Vehicular Ad-hoc Networks using slotted Aloha: Point-to-Point, Emergency and Broadcast Communications

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1 Vehicular Ad-hoc Networks using slotted Aloha: Point-to-Point, Emergency and Broadcast Communications Bartłomiej Błaszczyszyn Paul Muhlethaler Nadjib Achir INRIA/ENS INRIA Rocquencourt INRIA Rocquencourt and Paris XIII Abstract: The aim of this aer is to analyze the Aloha medium access (MAC scheme in one-dimensional, linear networks, which might be an aroriate assumtion for Vehicular Ad-hoc NETworks (VANETs. The locations of the vehicles are assumed to follow a homegeneous Poisson oint rocess. Assuming owerlaw mean ath-loss and indeendent oint-to-oint fading we study erformance metrics based on the signal-over-interference and noise ratio (SINR. In contrast to revious studies where the receivers are at a fixed distance from the transmitter, we assume here that the receivers are the nearest neighbors of the transmitters in the Poisson rocess and in a given direction. We derive closed formulas for the cature robability and for the density of rogress of a acket sent by a given node. We comute the mean delay to send a acket transmitted at each slot until successful recetion. We also evaluate an uer bound to discover the neighborhood within a given sace interval. We show that we can include noise in these models. Index Terms VANETs, slotted Alohaa, MAC (Medium Access Control Layer Otimization, erformance evaluation I. INTRODUCTION VANETs may well be the most romising alication of Mobile Ad hoc NETworks (MANETs. Promotors of VANETs believe that these networks will both increase safety on the road and rovide value-added services. Numerous challenging roblems must however be solved in oder to be able to roose these services e.g. efficient and robust hysical layers, reliable and flexible medium access rotocols, routing schemes and otimized alications. Although the medium access rotocols envisioned for VANETs are mostly based on Carrier Sense Multile Access (CSMA schemes e.g; IEEE 82., techniques using carrier sensing lead to comlex atterns of simultaneously transmitting nodes. Thus it is difficult to build analytical models for such schemes. Moreover carrier sensing introduces many arameters that must be otimized, such as the interframes, the defer threshold, the back-off window, etc. In contrast, slotted Aloha has only one arameter: the transmission robability (Medium Access Probability MAP and the attern of simultaneously transmitting nodes is very simle. Thus we chose to use Aloha as a medium access scheme. Aloha has received a lot of attention in the literature and most of the studies are for two-dimensional networks. Moreover these studies usually use the biolar model in which each node which can transmit has a receiver randomly ositioned at a fixed distance from the transmitter. In contrast to this assumtion, we are able to study a Signal-over-Interference and Noise Ratio (SINR model where both the transmitter and the receiver are in the intial Poisson oint rocess which models the locations of the vehicles. A. Main contributions The main contributions of this aer are the following For the two receiving models in this aer i.e. the nearestneighbor and the nearest-receiving neighbor in a given direction, we evaluate the robability of cature of a acket sent by a fixed node. With the same assumtion, we comute the density of rogress of such a acket and we study how to otimize this density. We then study the transmission delay for a safety acket which is sent by a given node and at each slot. We also derive an uer-bound of the delay to send a broadcast acket in a given sace interval. We show that the above analyses can take noise into account. B. Related Work There is abundant literature on the erformance evaluation of general MANETs under CSMA and Aloha. However, very few aers, such as [8, focus secifically on VANETs. To the best of our knowledge, most of these studies use simulations to evaluate the erformances of these networks and/or assume simlified interference models. Aloha can be analyzed more easily than CSMA, even in quite a realistic SINR scenario [5, 6, 9. However these studies assume lanar (two-dimensional networks and a Poisson oint rocess to model the location of the network nodes. This aroach was artially adated to linear VANETs in a study [7 where it is assumed that the nodes use slotted Aloha. Numerous studies [,, evaluate beaconing algorithms in VANETs. Another study [2 considers disconnected VANETs and evaluates DTN schemes to roagate the ackets within the network. In contrast to revious aers [5 7, 9, we comute the network erformances for receivers within the Poisson oint rocess which models the location of the network nodes, whereas the revious studies assume that each transmitter has its own receiver at given distance from it. C. Organization of the aer The remaining art of this aer is organized as follows: The network and roagation models are described resectively in Sections II-A, II-B. In Section II-C the SINR cature (nonoutage condition is described. In Section II-D the nearest neighbor models used throughout this aer are roosed. With these models we comute the robability of cature in Section II-E and the density of rogress in Section II-F. In /2/$3. 22 IEEE

2 section II-G we comute the nearest neighbor warning delay time and in Secion II-H we comute the neighborhood discovery time. We extend the results obtained to the case where we have constant noise. Section IV discusses the numerical examles, followed by the conclusion in Section V. II. LINEAR VANET MODEL WITH SLOTTED ALOHA AND NEAREST-NEIGHBOR TRANSMISSIONS A. Network model We assume a linear network toology, where (at a given time nodes (vehicles are randomly located on a straight line. The nodes are erfectly synchronized on time slots of the same duration and they try to access the channel according to the slotted Aloha scheme. For every time slot, each node, indeendently tosses a coin with some bias for heads, which will be referred to as the Aloha medium access robability (Aloha MAP. Nodes whose outcome is heads transmit their ackets in the time slot, and the others do not transmit. The locations of the nodes are assumed not to change during the transmission since the duration of a transmission is so short. The above situation will be modeled by the marked Poisson oint Φ = {(X i, e i } with intensity λ on the line R, where {X i } denotes the (instantaneous locations of vehicles; λ is the mean density of nodes er unit of (route distance, {e i } i is the medium access indicator of node i; e i = for the node which is allowed to emit and e i = for the node which is not allowed to emit. The random variables e i are indeendent, with P(e i = =. Sometimes we will use the notation e Xi = e i. Let us slit Φ into two oint rocesses: Φ = {X i : e i = } reresenting transmitters, and Φ = {X i : e i = } reresenting silent nodes. The latter are otential receivers. A basic fact from the theory of Point rocesses is that Φ and Φ are indeendent Poisson oint rocesses of intensity λ and λ( resectively. B. Signal roagation Each transmitting vehicle uses the same transmission ower S. The signal-ower ath-loss is modeled by the ower-law function l(r = (Ar β where r is the distance between the transmitter and the receiver; we will secify the receiver in section II-D. Our mathematical linear model of the network requires β > in order for the sum of all owers received at a given location to have a finite mean. Signal-ower is also erturbed by random fading F that is indeendently samled for each transmitter-receiver air. Thus, the actual signal-ower received at y from x is equal to SF (x,y /l( x y. If not secified, F is assumed to have a general distribution with finite moment E[F /β. In this section we will restrict ourselves to an imortant secial case when F is exonentially distributed, which corresonds to the situation of indeendent Rayleigh fading. By eventual renormalization of S we can assume without loss of generality that F has a mean of. Moreover in this section we do not consider external noise (W = but we kee W in the notation for the remaining art of this aer. C. Recetion When a vehicle located at x transmits a signal with ower S and this signal is being received by a vehicle located at y, then the success of this recetion deends on the signal-tointerference-and-noise ratio (SINR SINR (x,y = SF (x,y/l( x y, (2. W + I Φ (y where I Φ is the shot-noise rocess of Φ : I Φ (y = X i Φ SF (y,x i/l( y X i. In this aer we assume a fixed bit-rate coding, i.e., y successfully receives the signal from x if SINR (x,y T, (2.2 where SINR (x,y is given by (2. and T is the SINR-threshold related to the bit-rate given some articular coding scheme. D. Nearest neighbor receiver models A articular choice of receivers deends on the routing scheme imlemented in the network. Motivated by the usual routing mechanisms which tyically select relay nodes in the vicinity of the transmitters, in this section we consider the two following receiver models: Nearest Neighbor in a given Direction (NND. We assume that each transmitter in Φ (cf. the network model in Section II-A icks the nearest node to itself in Φ in the randomly selected direction (to the right or to the left on the line as its receiver, regardless of whether or not the latter is authorized to transmit at the same time. Successful recetion requires that this selected node is not authorized by Aloha to transmit (i.e, is in Φ, and that the SINR condition (2.2 for this transmitter-receiver air is satisfied. Nearest Receiver in a given Direction (NRD. Each transmitter in Φ icks the nearest node in Φ in a randomly selected direction, i.e., it icks a node that is not authorized by Aloha to transmit. Remark that the NND model corresonds to the usual searation of routing and MAC layers, while the NRD model might be seen as corresonding to an oortunistic routing. As we will see, in linear networks both models allow for quite exlicit erformance analysis and, in articular, an otimal tuning of the Aloha arameter, rovided the external noise can be ignored (W =. E. Cature robability For arbitrary a, b > denote C(a, b = a /(ub + du and π C(b = C(, b = b sin(π/b. (2.3 Moreover, for given T and β let us denote: C = C (T, β = T /β (C(T /β, β + C(β, C 2 = C 2 (T, β = 2T /β C(β = 2T /β π β sin(π/β. Proosition 2.: The robability of successful transmission for slotted Aloha with MAP in the NND receiver model with

3 exonential fading and negligible noise (W is equal to: P NND ( = + C (2.4 and in the NRD receiver model P NRD ( = + ( C 2. (2.5 Proof: Note directly from the form of the SINR in (2. that P NND ( and P NRD ( do not deend on S and A. Hence in the remaining art of the roof we take S = A =. Let us first consider the NND model. By symmetry, let us assume without loss of generality that a given transmitter (located at the origin sends a acket to its nearest neighbor to the right on the real line without knowing if this node is silent or not. By the known roerty of the Poisson oint rocess, the distance R to this node has an exonential distribution with arameter λ. Moreover given R = r, all other nodes form a Poisson oint rocess of intensity λ on (, (r,. Conditioning on R = r the robability of the SINR condition (2.2 is equal to: P{ F S T I Φ l(r R = r } = e st l(r dp(i Φ s R = r = ψ IΦ R=r(T l(r, where ψ IΦ R=r denotes the Lalace transform of the shot noise generated at the location of the receiver by a Poisson oint rocess of intensity λ on (, (r,. It is know (it can be derived from the formula for the Lalace functional of the Poisson.. (see e.g. [3: { ( ( } ψ IΦ R=r(ξ = ex λ + E[e ξf/l(s ds. r Plugging in the conditional robability of the SINR condition and unconditioning with resect to R we obtain: P NND ( = λ( { ( e λr ex λ + ( [ r E e l(rt F/l(s } ds dr, where ( in front of the first integral is due to the fact that successful recetion is ossible only if the nearest node is not authorized to transmit at the given time slot. Assuming exonential fading F and the ower law ath-loss l( we evaluate E[e l(rt F/l(s = and obtain +(s/r β /T ( P NND ( = λ( e λr +T /β (C(T /β,β+c(β dr, which boils down to the right hand side of (2.4. In order to obtain exression (2.5 for the N RD model we follow the same arguments, with the following modifications. The distance to the nearest receiver to the right has the exonential distribution of arameter λ(. Moreover the distribution of the oint rocess of emitters is Poisson with arameter λ and indeendent of the location of this receiver. Note also that by the very choice of the receiver, it is not authorized to emit during the given time slot, hence there is no ( factor in the numerator. Remark 2.2: It is easy to see that C 2 C and hence P NRD P NND, i.e., the oortunistic choice of the receiver ays off regarding the likelihood of successful transmission. F. Density of rogress Now we will calculate and otimize in the mean density of rogress d defined as the exected total rogress of all the successful transmissions er unit of road length; [ d = E X i Y i I(e Xi = I(e Yi = I(SINR (Xi,Y i T, X i [, where Y i is the receiver chosen by the transmitter X i in the given model (NND or NRD. The density of rogress, can also be seen as quantifying the number of bit-(or acket-meters umed er unit length of the VANET network. Thus we see it as a generic erformance metric of value-added services generating oint-to-oint traffic. Proosition 2.3: Under the assumtions of Proosition 2. the density of rogress in the NND receiver model is equal to: d = d NND ( = and is maximized for equal to: NND = C + C 2 2 C. ( ( + C 2 (2.6 Moreover, < NND <. Proof: Using Cambell s formula, the density of rogress in the NDD model can be exressed as: [ d NND ( = λ( E RP{ I(F S T I Φ l(r R = r } with the notation as in the roof of Proosition 2.. Following the same arguments as in this latter roof we obtain ( d NND ( = λ 2 ( re λr + C dr, which is equal to the right-hand side of (2.6. Taking the derivative of the latter exression in we find that its sign is equal to that of the olynomial P ( = (2+ C 2 2 C C 2. Note that P ( =, P ( =, P ( = C < and P ( =. Hence in the interval (,, P ( has a unique root: NND which maximizes d NND(. The exlicit exression for this root follows from the general formulas for the roots of cubic equations. Proosition 2.4: Under the assumtions of Proosition 2. the density of rogress in the NRD receiver model is equal to: d NRD ( = and is maximized for equal to: Moreover, < NRD <. ( ( + ( C 2 2. (2.7 NRD = C 2 + C C 2. I(Condition A is the indicator function. The value of this function is if Condition A is satisfied and otherwise.

4 Proof: Using the same reasoning as in the roof of Proosition 2.3 we find exression (2.7 and its otimization in goes along the same lines as for the NND model relacing C by C 2. G. Nearest neighbor warning delay time In this section we are interested in emergency broadcasting in VANETs. Suose that a tyical (tagged node needs to warn its nearest neighbor in a given direction by sending it a secific acket. Then P NND ( given by (2.4 is the robability that this task can be accomlished with a single transmission. However, a single transmission might not be sufficient. Hence, we assume that the tagged node does not have to resect the Aloha scheme and after the first unsuccessful transmission it is allowed to retransmit the acket in successive slots, ossibly several times, until it has been received successfully. Regarding the geograhic dislacement of nodes during this rocess, we assume that it is small enough to only imact on the fading conditions and not the average ath-loss conditions. In other words, we assume Poisson locations of nodes X i Φ unchanged and the fading F(x,y n, indeendently re-samled for every consecutive retransmission slot n =, 2,..., for every transmitter-receiver air (x, y. Let us denote by L the number of consecutive time slots needed for the tagged node to successfully transmit the emergency message to its nearest neighbor in Φ in a given direction. We call L emergency delay at the tagged node. Since we have assumed that fading is indeendent across the time slots, given Φ, L is a geometric random variable with mean /π(φ, where π(φ is the robability of successful recetion in a given time slot, given the location of nodes Φ. (This latter conditional robability regards only uncertainty due to fading conditions and MAC statuses of nodes in the network. In what follows we will give the exression for the unconditional exected emergency delay E[L (Φ, i.e.; average emergency delay at the tagged node, where the averaging regards all ossible Poisson configurations of other nodes Φ. As we will see, this exectation is finite only if is sufficiently small for any given T and β. Let us denote and D ( = D (; T, β = T /β( + T /β D 2 ( = D 2 (; T, β = 2T /β u β + du u β + du u β + du. Proosition 2.5: Under the assumtions of Proosition 2., the average emergency delay in the MMD model is equal to rovided E[L = and E[L = otherwise. ( ( D ( D ( < (2.8 Proof: We need to evaluate π(φ, i.e. the robability of successful recetion in a given time slot, given Φ. As reviously in the NND model, we suose that the receiver (nearest neigbour is at distance r from the tagged node. Following the same arguments as in the roof of Proosition 2., we find that π(φ given the receiver at distance r, which we denote by π(r, Φ is equal to ( E[e l(rt I(r,Φ Φ, where I(r, Φ is the interference at the receiver. Hence π(φ can be exressed as a conditional Lalace transform of the shot-noise given Φ. An exlicit exression for this is known (cf. the roof of [4, Lemma 7.3 and in our case gives { ( } π(r, Φ = ( ex log /T ( X i /r β. + X i Φ Taking the exectation of the inverse of the latter exression with resect to the distribution of the homogeneous Poisson rocess of intensity λ truncated to (, (r,, again by the known exression for the Lalace transform of a shotnoise, we obtain [ E ( /T (s/r β + R = r = { ( π(r, Φ ex λ + r } ds = ex{λr D (} Consequently E[L = = λ λ e λr e λr D( dr e λr( D( dr It is easy to see that this integral is infinite if D ( and /(( ( D ( otherwise. This comletes the roof. Remark 2.6: Note that D ( C with the equality reached when. Hence, for small we have D ( C, and the critical for which the average exected delay exlodes is aroximately / C. H. Neighborhood discovery time Let us now consider another imortant tye of communication in VANETs, related to osition broadcasting. Assume that each node uses a fraction of the time slots in which it is authorized by Aloha to transmit, to broadcast a secial acket to inform every other node being able to receive it about the transmitter s osition (and erhas other nodes arameters. More recisely, for each transmission slot, a given node transmits such a secial acket (called the localisation acket indeendently, with robability and other (usual traffic ackets sent with robability. We are interested in the mean time required for a given (tyical node to discover all nodes within a given distance R from it by receiving at least one localisation acket from each of them. We denote by D i the time (number of time slots until the recetion of the first localisation acket from node X i by the tyical node located at the origin. There is no exlicit exression for (the mean of the maximum max Xi [ R,R D i.

5 however, we can give an exlicit exression for the exectation of the sum L disc (R := X D i [ R,R i, which is in uer bounded by the aforementioned maximum. Proosition 2.7: Assume exonential fading and negligible noise (W =. Then for any R > E[L dics (R = 2 ( 2 D 2 ( R ( e λrd2(, where D 2 ( = D 2 (; T, β is as defined in Section II-G. Proof: By Cambell s formula R R [ E[L disc (R = λ E[D r dr = λ E R π(r, Φ dr, where π(r, Φ is given in the roof of Proosition 2.5. Consequently λ R E[L disc (R = e λrd2( dr. ( R Evaluation of the above integral concludes the roof. III. IMPACT OF EXTERNAL NOISE In the revious section we assumed that the external noise was negligible (W =. In this section we study the imact of a non-null external noise W. It is easy to extend the results of Proositions 2., 2.3 and 2.4 to the case with non-null noise. The otimisation of the density of rogress is however not exlicit. Proosition 3.: Under the assumtions of Proosition 2. for the NND receiver model, with an arbitrary noise distribution W, we have the followin robability of successful recetion: NND = λ( and the density of rogress: d NND = λ( e rλ(+ C T W (Arβ dr (3.9 re rλ(+ C T W (Arβ dr. (3. Relacing C by C in the two above exressions we obtain, resectively, the robability of successful transmission NRD and the density of rogress d NRD in the NRD receiver model with an arbitrary noise W. Proof: The roof follows the same lines as in the roofs of Proositions 2., 2.3 and 2.4. The introduction of noise into the comutation of the delay and the neighborhood discovery time is similar and is not carried out in this aer for reasons of sace. IV. NUMERICAL EXAMPLES The default arameters are λ =. vehicles er meter (the mean inter-vehicle distance is equal to meters, β = 4, T = and A =. The fading is exonentially distributed with rate. In Figure we show the cature robabilities for P NND and P NRD. We observe that the results obtained are close; for the same transmission robability, P NND is sligthly greater than P NRD. We also notice that these robabilities are greater for β = 4 than for β = 2. This is because when β is larger, then interference decays more raidly. In Figure 2 we resent the densities of rogress d NND and d NRD. We note that for the same transmission robability we have d NRD > d NND which confirms the inequality we show in general. For β = 2 the maximum of d is around 2.5% greater for the NRD scheme than for the NND scheme. For β = 4 the maximum of d is around 5% greater for the NRD scheme than for the NND scheme. Thus we can conclude that the oortunism of the NRD scheme has a limited imact. In Figure 3 we resent the value of which otimizes the density of rogress for both the NND and NRD schemes. We observe that there is a significant difference for this value of for the two schemes and this difference increases with β. In Figure 4 we resent the value of the average delay with resect to for β = 2, 3, 4 and 5 and T =. We note that there is a critical value of ( which deends on β for which this delay becomes infinite. Logically this critical value of is an increasing function of β. With less interference the transmission robability can be higher. Figure 5 also gives the value of the average delay with resect to for T =, 2, 5,. Of course for larger T the critical value of is smaller. In Figure 6 we resent the density of rogress d NND with resect to for W =,, 9, 8, 7, 6, 5. We note that the noise has a very great imact on the network erformances. For W 7 the density of rogress is more than times smaller than without noise. P NND,P NRD Cature robability P NND β=2 P NRD β=2 P NND β=4 P NRD β= Fig.. Cature robabilities P NND and P NRD for β = 2 and β = 4 P NND,P NRD Density of rogress d NND β=2 d NRD β=2 d NND β=4 d NRD β= Fig. 2. Density of successful transmissions for β = 2 and β = 4

6 .3.25 otimal value of for the density of rogress * NND T= * NRD T= Mean waiting time until successful recetion T= T=2 T=5 T=.2 E[L β Fig. 3. Otimal value of for the density of successful transmissions for 2 β 5 Fig. 5. Exected delay until successful recetion E[L for T =, 2, 5, Mean waiting time until successful recetion β=2 β=3 β=4 β= Density of rogress W= W= - W= - W= -9 W= -8 W= -7 W= -6 W= -5 E[L 8 d NND Fig. 4. Exected delay until successful recetion E[L for β = 2, 3, 4, Fig. 6. Exected delay until successful recetion E[L for various values of the noise. V. CONCLUSION In this aer we have studied the erformance of two simle NND and NRD relaying schemes in linear networks. We have derived closed formulas for the cature robability, the density of rogress and the average delay. These formulas allow the influence of the models arameters to be easily studied. We show how to introduce noise in our evaluation and we obtain other formulas that can be numerically evaluated. REFERENCES [ Vinel A, Vishnevsky V, and Koucheryavy Y. A simle analytical model for the eriodic broadcasting in vehicular ad-hoc networks. GLOBECOM Workshos, ages 5, 3 Dec 28. [2 Emmanuel Baccelli, Philie Jacquet, Bernard Mans, and Georgios Rodolakis. Highway Vehicular Delay Tolerant Networks: Information Proagation Seed Proerties. IEEE Transactions on Information Theory. Vol 58, No 3 ages , March 22. [3 F. Baccelli and B. Błaszczyszyn. Stochastic Geometry and Wireless Networks, Volume I Theory, volume 3, No 3 4 of Foundations and Trends in Networking. NoW Publishers, 29. [4 F. Baccelli and B. Błaszczyszyn. Stochastic Geometry and Wireless Networks, Volume II Alications, volume 4, No 2 of Foundations and Trends in Networking. NoW Publishers, 29. [5 F. Baccelli, B. Blaszczyszyn, and P. Mühlethaler. An Aloha Protocol for Multiho Mobile Wireless Networks. In Proceedings of the Allerton Conference, University of Illinois, Urbana Chamaign, November 23. And in IEEE Transactions on Information Theory, Vol 52, No 2: ages , February 26. [6 F. Baccelli, B. Błaszczyszyn, and P. Mühlethaler. Stochastic analysis of satial and oortunistic Aloha. IEEE JSAC, secial issue on Stochastic Geometry and Random Grahs for Wireless Networks, ages 5-9, 29. [7 B. Blaszczyszyn, P. Muhlethaler, and Y. Toor. Maximizing Throughut of Linear Vehicular Ad-hoc NETworks (VANETs - a Stochastic Aroach. In Proc. of Euroean Wireless 29, May 29. Aalborg, Denmark. [8 H.A. Cozzetti, R.M. Scoigno, L. Casone, and G. Barba. Comarative analysis of IEEE 82. and MS-Aloha in VANET scenarios. In Proc. of Services Comuting Conference (APSCC, ages 64 69, Singaore, Dec 29. [9 M. Haenggi. Outage, local throughut, and caacity of random wireless networks. IEEE Trans. Wireless Comm., Vol 8: ages , August 29. [ R. Reinders, E. M. van Eenennaam, G. Karagiannis, and G. J. Heijenk. Contention window analysis for beaconing in vanets. In Seventh IEEE International Wireless Communications and Mobile Comuting conference, IWCMC 2, Istanbul, Turkey, ages , Istanbul, July 2. IEEE Comuter Society. [ Razvan Stanica, Emmanuel Chaut, and Andr-Luc Beylot. Why VANET Beaconing is More than Simle Broadcast. In IEEE Vehicular Technology Conference (VTC, San Francisco, ages 5, Setember 2.