An Analytical Framework for Coverage in Cellular Networks Leveraging Vehicles

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1 1 An Analytical Framework for Coverage in Cellular Networks Leveraging Vehicles Chang-Sik Choi and François Baccelli arxiv: v2 [cs.it 3 Nov 217 Astract This paper analyzes an emerging architecture of cellular network utilizing oth planar ase stations uniformly distriuted in Euclidean plane and ase stations located on roads. An example of this architecture is that where, in addition to conventional planar cellular ase stations and users, vehicles also play the role of oth ase stations and users. A Poisson line process is used to model the road network and, conditionally on the lines, linear Poisson point processes are used to model the vehicles on the roads. The conventional planar ase stations and users are modeled y independent planar Poisson point processes. The joint stationarity of the elements in this model allows one to use Palm calculus to investigate statistical properties of such a network. Specifically, this paper discusses two different Palm distriutions, with respect to the user point processes depending on its type: planar or vehicular. We derive the distance to the nearest ase station, the association of the typical users, and the coverage proaility of the typical user in terms of integral formulas. Furthermore, this paper provides a comprehensive characterization of the performance of all possile cellular transmissions in this setting, namely vehicle-to-vehicle (V2V, vehicle-to-infrastructure (V2I, infrastructure-to-vehicle (I2V, and infrastructure-to-infrastructure (I2I communications. I. INTRODUCTION The ongoing changes in the automotive industry are expected to e disruptive in several ways. Today s vehicles have evolved from mere means of transportation to platforms that provide multiple services, including ride sharing, autonomous driving, data storage/processing, and Internet access. When the next generation vehicle technology will ecome commercially availale, roads will e flooded with new vehicles equipped with aanced sensors, GPS with Chang-sik Choi is with Department of ECE, The University of Texas at Austin, TX, USA ( chang-sik.choi@utexas.edu. François accelli is with the Department of Mathematics and the Department of ECE, The University of Texas at Austin, TX, USA ( accelli@math.utexas.edu

2 2 great accuracy, high-definition cameras, as well as long and and short-range communication devices with multiple radios. Undoutedly, a major transformation of the communication industry and its ecosystem will follow. For instance, the IEEE WAVE standard was estalished to enale vehicle-to-vehicle or vehicle-to-infrastructure communications with a low rate [1. The 3GPP has investigated vehicle to everything communications in the context of side link communications [2. New vehicular applications will require ultra fast and ultra reliale communications for vehicles to everything networking. The existing network architecture will e fundamentally altered, prompting the idea of cellular networks incorporating vehicles on roads serving as relays and/or ase stations [3 [6. To understand the role of vehicles in the context of future cellular architectures, we propose a framework allowing one to analyze the performance of a cellular network featuring vehicular network elements in addition to the classical planar elements (users and ase stations. The framework consists of a novel network model comprised of roads, vehicular ase stations and macro ase stations. It allows for the mathematical analysis of the performance experienced y users in this context. A. Related Work Studies of vehicular communications were mainly centered on ad hoc networking. A vehicle on a road was primarily identified as an apparatus to expand the performance of ad hoc networks. The existing literature thoroughly analyzed capacity and throughput [7 [9, delay [1, and routing protocols [11, [12 in this context. Stochastic geometry [13 provides a systematic approach to the performance analysis of large scale wireless networks [14 [18. The stationary framework of stochastic geometry allows one to investigate the performance of the typical user. It was used to derive the performance of cellular networks [19 [21. The Poisson network models were expanded to study multi-antenna techniques [22, [23 and new radio technologies [24, [25. More recently, this tool ecame essential to analyze heterogeneous cellular networks that are comprised of multiple layers of ase stations distinct in their transmit powers, locations, and ackhaul capailities. The literature on heterogeneous cellular networks first used Poisson models to study user association, coverage, and throughput [26 [3. Some more recent work, including [31 [34, proposed new spatial models for ase stations and users. As discussed in [35, some wireless network systems require non-poisson spatial models particularly for vehicular networks

3 3 since vehicles are typically on roads. The present paper can e seen as an extension of the classical Poisson cellular model to the scenario where ase stations are either Poisson or Cox on Poisson lines. [36 [39 investigated the performance of wireless networks with vehicular transmitters using stochastic geometry, y focusing on the spatial distriution of the vehicles. In [36, [37, one dimensional spatial network models were used to analyze the scheduling policy and reliaility of communication links on roads. Two-dimensional spatial models ased on Cox point process on lines were proposed in the past in [35 and recently as well to analyze the coverage proaility [38, [39. As in these two studies, the aim of the present paper is to analyze the coverage proaility perceived y typical users on roads. The main novelty of the present paper is the comprehensive architecture incorporating with vehicular ase stations, vehicular users, planar ase stations, and planar users. To the est of our knowledge, this framework is new and the systematic analysis of the proaility of coverage of typical users of oth categories which is proposed here has not een discussed so far. B. Technical Contriutions A systematic approach to characterize a heterogeneous cellular network with vehicular ase stations on roads. We adopt a model where the locations of vehicles are restricted to roads. To produce a tractale model for heterogeneous cellular network with vehicles, we propose the superposition of a stationary Cox point process on lines and a homogeneous planar Poisson point process to model two different layers of ase stations, namely vehicular ase stations and planar ase stations. In a similar way, a stationary planar Poisson point process and a stationary Cox point process on lines are used to model planar users and vehicular users, respectively. As it will ecome clear, the analysis will e ased on a snapshot of the network and the dynamics of vehicles is hence not taken into account. Therefore, the Cox model proposed in this paper also captures the situation with static ase stations (users deployed along the roads ecause they are topologically equivalent to vehicular ase stations (users of our model. These two Cox point processes are uilt on the same stationary Poisson line process. The proposed joint stationary framework enales a systematic approach to analyze the network performance seen y oth vehicular users and planar users. It will e evident from this paper that the type of a user (i.e., vehicular or planar fundamentally alters the statistical properties of its link, including the distance to the nearest ase station, the interference, and the coverage proaility.

4 4 Derivation of association, interference and coverage proaility using Palm calculus. To derive the performance of the proposed model, this paper considers the classical assumptions that users are associated with their nearest ase station, independently of their type, and that the received signal power attenuates according to a path loss model with Rayleigh fading. Statistical properties and fundamental metrics are otained using the joint stationarity framework and Palm calculus. In particular, we derive exact integral expressions for the association proaility and coverage proaility under the Palm proaility of each user point process. The integral expressions can e extended to interesting scenarios such as non-exponential fading or multiinput multi-output transceiver. Comprehensive characterization heterogeneous wireless transmissions in cellular networks with vehicles: We use the coverage and association expressions to inspect the performance of all wireless links in a network with oth planar and vehicular transmissions. For instance, the performance of vehicle-to-vehicle (V2V communications is captured through the coverage proaility of the typical vehicular user y vehicular ase stations, which involves the Palm distriution of the vehicular user point process. In the same vein, the performance of infrastructure-to-infrastructure (I2I communications is characterized y the coverage proaility of the typical planar user y planar ase stations. Using the Palm proaility, the paper gives a comprehensive analysis of all wireless links occurring in cellular networks with vehicles, namely V2V, V2I, I2V, and I2I communications. II. SYSTEM MODEL A. Spatial Modeling This paper proposes a novel heterogeneous cellular network with oth planar and vehicular ase stations. Homogeneous planar Poisson point processes Φ, Φ u, with intensities λ, λ u are used to model macro ase stations and users, respectively. This paper calls them planar ase stations and planar users respectively, to distinguish them from their vehicular counterparts. Both Φ and Φ u are assumed stationary, homogeneous, and independent. The vehicular ase stations and users are modeled y specific Cox point processes. To egin with, a road system is defined y an independent Poisson line process Φ l R 2 produced y a homogeneous Poisson point process Ξ on the cylinder C := R [, π. More precisely, a point of Ξ, denoted y (r i, θ i, descries the line l i R 2 of equation l(r i, θ i = {(x, y R 2 x cos(θ i + y sin(θ i = r i }, (1

5 5 Fig. 1. (dashed. An illustration of roads (line, planar ase stations (diamonds, and vehicular ase stations (circles with oundaries where the parameters r and θ correspond to the shortest distance from the origin to the line and the angle etween the positive X axis and line l, respectively. For the isotropic and stationary Poisson line process, the intensity measure of Ξ C is given y [4 Λ Ξ (dr dθ = λ l π dr dθ. Then, conditionally on the lines, vehicular ase stations and users are modeled y independent 1-D homogeneous Poisson point processes with intensity µ and µ u on each line, respectively. In other words, for each undirected line produced y a point of Ξ, conditionally independent copies of Poisson processes on R with parameters µ and µ u are used to descrie vehicular ase stations and vehicular users, respectively. Fig. 1 illustrates the planar ase stations Φ and the vehicular ase stations Ψ in a simulation all of radius 1km with λ l = 1/km and µ = 1/km. Notice that the proposed network model is flexile in creating road systems with different topologies. Fig. 2 illustrates the deployment of planar and vehicular ase stations when the angular component intensity measure of Ξ is limited to two simple values: {, π/2}, which admits only horizontal and vertical roads. It has a special name: a Poisson Manhattan Co x point process. The isotropic case, where angles are uniformly distriuted, is depicted in Fig. 1. Remind that this paper does not study the motion of vehicles. Therefore, the vehicular ase stations are

6 6 Fig. 2. An illustration of roads (line, planar ase stations (diamonds, and vehicular ase stations (circles with oundaries (dashed. Here the intensity over angular axis is concentrated over {, π/2} TABLE I NOTATION TABLE Symols Description Φ and Φ u Planar ase stations (spatial intensity and planar users (spatial intensity Ξ(Φ l Poisson point process on cylinder set C (Poisson line process on R 2 l(r, θ A line in Euclidean plane produced y a point (r, θ C l(r, θ A line that contains the origin φ l Ψ and Ψ u Poisson process on line l Vehicular ase stations and vehicular users P Φ u and P Ψ u Palm proailities with respect to Φ u and Ψ u, respectively X The ase station closest to the user at the origin topologically equivalent to static ase stations deployed along the roads. We will still call them vehicular ase stations in order to distinguish them from planar ase stations. B. Transmission Model and Performance Metrics This paper analyzes downlink cellular communications, namely links from ase stations to users, under the assumption of a distance-ased path loss and Rayleigh fading. The received

7 7 signal power P r at a distance R is given y P r = phr α, where α > 2, p is the transmit power, and H is an exponential random variale with mean one representing the Rayleigh fading. Since oth receive signal power and interference fluctuate etween users, spatially averaged performance metrics should e considered. In fact, all ase stations and users are jointly stationary, i.e., Φ, Ψ, Φ u, Ψ u are {θ t }-compatile on a stationary framework, and therefore, we can use the Palm proaility to study the network performance seen y a typical user. For instance, for a given constant T >, the coverage proaility of the typical user is defined y under the Palm proaility of Φ u + Ψ u y p c = P Φ u+ψ u (SINR > T, where the signal to interference-plus-noise ratio (SINR is given y the receive signal power divided y the interference-plus-noise power and where P Φ is the Palm proaility of the point process Φ. Similarly, the coverage proaility of the typical planar (resp. vehicular user is simply P Φ u (SINR > T (resp. P Ψ u (SINR > T. Assuming that noise power is negligile compared to the signal and interference powers, the SINR is equal to the signal to interference ratio and the coverage proaility of the typical user is given y p c = P Φ u+ψ u where X denotes the nearest ase station given y [ X := arg max X i Φ +Ψ E Φ u+ψ u ph X α X i Φ +Ψ \X ph X i α > T ph Xi X i α, = arg min X i Φ +Ψ X i. (2 The expression is similar for the typical planar (resp. vehicular user. There are simple connections etween these quantities which are discussed elow. C. Preliminaries: Properties of the Proposed Cox and its Joint Typicality In the following, the spatial intensity of the Cox point process is evaluated. It is classical. Yet we give its proof to etter explain the proofs of later results. Fact 1. Stationarity and spatial intensity of Cox point processes The vehicular ase stations and vehicular users have spatial intensities λ l µ and λ l µ u, respectively [38.

8 8 Fact 2. Consider two jointly stationary and independent point processes Φ 1 and Φ 2. Then, the Palm proaility of Φ 1 + Φ 2, is given y P Φ = λ 1 λ 1 + λ 2 P Φ 1 + λ 2 λ 1 + λ 2 P Φ 2, (3 where λ 1 and λ 2 are the intensity parameters of Φ 1 and Φ 2, respectively. Proof: Given in [16. We will apply the aove lemma to derive the coverage proaility of the typical user P Φ u+ψ u (SINR > T = λ u P Φ λ u + λ l µ u (SINR > T + λ lµ u P Ψ u }{{} λ u + λ l µ u (SINR > T, (4 u }{{} where (a is under the Palm proaility of Φ u and ( is under the Palm proaility of Ψ u. Therefore, in order to compute the coverage proaility of the typical user, (a and ( need to e derived separately. In particular, Section III considers the Palm proaility with respect to Φ u and Section IV considers it with respect to Ψ u. (a ( III. COVERAGE: PALM PROBABILITY OF Φ u In this section, we work under the Palm distriution P Φ u. Under the latter, there exists a typical planar user at the origin. The suscript of the Palm P Φ u is often omitted in the derivation. Remark 1. Numering of the proposed Cox points We assign two indexes (i, j to each point; the lines are numered y indexes {i} Z according to their distances from the origin; the points of line i are numered y indexes {j} Z according to their distances from {} i which is the point of line i closest to the origin. We utilize the classical numering in [41, Fig Fig. 3 illustrates the proposed counting of this paper. This numering allows one to descrie any Cox points as X i,j = (t j(i cos(θ i r i sin(θ i sgn(θ π/2, t j(i sin(θ i + r i cos(θ i, for all i, j Z 2. To visualize the points on line i, consider a Poisson point process {t j(i } on the X axis. These points are rotated y θ i and and translated y v = (r i sin(θsgn(θ π/2, r i cos(θ, respectively. Since an independent copy of the Poisson process is considered on each line, we simply write t j(i y t j. X i,j = t 2 j + r2 i. A. Association of the Typical Planar User Under the Palm distriution P Φ u, the typical user is associated with its closest ase station: either a planar ase station or a vehicular ase station. We denote these events y X Φ and

9 9 l 2 X2, 1 2 l 1 l 1 X,1 l X2,1 X 1, X, 1 X 1,1 X 2, 2 X 2,1 l 2 Fig. 3. Illustration of the proposed numering for the Cox point process. Notice that k are points on the lines that are closest to the origin. X Ψ, respectively. The user association is an important metric ecause it plays a key role to derive the coverage proaility. Proposition 1. The proailities that the typical planar user is associated with a planar ase station and a vehicular ase station are given y P Φ u (X Φ = 2πλ re πλ r 2 2λ r l 1 e 2µ r 2 t 2 dt dr, (5 respectively. P Φ u (X Ψ = 1 2πλ re πλ r 2 2λ r l 1 e 2µ r 2 t 2 dt dr, (6 Proof: Recalling the user at the origin is a planar user, let us define two random variales R p = inf X i and R v = inf X i. X i Φ X i Ψ They capture the distances from the origin to the nearest planar ase station and to the nearest vehicular ase station, respectively. By recalling the association principle in Eq. (2, the association proaility is given y P Φ u (X Φ = E Φ u [ 1Rp<R v = E [ E [ 1rp<R v R p = r p, where we used the independence assumption. The integrand of the aove conditional expectation is given y 1 Rv>r p X i,j Ψ 1 Xi,j >r p.

10 1 Therefore, the association proaility of the typical planar user is given y E [ E [ 1 Rv>rp R p = E E = E E (a = E E ( = E E X i,j Ψ 1 Xi,j >r p i,j Z 2 1 r 2 i +t 2 j >r2 p r i r p (r i,θ i Ξ r i r p (r i,θ i Ξ E [ (c = E e 2λ rp l 1 e 2µ t j φ l(ri exp ( µ r 2 p t2 dt 1 t 2 j >r Φ p 2 r2 l i 1 1 t 2 >r dt p 2 r2 i, (7 where (a follows from the fact that, conditionally on Φ l, the Poisson point processes on lines are independent. We get ( y using the PGFL formula on the Poisson point process φ l(ri,θ i, and (c y utilizing the PGFL formula of the Poisson point process Ξ on the cylinder. The formula (7 is integrated with the density function of R p, given y f Rp (w = w (1 e πλ w 2 = 2πλ we πλ w 2, where f X (x denotes the proaility distriution function of random variale X. As a result, the association proaility is given y P (X Φ = 2πλ we πλ w 2 2λ l w 1 e 2µ w 2 t 2 dt dw. Notice that P (X Ψ = 1 P (X Φ since (X Ψ = (X Φ c. Eq. (5 or (6 give the spatial average of the associations for Poisson distriuted users. Fig. 4 illustrates the proaility of the typical planar user eing associated with a vehicular ase station. All three curves monotonically increase as µ grows. Notice the horizontal axis is log-scaled. The spatial intensity, λ l µ, varies from 1/km 2 to 1/km 2. Interestingly, when the spatial intensities of the planar and vehicular ase stations are exactly the same, i.e., in the middle curve, where λ = λ l µ = 1/km 2, the typical planar user is slightly more likely to e associated with a planar ase station.

11 λ l =2/km,λ =1/km 2 λ l =1/km,λ p =1/km 2 λ l =1/km,λ p =2/km 2.7 Pr (X* Φu Ψ µ (unit: x/km Fig. 4. The proaility that the typical planar user is associated with a vehicular ase station on road. The horizontal axis is log-scaled. Corollary 1. Under the Palm distriution of Φ u, the distriution of the distance from the origin to the nearest ase station is given y f R (r =2πλ re πλ r (1 2 e 2λ r l 1 e 2µ r 2 t 2 dt + e πλ r 2 r 2λ r l 1 e 2µ r 2 u 2 du 4λ l µ re 2µ r 2 u 2 du. (8 r2 u 2 Proof: The proof immediately follows from the fact that the distance from the origin to the nearest ase station is given y the minimum of independent random variales R p and R v. B. Coverage Proaility of the Typical Planar User The coverage proaility under the Palm proaility of Φ u is given y P Φ u (SINR > T = P Φ u (SINR > T, X Φ + P Φ u (SINR > T, X Ψ. In the following, each term of the aove expression is otained in the form of an integral.

12 12 Lemma 1. The coverage proaility y a planar ase station is given y P Φ u (SINR > T, X Φ = 2πλ re πλ r 2 e 2πλ r exp ( 2λ l r T r α u 1 α 2µ T r α (v 2 +u 2 α 2 1+T r α u α du e 2λ l r 1 e 1+T r α (v 2 +u 2 α 2 1 e 2µ r 2 v 2 2µ T r r α (v 2 +u 2 α 2 2 v 2 1+T r α (v 2 +u 2 α 2 Proof: The coverage proaility y a planar ase station is given y P Φ u (SINR > T, X Φ ( = P ph X α Φ u X k Φ +Ψ \{X ph X } k > T, X α Φ du du dr. (9 [ ( = E Ξ P Φ u H > T X α H X k α, X Φ Ξ X k Φ +Ψ [ ( = E Ξ P Φ u H > T r α H X k α, X Φ X = r, Ξ f X,X Φ Ξ(r dr, X k Φ +Ψ where the integrand of Eq. (1 is ( P Φ u H > T r α H X k α, X Φ X = r, Ξ X k Φ +Ψ ( ( = P H > T r α H X k α 1 Xk >r + H X k α 1 Xk >r X, Ξ. X k Φ X k Ψ Therefore, we have P Φ u (SINR > T, X Φ T rα H X k X = E Ξ E [e 1 Xk >r k Φ } {{ } (a T rα E [e X k Ψ H X k α 1 Xk >r } {{ } ( (1 Ξ f X,X Φ Ξ(r dr. For expression (a, the Laplace transform of the Poisson point process Φ gives E [e T rα H X k α 1 Xk >r X k Φ = e 2πλ T r α u 1 α r 1+T r α u α du. (11

13 13 For expression (, y using the explicit expressions for locations of the Cox points, X i,j = (t j cos(θ i r i sin(θ i, t j sin(θ i + r i cos(θ i, we have [ T r α H X k α 1 Xk >r X E e k Ψ \B(r Ξ X i,j >r = E T rαhi,j Xi,j α e X i,j Ψ Ξ r i r t 2 j >r2 ri 2 r = E e T rα H i,j (ri 2 α i >r +t2 j 2 Ξ E = r i r r i >r exp t j φ l(ri,θ i ( 2µ r 2 r 2 i exp ( 2µ T r α (r 2 i + v 2 α T r α (r 2 i + v2 α 2 T r α (r 2 i + v 2 α T r α (r 2 i + v2 α 2 t j φ l(ri,θ i e T rα H j (r 2 i +t2 j α 2 Ξ, (12 where we used the PGFL formula of the Poisson point processes on lines; lines meeting the all B(r and lines avoiding this all produce different expressions. Finally, the distriution function of the distance from the origin to the nearest ase station with no other ase stations exist inside B(r is given y f X (r, X Φ Ξ = P( X = r, Φ + Ψ (B(r = Ξ ( = r (1 P P(Ψ (B(r = Ξ r i <r = 2πλ re πλ r 2 X i Φ 1 X r r i <r = 2πλ re πλ r 2 E t j φ l(ri,θ i 1 r 2 i +t 2 j >r2 e 2µ r 2 r 2 i, (13 y independence of Φ and Ψ. Comining Eqs. (11, (12, and (13, we get P Φ u (SINR > T, X Φ = E Ξ e 2πλ r T r α u 1 α r i r 1+T r α u α du r i <r 2πλ re πλ r 2 e 2µ r 2 r i 2 T r α (r 2 i +t2 α 2 1+T r α (r 2 i +t2 α 2 dt r i >r e 2µ T r α (r 2 i +t2 α 2 1+T r α (r 2 i +t2 α 2 e 2µ r 2 r 2 i dr. (14 dt

14 14 Using Fuini s theorem, we have r i r r 2 ri 2 2µ r 2 r i 2 E Ξ e 2µ T r α (r 2 i +t2 α 2 1+T r α (r 2 i +t2 α 2 2µ = e 2λ r 2 v 2 2µ T r α (v 2 +u 2 α 2 r l 1 e r 2 v 2 1+T r α (v 2 +u 2 α 2 Incorporating it to Eq. (14 completes the proof. dt r i >r e 2µ T r α (r 2 i +t2 α 2 1+T r α (r 2 i +t2 α 2 du 2µ T r α (v 2 +u 2 α 2 e 2λ l r 1 e 1+T r α (v 2 +u 2 α 2 Lemma 2. The coverage proaility y a vehicular ase station is given y P Φ u (SINR > T, X Ψ = 4λ l µ re πλ r 2 e 2πλ r π/2 exp exp ( ( e 2µ r sin(θ 2µ r sin(θ r 2λ l 1 e 2µ 2λ l r 1 e 2µ T r α u 1 α 1+T r α u α du T r α (r 2 cos 2 (θ+v 2 α 2 1+T r α (r 2 cos 2 (θ+v 2 α 2 dθ r 2 u 2 e 2µ T r r α (u 2 +v 2 α 2 2 u 2 T r α (u 2 +v 2 α 2 1+T r α (u 2 +v 2 α 2 1+T r α (u 2 +v 2 α 2 du dt du du. dr. (15 Proof: Notice that under the event X Ψ, the nearest ase station is on a line. Throughout the paper, we will denote this line and the point of Ξ giving this line y l and (r, θ, respectively. To derive the coverage formula, let us first condition the coverage proaility with respect to the Poisson line process Ξ, then to the line l, and then to the distance to the nearest ase station X, sequentially. Then, we have P Φ u (SINR > T, X Ψ = E Ξ E l P Φ u H > T r α X k Φ +Ψ +φ l (z H X k α, X Ψ X = r, l, Ξ F (r dr, where F (r = f X,X Ψ l,ξ(r, the distriution of the distance from the origin to the nearest vehicular ase station on l given l and Ξ. In a similar way to the proof of Lemma 1, the (16

15 15 proaility inside the triple integrals of (16 is given y P Φ u H > T r α H X k α, X Ψ X k Φ +Ψ +φ l X = r, l, Ξ = P H > T r α H X k α 1 Xk >r X = r, l, Ξ. X k Φ +Ψ +φ l Therefore, it is descried y the product of the following expressions T rα H k X k X E [e α 1 Xk >r k Φ =e 2πλ T r α u 1 α r 1+T r α u α du. (17 T rα E [e X k Ψ H k X k α 1 Xk >r Ξ = r i r e 2µ r 2 r i 2 T r α (r 2 i +v2 α 2 1+T r α (r 2 i +v2 α 2 E z 2 +t 2 j >r2 e t j φ l (z T rα(z2+t2 j α 2 r i >r (r, θ = (z, θ = e 2µ e 2µ T r α (r 2 i +v2 α 2 1+T r α (r 2 i +v2 α 2 r 2 z 2 T r α (z 2 +v 2 α 2 1+T r α (z 2 +v 2 α 2. (18, (19 where Eq. (19 is otained y the Laplace transform of the Poisson point process on the line of parameter (z, θ with z < r. The distriution function F (r of Eq. (16 is the proaility that line l has a point at a distance r and the point process Ψ + Φ + φ l (z is empty of points inside B(r. In other words, f X,X Ψ l,ξ(r = P( X = r, Φ + Ψ (B(r = l, Ξ = P ( X = r, Φ (B(r =, Ψ!l (B(r = = r (1 P(φ l (z(b(r = l P(Φ (B(r = P(Ψ (B(r = Ξ r = r (1 e 2µ i <r r 2 z 2 e πλ r 2 e 2µ r 2 ri 2 = 2µ re 2µ r 2 z 2 r2 z 2 r i <r e πλ r 2 where we used independence and Slivnyak s theorem. e 2µ r 2 r 2 i, (2

16 16 Finally, we apply Fuini s theorem to evaluate the triple integral in (16. By changing the order of integrals, the expectation with respect to l is evaluated first. By Campell s averaging formula, we have [ 2µ re 2µ r 2 z 2 E l (z exp ( 2µ r2 z 2 r 2 z 2 = = r π/2 4λ l µ re 2µ r 2 z 2 exp ( 2µ r2 z 2 r 2 z 2 4λ l µ e 2µ r sin(θ 2µ r sin(θ T r α (r 2 cos 2 (θ+v 2 α 2 1+T r α (r 2 cos 2 (θ+v 2 α 2 T r α (z 2 + v 2 α T r α (z 2 + v 2 α 2 T r α (z 2 + v 2 α T r α (z 2 + v 2 α 2 Then, the integral with respect to the Poisson point process Ξ gives r i r e 2µ r 2 ri 2 2µ T r α (r i 2 +v2 α 2 e r 2 r i 2 1+T r α (r i 2+v2 α r i >r 2 E Ξ = exp exp ( ( r 2λ l 1 e 2µ 2λ l r 1 e 2µ r 2 u 2 e 2µ T r r α (u 2 +v 2 α 2 2 u 2 T r α (u 2 +v 2 α 2 1+T r α (u 2 +v 2 α 2 1+T r α (u 2 +v 2 α 2 du dz dθ. (21 e 2µ du T r α (r 2 i +v2 α 2 1+T r α (r 2 i +v2 α 2, (22 where we use Slivnyak s theorem and the PGFL formula. Comining Eqs. (17, (21 and (22 completes the proof. Another version of proof is provided in Appendix A Fig. 5 illustrates the coverage proaility of the typical planar user. We first consider parameters λ l = 5.34/km, λ = 6.15/km 2, and µ = 5/km. They are proposed to illustrate the 3GPP LTE uran V2X scenario where inter site distance of hexagonal ase station is 5 meters and road lock of Manhattan grid is 25 meters y 433 meters 1 [42. In this setting, the coverage proaility y vehicular ase stations significantly higher than the one provided y planar ase stations. It elaorates the use of vehicle-to-planar user (i.e., planar pedestrians communications since those users have higher coverage y vehicular ase stations than y planar ase stations. We also evaluate the coverage proaility when the densities of vehicular ase stations and planar ase stations are the same: λ l = 5/km, µ = 5/km, and λ = 25/km 2. In this case, the impact of topological differences on the coverage proaility is emphasized. Unexpectedly, planar users still have a slightly higher coverage proaility y vehicular ase stations. In the 1 In the proposed model, the mean area of surface surrounded y Poisson lines is π/λ 2 l. On the other hand, assuming no vehicular ase stations, the mean area of Poisson-Voronoi cell of macro ase station is 1/λ [13. Therefore, inter site distance of 5 meters is translated into λ = 6.15 while the road lock of size 25 meters y 433 meters is translated into λ l = 5.3.

17 coverage y planar λ l =5.34,µ =5, λ =6.15 coverage y vehicular λ l =5.34,µ =5, λ =6.15 coverage λ l =5.34µ =5λ =6.15 coverage y planar λ l =5, µ =5, λ =25 coverage y vehicular λ l =5, µ =5, λ =25 coverage λ l =5, µ =5, λ =25 Coverage proaility SINR threshold (db Fig. 5. The coverage proaility of the typical planar user under the Palm distriution of Φ u. specific scenario, the average distance from the typical planar user to its nearest vehicular ase station is 85 meters while it increases up to 125 meters with its nearest planar ase station. IV. COVERAGE: PALM PROBABILITY OF Ψ u In this section, we derive the user association and coverage proailities under the Palm distriution P Ψ u. Note that under this proaility, the typical vehicular user is located at the origin and there is a road passing through the origin and containing the typical user. As in Section III, Slivnyak s theorem is very useful to produce integral formulas. A. Association of the Typical Vehicular User The typical user is associated with either a planar ase station of Φ or a vehicular ase station of Ψ. These events are denoted y X Φ and X Ψ, respectively. Proposition 2. Under the Palm distriution P Ψ u, the proailities of the typical vehicular user to e associated with a planar ase station and a vehicular ase station are given y respectively. P Ψ u (X Φ = P Ψ u (X Ψ = 1 2πλ we πλ w 2 2µ w 2λ l w 2πλ we πλ w 2 2µ w 2λ l w 1 e 2µ w 2 t 2 dt dw, (23 1 e 2µ w 2 t 2 dt dw, (24

18 18 Proof: Let us denote the line containing the origin and the Poisson process on the line y l and φ l, respectively. Under the Palm distriution of Ψ u, let us define ˆR p = inf X k and ˆRv = inf X k X k Φ X k Ψ +φ l in order to descrie the distances to the nearest points of Φ Proposition 1, the association proaility is given y P Ψ u (X Φ = E Ψ u [1 ˆ Rp< ˆ R v = where the integrand is given as follows: P Ψ u ( ˆR v > r p = E = E X k Ψ +φ l 1 Xk >r t j φ l 1 tj >r p r i <r E E and Ψ, respectively. As in P Ψ u ( ˆR v > rf ˆ Rp (r dr, (25 t j φ l(ri,θ i 1 t 2 j >rp r 2 i 2 Ξ = e 2µ r p e 2λ rp l 1 e 2µ rp 2 t2 dt. (26 Due to the independence property of Poisson point processes, the distriution function of ˆR p is equal to f ˆ Rp (w = 2πλ we πλ w 2. (27 Comining Eqs. (25, (26 and (27 gives the result. The proof is concluded y oserving that P Ψ u (X Ψ = 1 P Ψ u (X Φ. Corollary 2. Under the Palm distriution P Ψ u, the distriution of the distance from the origin to the nearest ase station is given y ( f R (r =2πλ re πλ r 2 e 2µ r 1 e 2λ r l 1 e 2µ r 2 u 2 du ( + e πλ r 2 2µ e 2µ r 1 e 2λ r l 1 e 2µ r 2 u 2 du + e πλ r 2 e 2µ r e 2λ r r l 1 e 2µ r 2 u 2 du 4λ l µ re 2µ r 2 u 2 du. (28 r2 u 2 Proof: The proof immediately follows from the fact that, under the Palm distriution of Ψ u, the distance from the origin to the nearest point of Φ + Ψ is the minimum of the two independent random variales ˆR p and ˆR v. Note that the distriution functions of the nearest distance are different under P Φ u and P Ψ u.

19 19 B. Coverage Proaility of the Typical Vehicular User Since (X Ψ = (X Φ c, the coverage proaility is descried as follows. P Ψ u (SINR > T = P Ψ u (SINR > T, X Φ + P Ψ u (SINR > T, X Ψ. In the following, each term is derived as an integral formula. Lemma 3. Under the Palm distriution of Ψ u, the coverage proaility y a planar ase station is given y P Ψ u (SINR > T, X Φ = e 2µ r T r α u α 1+T r α u α du e 2πλ r 2πλ re πλ r 2 2µ r exp ( T r α u 1 α 2µ T r α (v 2 +u 2 α 2 1+T r α u α du e 2λ r l 1 e r 2 v 2 1+T r α (v 2 +u 2 α 2 2λ l r 1 e 2µ r 2 v 2 2µ T r r α (v 2 +u 2 α 2 2 v 2 1+T r α (v 2 +u 2 α 2 du du Proof: Conditionally on the Poisson point process Ξ and on the distance to the nearest ase station X, the coverage proaility under the Palm distriution of Ψ u is given y P Ψ u (SINR > T, X Φ ( = E P Ψ u H > T r α H X k α, X Ψ X = r, Ξ f X X Ψ Ξ(r dr X k Φ +Ψ = E P H > T r α H X k α, X Ψ X k Φ +Ψ +φ l X = r, Ξ f X X Ψ Ξ(r dr T rα H X k X = E E [e α 1 Xk >r T rα H X k k Φ X E [e α 1 Xk >r k Ψ Ξ }{{}}{{} (a ( [ E e T r α X k φ l H X k α 1 Xk >r dr. (29 f X,X Ψ Ξ(r dr, (3 where (a and ( are given y Eq. (17 and (18, respectively. We have [ E e T r α X k φ l H X k α 1 Xk >r =e 2µ r r T r α v α 1+T r α v α, (31 using the Laplace transform of the Poisson point process φ l, which is independent of Ξ.

20 2 Under the Palm distriution of Ψ u, the distriution of the distance from the origin to the nearest ase station is given y f X,X Φ Ξ(r = P Ψ u ( X = r, X Φ Ξ = r (1 P Ψ u (Φ (B(r = P Ψ u (Ψ (B(r = Ξ = 2πλ re πλ r 2 P(Ψ l (B(r = = 2πλ re πλ r 2 e 2µ r i <r e 2µ r 2 r 2 i, (32 where the last expression is given y Slivnyak s theorem and the independence property: P(Ψ l (B(r = Ξ = P(Ψ!l (B(r = Ξ P(φ l (B(r = Ξ = e 2µ r r i <r e 2µ r 2 r 2 i. Comining Eqs. (17, (18, (31, and (32 and applying Fuini s theorem allows one to complete the proof. Lemma 4. Under the Palm proaility of Ψ u, the coverage proaility y a vehicular ase station is given y P Ψ u (SINR > T, X Ψ = P Ψ u (SINR > T, X Ψ, l l }{{} (a + P Ψ u (SINR > T, X Ψ, l l, (33 }{{} ( where (a and ( are given y Eqs. (36 and (38, respectively. More precisely, part (a gives the proaility that the typical vehicular user is covered and its nearest vehicular ase station is on its road, whereas part ( gives the proaility that the typical vehicular user is covered and its nearest vehicular ase station is not on its road. Proof: Under the Palm proaility and under the event {X Ψ }, we can consider two lines that are not necessarily distinct: the one containing the origin, l, and the one containing the nearest ase station l. The coverage proaility is then partitioned into the events {l l } and {l l }, which are denoted y E 1 and E 2, respectively. Then, we have P Ψ u (SINR > T, V = P Ψ u (SINR > T, V, E 1 + P Ψ u (SINR > T, V, E 2,

21 21 where V denotes the event X Ψ. In order to get an expression for (a, we condition on the Poisson line process. Then, the coverage proaility is given y P Ψ u (SINR > T, V, E 1 = E Ξ T rα H k X k X E [e α 1 Xi >r T rα k Φ E [e E [ e T r α X k φ l H k X k α 1 Xi >r X k Ψ H k X k α 1 Xk >r Ξ f X,V,E 1 Ξ(r dr, (34 where the integrand is already given as the product of Eqs. (17, (18 and (31. Analogous to the distriution function given in the proof of Lemma 3, the aove distriution function is f X,V,E 1 Ξ = P Ψ u ( X = r, X Ψ, Φ + Ψ (B(r = Ξ = P l Ψu ( X = r, X φ l, Φ + Ψ!l (B(r = Ξ = r P 1 P l Ψu (Φ + Ψ!l (B(r = Ξ X i φ l 1 Xi r = r (1 exp( 2µ r P(Φ (B(r = Ξ P(Ψ (B(r = Ξ r i <r = 2µ e 2µ r e πλ r 2 e 2µ r 2 r 2 i. (35 Incorporating (17 (18, (31, and (35 into Eq. (34 and applying Fuini s theorem gives P Ψ u (SINR > T, X Φ, l l = e 2πλ r T r α u 1 α 1+T r α u α du 2µ r 2µ e 2µ r πλ r 2 exp ( 2λ l r T r α u α 1+T r α u α du 2λ l r 2µ T r α (v 2 +u 2 α 2 1 e 1+T r α (v 2 +u 2 α 2 1 e 2µ r 2 v 2 2µ T r r α (v 2 +u 2 α 2 2 v 2 1+T r α (v 2 +u 2 α 2 du du dr. (36

22 22 On the other hand, for expression ( of Eq. (33, conditionally on Ξ and on l, the coverage proaility is given y P Ψ u (SINR > T, V, E 2 = E Ξ E l P H > T r α where the integrands are given y P H > T r α = E [e T rα [ E e X k Φ +Ψ +φ l (z +φ l X k Φ +Ψ +φ l (z +φ l T r α X k Φ H k X k α 1 Xk >r X k φ l H k X k α 1 Xk >r H k X k α 1 Xk >r, V, E 2 X, l, Ξ F (r dr, (37 H k X k α 1 Xk >r, V, E 2 X, l, Ξ E [e T rα [ E e T r α X k Ψ H k X k α 1 Xk >r X k φ l H k X k α 1 Xk >r Ξ l, that are already given y Eqs. (17, (18, (31, and (19, respectively. The function F (r is F (r = P Ψ u ( X = r, X φ l, Φ + Ψ (B(r = Ξ, l = P Ψ u ( X = r, X φ l l (z, θ P Ψ u (Φ + Ψ (B(r = Ξ = r 1 P l (z, θ P l Φu (φ l + Φ + Ψ!l (B(r = Ξ X i φ 1 Xi r = 2µ re 2µ r 2 z 2 P(φ l (B(r = P(Φ (B(r = P(Ψ (B(r = Ξ r2 z 2 = 2µ re 2µ r 2 z 2 r2 z 2 r i <r e 2µ r e πλ r 2 e 2µ r 2 r 2 i. Therefore, we comine the aove integrands and the distriution function and apply Fuini s

23 23 theorem to have P Φ u (SINR > T, X Ψ, l l = 4λ l µ re 2µ r πλ r 2 2πλ r π/2 exp exp ( ( e 2µ r sin(θ 2µ r sin(θ r 2λ l 1 e 2µ 2λ l r 1 e 2µ The aove expression gives ( of Eq. (33. T r α u 1 α 1+T r α u α du 2µ r T r α (r 2 cos 2 (θ+v 2 α 2 1+T r α (r 2 cos 2 (θ+v 2 α 2 T r α u α 1+T r α u α du dθ r 2 u 2 e 2µ T r r α (u 2 +v 2 α 2 2 u 2 T r α (u 2 +v 2 α 2 1+T r α (u 2 +v 2 α 2 1+T r α (u 2 +v 2 α 2 du du dr. (38 As a result, adding (36 and (38 gives the proaility that the typical vehicular user is associated with a vehicular ase station and is covered under the Palm distriution of Ψ u. Fig. 6 illustrates the coverage proaility of the typical user under the Palm distriution of Ψ u. Even when the spatial densities of planar ase stations and vehicular ase stations are the same, the coverage proaility provided y planar ase stations is much smaller than the one provided y vehicular ase stations. We can interpret this as follows. In general, under the Palm distriution of Ψ u, most vehicular users are associated with vehicular ase stations and only a small fraction of vehicular users are associated with planar ase stations (Proposition 1. When the vehicular users are associated with planar ase stations, they are easily located at the cell oundaries where interference from neary vehicular ase stations is significant. Note that even with twice more planar ase stations, λ l µ = 5 and λ = 1, the spatial average of the SINR provided y planar ase stations is smaller than that provided y vehicular ase stations. This phenomenon is related to the topology of lines and it should have an impact on the architecture of future cellular networks. V. COVERAGE: PALM PROBABILITY OF Φ u + Ψ u Here, we summarize Lemmas 1 to 4 to evaluate the coverage proaility of the typical user under the Palm distriution of Φ u + Ψ u.

24 coverage y planar λ l =5µ =1λ =1 coverage y vehicular λ l =5µ =1λ =1 coverage λ l =5µ =1λ =1 coverage y planar λ l =5µ =1λ =5 coverage y vehicular λ l =5µ =1λ =5 coverage λ l =5µ =1λ =5 Coverage proaility SINR Threshold (db Fig. 6. Coverage proaility of the typical user under the Palm distriution of Ψ u. A. Coverage Proaility of the Typical User: Deconditioning Theorem 1. The coverage proaility of the typical user is given y p c = λ u P Φ λ u + λ l µ u (SINR > T, X Φ + P Φ u }{{} λ u + λ l µ u (SINR > T, X Ψ u }{{} (a ( λ u + λ lµ u P Ψ λ u + λ l µ u (SINR > T, X Φ + λ lµ u P Ψ u }{{} λ u + λ l µ u (SINR > T, X Ψ, u }{{} (c where (a, (, (c, and (d are given y (9, (15, (29, and (33, respectively. Proof: The proposed model is jointly stationary. Using (3, the coverage proaility is p c = λ u P Φ λ u + λ l µ u (SINR > T + λ lµ u P Ψ u λ u + λ l µ u (SINR > T. u As a result, comining Lemmas from 1 to 4 gives the proof of the theorem. (d B. Interpretation of the Palm Coverage Proailities In a vehicular cellular network, downlink transmissions are divided into four groups: vehicleto-vehicle, vehicle-to-infrastructure, infrastructure-to-vehicle, and infrastructure-to-infrastructure. Each group has different service requirements and operation protocols. For instance, the vehicleto-vehicle communication requires higher reliaility for a shorter distance than the other groups do ecause it might e used in safety applications [1. In the course of this paper, the coverage expression of each communication group was actually derived 2. 2 Here, we consider the Poisson distriuted planar users as infrastructure users y noting their topological equivalence.

25 I2I λ l =5.34, µ =, λ =6.15, α=4 I2I λ =5.34, µ =5, λ =6.15, α=4 l.9 V2I λ l =5.34µ =5λ =6.15, α=4 V2I λ =5.34µ =1λ =6.15, α=4 l.8 I2I λ l =5.34, µ =1, λ =6.15, α=4 I2I λ =5.34µ =λ =6.15α=3 l.8 V2I λ l =5.34µ =15λ =6.15, α=4 V2I λ =5.34, µ =5, λ =6.15α=3 l I2I coverage proaility I2I λ l =5.34, µ =5, λ =6.15, α=3 I2I λ =5.34, µ =1, λ =6.15, α=3 l V2I coverage proaility V2I λ l =5.34, µ =1, λ =6.15α=3 V2I λ =5.34, µ =15, λ =6.15α=3 l SINR threshold (db SINR threshold (db I2V sparse uran areaλ l =3.17, µ =3, λ =3.7, α=3 I2V uran area λ l =5.34, µ =5, λ =6.15, α=4.9 V2V λ =5.34, µ =5, λ =6.15, α=4 l V2V λ l =5.34, µ =1, λ =6.15, α=4.8 I2V dense uran areaλ l =1.68, µ =1, λ =13.3, α=4.8 V2V λ l =5.34, µ =15, λ =6.15, α=4 V2V λ l =7.55, µ =5, λ =6.15, α=4 I2V coverage proaility V2V coverage proaility λ l increases V2V λ l =1.68, µ =5, λ =6.15, α=4 µ increases SINR threshold (db SINR threshold (db Fig. 7. Illustration of the coverage proailities of all typical links presented in the proposed model. The coverage proaility of link from vehicle ase station-to-vehicle user (V2V is P Ψ u (SINR > T X Ψ = P Ψ u (SINR > T, X Ψ P. (39 Ψ u (X Ψ The coverage proaility of link from infrastructure ase stations-to-vehicle users (I2V is P Ψ u (SINR > T X Φ = P Ψ u (SINR > T, X Φ P. (4 Ψ u (X Φ The coverage proaility of link from vehicle ase station-to-infrastructure user (V2I is P Φ u (SINR > T X Ψ = P Φ u (SINR > T, X Ψ P. (41 Φ u (X Ψ The coverage proaility of link from infrastructure ase station-to-infrastructure user (I2I is P Φ u (SINR > T X Φ = P Φ u (SINR > T, X Φ P. (42 Φ u (X Φ All numerators and denominators in Eq. (39-(42 are already provided in previous sections.

26 26 Fig. 7 illustrates the coverage proaility of all possile links present in the proposed network. Let us first investigate the coverage of infrastructure users. The upper left figure illustrates the I2I coverage proaility. As the numer of vehicular ase stations increases, µ = 5, 1, 15/km, the typical infrastructure user experiences a lower coverage proaility. Similarly, as illustrated in the upper right figure, the V2I coverage proaility diminishes as the density of vehicular ase stations grows. Then, let us investigate the coverage of vehicular users. In the lower right figure, we oserve that as the numer of vehicular ase station increases the V2V coverage proaility increases. This is ecause the typical vehicle is more likely to have an increased desired signal power. In contrast, as the size of road locks shrinks, λ l = 5.34, 7.55, 1.88/km, the V2V coverage proaility decreases. This happens ecause the typical vehicular user is associated with vehicular ase stations on the same line at the same average distance; nevertheless, it is exposed to an increased interference from vehicular ase stations on the other lines. Notice that these trends are sensitive to network parameters and path loss model. For instance, when µ is very small, the trend explained aove could e reversed; with a high proaility, the typical vehicle might e associated with vehicular ase stations on the other lines since there might e no close vehicular ase station on its own line. In the lower left figure, we consider sparse, normal, and dense scenarios where the density of road locks and planar ase stations vary. This figure shows that the I2V coverage proaility is higher in dense areas than in sparse areas. VI. CONCLUSION This paper provides a representation of heterogeneous cellular networks consisting of vehicular ase stations, vehicular users, planar ase stations, and planar users which takes their topological characteristics into account. Vehicular ase stations and users on roads are modeled y Cox point processes given y independent Poisson point processes conditionally on a Poisson line process; on the other hand, planar ase stations and users are modeled y independent Poisson point processes. We characterized the network performance seen y typical users of oth kinds y deriving the association proaility and coverage proaility under the Palm distriution of each user point process. This allows us to otain analytical expressions for the coverage proailities of all possile links present in the proposed network architecture. This rings a first understanding on the asic performance metrics such as user association proaility, interference power distriution, and coverage proaility, ut also provides a

27 27 framework for future research on practical scenarios in future cellular architecture ased on vehicles. For instance, this framework could e used to address resource allocation prolems in the presence of simultaneous vehicular and planar links with possily different quality of service requirements. ACKNOWLEDGMENT This work is supported in part y the National Science Foundation under Grant No. NSF-CCF and an award from the Simons Foundation (#197982, oth to the University of Texas at Austin. REFERENCES [1 G. Karagiannis, O. Altintas, E. Ekici, G. Heijenk, B. Jarupan, K. Lin, and T. Weil, Vehicular networking: A survey and tutorial on requirements, architectures, challenges, standards and solutions, IEEE Commun. Surveys Tuts, vol. 13, no. 4, pp , 211. [2 3GPP TS , Evolved universal terrestrial radio access (E-UTRA; physical channels and modulation. [3 G. Araniti, C. Campolo, M. Condoluci, A. Iera, and A. Molinaro, LTE for vehicular networking: a survey, IEEE Commun. Mag., vol. 51, no. 5, pp , 213. [4 H. Seo, K.-D. Lee, S. Yasukawa, Y. Peng, and P. Sartori, LTE evolution for vehicle-to-everything services, IEEE Commun. Mag., vol. 54, no. 6, pp , 216. [5 Veniam: An Internet of moving things, accessed: [6 K. Zheng, Q. Zheng, P. Chatzimisios, W. Xiang, and Y. Zhou, Heterogeneous vehicular networking: a survey on architecture, challenges, and solutions, IEEE Commun. Surveys Tuts., vol. 17, no. 4, pp , 215. [7 P. Gupta and P. R. Kumar, The capacity of wireless networks, IEEE Trans. Inf. Theory, vol. 46, no. 2, pp , 2. [8 M. Grossglauser and D. Tse, Moility increases the capacity of ad-hoc wireless networks, in Proc. IEEE INFOCOM, vol. 3, 21, pp [9 J. Li, C. Blake, D. S. De Couto, H. I. Lee, and R. Morris, Capacity of ad hoc wireless networks, in Proc. ACM MoiCom, 21, pp [1 J. Zhao and G. Cao, VADD: Vehicle-assisted data delivery in vehicular ad hoc networks, IEEE Trans. Veh. Technol., vol. 57, no. 3, pp , May 28. [11 Y.-C. Tseng, S.-Y. Ni, Y.-S. Chen, and J.-P. Sheu, The roadcast storm prolem in a moile ad hoc network, Wireless Networks, vol. 8, no. 2/3, pp , 22. [12 C. Perkins, E. Belding-Royer, and S. Das, Ad hoc on-demand distance vector (AODV routing, Tech. Rep., 23. [13 S. N. Chiu, D. Stoyan, W. S. Kendall, and J. Mecke, Stochastic geometry and its applications. John Wiley & Sons, 213. [14 F. Baccelli and B. Błaszczyszyn, On a coverage process ranging from the Boolean model to the Poisson Voronoi tessellation, with applications to wireless communications, A. in Appl.Pro. (SGSA, vol. 33, no. 2, pp , 21. [15 F. Baccelli, B. Blaszczyszyn, and P. Muhlethaler, An aloha protocol for multihop moile wireless networks, IEEE Trans. Inf. Theory, vol. 52, no. 2, pp , Fe. 26.

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