Highway Vehicular Delay Tolerant Networks: Information Propagation Speed Properties

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1 Highay Vehicular Delay Tolerant Netorks: Information Propagation Speed Properties Emmanuel Baccelli, Philippe Jacquet, Bernard Mans, Georgios Rodolakis INRIA, France, Macquarie Uniersity, Australia, Abstract In this paper, e proide a full analysis of the information propagation speed in bidirectional ehicular delay tolerant netorks such as roads or highays. The proided analysis shos that a phase transition occurs concerning the information propagation speed, ith respect to the ehicle densities in each direction of the highay. We proe that under a certain threshold, information propagates on aerage at ehicle speed, hile aboe this threshold, information propagates dramatically faster at a speed that increases quasi-exponentially hen the ehicle density increases. We proide the exact expressions of the threshold and of the aerage information propagation speed near the threshold, in case of finite or infinite radio propagation speed. Furthermore, e inestigate in detail the ay information propagates under the threshold, and e proe that delay tolerant routing using cars moing on both directions proides a gain in propagation distance, hich is bounded by a sub-linear poer la ith respect to the elapsed time, in the referential of the moing cars. Combining these results, e thus obtain a complete picture of the ay information propagates in ehicular netorks on roads and highays, hich may help designing and ealuating appropriate VANET routing protocols. We confirm our analytical results using simulations carried out in seeral enironments (The One and Maple). I. INTRODUCTION The limits of the performance of multi-hop packet radio netorks hae been studied for more than a decade, yielding fundamental results such as those of Gupta and Kumar [2] on the capacity of fixed ad hoc netorks. These studies assume that either end-to-end paths are aailable or packets are dropped on the spot. Folloing seminal orks such as [] ealuating the potential of mobility to increase capacity, recent research studies focussed on the limits of the performance beyond the end-to-end hypothesis, i.e., hen end-to-end paths may not exist and communication routes may only be aailable through time and mobility. In this context nodes may carry packets for a hile until adancing further toards the destination is possible. Such netorks are generally referred as Intermittently Connected Netorks (ICNs) or Delay Tolerant Netorks (DTNs). Interest in DTN modeling and analysis has risen as noel netork protocols and architectures are being elaborated to accommodate arious forms of ne, intermittently connected netorks, hich include ehicular ad hoc netorks (VANETs), poer-saing sensor netorks, or een Interplanetary Internet [7]. In this paper, e study the information propagation speed in the typical case of bidirectional ehicular DTNs, such as roads or highays (e.g., about 75% of the total statute miles in the USA [8]). Our objectie consists in determining the maximum speed at hich a packet (or beacon) of information can propagate in such a bidirectional ehicular netork. Our analysis shos that a phase transition occurs concerning information propagation speed, ith respect to the ehicle density. We proe that under a certain threshold, information propagates on aerage at ehicle speed, hile aboe this threshold, information propagates much faster. We proide the exact expressions of the threshold and of the aerage propagation speed near the threshold. With applications such as safety, ad hoc ehicular netorks are receiing increasing attention (see recent sureys [6], [7]). Delay tolerant architectures hae thus been considered in this context in recent studies, and arious analytical models hae been proposed. In [9], the authors study ehicle traces and conclude that ehicles are ery close to being exponentially distributed on highays. The authors of [5] also base themseles on traces gathered in DieselNET (the experimental ehicular netork deployed by UMass) to elaborate and ealuate a noel DTN routing algorithm. In [], the authors proide a model for critical message dissemination in ehicular netorks and derie results on the aerage delay in deliery of messages ith respect to ehicle density. The authors of [2] propose an alternatie model for ehicular DTNs and deried results on node connectiity, under the hypothesis that ehicles are exponentially distributed. The study is based on queuing theory techniques and characterizes the relationship beteen node connectiity and seeral parameters including speed distribution and traffic flo. In [2], the authors model ehicles on a highay, and study message propagation among ehicles in the same direction, taking into account speed differences beteen ehicles, hile in [6] authors study message dissemination among ehicles in opposing directions and conclude that using both directions increases dissemination significantly. Seeral studies focus on characterizing the packet propagation delay in DTNs: [8] hich models DTNs as Erdös- Rényi random graphs to derie results concerning packet propagation delay, [22] hich uses fluid limit techniques to derie relationships beteen buffer space, packet duplication and dissemination delay. Other studies focus on information propagation speed in DTNs. In [5] the authors sho that hen a to-dimensional netork is not percolated, the latency scales linearly ith the Euclidean distance beteen the sender and the receier, hile in [3], the authors obtained analytical estimates of the constant bounds on the speed at hich information can propagate in to-dimensional DTNs. Studies

2 such as [], [2], [3] are the closest related ork, also focusing on information propagation speed in one-dimensional DTNs. These studies introduce a model based on space discretization to derie upper and loer bounds in the highay model under the assumption that the radio propagation speed is finite. Their bounds, although not conerging, clearly indicates the existence of a phase transition phenomenon for the information propagation speed. Comparatiely, e introduce a model based on Poisson point process on continuous space, that allos both infinite and finite radio propagation speed, and derie more fine-grained results aboe and belo the threshold (some of the ork described in the folloing as presented in [4]). Using our model, e proe and explicitly characterize the phase transition. In this context, our contributions are as follos: () e deelop a ne ehicule-to-ehicule model for information propagation in bidirectional ehicular DTNs in Section II; (2) e sho the existence of a threshold (ith respect to the ehicle density), aboe hich the information speed increases dramatically oer the ehicle speed, and belo hich the information propagation speed is on aerage equal to the ehicle speed, and (3) e gie the exact expression of this threshold, in Section III; (4) in Section IV, e proe that, under the threshold, een though the aerage propagation speed equals the ehicle speed, DTN routing using cars moing on both directions proides a gain in the propagation distance, and this gain follos a sub-linear poer la ith respect to the elapsed time, in the referential of the moing cars; (5) e characterize information propagation speed as increasing quasi-exponentially ith the ehicle density hen the latter becomes large aboe the threshold, in Section V; (6) e coer both infinite radio propagation speed cases, then finite radio propagation speed cases in Section VI, and (7) e alidate the proided analysis ith simulations in multiple enironments (The One and Maple), in Section VII. Eastbound Fig.. II. MODEL AND RESULTS Westbound cluster Eastbound cluster Westbound Model of a bidirectional ehicular netork on a highay. In the folloing, e consider a bidirectional ehicular netork, such as a road or a highay, here ehicles moe in to opposite directions (say east and est, respectiely) at speed, as depicted in Figure. Let us consider eastbound ehicle density as Poisson ith intensity λ e, hile estbound ehicle density is Poisson ith intensity. We note that the Poisson distribution is indeed a reasonable approximation of ehicles moing on non-congested highays [9]. Furthermore, e consider that the radio propagation speed (including store and Fig. 2. Information propagation threshold ith respect to (λ e, ). Belo the cure, the aerage information propagation speed is limited to the ehicle speed (i.e., the propagation speed is in the referential of the eastbound cars), hile aboe the cure, information propagates faster on aerage. forard processing time) is infinite, and that the radio range of each transmission in each direction is of length R. The main result presented in this paper is that, concerning the information propagation speed in such an enironment, a phase transition occurs hen λ e R and R coincide on the cure y = xe x, i.e., λ e Re λer = Re λr. () Figure 2 shos the corresponding threshold cure for R = (in the folloing, e ill alays consider the case R =, ithout loss of generality). We sho that belo this threshold, the aerage information propagation speed is limited to the ehicle speed, hile aboe the cure, information propagates faster on aerage. We focus on the propagation of information in the eastbound lane. Our aim is the ealuation of the maximum speed at hich a packet (or beacon) of information can propagate. An information beacon propagates in the folloing manner, illustrated in Figure 3: it moes toard the east jumping from car to car until it stops because the next car is beyond radio range. The propagation is instantaneous, since e assume that radio routing speed is infinite. The beacon aits on the last eastbound car until the gap is filled by estbound cars, so that the beacon can moe again to the next eastbound car. We denote T i the duration the beacon aits hen blocked for the ith time and D i the distance traeled by the beacon just after. The random ariables T i and D i are dependent but, due to the Poisson nature of ehicle traffic, the tuples in the

3 Theorem 3. When λ > λ e (case p = ), hen t, E(L(t)) = B(λ e, )(2t) λe λ + O(t 2 λe λ ). (7) (a) for some B(λ e, ), function of (λ e, ). III. PHASE TRANSITION: PROOF OF THEOREM (b) Fig. 3. Eastbound information propagation: the beacon aits on the last eastbound car (a), until the gap is bridged by estbound cars so that the beacon can moe again (b). sequence (T i, D i ) are i.i.d.. From no on, e denote (T, D) the independent random ariable. We denote L(t) the distance traeled by the beacon during a time t on the eastbound lane. We consider the distance traeled ith respect to the referential of the eastbound cars. We also define the aerage information propagation speed p as: E(L(t)) p = lim. (2) t t By irtue of the reneal processes, e hae p = E(D) E(T). (3) We proe that p is ell-defined because E(D) is finite, in Section III-F. For the remainder of the paper, for x >, e denote x the conjugate of x ith respect to the function xe x : x is the alternate solution of the equation x e x = xe x. Notice that x = x and =. We proe the folloing theorems, concerning the information propagation phase transition threshold (Theorem ), the distribution of the aiting time spent by information packets in the phase depicted in Figure 3a (Theorem 2), and the total propagation distance achieed (under the phase transition threshold) because of multi-hop bridging using nodes in both traffic directions, as depicted in Figure 3b (Theorem 3). Theorem. For all (λ e, ), the information propagation speed p ith respect to the referential of the eastbound cars is p <, and, Theorem 2. When t, λ e < λ p =, (4) λ e > λ p >. (5) P (T > t) = A(λ e, )(2t) λe λ ( + o()), (6) for some A(λ e, ), function of (λ e, ). Notice that Theorem is in fact a corollary of Theorem 2. A. Proof Outline We call cluster a maximal sequence of cars such that to consecutie cars are ithin radio range. A estbound (respectiely, eastbound) cluster is a cluster made exclusiely of estbound (respectiely, eastbound) cars. A full cluster is made of estbound and eastbound cars. We define the length of the cluster as the distance beteen the first and last cars augmented by a radio range. We denote L a estbound cluster length. We start by proing in Section III-B that the Laplace transform of L : f (θ) = E(e θl ) equals: f (θ) = ( + θ)e λ θ θ + e λ θ, (8) and the exponential tail of the distribution of L is gien by P (L > x) = Θ(e λ x ). (9) To ealuate ho information ill propagate according to Figure 3, e compute the distribution of the gap length G e beteen the cluster of eastbound cars on hich the beacon is blocked and the next cluster of eastbound cars. We sho in Section III-D that P (G e > x) = O(e λex ). No, let T(x) be the time needed to meet a estbound cluster long enough to fill a gap of length x (i.e., a estbound cluster of length larger than x). We sho in Section III-C that: E(T(x)) = Θ( P (L > x) ) = Θ(e x ). () Therefore, the aerage time T to get a bridge oer all possible gaps is E(T) = = 2 E(T(x))e xλe dx Θ(exp((λ λ e )x))dx. () As a result, the threshold ith respect to (, λ e ) here E(T) dierges is clearly hen e hae: λ = λ e, (2) or, in other ords, since λ e λ = λ e λ, hen e hae: e λ = λ e e λe. (3)

4 B. Cluster Length Distribution Lemma. The Laplace transform of the estbound cluster length f (θ) = E(e θl ) satisfies: L3 > x L < x L2 < x f (θ) = ( + θ)e λ θ θ + e λ θ. (4) Unbridged gap length x R = Proof: The length of the cluster is counted from the first car. The random ariable L satisfies: L =, ith probability e λ, hen the first car has no car behind ithin the radio range; L = g + L, ith probability e λ, here g is the distance to the next car and g <. Translating this in terms of Laplace transforms yields: L3 > x Road length to bridge gap B(x) (starting from arbitrary cluster) (a) f (θ) = e λ θ + f (θ) e (λ+θ)x dx (5) = e λ θ + f (θ) + θ ( e λ θ ). (6) In passing, e get E(L ) = f () = eλ. Lemma 2. We hae the asymptotic formula: P (L > x) = ( λ )e λ λ ( λ )λ e λ x ( + o()) (7) Proof: The asymptotics on P (L > x) are gien by inerse Laplace transform: P (L > x) = ε+i 2iπ = 2iπ ε i ε+i ε i f (θ) e θx dθ θ ( + θ)e λ (θe θ + e λ )θ eθx dθ, for some ε > small enough. For R(θ) < the denominator (θe θ + e λ ) has to simple roots at θ = λ and θ = and is absolutely integrable elsehere. The root does not lead to a singularity since it is canceled by the numerator + θ. The residues theorem neutralizes the pole at, therefore for some ε > : P (L > x) = ( λ )e λ λ ( λ )λ e λ x + O(e (λ +ε)x ). C. Road Length to Bridge a Gap No, let us assume that e ant to fill a gap of length x. We ant to kno the aerage length of estbound road until the first cluster that has a length greater than x. Figure 4 depicts a gap of length x, and the length of estbound road until a cluster is encountered hich can bridge the gap. Let f (θ, x) be the Laplace transform of the cluster length, under the condition that it is smaller than x: f (θ, x) = E( (L<x)e θl ). R = (b) Fig. 4. Illustration of the road length B (x) until a gap x is bridged: (a) smaller clusters cannot bridge the gap, (b) until a estbound cluster of length at least x is encountered. Lemma 3. The Laplace transform of the road length B (x) to bridge a gap of length x, starting from the beginning of an arbitrary cluster, is: β (θ, x) = E(e θb(x) P (L > x ) ) = λ λ f +θ (θ, x ), (8) and, E(B (x)) = ( + O(e εx ) ) e λ ( λ )λ ( λ )e λ e(x )λ λ. (9) Proof: Before a cluster of length greater than x appears there is a succession of clusters L, L 2,... each of length smaller than x (see Figure 4b). Seeral cases are possible: the first cluster to come is a cluster of length greater than x, ith probability P (L > x ); the road length to this cluster is equal to. the first cluster greater than x is the second cluster; in this case, the road length Laplace transform is equal to f (θ, x) λ +θ, i.e., the Laplace transform of a cluster multiplied the Laplace transform of the exponentially distributed inter-cluster distance, namely λ +θ (as Laplace transform multiplication represents the addition of the corresponding independent random ariables). or, in general, the first cluster greater than x is the kth cluster; in ( this case, the road ) length Laplace transform is k equal to f (θ, x) λ +θ Therefore, the Laplace transform of the road length to the cluster of length greater than x is equal to the sum of the Laplace transforms of the preious cases, i.e., P (L > x ) ( ) k. k= f (θ, x) λ +θ

5 Thus, the aerage is (combining ith Lemma 2): E(B (x)) = θ β (, x) ( = θ f (, x ) ) f (, x ) = D. Gap Distribution P (L > x ) ( e ) + O(e (x )λ ) P (L > x ) = ( + O(e εx ) ) e λ ( λ )λ ( λ )e λ e(x )λ λ Let us call G e an eastbound gap hich is not bridged (see Figure 5). As illustrated in Figure 6, G e can be decomposed into a estbound cluster length L ithout eastbound cars, plus a random exponentially distributed distance I e to the next eastbound car. Westbound The Laplace transform of the estbound cluster length ithout eastbound cars, defined for all R(θ) > λ +λ e, equals: E(e θl ) = p (x)e λex e θx dx P r(no eastbound) = f (θ + λ e ). (23) f (λ e ) and the Laplace transform of G e = L + I e follos, as ell as its aerage estimate. Lemma 5. The probability density p e (x) of G e is: p e (x) = λ e f (λ e ) e λex ( + O(e εx )). (24) Proof: The proof comes from a straightforard singularity analysis on the inerse Laplace transform. E. Distribution of Waiting Time T Lemma 6. We hae 2T = L + I + B (here I is a random exponentially distributed distance to the next estbound car), and, therefore, 2E(T) = E(L ) + + E(B (x))p e (x)dx. (25) Distance to next estbound car Road length to bridge gap B Eastbound Fig. 5. Bridged eastbound gap Ge Unbridged eastbound gap Ge Illustration of a bridged gap Ḡe, and an unbridged gap Ge. L * R = R = Total distance to bridge gap: 2T Fig. 7. Waiting time T: the total distance to bridge a gap is L +I+B. L * R Gap length Ge Distance to next eastbound car Fig. 6. Unbridged gap G e model; L corresponds to a estbound cluster length ithout eastbound cars. Lemma 4. The distribution of G e satisfies E(e θge ) = f (θ + λ e ) f (λ e ) hich is defined for all R(θ) > λ e, and λ e λ e + θ, (2) E(G e ) = f (λ e ) f (λ e ) + λ e. (2) Proof: Let p (x) be the probability density of a estbound cluster length L. The probability that a estbound cluster has no eastbound cars is: p (x)e λex dx = f (λ e ). (22) Proof: The total distance to bridge a gap, as depicted in Figure 7, equals the distance to the beginning of the first estbound cluster (L + I ) plus the road length to bridge a gap starting from an arbitrary cluster (B ) minus, since communication can start at exactly one radio range. Since the distance is coered by cars moing in opposite directions, e hae 2T = L + I + B. We complete the proof by taking the expectations, and aeraging on all possible gap lengths x. Corollary. The quantity E(T) conerges hen λ e > λ and dierges hen λ e < λ. Proof: The proof comes from the leading terms of E(B (x)) and p e (x). F. Distance D Traeled after Waiting Time T We denote C e the distance traeled beyond the first gap. As depicted in Figure 8, e hae D = G e + C e. Lemma 7. The Laplace transform E(e θce ) is defined for all R(θ) > (λ e + ). Proof: The random ariable C e is smaller in probability than a full cluster.

6 Gap Ge Distance traelled in bridging D Bridged distance after gap Ce Fig. 8. Total distance D traeled hen a bridge is created D = G e + C e. Lemma 8. The aerage alue of C e satisfies: E(C e ) = λ e f (λ e ) f (λ e ) + f (λ e ) f (λ e ). (26) Proof: From (22), the probability that an eastbound car is not bridged to the next eastbound car equals f (λ e ). The unconditional gap length is λ e. We define Ḡe as an eastbound gap, under the condition that the gap is bridged (see Figure 5). Therefore, the aerage gap length Ḡe satisfies the folloing: f (λ e )E(G e ) + ( f (λ e ))E(Ḡe) = λ e, (27) hich gies E(Ḡe) = λ e + f (λe) f (λ e). The distance C e traeled in bridging (beyond the first gap, and extended to the next cluster, hich is eentually bridged) satisfies: E(C e ) = ( f (λ e )) ( E(Ḡe) + E(C e ) ) (28) = λ e f (λ e ) f (λ e ) + f (λ e ) f (λ e ). (29) Corollary 2. The total distance D e traeled including the first gap satisfies E(D e ) = E(G e ) + E(C e ) = λ, hich ef (λ e) remains finite for all ehicle densities. Since E(D e ) is finite (Corollary 2) and E(T) conerges hen λ e > λ, and dierges hen λ e < λ (Corollary ), e obtain the proof of Theorem. IV. POWER LAWS: PROOF OF THEOREMS 2 AND 3 A. Waiting Time Distribution In this section, e are interested in finding an ealuation of the aiting time distribution P (T > y), hen y, in case λ e < λ, i.e., hen the information propagation speed is on aerage. Lemma 9. When y tends to infinity, ith P (B > y) = A(λ e, )y λe λ ( + o()), A(λ e, ) = λ ( ) ee λe λ f (λ e ) Γ λe λ β λe λ, here Γ(.) is the Euler Gamma function, β = ( λ )eλ 2λ λ. Proof: See appendix. Lemma. When t : P (T > t) = A(λ e, ) (t2) λe λ ( + o()). (3) Proof: We hae the relation T 2 = G e + B, ith G e = L + I. We kno that G e, in analogy ith G e, has an exponential tail, i.e., E(e θge ) < for all θ >. In other ords P (G e > y) = O(exp( θy)). The other particularity is that G e and B are dependent. First e hae the inequality for all y P (G e + B > y) P (B > y), therefore, e hae P (T > t) A(λ e, ) (t2) λe λ ( + o()). Second, e hae the other inequality for all (y, z): P (G e + B > y) P (G e > z) + P (B > y z). Thus, by selecting z = O(log t) such that P (G e > z) = o(t λe λ ), e get P (T > t) A(λ e, ) (t O(log t)2) λe λ + o(t λe λ ) Therefore, e conclude the proof of Theorem 2. B. Traeled Distance Distribution No, e focus on the reneal process made of the arious aiting time interals T, experienced by the information beacon. Considering the sequence of aiting phases, T, T 2,..., T n,...: the beacon moes at time T, then at time T +T 2, T +T 2 +T 3, etc. We denote n(t) the number of phases achieed before time t: i=n(t) i= T i t < i=n(t)+ i= T i. Since the T i are independent, this is a reneal process, and e hae the identity: P (n(t) n) = P (T + + T n t). We can get a precise estimate of the aerage number of reneals E(n(t)) during time t. Lemma. There exists b > such that, hen t, the folloing estimate is alid: λ ) E(n(t)) = sin2 (π λe bπ 2 Γ( λ e λ )t λe λ + O(t 2 λe λ ). (3) Proof: See appendix. In parallel to the sequence of aiting interals {T i } i, e hae the sequence {D i } i the distances traeled by the beacon after eery aiting interal T i. No e denote L(t) = i=n(t) i= D i hich is the total distance traeled by the beacon until time t. Lemma 2. E(L(t)) = E(n(t))E(D). Proof: We hae the identity: E(L(t)) = n> E( n(t) n D n ), (32)

7 here n(t) n is the indicatie function of the eent n(t) n. Since, from the definition of n(t), e hae the equialence: n(t) n T + +T n < t, then n n(t) and D n are independent random ariables and, therefore, E( n(t) n D n ) = P (n(t) n)e(d). Quantity E(D) has a closed expression (E(D) = λ, ef (λ e) from Corollary 2. Substituting and using Lemma, the poer la for E(L(t)) in Theorem 3 is shon. A. Near the Threshold V. ASYMPTOTIC ESTIMATES First, e inestigate the case here (λ e, ) is close to the threshold boundary. Corollary 3. When λ e (λ ) + : p 2 ( λ ) λ 2 e( λ )λ (λ e λ )e λ +λe 2λ. (33) Proof: Using Lemma 6, e hae: 2E(T) = f (λ e ) f (λ e ) + e λ E(B (x))p e (x)dx ( λ )λ λ e λ λ e (x )λ λ e f (λ e ) e λex dx. Integrating, e obtain p = E(D) E(T). B. Large Densities Corollary 4. When the ehicle densities become large, i.e., λ e, : p 2 e λe+λ + λ λ e Proof: According to Lemma 4, e hae: E(L ) = + + λe. (34) λ e ( + λ e ), (35) and the expected gap length tends to. The aerage road length to bridge such a gap tends to. From Lemma 6: 2E(T) E(L ) +. (36) From corollary 2, the aerage distance traeled in bridging is: E(D) = λ e f (λ e ) eλe+λ λ e +, (37) and e obtain p = E(D) E(T). Note that the information propagation speed gros quasiexponentially ith respect to the total ehicle density. VI. FINITE RADIO PROPAGATION SPEED If the radio propagation speed (including store and forard timings) is finite and constant, equal to r, then the aerage information propagation speed becomes: s = E(T ) + E(D e ) E(T ) + r E(D e ). (38) But the speed r impacts the random ariables G e and T. The main impact is that, to fill an eastbound gap of length x, one needs a estbound cluster of length at least x(+γ) γ ith γ = r, otherise the message ill be off the gap hen arriing at the end of the cluster. Thus, E(T (x)) = O(e (+γ)θ γ x ). The gap length is also modified, since e must consider γ +γ L : E(e θge ) = f ( γ +γ (θ + λ e)) λ e f ( γ +γ λ e) λ e + θ. (39) But, this does not change the exponential term e λex in the asymptotic expression of p e (x). Therefore, the threshold condition becomes: e λ = γ + γ λ ee γ +γ λe. (4) +γ This corresponds to a dilatation by a factor γ of the horizontal axis in the diagram of Figure 2. Notice ho the diagram then loses its symmetry ith respect to λ e ersus, hich can be obsered in Figure 9. Notice also that, hen r decreases to, the threshold limit tends to infinity. Similarly, in correspondance to Theorem 3, e hae E(L(t)) = Ω(t γ λe +γ +γ γ. λ ), due to the dilatation by the factor VII. SIMULATIONS In this section, e present simulation results obtained ith Maple on one hand, and the Opportunistic Netork Enironment (ONE [4]) simulator on the other hand. A. Maple Simulations We first compare the theoretical analysis ith measurements performed using Maple. In this case, the simulations follo precisely the bidirectional highay model described in Section II: e generate Poisson traffic of eastbound and estbound traffic on to opposite lanes moing at constant speed, hich is set to = m/s. The radio propagation range is R = m, and radio transmissions are instantaneous; the length of the highay is sufficiently large to proide a large number of bridging operations (of order at least 3 ) for all considered traffic densities. We measure the information propagation speed hich is achieed using optimal DTN routing, by selecting a source and destination pairs at large distances, taking the ratio of the propagation distance oer the corresponding delay, and aeraging oer multiple iterations of randomly generated traffic. We ary the total traffic density, and e plot the resulting

8 Fig. 9. Threshold zones for (λ e, ) for arious γ = r, γ = red, γ =. blue, γ =.5 green, γ =.8 yello. Fig.. Maple simulations. Information propagation speed p for λ e =, ersus the total ehicle density λ e +. information propagation speed. Figures and sho the eolution of the information propagation speed near the threshold ersus the total ehicle density, hen λ e =, in linear and semilogarithmic plots, respectiely. We can obsere the threshold at λ e + = 2 in Figure, hich confirms the analysis presented preiously in Section III, and corresponds to λ e = = in Figure 2. In semilogarithmic scale (Figure ), e obsere that the simulation measurements quickly approach a straight line, and therefore are close to the theoretically predicted exponential groth aboe the phase transition threshold, in Section V. In Figure 2, e present a 3-dimensional plot of the eastbound information propagation speed p by arying the ehicle densities in both eastbound (λ e ) and estbound ( ) traffic. Finally, e perform detailed measurements of the aiting time T that each packet of information spends in the buffer of an eastbound car until it encounters a estbound cluster hich allos it to propagate faster to the next eastbound car. For the measurements, e set the ehicle densities to λ e = =.9; thus the conjugate λ = In Figure 3, (i.e., belo the phase transition threshold), e plot the distribution of the aiting time T, and e compare it to the predicted poer la in Theorem 2: P (T > t) = t λe λ. B. ONE Simulations In this section, e depart from the exact Poisson model simulations in Maple, and e present simulation results obtained ith the Opportunistic Netork Enironment (ONE [4]) simulator. Vehicles are distributed uniformly on the length of both lanes of a road, and moe at a constant unit speed. The total Fig.. Maple simulations. Information propagation speed p for λ e =, ersus the total ehicle density λ e +, in semi-log scale, compared to the theoretically predicted asymptotic exponential groth.

9 speed Le+L Fig. 4. ONE simulations. Information propagation speed for λ e =, ith respect to λ e +. Fig. 2. Maple simulations. Eastbound information propagation speed p for different alues of the ehicule densities, λ e and. Fig. 3. Maple simulations. Cumulatie probability distribution P (T > t) of the aiting time T, compared to the poer la t λe λ, for λ e = =.9. number of ehicles aries from to 5. Similarly to the preious section, e measure the fastest possible information propagation speed hich is achieed using epidemic broadcast, assuming that radio transmissions are instantaneous and that there are no buffering or congestion delays, ith a radio range R = m. Again, e ary the ehicle densities λ e and, hich are gien in ehicles per radio range, and e perform seeral simulation iterations of randomly generated traffic. As shon in Figure 4, similarly to the Maple simulation results and to the analysis, e obsere the threshold phenomenon at λ e = = : the information propagation speed remains almost constant belo the threshold but increases dramatically beyond it. We also obsere an exponential groth aboe the threshold in Figure 5. We remark that measurements belo the phase transition threshold yield an aerage information propagation speed hich is slightly larger than the ehicle speed. This is due to the finite duration of the simulations and in the computations of the expectations. This phenomenon can also be explained from the theoretical analysis in Section IV: een belo the threshold, DTN routing using cars moing on both directions still proides a gain in the propagation distance, hich follos a sub-linear poer la ith respect to the elapsed time (in the referential of the moing cars). Figure 6 confirms the predicted poer la t λe/λ for E(L(t)), the aerage distance traeled by information ith respect to time (in Theorem 3), hich is shon in Figure 6 for λ e = and =.9, therefore, a groth of order t.93. VIII. CONCLUDING REMARKS In this paper, e proided a detailed analysis for information propagation in bidirectional ehicular DTNs. We proed the existence of a threshold, concerning ehicle density, aboe hich information speed increases dramatically oer ehicle speed, and belo hich information propagation speed

10 speed is on aerage equal to ehicle speed (in Theorem ), and e computed the exact expression of this threshold. We exactly characterized the information speed near the threshold (Corollary 3), and e shoed that, aboe the threshold, the information propagation speed increases quasi-exponentially ith ehicle density (Corollary 4). We also analyzed in detail the ay information propagates under the threshold, and e shoed that DTN routing using bidirectional traffic proides a gain in the propagation distance, hich follos a sub-linear poer la ith respect to the elapsed time (Theorems 2 and 3). Combining all these different situations, e obtain a complete image of the ay information propagates in ehicular netorks on roads and highays, hich is useful in determining the performance limits and designing appropriate routing protocols for VANETs. All our theoretical results ere alidated ith simulations in seeral enironments (The One and Maple). Our analysis can be extended to inestigate other models of ehicle traffic and radio propagation. In future orks, e intend to proide a detailed expression of this threshold in specific VANET models (e.g., intersections). Finally, an interesting direction for further research consists in collecting large traces of real traffic on roads and highays, and ealuating the information propagation properties in this context. REFERENCES Le+L Fig. 5. ONE simulations. Information propagation speed for λ e =, ith respect to λ e +, in semi-log scale, compared to the theoretically predicted asymptotic exponential groth. distance / t Le/L* time Fig. 6. ONE simulations. Aerage distance traeled by information ith respect to elapsed time E(L(t)), diided by the predicted poer la t λe/λ for λ e = and =.9, compared to the constant alue.8 (dash). [] A. Agaral, D. Starobinski and T. Little, Phase Transition Behaior of Message Propagation in Delay Tolerant Vehicular Ad Hoc Netorks. MCL Technical Report No , 28. [2] A. Agaral, D. Starobinski and T. Little, Analytical Model for Message Propagation in Delay Tolerant Vehicular Netorks. Proceedings of the Vehicular Technology Conference (VTC). Singapore, 28. [3] A. Agaral, D. Starobinski and T. Little, Exploiting Mobility to Achiee Fast Upstream Propagation. Proceedings of Mobile Netorking for Vehicular Enironments (MOVE) at IEEE INFOCOM. Anchorage, 27. [4] E. Baccelli, P. Jacquet, B. Mans and G. Rodolakis, Information Propagation Speed in Bidirectional Vehicular Delay Tolerant Netorks, Infocom, 2. [5] J. Burgess, B. Gallagher, D. Jensen and B. Leine, MaxProp: Routing for Vehicle-Based Disruption-Tolerant Netorks. In Infocom, 26. [6] A. Casteigts, A. Nayak and I. Stojmenoic, Communication protocols for ehicular ad hoc netorks. Wireless Communications and Mobile Computing, [7] V. Cerf, S. Burleigh, A. Hooke, L. Torgerson, R. Durst, K. Scott, K. Fall and H. Weiss, Delay-Tolerant Netorking Architecture, IETF Request For Comment RFC 4838, Internet Engineering Task Force, 27. [8] F. De Pellegrini, D. Miorandi, I. Carreras and I. Chlamtac, A Graphbased model for disconnected ad hoc netorks, Infocom, 27. [9] P. Flajolet, A. Odlyzko, Singularity analysis of generating functions, SIAM Journal Disc. Math.,Vol. 3, No. 2, pp , May 99. [] R. Fracchia and M. Meo, Analysis and Design of Warning Deliery Serice in Inter-ehicular Netorks. IEEE Transactions on Mobile Computing. 28. [] M. Grossglauser and D. Tse, Mobility increases the capacity of ad hoc ireless netorks, Infocom, 2. [2] P. Gupta and P. R. Kumar, The capacity of ireless netorks, IEEE Trans. on Info. Theory, ol. IT-46(2), pp , 2. [3] P. Jacquet, B. Mans and G. Rodolakis, Information propagation speed in mobile and delay tolerant netorks, Infocom, 29. [4] A. Keranen, J. Ott and T. Karkkainen, The ONE Simulator for DTN Protocol Ealuation. SIMUTools 9: 2nd International Conference on Simulation Tools and Techniques. Rome, 29. [5] Z. Kong and E. Yeh, On the latency for information dissemination in Mobile Wireless Netorks, MobiHoc, 28. [6] T. Nadeem, P. Shankar, and L. Iftode, A Comparatie Study of Data Dissemination Models for VANETs, in Proc. of MOBIQUITOUS, 26. [7] Y. Toor, P. Mühlethaler, A. Laouiti, A. de la Fortelle, Vehicle ad hoc netorks: Applications and related technical issues. IEEE Communications Sureys and Tutorials (-4), pp (28) [8] US-Department of Transportation. National Transportation Statistics, 2. US Goernment printing Office, Washington, DC. [9] N. Wisitpongphan, F. Bai, P. Mudalige, and O. Tonguz, On the Routing Problem in Disconnected Vehicular Ad-hoc Netorks, in Infocom, 27. [2] H. Wu, R. Fujimoto, and G. Riley, Analytical Models for Information Propagation in Vehicle-to-Vehicle Netorks, in Proc. Fall VTC 4, Los Angeles, CA, USA, 24. [2] S. Yousefi, E. Altman, R. El-Azouzi and M. Fathy, Analytical Model for Connectiity in Vehicular Ad Hoc Netorks. IEEE Transactions on Vehicular Technology. No. 28. [22] X. Zhang, G. Neglia, J. Kurose and D. Tosley, Performance modeling of epidemic routing, Computer Netorks, Vol. 5, 27.

11 A. Proof of Lemma 9 APPENDIX Proof: We ant to find an ealuation of P (T > y) hen y. For this e ill ealuate for x gien P (B (x) > y). We kno that E(e θb(x+) ) = β (θ, x + ) = P (L > x) θ (f (θ, x + ) ) Since P (L > x) = α λ e λ x ( + o(e εx )), ith α = ( λ )eλ λ λ, e hae f (θ, x) = f (θ) αe λ x θ + λ ( + o(e εx )) (4) From here e drop the o(e εx ) for simplicity, since it ill just bring an exponentially small factor. Therefore e hae β (θ, x + ) = λ We hae P (B (x + ) = y) = 2iπ αe λ x θ (f (θ) αe λ x θ+λ ) β (θ, x + )e yθ dθ. (42) Let θ(x) be the root of θ (f (θ) αe λ x θ+λ ). Straightforard analysis gies θ(x) = βe λ x +O(e 2λ x ), ith β = θ f() α λ P (B (x + ) = y) = αe λ x. Via singularity analysis e hae Omitting the O( ) terms e get Or λ e θ(x)y θ f (θ(x)) +O(e (θ(x) ε)y ) αe λ x (θ(x)+λ )2 P (B (x + ) = y) = αe λ x e βe λ x y λ θ f () = βe θx exp( βe λ x y), P (B (x + ) > y) = exp( βe λ x y). (43) Therefore stating P (B > y) = p e (x)p (B (x) > y)dx e get, omitting O( ) terms λ e P (B > y) = f (λ e ) e (x+)λe exp( βe λ x y)dx. (44) ith the change of ariable u = e λ x e get λ e e λe P (B > y) = λ f (λ e ) u λe λ exp( βyu)du ( ) λ e e λe = λ f (λ e ) Γ λe (βy) λe λ, hich is in poer la as claimed. λ B. Proof of Lemma We first proe the property for an hypothetic reneal process based on the B i s on the real line. Let b(x) be this reneal process at time t: E(b(y)) = P (B < y) + P (B + B 2 < y) + + P (B + + B n < y) + Let us define β = E(e θb ). We also define N B (θ) = E(b(y))e yθ the Laplace transform of E(b(y)). We hae: N B (θ) = B(θ) θb(θ) Lemma 3. When θ, then B(θ) = b πθa sin(πa) and a = λe λ. Proof: We start from and B(θ) = We denote β(θ, x + ) = P (L > x + ) θ (f (θ, x + ) ), p e (x)β(θ, x)dx = β(θ, x) = θ g(θ, x), θ + θ(x). for some b p e (x)( β(θ, x))dx. ith g(θ, x) bounded and uniformly integrable hen R(θ) remains in a compact set and x. Indeed e hae Thus B(θ) = g(θ, x) = + O(θ(x)) θ θ + θ(x) g(θ, x)p e(x). By change of ariable y = θ(x) e get B(θ) = θ() θ θ + y g(θ, θ (y))p e (θ (y))(θ (y)) dy. We use together the folloing estimates θ(x) = βe λ x + O(e 2λ x ), and p e (x) = λe f (λ e) e λex ( + O(e εx )) to state p e (θ (y))(θ (y)) = for some ε >. Then, e use the fact that and, hen θ, λ e f (λ e ) ( y β ) λe λ ( + O(y ε )), y a πθa dy = θ + y sin(πa), θ() y a dy = O() θ + y Therefore, B(θ) = b πθa λe sin(πa) + O(θ) for some b and a = λ hen θ. This is also true for complex θ. We can no proe Lemma for the original reneal process.

12 Proof: From the preious lemma, e hae that: N B (θ) = sin(πa) bπ θ a + O(θ + θ 2a ), hen θ. The inersion of the Laplace transform yields E(b(y)) = 2iπ c+i c i N B (θ)e θy dθ. for any c >. e bend the integration path so that it resemble the path described in figure 7 ith c <. Since far from its singularities on the line θ(x) for x > (hich corresponds to the negatie real axis), the functions β(θ, x) is uniformly in θ, therefore N B(θ) = O( θ ) and the integral 2 on the ertical part gies a bounded contribution (in fact exponentially decreasing in exp(c y)). Fig. 7. c ʹ The Flajolet-Odlyzko cranted integral path. The cranted part of the integral, using the Flajolet Odlyzko theorem on continuous functions [9], gies a contribution hich is sin 2 (πa) bπ 2 Γ( a)t a + O( + t 2a ). Therefore, e obtain that, for y : E(b(y)) = sin2 (πa) bπ 2 Γ( a)y a + O( + y 2a ). To terminate the argument e ill proe that E(n(t)) = E(b(2t)) + o(). We hae 2T = B + H, here H = L +I, and H has distribution ith an exponentially decreasing tail. Hoeer, e stress the fact that B and H are not independent. We denote H i the successie alues of the H. We hae therefore 2T i = B i + H i. Since T i 2B i, e already hae E(n(t)) E(b(2t)). We also hae for any t [, t] and for any integer i: c Since log E(e θh ) = θe(h) + O(θ 2 ), it is sufficient that x k E(H) > γ, to find θ > such that log P (H + + H i > x) < kγθ. On the other side, e also hae y > : P (B + +B i < y) i k P (B + +B i < y) = E(b(y)) i>k We hae P (B + + B i < y) (P (B < y)) i We use the fact that P (B > y) Ay a for a = λe λ and for some A > to state that P (B + + B i < y) exp( iay a ) and, finally, P (B + + B i < y) exp( kay a ) Ay a. ith i>k Collecting all results yields: E(n(t)) E(b(2(t t ))) G k (2t, 2t, θ), G k (x, y, θ) = exp( ka(y x) a ) A(y x) a + ke(exp(kθh xθ)). If e take t = t a ith > a > a, k = 2t γ+e(h) = Ω(ta, then e hae G( k (2t, 2t, θ) exp( k(2(t t )) a ) A(2(t t )) α + k exp( kγθ) hich is exponentially small hen t. Therefore, E(n(t)) E(b(t)) + o(). P (T + + T i < t) P (B + + B i < 2(t t )) P (H + + H i > 2t ). Using Chernoff bounds, e hae for all θ > such that E(e θh ) exists, i k and x > : P (H + + H i > x) E(exp(kθH xθ))