for a particular snapshot. Also, it would be worth demonstrating whether the summary statistics for a snapshot fall within the envelope of the fitted hardcore process.

(a) Left lane
(b) Middle lane
(c) Right lane
Fig. 7: Empirical CDF of inter-vehicle distances at the 1000-th snapshot of the busy hour along with approximations.
(a) Intensity
(b) Hardcore
(c) Product
Fig. 8: Estimation of intensity and hardcore distance per lane using synthetic traces between 8:40 a.m. and 9:00 a.m.
(a) Intensity
(b) Hardcore
(c) Product
Fig. 9: Estimation of intensity and hardcore distance per lane using synthetic traces between 11:40 a.m. and 12:00 p.m.

Let us denote the inter-vehicle distances for the -th lane by , where is the sample size. For the PPP, the intensity will be estimated as being equal to the inverse of the mean inter-vehicle distance obtained from the sample, i.e., . Note this is also the Maximum Likelihood Estimate (MLE) for the PPP intensity. For the hardcore process, the parameterization of the intensity and the hardcore distance can be done with various methods: (i) The method of moments matches the mean, , and the variance, , of the shifted-exponential distribution to the sample mean and variance. (ii) The MLE is the minimum inter-vehicle distance obtained from the sample , and . (iii) The (non-linear) least-squares iteratively estimates the that minimize the square difference between the empirical CDF and the shifted-exponential CDF, , where are the bins of the empirical CDF. In order to reduce computational complexity, we may set first , then estimate a single parameter . In that case, the PPP and the hardcore process are forced to have the same intensity. In either case, we must constrain the estimates as .

In Fig. 7 we have selected the 1000-th snapshot, and plot the empirical CDF of inter-vehicle distances for the three lanes along with approximations. The five fitting methods perform similarly for other snapshots too. The exponential CDF cannot capture at all the repulsion between successive vehicles. The method of moments may give a negative estimate for the hardcore distance, see Fig. (b)b and Fig. (c)c. The MLE for the shifted-exponential distribution fits very well the lower tail but it completely fails elsewhere. The least-squares estimation provides relatively good fit over the full range. When is fixed equal to the MLE of PPP, the fit of least-squares becomes slightly worse.

In Fig. 8, we depict the estimates for the intensity and the hardcore distance over snapshots using least-squares fitting both for and . We use the curve fitting toolbox in MatLab. The right lane is characterized by the highest intensity and the left lane, due to the high speeds, gives the highest values for the hardcore distance and for the product . During off-peak, see Fig. 9, the discrepancy of between the lanes becomes less prominent. This is in accordance with the behavior of the envelope of the J-function during busy hour and off-peak, see Fig. 4 and Fig. 5.

Finally, for a snapshot of vehicles, we illustrate in Fig. 10 that the envelope of J-function for the fitted hardcore process contains the J-function generated by the snapshot. Only for the right lane, the snapshot may fall slightly outside the envelope. As illustrated in Fig. 4 Fig. 9, it is the left lane characterized by the highest repulsion. Cowan M2 best describes the distribution of vehicles in the left lane followed by the other two lanes. On the other hand, the envelope of J-function using the fitted PPP cannot capture at all the J-function of the snapshot for all lanes and distances m. This is one more evidence about the suitability of the hardcore process to model motorway traffic in comparison with PPP. Even though the calculated for the fitted process matches that of the snapshot only in the initial increasing part, we will illustrate in the next section that Cowan M2 considerably improves the outage probability predictions of PPP.

(a) Left lane
(b) Middle lane
(c) Right lane
Fig. 10: Empirical J-function from the 1000-th snapshot of the busy hour (solid line), calculated , see (3), for the fitted hardcore point process (dashed-line), simulated envelope of J-function for the fitted hardcore process over runs (gray-shaded area), and simulated envelope of J-function for fitted PPP over runs (lightly gray-shaded area).

V Outage probability under reduced Palm

Recall from Section II that the interfering vehicles follow the hardcore point process behind the transmitter. Since we condition on the location of a point of the process (the transmitter), the calculation requires the reduced Palm probabilities.

The Probability Generating Functional (PGFL) of the hardcore point process, needed to calculate the LT of interference, is not available. Also, due to the correlated locations of vehicles, only the first few terms of the factorial moment expansion for the PGFL, see [32], can be approximated. The simplest way to get around these issues, is to calculate a few moments and fit the interference distribution to well-known functions with simple LTs. The method of moments have been widely-used in wireless communications research community to model signal-to-noise ratio in composite fading channels [33], aggregate interference [34] and spectrum sensing channels [35].

In order to select appropriate candidate distributions, we first note that the interference CDF decays exponentially fast near the origin because the point process is stationary [36, Theorem 4]. In addition, for bounded propagation pathloss (the hardcore distance essentially makes the pathloss function bounded), the decay at the tail is dominated by the fading [36, Theorem 3], thus this is exponential too. Popular distributions for interference modeling in wireless networks with irregular geometry can be found in [37, 38] including gamma, inverse Gaussian and Weibull. Even though the inverse Gaussian distribution seems the best candidate because it decays exponentially fast near the origin and at the tail, it has only two free parameters, and it does not provide very good fit via moment matching in our system set-up. Its generalized counterpart has three free parameters, but numerical methods are required to calculate them. We will use the shifted-gamma distribution that allows to express its parameters as simple functions of the mean, the variance and the skewness of interference. The gamma distribution decays polynomially at the origin and thus, we expect to see some discrepancy in the upper tail of the Signal-to-Interference Ratio (SIR) CDF. In some recent work [39], the parameters of the approximating distribution are calculated using a combination of moment matching and MLE. An apparent advantage of this method is its extention to mixture models using iterative expectation maximization algorithm which provides a very good fit. Nevertheless, the major drawback is the requirement for interference samples. In our system set-up the interference depends on the link distance , the road traffic parameters , the activity and the channel models. Therefore extensive measurements (or simulations) are needed before regression analysis.

In order to approximate the mean, the variance and the skewness of interference, we may approximate the PCF of the hardcore process by the PCF of PPP for distance separations larger than . The same approach has been followed in [23, 24], however, without conditioning on the location of a point. The calculation details under reduced Palm are given in the supplementary material (optional reading). Over there, we see that the resulting expressions can be evaluated numerically but they are quite complex, not providing us with enough insight about the impact of different parameters on the interference. Because of that, we have also included the expressions after approximating the PCF of by the PCF of PPP for distance separation larger than instead of . Even with this simplification, the correlated locations of vehicles are still retained by the model. Finally, we get

(5)

In Fig. 11 we depict the mean, the standard deviation and the skewness of interference for . Approximating the PCF for distance separation larger than seems very accurate in the estimation of moments. Fortunately, the simpler approximation (, see (5)) captures the general trend in the behavior of interference statistics.

(a) Mean
(b) Std. dev.
(c) Skewness
Fig. 11: Interference statistics with respect to the hardcore distance . Intensity of vehicles , pathloss exponent , activity probability , and useful link distance . simulations per marker.

The parameters of the shifted-gamma distribution, , as functions of the link distance can be estimated as , and . The LT of interference evaluated at is . Finally, the calculation of the outage probability requires to average the LT over the distance .

(6)

In Fig. (a)a we see that the shifted-gamma approximation is somewhat off in the upper tail due to the polynomial, , decay of the gamma distribution near . On the other hand, the PPP fails in the high reliability regime (lower tail), and it significantly underestimates the outage probability in the upper tail. For presentation completeness, the outage probability due to PPP has been calculated as

It is worth to mention that under reduced Palm, the outage probability due to PPP does not depend on the intensity of vehicles . The link distance follows the exponential distribution with rate , equal to the intensity of PPP interferers. This resembles the calculation of downlink coverage probability in PPP cellular networks, which is also independent of the intensity of base stations in the interference-limited regime with nearest base station association [40, equation (14)]. For the hardcore process, the outage probability in (6) depends on the intensity through the parameter and also through the parameters of the shifted-gamma distribution, see (5).

(a) Own lane only,
(b) Other lane only,
(c) Both lanes,
Fig. 12: Probability of outage at the receiver when the interference originates from (a) the lane containing the transmitter-receiver link , (b) another lane parallel to , and (c) both lanes. Intensity of vehicles , pathloss exponent , acitivity probability , hardcore distance and guard zone m for . simulations.

Vi Multi-lane networks

Let us consider another lane, , with same road traffic parameters and same activity probability for the vehicles. Extension to more lanes with different intensities and hardcore distances is straightforward, and it will be discussed in the next section. The transmitter and receiver of the considered link are still located at ; is just an extra source of interference. Let us denote by the lane separation and by the beamwidth of the vehicle antenna. Then, the interfering vehicles from are located at distances larger than from the receiver and behind it, see Fig. 13 for an illustration. Note that the guard zone is expected to be much larger than the lane separation for practical values of and .

Unlike , the interference analysis for does not require conditioning. Keeping in mind that the point process is stationary, the mean interference due to does not depend on the correlation properties but only on the intensity . After neglecting the impact of lane separation on the distance-based pathloss, the variance of interference due to has been calculated in [23, Equation (14)] and the skewness in [24, Lemma 1]. After scaling all moments by half to consider the vehicles behind the receiver, the approximations for the mean, the variance and the skewness become

(7)

We will use a shifted-gamma approximation for the distribution of interference from . The moments of interference and subsequently the gamma parameters for , , do not depend on the link distance as in (5) and (6). The outage probability due to only can be calculated by integrating the LT of interference over the link distance.

(8)

where .

In Fig. (b)b we have validated (8) against simulations. In order to calculate the outage probability under aggregate interference from and , we need to multiply the LTs of interference from the two lanes before integrating over the link distance. The results are depicted in Fig. (c)c.

Fig. 13: The vehicles at lane at distances larger than from the receiver and behind it (red disks) are potential sources of interference. The receiver (hollow disk) is now placed at the origin to facilitate interference analysis due to . See also the caption of Fig. 2.

Vii Probability of outage synthetic traces

The synthetic traces in [5, 6] consist of three-lane traffic and 1200 snapshots. We denote by the left lane characterized by the highest average speed, by the middle lane and by the right lane. In order to validate the shifted-exponential deployment along with the shifted-gamma interference models, we will carry out simulation runs for the outage probability per snapshot. For each snapshot, we construct the empirical CDF of inter-vehicle distances for every lane. In each simulation run, we sample the empirical CDFs (using linear interpolation), generating enough samples to cover a road segment of km. The link is located, without loss of generality, in the middle lane, . The link distance is generated by sampling the empirical CDF of . In order to simulate the interference originated from , we place the transmitter at the origin, and we deploy behind it vehicles using the samples generated for . In order to simulate the interference from , we place the receiver at the origin, and we deploy vehicles behind it using the inter-vehicle distance samples for . We place the first vehicle of at km behind the receiver, and we keep on adding vehicles till we reach at the receiver. The vehicles of within a distance from the receiver are filtered out. The procedure generating the vehicle locations for is repeated for . Independent samples for the fading and the activity for each vehicle are also generated. Finally, the interference levels from the three lanes are aggregated at the receiver. Using m for lane separation and for antenna beamwidth, we calculate m for and . Since , we may ignore the impact of in the attenuation without introducing much error. If we fix the location of the link at (or ), the guard zone for is but for (or ) it is .

For the outage probability predictions using the hardcore point processes, we use the estimated by least square fitting, see Section IV, to generate the LT of interference for each lane. We integrate the product of LTs over a shifted-exponential distribution with parameters .

(9)

where is the LT of interference for the -th lane. Particularly, , and . The parameters for ( for ) do not depend on and they are calculated via moment matching from (7) using the estimates (). The parameters are calculated via moment matching using (5) with estimates .

For the outage probability predictions using PPP, we use the MLE for PPP, and we integrate the product of LTs over an exponential distribution with parameter .

where and .

For illustration purposes, we select the 1000-th snapshot. In the busy hour, the estimated intensities for the hardcore processes are and . The estimated hardcore distances are and . In Fig. 14, we see that our model predicts very well the simulated outage probability using the sampled point set. On the other hand, the prediction using PPPs with estimates and fails. Note that the simulated outage probability at dB becomes slightly higher during off-peak. During off-peak, the estimated intensities for the hardcore processes decrease and , but apart from the interference level, it is also the useful signal level becoming less in probability. In order to further assess the quality of the two approximations, we calculate the maximum vertical difference between the simulated CDF and the CDFs obtained from the two models (this is the metric used in Kolmogorov-Smirnov test). During the busy hour, we get for our model and for PPP. In addition, the absolute difference between simulations and models at dB is for our model and for PPP. In Fig. 15, we depict the goodness-of-fit metrics over 30 snapshots. We see that our model consistently gives much closer predictions to the empirical CDF than PPP. The studies in [5, 6] have made available traces for three more days and for another motorway. For these traces, we have seen similar behavior to Fig. 14 and Fig. 15, but we do not depict the results due to lack of space.

(a) Busy hour
(b) Off-peak
Fig. 14: Probability of outage due to interference originated from three lanes with synthetic traces. simulation runs. The simulations converged already from runs. , .
(a) Busy hour
(b) Off-peak
Fig. 15: Goodness-of-fit metrics for the approximations of the outage probability over 30 snapshots. , , simulation runs per snapshot.

The estimates for the hardcore point processes could have been obtained by directly fitting them to minimize the square difference between the estimated and empirical outage probabilities. This is known as the method of minimum contrast, see [27, Section IV] for fitting the Strauss and Poisson hardcore processes to snapshots of macro base stations. This method requires extensive numerical search because there are six parameters to optimize, and the outage probability is also a complicated function of these parameters. On the other hand, estimating from the traces, as discussed in Section IV, and plugging these estimates into the interference model has much lower complexity, while still providing a very good estimate for the outage probability.

Viii Conclusions

In this paper we developed a low-complexity model for the probability of outage in multi-lane VANETs. The model consists of two parts: Firstly, it borrows from transportation (and statistical mechanics) literature a simple and realistic extension to the PPP for the deployment of vehicles along a lane. The extension does not come without cost because the hardcore distance makes the locations of vehicles correlated. Secondly, it applies the method of moments using a shifted-gamma distribution to approximate the Laplace transform of generated interference per lane. We constructed simple but accurate approximations for the first three moments of interference under Rayleigh fading with and without conditioning. The main contribution of this work is the validation of the deployment and interference models with synthetic traces [5, 6]. We have seen very good prediction of the outage probability in realistic motorway vehicular set-ups, while the widely-accepted PPP fails. Instead of running time-consuming simulations, the system designer should estimate the intensity of vehicles and the hardcore distance from the available traces, and use a single numerical integration, see (9), to assess the probability of outage. Generalization to more than three lanes is straightforward. The model should be particularly useful in cases with high transmission probability because, with strong thinning, the hardcore process converges to PPP [20]. Potential direction for future work is the application of more realistic propagation functions and fading channels for vehicle-to-vehicle communication. In addition, while a hardcore process has fitted well the available traces for small distances, the development of point processes tailored to strong clustering of vehicles might be needed to model different traffic conditions.

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