VEHICULAR ad-hoc network (VANET) is a type of

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1 1 Analysis of Aess and Connetivity Probabilities in Vehiular elay Networks Seh Chun Ng, Student Member, IEEE, Wuxiong Zhang, Yu Zhang, Yang Yang, Senior Member, IEEE, and Guoqiang Mao, Senior Member, IEEE Abstrat IEEE p and 1609 standards are urrently under development to support Vehile-to-Vehile and Vehileto-Infrastruture ommuniations in vehiular networks. For infrastruture-based vehiular relay networks, aess probability is an important measure whih indiates how well an arbitrary vehile an aess the infrastruture, i.e. a base station (BS). On the other hand, onnetivity probability, i.e. the probability that all the vehiles are onneted to the infrastruture, indiates the servie overage performane of a vehiular relay network. In this paper, we develop an analytial model with a generi radio hannel model to fully haraterize the aess probability and onnetivity probability performane in a vehiular relay network onsidering both one-hop (diret aess) and two-hop (via a relay) ommuniations between a vehile and the infrastruture. Speifially, we derive lose-form equations for alulating these two probabilities. Our analytial results, validated by simulations, reveal the tradeoffs between key system parameters, suh as inter-bs distane, vehile density, transmission ranges of a BS and a vehile, and their olletive impat on aess probability and onnetivity probability under different ommuniation hannel models. These results and new knowledge about vehiular relay networks will enable network designers and operators to effetively improve network planning, deployment and resoure management. Index Terms Vehiular Ad Ho Network (VANET), Wireless Aess in Vehiular Environments (WAVE), IEEE p, IEEE 1609, aess probability, onnetivity, relay. I. INTODUCTION S. C. Ng and G. Mao are with Shool of Eletrial and Information Engineering, University of Sydney, Australia and National ICT Australia (NICTA), Sydney ( sehhun,guoqiang}@ee.usyd.edu.au). W. Zhang is with Shanghai esearh Center for Wireless Communiations (WiCO), Chinese Aademy of Sienes, China ( wuxiong.zhang@shrw.org). Y. Zhang is with Dept. of Computing, Imperial College ondon, UK ( yuzhang@do.i.a.uk). Y. Yang (orrespondene author) is with Dept. of Eletroni and Eletrial Engineering, University College ondon, UK and Shanghai esearh Center for Wireless Communiations (WiCO), Chinese Aademy of Sienes, China ( y.yang@ee.ul.a.uk). This work is partially supported by the Australian esearh Counil (AC) under the Disovery projet DP and the National Natural Siene Foundation of China (NSFC) under the grant This work is partially supported by National ICT Australia (NICTA), whih is funded by the Australian Government Department of Communiations, Information Tehnology and the Arts and the Australian esearh Counil through the Baking Australia Ability initiative and the ICT Centre of Exellene Program, and by the Air Fore esearh aboratory, under agreement number FA The U.S. Government is authorized to reprodue and distribute reprints for Governmental purposes notwithstanding any opyright notation thereon. The views and onlusions ontained herein are those of the authors and should not be interpreted as neessarily representing the offiial poliies or endorsements, either expressed or implied, of the Air Fore esearh aboratory or the U.S. Government. VEHICUA ad-ho network (VANET) is a type of promising appliation-oriented network deployed along a highway for safety and emergeny information delivery (for drivers), entertainment ontent distribution (for passengers), and data olletion and ommuniation (for road and traffi managers). VANET is a hybrid wireless network that supports both infrastruture-based and ad ho ommuniations. Speifially, vehiles on the road an ommuniate with eah other through a multi-hop ad ho onnetion. They an also aess the Internet and other broadband servies through the roadside infrastruture, i.e. base stations (BSs) or aess points (APs) along the road. When a vehile moves out of the radio overage area of a BS, e.g. it is loated in the overage gap between two adjaent BSs, it will identify and use its neighboring vehiles (if any) as relays to aess the roadside infrastruture. These types of Vehile to Vehile (V2V) and Vehile to Infrastruture (V2I) ommuniations have reently reeived signifiant interests from both aademia and industry [1], [2], [3]. V2V ommuniation has so far been envisioned for supporting safety and traffi management appliations. With better sensing and data ommuniation tehniques, drivers an share the information suh as slippery road, poor visibility, sudden stop and road ongestion with eah other. Hene warnings are provided to prevent aidents and improve road safety. Fig. 1. Arhiteture of IEEE 1609 standards family As shown in Fig. 1, IEEE p standard ooperates with the IEEE 1609 standard family, whih is developed to support Wireless Aess in Vehiular Environment (WAVE) and to deliver safety and infotainment appliations to vehiles on the road [4], [5]. Speifially, IEEE p is a draft amendment (with WAVE apability) to the IEEE standards, whih is expeted to be finalized and approved in The goal of

2 p standard is to provide V2V and V2I ommuniations, up to a range of 1Km, at an average data rate of 6 Mbps over the dediated 5.9 GHz ( GHz) liensed frequeny band. IEEE p uses an amended a physial-layer speifiation with Orthogonal Frequeny-Division Multiplexing (OFDM) tehnique. On Medium Aess Control (MAC) layer, it adopts the Enhaned Distributed Channel Aess (EDCA) protool from the e standard to support Quality of Servie (QoS). The following standards are inluded in the IEEE 1609 standard family: IEEE P1609.0, IEEE P1609.1, IEEE P1609.2, IEEE P1609.3, IEEE P Some new standards have reently been added to IEEE 1609 family, suh as (Communiations Management), (Failities) and (Eletroni Payment Servie). Their funtions and relationships with other 1609 standards are shown in Fig. 1. IEEE p and 1609 standards are still in the draft stage. The harsh vehiular ommuniation environment, aused by variable vehile speed, high mobility, and dynami network topology, brings many tehnial hallenges in developing and deploying WAVE appliations and servies. From a user/vehile s perspetive, the first, and probably the most important, servie requirement is to be able to aess the roadside infrastruture, diretly or indiretly via a relay vehile. From the perspetive of a network operator or servie provider, it is important to guarantee satisfatory and profitable servie overage while minimizing the deployment and maintenane osts of the roadside infrastruture. To improve user satisfation and servie overage of future IEEE 1609 based WAVE systems and appliations, this researh develops an analytial model to fully haraterize the aess probability (for user satisfation analysis) and the onnetivity probability (for servie overage analysis) for infrastruture-based vehiular relay networks, wherein both one-hop (diret aess) and two-hop (via a relay) ommuniations between a vehile and the infrastruture (i.e. a BS) are supported. In this paper, a generi onnetion model is used to investigate the impat of different system parameters, i.e., inter-bs distane (or BS density), vehile density, radio overage ranges of BSs and vehiles, on key performane metris, i.e. user aess probability and servie onnetivity probability. The analysis is then applied to two widely used ommuniation hannel models as speifi examples of the generi onnetion model. This researh enables us to improve aess probability and onnetivity probability in vehiular relay networks, and therefore support reliable V2I and V2V data transmissions in different ommerial appliations and servies, suh as emergeny messaging servie, mobile Internet aess and on-road entertainments. The rest of this paper is organized as follows. In Setion II we introdue related work. In Setion III we define the system model. In Setion IV we present the analysis of the aess and onnetivity probabilities under a generi radio hannel model. In Setion V we fous on two widely used radio hannel models, i.e. the unit disk ommuniation model and the lognormal shadowing model, and their analysis as speial ases of the generi hannel model. In Setion VI we disuss the analytial and simulation results, followed by onlusions in Setion VII. II. EATED WOK eently, signifiant researh on VANET, WAVE, IEEE p and 1609 standards has been undertaken to measure, estimate and haraterize wireless vehiular hannels [6], [7], [8], to model and analyze system performane [9], [10], [11], [12], [13], [14], to design and evaluate MAC protools [15], [16], [17], [18], [19], and to develop VANET simulator [20] and testbed [21]. Speifially, in [9], it is found a ommuniation distane of 1000m, whih is speified in the IEEE802.11p Projet Authorization equest (PA), annot be ahieved by an Equivalent Isotropially adiated Power (EIP) of 2W in a vehile-to-vehile (V2V) highway senario. The impats of vehile density (or V2V distane) and ine-of-sight (OS) ommuniation link on different system performane metris, suh as throughput, average delay, paket loss, and ollision probability, are investigated in [10], [11], [12]. In [15], it is pointed out that the performane of IEEE p standard is not satisfatory in the infrastruture data olletion mode with a stati bakoff sheme. The apability of p MAC protool is further evaluated and enhaned to support both safety appliations (i.e. emergeny message dissemination with strit time onstraints) and non-safety appliations [16], [17], [22], [18], [19]. In [23], Salhi et al. presented a novel data gathering and dissemination arhiteture based on hierarhial and geographial mehanisms for vehiular sensor networks. In [24] and [25] pratial traffi prioritization and power ontrol shemes are proposed and evaluated respetively, to support real-time delivery of safetyritial emergeny information. In [26], Shrestha et al. develop a new sheme using the BitTorrent tool and bargaining game to effiiently distribute a large amount of data over V2V and V2I ommuniation links. Aess and onnetivity probabilities have been studied in the literature for one-dimensional (1-D) [27], [28] and two-dimensional (2-D) [29], [30], [31] multi-hop wireless networks. In [27], Wu fouses on V2V ommuniations and derives a lose-form expression of onnetivity probability in a linear VANET with high-speed vehiles and time-varying vehile populations, i.e. dynami network topology and vehile density. The impats of some key network parameters, suh as vehile arrival rate, random vehile speed and transmission range, are onsidered in his work. Based on a Poisson assumption of node distribution, Dousse, Thiran and Hasler study a 1-D network with equally-spaed BSs and Poissonly distributed vehiles in [28]. Considering a unit disk ommuniation model, they derive the onnetivity probability (defined as aess probability in this paper) that an arbitrary vehile an reah at least one BS over multiple hops. For a 2-D multi-hop ellular network, where nodes are uniformly distributed in a irular area of unit radius, Ojha et al. obtain the minimum transmission range required for these nodes to be able to aess a BS loated at the enter of this irular area over multiple hops under the unit disk ommuniation model as the number of nodes goes to infinity [29]. When both BSs and nodes are Poissonly distributed in a 2-D area and a log-normal shadowing ommuniation model is onsidered, a lower bound on the probability that a node

3 3 annot aess any BS within a designated number of hops has been derived in [30]. The result relies on the independene assumption that the event that a node annot aess to a BS in a speifi number of hops, say t, and the event that another node annot aess to a BS in t hops are independent, whih is not always valid for some ases (to be shown in this paper). In [31], the probability that a wireless ad-ho network with randomly and uniformly distributed nodes form a onneted network is studied. It is shown that the probability of having a onneted network and the probability of having no isolated node asymptotially onverges to the same value as the number of nodes in the network goes to infinity. Different from previous work arried out mainly under the unit disk ommuniation model and onsidering no restrition on the maximum hop ount for paket transmission, this researh fouses on vehiular relay networks with a maximum hop ount of two hops to ensure high ommuniation quality and reliability between a node and a BS, i.e. two-hop V2I ommuniations, whih is more pratial for real-world VANET appliation senarios. In addition, our analytial approah uses a generi ommuniation hannel model and derives the exat lose-form equations of aess and onnetivity probabilities, not the asymptoti results that are valid only when the number of nodes (or node density) in a network is very large. The results obtained under the generi hannel model are then applied to two widely used models, i.e. the unit disk model and the log-normal model, as speial ases. Finally, we investigate the impats of some key system parameters, suh as inter- BS distane (or BS density), vehile density, transmission ranges of a BS and a vehile, on user aess probability and servie onnetivity probability performane under different ommuniation hannel models. Fig. 2. III. SYSTEM MODE An Infrastruture-based Vehiular elay Network. We onsider an infrastruture-based vehiular relay network, as shown in Fig. 2, wherein a number of BSs are uniformly deployed along a long road, while other vehiles are distributed on the road randomly aording to a Poisson distribution. We analyze the aess probability, i.e. the probability that an arbitrary vehile an aess its nearby BSs within two hops, and the onnetivity probability, i.e. the probability that all vehiles an aess at least one BS within two hops, of the network by investigating a subnetwork bounded by two adjaent base stations. et be the Eulidean distane (in meters) between two adjaent BSs and ρ be the vehile density measured in vehiles per meter (vpm). Sine the vehiles are assumed to be Poissonly distributed with density ρ, disussion on the distribution of the number of vehiles on the road is only meaningful if we restrit to the (random) number of vehiles in a speifi setion of the road, and we all any setion of the road a road segment. For a road segment with length x, the number of vehiles on that road segment is then a Poisson random variable with mean ρx. So the probability that there are k vehiles on a road segment of x meters is given as f(k, x) (ρx)k e ρx, k 0. (1) k! Sine we investigate a subnetwork bounded by two adjaent BSs, the probability that there are k vehiles on the road segment bounded by two adjaent BSs is then f(k, ). Assuming a generi hannel model C, let gv C (x) be the probability that two vehiles separated by an Eulidean distane x are diretly onneted. Similarly, denote by gb C (x) the probability that a vehile and a BS separated by an Eulidean distane x are diretly onneted. We assume that the event that two vehiles (or a vehile and a BS) are diretly onneted is independent of the event that another two vehiles (or a vehile and a BS) are diretly onneted. That is, the event that two vehiles (or in the similar ase, between a vehile and a BS) are diretly onneted is only determined by the loations of the two vehiles and is not affeted by the presene or absene of onnetions between other pairs of vehiles 1. We also assume that gb C(x) gc v (x). This assumption is justified beause it is often the ase that a BS an not only transmit at a larger transmission power than a vehile, it an also be equipped with more sophistiated antennas, whih make it more sensitive to the transmitted signal from a vehile. IV. ANAYSIS OF ACCESS AND CONNECTIVITY POBABIITIES Assume that the subnetwork being onsidered is plaed at [0, ]. The two BSs at both ends of the subnetwork are labeled as BS1 and BS2 and are at 0 and respetively. Denote by G(, ρ, C) the subnetwork with length, vehile density ρ and hannel model C. We investigate the aess probability p a that an arbitrary vehile in G(, ρ, C) an aess either BS (either BS1 or BS2). We also investigate the probability p that all vehiles in G(, ρ, C) are onneted to at least one of the BSs at both ends of the subnetwork. A vehile is said to be loated at x if its Eulidean distane to BS1 is x. The probability that a vehile loated at x is not diretly onneted to BS1 and BS2 are 1 gb C (x) and 1 gb C ( x) respetively. Beause the event that a vehile is not diretly onneted to BS1 and the event that the same vehile is not diretly onneted to BS2 are independent, the probability that the vehile is diretly onneted to either BS1 or BS2 is then p 1 (x) 1 (1 g C b (x))(1 g C b ( x)). (2) 1 Although field measurements in real appliations seem to indiate that the onnetivity between different pairs of geographially / frequeny proximate wireless nodes are orrelated [32], [33], [34], the independene assumption is generally onsidered appropriate for far-field transmission and has been widely used in the literature under many hannel models inluding lognormal shadowing model [35], [36], [30], [37]. Note that the unit disk model is a speial hannel model whih fulfills the independene assumption by nature [38, pg. 12]. This is beause for the unit disk model, two vehiles are diretly onneted if and only if their Eulidean distane is smaller than the transmission range, and is not affeted by the presene or absene of other onnetions (vehiles).

4 4 In order to derive p a and p we need the following lemmas. emma 1. et K 1 be the set of vehiles in the subnetwork G(, ρ, C) whih are diretly onneted to either BS1 or BS2, then K 1 has an inhomogeneous Poisson distribution with density ρp 1 (x) where p 1 (x) is given by Eq. (2). Proof: et K denotes the set of vehiles in G(, ρ, C). Then K has a homogeneous Poisson distribution with density ρ over the segment [0, ]. Consider a realization of K and remove a vehile loated at x from this realization with probability 1 p 1 (x), independent of the removal probability of other vehiles. The remaining set of vehiles an be effetively viewed as a realization of K 1. Note that the above proedure whih removes/retains vehiles independently with some probabilities is alled thinning [38]. et N (K 1 ) be the number of vehiles in K 1. Then following the law of total probability Pr(N (K 1 ) j) Pr(N (K) i) Pr(N (K 1 ) j N (K) i). (3) ij For a randomly hosen vehile in K, the vehile is known to be uniformly distributed in [0, ]. Hene, the probability that a vehile is in K 1 given that the vehile is in K is q 1 0 p 1 (x)dx. (4) Sine the probability of a randomly hosen vehile in K being diretly onneted to either BS1 or BS2 are identially and independently distributed, the probability that among i vehiles in K there are j vehiles in K 1 follows the binomial distributed B(i, q). Thus ( ) i Pr(N (K 1 ) j N (K) i) q j (1 q) i j. (5) j Applying Eq. (1) and (5) into Eq. (3) we have (ρ) i ( ) i Pr(N (K 1 ) j) e ρ q j (1 q) i j i! j ij ( 0 ρp 1(x)dx) j j! e 0 ρp1(x)dx. (6) Furthermore, denote by N (K 1 (l)) the number of vehiles in a road segment l within [0, ] whih are diretly onneted to at least one BSs. Using the above proedure, it is trivial to show that Pr(N (K 1 (l)) j) ( x l ρp 1(x)dx) j e x l ρp1(x)dx. (7) j! For n mutually disjoint road segments l 1, l 2,, l n in [0, ], the random variables N (K 1 (l 1 )),, N (K 1 (l n )) are mutually independent. This is beause the event that one vehile is diretly onneted to either BS1 or BS2 is not affeted by the loations of other vehiles, and whether or not those vehiles are diretly onneted to either BS1 or BS2. Consequently, the existene and loations of vehiles in one road segment will not affet the number of vehiles diretly onneted to BS1 or BS2 in another disjoint road segment. With the above independene property, Eq. (6) and (7), the proof is then omplete. Note that some parts of the proof are similar to the arguments used in [38]. emma 2. et p 2 (x) be the probability that a vehile loated at x in G(, ρ, C) is diretly onneted to at least one vehile in K 1, then p 2 (x) 1 e 0 gc v ( x y )ρp1(y)dy (8) where p 1 (y) is given by Eq. (2) and. denotes the Eulidean norm. Proof: Imagine we partition [0, ] into /dy nonoverlapping intervals of differential length dy. Sine dy is a very small value, the probability that there exist more than one vehile within eah interval of length dy an be ignored and the probability that there exists exatly one vehile within dy is ρdy. The probability that there exists a vehile in [y, y + dy] whih is also in K 1 is then given by ρp 1 (y)dy. Note that the vehiles at x and y are diretly onneted to eah other with probability gv C ( x y ). Therefore, the probability that a vehile at x is diretly onneted to a vehile in K 1 and is loated in [y, y + dy] is gv C ( x y )ρp 1 (y)dy. et h(x, y) denotes the probability that the vehile at x is not diretly onneted to any of the vehiles in K 1 loated within [0, y]. Beause the events that distint pairs of vehiles are diretly onneted are independent, the event that the vehile at x is not diretly onneted to any of the vehiles in K 1 loated within [0, y] is independent of the event that the same vehile is not diretly onneted to the vehile in K 1 loated within [y, y + dy] (if there is any). We have h(x, y + dy) h(x, y)(1 g C v ( x y )ρp 1 (y)dy) (9) where the seond term on the right hand side of the equation is the omplement of the probability that a vehile at x is diretly onneted to a vehile in K 1 and loated in [y, y+dy]. Eq. (9) leads to dh(x, y) h(x, y)g C v ( x y )ρp 1 (y)dy. (10) Therefore the probability that a vehile at x is not diretly onneted to any vehile in K 1 is h(x) e 0 gc v ( x y )ρp1(y)dy. (11) The result follows immediately. The following two theorems give the aess probability p a and the onnetivity probability p respetively. Theorem 1. Denote by p a (x) the aess probability of a vehile at x, i.e. the probability that the vehile at x is onneted to either BS1 or BS2 in at most two hops. Then p a (x) 1 (1 p 1 (x))(1 p 2 (x)) (12) where p 1 (x) is given by Eq. (2) and p 2 (x) is given by Eq. (8). Proof: The result follows immediately from the observation that the event that a vehile at x is diretly onneted to either BS1 or BS2 is independent of the event that the same vehile is diretly onneted to at least one vehile in K 1.

5 5 Theorem 2 (Approximate result). Denote by p the onnetivity probability of G(, ρ, C), i.e. the probability that all vehiles in the subnetwork G(, ρ, C) are onneted to either BS1 or BS2 in at most two hops. Assume that the event that a vehile is onneted to either BS1 or BS2 in at most two hops is independent of the event that another vehile is onneted to either BS1 or BS2 in at most two hops. Then where p a (x) is given by Eq. (12). p e 0 ρ(1 pa(x))dx (13) Proof: et K 2 be the set of vehiles in G(, ρ, C) whih are onneted to either BS1 or BS2 in exatly two hops. Together with the definition of K 1 in emma 1, let K 1 + K 2 K\(K 1 +K 2 ) be the set of vehiles in G(, ρ, C) whih are not onneted to either BS1 or BS2 in at most two hops. Apply the thinning proedure for K, i.e. onsider a realization of K and remove eah vehile loated at x independently from this realization, with probability p a (x). The resulting set of vehiles an be viewed as a realization of K 1 + K 2 under our assumption that the event that one vehile is onneted to either BS in two hops is independent of the event that another vehile is onneted to either BS in two hops, and the probability that vehile at x is onneted to either BS in two hops is p a (x). Using the same tehnique as that used in the proof of emma 1, it an be readily shown that K 1 + K 2 has an inhomogeneous Poisson distribution with density ρ(1 p a (x)). Then all vehiles G(, ρ, C) are onneted to either BS1 or BS2 in at most two hops if and only if N (K 1 + K 2 ) 0. The result follows. Note that Theorem 2 only gives an approximate result for the onnetivity probability beause of the independene assumption. The following lemma proves, in a way, that the event that a vehile is onneted to either BS1 or BS2 in at most two hops is not independent of the event that another vehile is onneted to either BS1 or BS2 in at most two hops. emma 3. et h(x) 1 p 2 (x) be the probability that a vehile at x is not diretly onneted to any vehile in K 1 ; let h(x 1, x 2 ) be the probability that two vehiles, at x 1 and x 2 respetively, are not diretly onneted to any vehile in K 1. Then, h(x 1, x 2 ) h(x 1 )h(x 2 ). Proof: et h(x 1, x 2 ; y) denotes the probability that two vehiles, at x 1 and x 2 respetively, are not diretly onneted to any vehile in K 1 loated in [0, y]. Using the similar argument in Eq. (9), we have h(x 1, x 2 ; y + dy) h(x 1, x 2 ; y)k(x 1, x 2 ; y) (14) where k(x 1, x 2 ; y) (1 g C v ( x 1 y ))(1 g C v ( x 2 y ))ρp 1 (y)dy + (1 ρp 1 (y)dy). The first term on the right hand side of k(x 1, x 2 ; y) is the probability there is a vehile in K 1 loated in [y, y +dy] and both vehiles in x 1 and x 2 are not diretly onneted to it. The seond term on the right hand side of k(x 1, x 2 ; y) is the probability that there is no vehile in K 1 loated in [y, y + dy]. Expanding the right hand side of k(x 1, x 2 ; y) we have k(x 1, x 2 ; y) 1 g C v ( x 1 y )ρp 1 (y)dy g C v ( x 2 y )ρp 1 (y)dy + g C v ( x 1 y )g C v ( x 2 y )ρp 1 (y)dy Using the same approah in emma 2 we obtain h(x 1, x 2 ) e 0 [g C v ( x1 y )+gc v ( x2 y )]ρp1(y)dy e 0 gc v ( x1 y )gc v ( x2 y )ρp1(y)dy e 0 [g C v ( x1 y )+gc v ( x2 y )]ρp1(y)dy h(x 1 )h(x 2 ) (from Eq. (11)) Before obtaining the exat result of the onnetivity probability, we introdue some properties in the following lemma. emma 4. et p (y) be the onnetivity probability of G(, ρ, C) onditioned on that the number of vehiles diretly onneted to either BS is n and they are loated at y y 1, y 2,, y n : 0 y i, 1 i n}; let p Y (y) be the probability density funtion (pdf) of y onditioned on that there are n vehiles diretly onneted to either BS. The following properties hold. (i) p Y (y) (ii) n i1 0 p 1 (y i ) p 1(x)dx (15) p (y) e 0 ρ(1 p1(x)) n i1 (1 gc v ( x yi ))dx (16) p1(y1) Proof: For n 1, p Y (y 1 ) is the probability p1(x)dx 0 that a vehile in K 1 is loated at y 1. Sine p 1 (y i ) and p 1 (y j ) are mutually independent for i j, the result follows for Eq. (15). For Eq. (16), note that a vehile at x is not onneted to any BSs in at most two hops if it is not diretly onneted to any BSs (the probability is 1 p 1 (x)) and it is not diretly onneted to vehiles whih are loated at y given that these vehiles are in K 1 (the probability is 1 gv C ( x y i ) for 1 i n). That is, vehile at x annot aess any BS in at most two hops with probability (1 p 1 (x)) n (1 gv C ( x y i )). (17) i1 Eq. (17) is valid when x / y. When x y j for arbitrary j, we assume that gv C (0) 1. This implies that p a (x y) 0. So Eq. (17) is still valid when x y. Applying the thinning proedure and the tehnique used in emma 1, we have the number of vehiles whih are neither diretly onneted to any BSs nor diretly onneted to any of the vehiles at y is an inhomogeneous Poisson random variable with density ρ(1 p 1 (x)) n i1 (1 gc v ( x y i )). The result follows immediately. Theorem 3 (Exat result). Denote by p the onnetivity probability of G(, ρ, C), i.e. the probability that all vehile in the subnetwork G(, ρ, C) is onneted to either BS1 or BS2

6 6 in at most two hops. Then [ ] p Pr(N (K 1 ) n) p (y)p Y (y)dy [0,] n n0 (18) where p (y) and p Y (y) are given by emma 4; Pr(N (K 1 ) n) is given by emma 1. When n 0, we delare p (y)p Y (y)dy p (y)p Y (y) [0,] n n0 e 0 ρ(1 p1(x))dx. n0 Proof: Eq. (18) diretly follows from the law of total probability, so the details are omitted here. Eq. (18) gives an exat formula for the onnetivity probability whih does not rely on the assumption that the event that a vehile is onneted to either BS in two hops and the event that another vehile is onneted to either BS in two hops are independent. However Eq. (18) is muh more ompliated than the approximate result in Eq. (13). In many situations, Eq. (13) provides a reasonably aurate result for the onnetivity probability. Therefore we inlude both results in this paper. V. PEFOMANCE EVAUATION UNDE SPECIFIC CHANNE MODES Based on the analysis in Setion IV, we further derive and ompare in this setion the aess probability and onnetivity probability performane under two speifi hannel models, i.e. unit disk model and log-normal shadowing model. A. Unit Disk Model In the unit disk model U, assume that two vehiles are diretly onneted if and only if their Eulidean distane is less than or equal to r; assume that a vehile and a BS are diretly onneted if and only if their Eulidean distane is not more than. In other words, g U v (x) 1 if x r 0 otherwise, g U b (x) 1 if x 0 otherwise. where r and are predetermined values, ommonly known as the transmission ranges. Typially we have > r. Applying the above equations into Eq. (2), (8) and (12) we obtain the aess probability under the unit disk model U: (I) For 0 < 2, we have p 1 (x) 1 implies that p a (x) 1 for x [0, ]. Hene, p a 1. (II) For 2 < 2 + r, we have p 1 (x) is 0 when x (, ), and 1 otherwise. When x (, ), Eq. (8) beomes p 2 (x) 1 e 0 gc v ( x y )ρdy gc v ( x y )ρdy 1 e x r ρdy x+r ρdy 1 e ρ(2+2r ). So substitute p 2 (x) into Eq. (12), p a (1 e ρ(2+2r ) ) 1 2 e ρ(2+2r ). (III) For 2 + r < 2 + 2r, we have for x (, ), Eq. (8) beomes p 2 (x) 1 e 0 gc v ( x y )ρdy gc v ( x y )ρdy 1 e x r ρdy x+r ρdy. So substitute p 2 (x) into Eq. (12), p a r r r (1 e ρ(+r x) )dx (1 e ρ(2+2r ) )dx + 1 (1 e ρ(+r+x ) )dx +r ρ (e ρr e ρ(2+2r ) ) 2 + 2r e ρ(2+2r ). (IV) For > 2 + 2r, From Eq. (8) 1 e x r ρdy 1 e ρ(+r x) when x (, + r], p 2 (x) 1 e x+r ρdy 1 e ρ(+r+x ) when x [ r, ), 0 when x ( + r, r). So substituting p 2 (x) into Eq. (12), p a r r r (1 e ρ(+r x) )dx (1 e ρ(+r+x ) )dx + 2(e ρr 1). ρ To derive the equations for the onnetivity probability (exat result), we first look at emma 4. For the unit disk model, p 1 (x) is 1 when x [0, ] [, ] and zero otherwise. Hene, Eq. (15) beomes p Y (y) 1 (min(2, )) n (19) when y i [0, ] [, ], y i, and zero otherwise. Eq. (16) beomes p (y) e ρ n i1 (1 gc v ( x yi ))dx. (20) Note that n i1 (1 gc v ( x y i )) is 1 when x y i > r for all y i. For 2, we an easily obtain p from Eq. (18) by substituting Eq. (19) and (20) into it (will be shown later). To obtain the result for > 2, the following transformation will simplify the arithmeti work.

7 7 et S a (and S b ) be the set of vehiles in [0, ] (and [, ]) whih, by definition, are also in K 1. et N (S a ) (and N (S b )) be the number of vehiles in S a (and S b ). Note that S a S b K 1 and N (S a )+N (S b ) N (K 1 ). et y a (and y b ) be the loation of the vehile in S a (and S b ) whih is furthest from BS1 (and BS2). That is, 0 if S a y a (21) y b maxy : y S a } miny : y S b } otherwise, if S b otherwise. Therefore, the umulative probability funtion of p a is Pr(y a y max ) Pr(y i y max, y i S a ) (22) ( y max )na for n a N (S a ) 1. With Eq. (21) defines y a 0 when N (S a ) 0, we have the pdf of y a as na f a (y a ; n a ) ( ya )na 1 if n a 1 δ(y a ) if n a 0. Similarly we have the pdf of y b as nb f b (y b ; n b ) ( y b )n b 1 if n b 1 δ( y b ) if n b 0. With y a and y b, we an rewrite Eq. (20) into p (y a, y b ) e min,y b r} max,ya+r} ρdx, (23) and Eq. (18) an be transformed into p Pr(N (S a ) n a ) Pr(N (S b ) n b ) n a0 n b 0 [ 0 p (y a, y b )f a (y a ; n a )f b (y b ; n b )dy b dy a ] (24) for > 2. Eq. (24) an be further simplified under different ases. For n a > 0 and n b > 0, Eq. (24) beomes p (na>0,n b>0) 0 p (y a, y b ) [ n a1 n b 1 Pr(N (S a ) n a ) Pr(N (S b ) n b )f a (y a ; n a )f b (y b ; n b )] dy b dy a [ (ρ) na p (y a, y b ) e ρ 0 n n a1 n b 1 a! (ρ) n b e ρ n a n b! (y a n )na 1 b ( y ] b )n b 1 dy b dy a p (y a, y b )ρ 2 e 2ρ 0 [ ] [ (ρy a ) na 1 (n a 1)! n a1 0 n b 1 ] (ρ( y b )) n b 1 dy b dy a (n b 1)! p (y a, y b )ρ 2 e 2ρ e ρya e ρ( y b) dy b dy a. (25) For n a 0 and n b > 0, Eq. (24) beomes p (na0,n b>0) n b 1 e ρ (ρ)n b n b! p (0, y b )ρe 2ρ [ e ρ p (0, y b ) n b ( y b n b 1 (ρ( y b )) n b 1 dy b (n b 1)! )n b 1 dy b p (0, y b )ρe 2ρ e ρ( y b) dy b. (26) With similar steps (omit here) we an obtain for n a > 0 and n b 0, Eq. (24) beomes p (na>0,n b0) 0 p (y a, )ρe 2ρ e ρya dy a. (27) Note that it an be shown that Eq. (27) equals to (26) by letting y b y a, then p (na>0,n b0) p ( y b, )ρe 2ρ e ρ( y b) dy b p (0, y b )ρe 2ρ e ρ( y b) dy b where p ( y b, ) p (0, y b ). Finally for n a 0 and n b 0, p (na0,n b0) e ρ e ρ p (0, ) e 2ρ e ρdx e 2ρ e ρ( 2) e ρ. (28) Using Eq. (25), (26), (27) and (28), we an obtain the onnetivity probability as follows. Due to the lengthy (but straightforward) steps involved to derive the results, we omit the intermediate steps and only inlude the results of Eq. (25) and (26) for readers onveniene. (I) For 0 < 2, p Y (y) 1 from Eq. (15) and p n (y) 1 from Eq. (16) implies that p Pr(N (K 1 ) n) 1. n0 (II) For 2 < 2 + r, p (na>0,n b>0) p (na0,n b>0) 1 + e ρ 2e ρ + e ρ(3+r ) + ( ρ( 2))e ρ(2+2r ) 2 e ρ(+r ) e ρ(+2r 2), e ρ + e ρ 1 2 e ρ(3+r ) eρ(+r ), p e ρ(+2r 2) + ( ρ( 2))e ρ(2+2r ). ]

8 8 (III) For 2 + r < 2 + 2r, p (na>0,n b>0) 1 + e ρ e ρ( 2) e ρ(+r ) + ( ρ(2 + 2r ))e ρ(2+2r ) e ρ(+2r 2) e ρ( r), p (na0,n b>0) e ρ e ρ(+r ) e ρ( r), p e ρ( 2) e ρ(+2r 2) (IV) For > 2 + 2r, + ( ρ(2 + 2r ))e ρ(2+2r ). p (na>0,n b>0) e ρ e ρ(+2r 2) e ρ(+r ) results under the unit disk model later. It an be shown that under the log-normal model gv (x) Pr(p rx p v th) Q( 10α σ log x 10 r ). where funtion Q(y) 1 x2 e 2 2π dx is the tail probability of the standard normal distribution. Similarly, gb (x) y Q( 10α σ log 10 x ). When σ 0, g v (x) Pr(x r), gb (x) Pr(x ) and the log-normal model beomes the unit disk model as expeted. The aess probability an then be obtained for different values of α and σ by omputing Eq. (12) using any numerial integration tehnique. The approximate and exat results for the onnetivity probability an be obtained by omputing Eq. (13) and (18) using any numerial integration tehnique. VI. ANAYTICA AND SIMUATION ESUTS A. Unit Disk Model e ρ( 2 2r) e ρ( r) e ρ( 2), p (na0,n b>0) e ρ e ρ(+r ) e ρ( r), p 1 4 e ρ(+2r 2) e ρ( 2 2r) e ρ( 2). B. og-normal Shadowing Model The log-normal model is ommonly used to model the real world signal propagation where the transmit power loss inreases logarithmially with the Eulidean distane between two wireless nodes and varies log-normally due to the shadowing effet aused by surrounding environment. In the lognormal model, we formulate the reeived power (in db) at a destination vehile as p rx p 0 10α log 10 l d 0 + N σ (29) where p rx is the reeived power (in dbmw) at the destination vehile; p 0 is the power (in dbmw) at a referene distane d 0 ; α is the path loss exponent; N σ is a Gaussian random variable with zero mean and variane σ 2 ; l is the Eulidean distane between the two vehiles (or a vehile and a BS depending on the ontext). A soure vehile an establish a diret onnetion to a destination vehile if the reeived power at the destination vehile p rx is greater than or equal to a ertain threshold power p v th. Similarly, a soure vehile an establish a two-way diret onnetion to a destination BS if the reeived power at the destination BS p rx is greater than or equal to a ertain threshold power p b th. In this paper, we assume that wireless onnetions between vehiles, and between vehiles and BSs, are symmetri. Note that when σ 0, the log-normal model redues to the unit disk model. Due to this fat, we assign p v th p r 0 10α log 10 d 0, p b th p 0 10α log 10 d 0 so that the results under log-normal model an be ompared with the Fig. 3. Aess probability with hanging under the unit disk model, 1000m, r 500m, ρ 1/5, 1/50, 1/500 vehiles/m respetively. Fig. 3 shows the aess probability given different values of and ρ. The analytial results are verified by the simulation results whih are obtained from randomly generated network topologies. As the number of instanes of random networks used in the simulation is very large, the onfidene interval is too small to be distinguishable and hene ignored in this plot as well as other plots. As shown in the figure, the aess probability dereases with when exeeds some limits. For small ρ, the aess probability dereases as soon as > 2. That is beause when the vehile density ρ (number of vehiles per meter) is low, a vehile is either diretly onneted to a BS or disonneted, i.e annot reah any BS in at most two hops. It is hard for the vehile to find a one-hop relay in its range via whih it an aess a BS if it is not within the transmission range of any BS. However for large ρ, it is easier for the vehile, whih is not within the transmission range of any BS, to find a one-hop relay to aess the BS. In general the aess probability inreases with an inrease in ρ, and the reason is that when the vehile density inreases, the probability inreases for vehiles in the gap of the transmission ranges of BSs to find a neighbor within the transmission range of a BS to at as a relay. Similarly, Fig. 4 shows the onnetivity probability for different values of and ρ. The exat analytial results are verified by the simulation results. The approximate analytial

9 9 again that the aess probability inreases with an inrease in ρ. Fig. 4. Connetivity probability with hanging under the unit disk model, 1000m, r 500m, ρ 1/5, 1/50, 1/500 vehiles/m respetively. result is shown to be reasonably lose to the exat analytial result. The figure shows that when 2+r 2500 meters, it is easy for all vehiles to be onneted to either BS in at most two hops, hene the onnetivity probability is high. As gets larger, it is harder for all vehiles to be onneted to the BSs due to the larger possible distanes between the vehiles and the BSs. This auses a drop in the onnetivity probability, and the onnetivity probability tends to zero as goes to infinity. The transition of the onnetivity probability from 1 to 0 gets sharper as the vehile density inreases. As ρ goes to infinity, the transition happens at the ritial distane 2 + 2r 3000 meters, below whih the network is disonneted with a high probability and above whih the network is onneted with a high probability. Furthermore, the networks with a larger ρ have a higher onnetivity probability than the networks with a smaller ρ when is small. This is beause when the vehile density is large, it is easier for vehiles not diretly onneted to a BS to find a vehile within its ommuniation range and is diretly onneted to a BS to at as a relay. When is large, the networks with a larger ρ have a lower onnetivity probability than the networks with a smaller ρ. This is beause at large values of when the vehile density is large it is easier to have at least one vehile whih is loated too far from the BSs to be onneted to a BS in at most two hops. Fig. 5. Aess probability with r hanging under the unit disk model, 1000m, 2500m, ρ 1/5, 1/50, 1/500 vehiles/m respetively. Fig. 5 shows how the transmission range of the vehiles r affet the aess probability. It shows that the aess probability inreases with r, and when ρ is large enough, the aess probability ould be quite lose to 1. And it shows Fig. 6. Connetivity probability with r hanging under the unit disk model, 1000m, 2500m, ρ 1/5, 1/50, 1/500 vehiles/m respetively. With a similar setup, Fig. 6 shows the sensitivity of the onnetivity probability to r. For a large ρ, around a ertain value of r a small inrease in r will inur a dramati inrease in the onnetivity probability from near 0 to near 1, i.e. the well-known phase transition phenomenon. From the figure it shows that suh phenomenon does not exist for small ρ. Fig. 6 also shows a senario where there may be a signifiant gap between the approximate and exat results for onnetivity probability. Fig. 7. Aess probability with ρ hanging under the unit disk model,,, r are onstants. Fig. 7 supported our onlusion that an inrease in ρ will improve the aess probability as it shows that the aess probability monotonially inreases with ρ. While ρ is relatively small, and the width of the gap region not diretly overed by any of the BSs is relatively large, the aess probability will be low, and thus, in this irumstane, network operator should onsider to deploy more BSs along the highway for better onnetivity and greater aess probability. B. og-normal Shadowing Model Fig. 8 shows the aess probability under the log-normal shadowing model. In general, it is easier for the vehiles in the subnetwork to get aess to any BS under the log-normal model. As σ inreases, the aess probability improves. The improvement in aess probability is more signifiant for high vehiular density. Fig. 9 shows the onnetivity probability under the lognormal model when the vehile density is low (ρ vpm).

10 10 Fig. 8. Aess probability with hanging under the log-normal model, 1000m, r 500m, ρ 1/5, 1/500 vehiles/m respetively under different values of σ. (r) is the transmission range of a BS (vehile) ignoring shadowing effet, i.e. σ 0. Fig. 10. Connetivity probability with hanging under the log-normal model, 1000m, r 500m, ρ 1/5, 1/50 vehiles/m respetively under different values of σ. (r) is the transmission range of a BS (vehile) ignoring shadowing effet, i.e. σ 0. Fig. 9. Connetivity probability with hanging under the log-normal model, 1000m, r 500m, ρ 1/500 vehiles/m under different values of σ. (r) is the transmission range of a BS (vehile) ignoring shadowing effet, i.e. σ 0. The exat analytial results are verified by the simulation results. As the vehile density inreases, the omputational omplexity involved in numerially omputing the exat result inreases very quikly. As suh, we only provide the exat analytial results for low vehile density. Furthermore, Fig. 9 shows that the approximate analytial results are reasonably lose to the true values when the vehile density is low. However, as shown in Fig. 10, the disrepany between the approximate results and the true values an be signifiant when the vehile density is high (ρ 1 50, 1 5 vpm). In general, the approximate analytial result always under-estimate the simulation result. Same situation an be observed for the result under the unit disk model. This an be explained by emma 3 that a vehile is more likely to be able to aess to any BS where there is another vehile nearby that an aess to the BSs. Beause of the independene assumption used in obtaining the approximate analytial result, the approximate result will under-estimate the true value. VII. CONCUSIONS In this paper, we analyzed the onnetivity probability and the aess probability for a given network bounded by two adjaent base stations, and vehiles in the network are Poissonly distributed with known density and eah vehile an ommuniate with a base station in at most two hops. Under a general onnetion model, and later on taking the unit disk ommuniation model and the log-normal shadowing model as the speifi examples, we derived losed-form formulas for the aess probability and onnetivity probability onsidering that the base stations and the vehiles have different transmission apabilities. These formulas haraterize the relation between these key parameters, i.e. the transmission ranges of the base stations and the vehiles, the distane between adjaent base stations, the vehile density and their impat on the aess and onnetivity probabilities. These results an be useful for a network operator to design a network with a given level of aess guarantee. In future, we plan to extend the urrent work on 1-D networks to 2-D networks. EFEENCES [1] S. Sai, E. Niwa, K. Mase, M. Nishibori, J. Inoue, M. Obuhi, T. Harada, H. Ito, K. Mizutani, and M. Kizu, Field evaluation of UHF radio propagation for an ITS safety system in an urban environment, IEEE Communiations Magazine, vol. 47, no. 11, pp , [2] C.-X. Wang, X. Hong, X. Ge, X. Cheng, G. Zhang, and J. S. Thompson, Cooperative mimo hannel models: A survey, IEEE Communiations Magazine, vol. 48, no. 2, pp , [3] C.-X. Wang, X. Cheng, and D. I. aurenson, Vehile-to-vehile hannel modeling and measurements: reent advanes and future hallenges, IEEE Communiations Magazine, vol. 47, no. 11, pp , [4] IEEE P task group p, eports/tgp update.htm. [5] IEEE family of standards for wireless aess in vehiular environments (WAVE), sheet.asp? f80, [6] G. Aosta-Marum and M. A. Ingram, Six time- and frequeny- seletive empirial hannel models for vehiular wireless ANs, IEEE Vehiular Tehnology Magazine, vol. 2, no. 4, pp. 4 11, [7] Y. Chang, M. ee, and J. A. Copeland, An adaptive on-demand hannel estimation for vehiular ad ho networks, in Consumer Communiations and Networking Conferene (CCNC), 2009, pp [8]. Cheng, B. E. Henty,. Cooper, D. D. Stanil, and D. F. Bai, A measurement study of time-saled a waveforms over the mobile-to-mobile vehiular hannel at 5.9 GHz, IEEE Communiations Magazine, vol. 46, no. 5, pp , [9]. Stibor, Y. Zang, and H.-J. eumerman, Evaluation of ommuniation distane of broadast messages in a vehiular ad-ho network using IEEE p, in IEEE Wireless Communiations and Networking Conferene (WCNC), 2007, pp [10] M. Wellens, B. Westphal, and P. Mahonen, Performane evaluation of IEEE based WANs in vehiular senarios, in IEEE Vehiular Tehnology Conferene (VTC), 2007, pp

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Hartmann, Connetivity of wireless multihop networks in a shadow fading environment, Wireless Networks, vol. 11, no. 5, pp , [36] D. Miorandi, E. Alman, and G. Alfano, The impat of hannel randomness on overage and onnetivity of ad ho and sensor networks, IEEE Transations on Wireless Communiations, vol. 7, no. 3, pp , [37] X. Ta, G. Mao, and B. D. Anderson, On the giant omponent of wireless multi-hop networks in the presene of shadowing, IEEE Transations on Vehiular Tehnology, vol. 58, no. 9, pp , [38] M. Franeshetti and. Meester, andom Networks for Communiation: From Statistial Physis to Information Systems. Cambridge University Press, Seh Chun Ng reeived the BEng degree in IT & Teleommuniation from University of Adelaide, Australia, and the MS degree in IT from Malaysia University of Siene and Tehnology (MUST), Malaysia. He is urrently working toward the PhD degree in Engineering at University of Sydney. His researh interest inlude wireless multi-hop networks and graph theory. Wuxiong Zhang reeived the BEng degree in Information Seurity from Shanghai Jiao Tong University (SJTU), Shanghai, P.. China, in He is now a PhD andidate of ommuniation and information system in Shanghai esearh Center for Wireless Communiations, SIMIT, Chinese Aademi of Sienes. His researh interests lie in the areas of ommuniation theory, information theory and networking. Current researh fouses on ad-ho, and vehiular networks. Yu Zhang reeived her Ph.D. degree in omputing from Imperial College ondon, UK, in She is urrently working at Shanghai Engineering esearh Center for Broadband Tehnologies and Appliations, China. Her researh interests inlude queueing theory and performane evaluation. Yang Yang reeived his PhD degree in Information Engineering from The Chinese University of Hong Kong in He is urrently working at Shanghai esearh Center for Wireless Communiations (WiCO), SIMIT, Chinese Aademy of Sienes. Prior to that, he served the Department of Eletroni and Eletrial Engineering at University College ondon (UC), United Kingdom, as a Senior eturer. His general researh interests inlude wireless ad ho and sensor networks, wireless mesh networks, B3G mobile ommuniation systems. Guoqiang Mao reeived PhD in teleommuniations engineering in 2002 from Edith Cowan University. He joined the Shool of Eletrial and Information Engineering, the University of Sydney in Deember 2002 where he is a Senior eturer now. His researh interests inlude wireless loalization tehniques, wireless multihop networks, graph theory and its appliation in networking, and network performane analysis. He is a Senior Member of IEEE.