Analysis of Multi-Hop Emergency Message Propagation in Vehicular Ad Hoc Networks

Transcription

1 Analysis of Multi-Ho Emergency Message Proagation in Vehicular Ad Hoc Networks ABSTRACT Vehicular Ad Hoc Networks (VANETs) are attracting the attention of researchers, industry, and governments for their otential of significantly increasing the safety level on the road. In order to understand whether VANETs can actually realize this goal, in this aer we analyze the dynamics of multiho emergency message dissemination in VANETs. Under a robabilistic wireless channel model that accounts for interference, we derive lower bounds on the robability that a car at distance d from the source of the emergency message correctly receives the message within time t. Besides d and t, this robability deends also on 1-ho channel reliability, which we model as a robability value, and on the message dissemination strategy. Our bounds are derived for an idealized dissemination strategy which ignores interference, and for two rovably near-otimal dissemination strategies under rotocol interference. The bounds derived in the first art of the aer are used to carefully analyze the tradeoff between the safety level on the road (modeled by arameters d and t), and the value of 1-ho message reliability. The analysis of this tradeoff discloses several interesting insights that can be very useful in the design of ractical emergency message dissemination strategies. 1. INTRODUCTION Vehicular Ad Hoc Networks (VANETs) have recently attracted the attention of researchers, automotive industry, and governments, for their otential of imroving driver s awareness of the surrounding environment through infrastructureless, vehicle-to-vehicle (V2V) wireless communications. The major driving factor for the investigation and deloyment of this new technology is safety: by collecting accurate and u-to-date information about the status of the surrounding environment, a driver assistance system can quickly detect otentially dangerous situations, and notify the driver about the aroaching eril. Since a relatively small reduction in the driver resonse time (in the order of a fraction of a second) can result in avoiding an accident, driver assistance systems based on V2V communications have the otential of significantly imroving the safety level on the road. While imroving the safety level on the road is unanimously considered the major driving factor for the deloyment of VANETs (see, e.g., [6]), the challenges related to im- Permission to make digital or hard coies of all or art of this work for ersonal or classroom use is granted without fee rovided that coies are not made or distributed for rofit or commercial advantage and that coies bear this notice and the full citation on the first age. To coy otherwise, to reublish, to ost on servers or to redistribute to lists, requires rior secific ermission and/or a fee. Coyright 200X ACM X-XXXXX-XX-X/XX/XX...$5.00. lementing safety alications (e.g., collision avoidance system, hazard warning, etc.) are the most difficult to solve in the area of vehicular networking. In fact, safety alications must rely on very accurate and u to date information about the surrounding environment, which, in turn, requires the use of accurate ositioning systems and smart communication rotocols for exchanging information. These smart communication rotocols should guarantee fast and reliable delivery of information to all vehicles in a neighborhood, in a network environment in which the communication medium is shared, highly unreliable, and with limited bandwidth. It is clear from the above descrition that a fundamental building block of any safety alication for VANETs is a reliable and time constrained 1-ho broadcast rimitive, which is not currently included in the emerging DSRC (Dedicated Short Range Communications) standard [5] (a variation of IEEE a). A recent study has shown that the DSRC standard rovides adequate erformance for what concerns delay, but it is defective in terms of reliability [17]. This exlains the many recent roosals for modifications of the MAC layer aimed at imroving either reliability, or reducing delay, or both, of 1-ho broadcast [2, 14, 15]. Another imortant building block of safety alications is rioritization of messages: safety alications will coexist with non-safety alications such as traffic information systems, commercial services, and with less critical safetyrelated messaging (beaconing). Although a searate control channel is used in DSRC for safety-related alications, these alications still have to cont the channel in case cars are equied with a single radio (as it will tyically be the case). As a consequence of this, it is imortant to ensure that, when a safety alication needs to access the channel, it gets a higher riority than non-safety related or less critical safety-related messages. Prioritization of safety-related messages is investigated in [14, 11, 12, 13]. In this aer, we are concerned with one secific safety alication: hazard warning dissemination 1. In this tye of alication, a certain vehicle issues a hazard warning message (also called emergency message in the following) in case a dangerous situation is detected (e.g., obstacle on the road, airbag exlosion, malfunctioning of the braking system, and so on). This emergency message should be roagated backward on the road as quickly and reliably as ossible, in order to enable the drivers of aroaching vehicles to undertake adequate countermeasures. Fast backward roagation is needed because the information contained in the emergency message has a very limited lifetime (i.e., it is useful only if received within a short time from the hazard warning incetion); reliable roagation is needed because a single vehicle missing the warning can become extremely dangerous for the 1 Indeed, the study reorted in this aer is relevant to all safety alications which require fast and reliable (multi-ho) dissemination of emergency messages.

2 other vehicles. Observe that, since the tyical transmit range for reliable communication in VANETs is in the order of a few hundred meters, multi-ho emergency message dissemination is needed in this alication scenario to increase the area of hazard warning delivery. The main goal of this aer is to investigate the fundamental tradeoff between safety-level and emergency message resource wastage encountered in hazard warning alications. A common feature of several techniques roosed in the literature to imrove reliability of 1-ho emergency message broadcast is allowing the tuning of the amount of bandwidth reserved for emergency message dissemination (e.g., by setting the number of MAC layer reetitions [14], or the transmit ower level [12, 13]): by devoting more resources to safety-related message dissemination, reliability of 1-ho broadcast can be increased. On the other hand, this comes at the exense of less available bandwidth for non safety-related alications, which is reduced accordingly. To the best of our knowledge, the question of how to adequately address this tradeoff has been evaded in current literature. Note that a major difficulty in the investigation of the above described tradeoff is analyzing the relation between 1-ho reliability (which is relatively easy to estimate see [11, 13]) and multi-ho, time-constrained reliability. In fact, multi-ho, time-constrained reliability is influenced by several factors (including, of course, 1-ho reliability), such as network toology, channel characteristics, interference, message dissemination strategy, and so on. As a first ste towards an in-deth investigation of this tradeoff, in this aer we fix the network toology, channel characteristics and interference model, and we study the deendence between 1-ho and multi-ho, time-constrained reliability under various message dissemination strategies. Our study uses a robabilistic aroach, and includes time constraints on hazard warning recetion. More secifically, we estimate the robability P ( d, t, ) that a vehicle located at distance d from the hazard warning initiator correctly receives the warning within time t, where is a arameter modeling 1-ho reliability. Based on this estimation, we gain several insights on the safety-level / emergency message resource wastage tradeoff. In articular, we find that the relative advantage of having an increased 1-ho reliability value (which might come at the exense of considerable resource wastage) tends to decrease as the distance from the emergency message initiator increases. Furthermore and more interestingly, the efficacy of increased 1-ho reliability values dislays a clear deendence on the traffic conditions: under heavy traffic conditions, relatively low values of 1-ho channel reliability can be used without considerably imacting multi-ho reliability; on the contrary, under light traffic conditions high values of 1-ho channel reliability turn out to be very useful to imrove multi-ho reliability. We also find that the emergency message dissemination strategy has a major imact on P ( d, t, ). Finally, we have verified through simulations that our derived bounds on P ( d, t, ) are qualitatively very accurate. To the best of our knowledge, the one reorted in this aer is the first attemt to systematically analyze timeconstrained, multi-ho reliability in vehicular ad hoc networks. A few other aers in the literature are concerned with multi-ho message roagation in VANETs, but they are either not concerned [3, 8, 10] or only artially concerned [16] with delay in message roagation. 2. NETWORK AND CHANNEL MODEL In this section, we introduce the network and channel model used in our analysis of multi-ho emergency message roagation in VANETs. Due to the inherent difficulties of this Probability of Message Recetion Prob. of Message Recetion Distance [m] Figure 1: Examle of how our channel model can be used to lower bound message recetion robability. Actual message recetion robabilities are comuted assuming a nominal communication range of 1000m, and Nakagami radio roagation (courtesy of M. Torrent-Moreno). analysis, we use a simlified network and channel model, which, however, cature relevant VANET features. We consider a vehicular ad hoc network in which nodes (cars) are equally saced along a line, where d is the searation distance between consecutive cars. When a car transmits a message, all nodes within distance r T (transmit range) from the sender have the same robability (0 < < 1) of correctly receiving the message in absence of interference. To ease resentation, in the following we normalize the transmit range and all the other relevant distances with resect to the inter-vehicle distance d (i.e, we assume d = 1). Since we are concerned with life critical message dissemination, which is of utmost imortance to all cars on the road, we assume that all messages are sent in broadcast mode. Note that the above channel model, although very simle, catures the most relevant feature of wireless transmission, i.e. uncertainty about correct message recetion. A very similar channel model is used in [16] to estimate the erformance of a cooerative collision warning rotocol. As shown in Figure 1, the inaccuracy of our channel model with resect to more realistic ones goes in the right direction, i.e., it underestimates correct message recetion robability. How to extend the analysis reorted in this aer to more accurate robabilistic channel models is matter of ongoing research. To model interference between concurrent transmissions 2, we use the rotocol model introduced in [7]: for a given node u, a message is correctly received by u with robability if and only if there exists a transmitter within distance r T from u, and no other node within distance r I r T (r I is the interference range) from u is concurrently transmitting. To analyze message roagation, we use the concet of token: initially, only the rightmost car in the network (the one ahead of the cloud of cars, which is called the initiator in the following) has the token (hazard warning to disseminate); the token is then disseminated in the network through a sequence of (ossibly concurrent) 1-ho broadcasts; when a node first receives the emergency message, it gets a coy of the token, and it becomes a otential forwarder of the token. To simlify the analysis, we assume that token roagation roceeds in rounds, and that transmissions are carefully scheduled in each round to seed-u message roagation as much as ossible. We are interested in estimating, for a given node u at a certain distance d from the initiator, the robability P ( d, t, ) 2 In this aer, we are not concerned with external interference.

3 that u gets the token within time t. Note that the above robability clearly deends on the distance d from the initiator, on the reference time t, and on the robability of correctly receiving a 1-ho broadcast message in absence of interference. The goal of our analysis is to gain insights on the interdeendence of these three arameters, which reresents the fundamental tradeoff between safety level on the road and resources devoted to the emergency message dissemination mechanism: by tuning arameters d and t, we can set our required safety level, which can be exressed as a set of (distance, time) airs (e.g., cars at distance 200m from the dangerous event should receive the message within 0.5sec, cars at distance 300m should receive the message within 0.8sec, and so on). On the other hand, the value of arameter is determined by the amount of wireless resources (bandwidth) which are allocated to the emergency message dissemination mechanism: the more resources are allocated for emergency message dissemination, the higher the value of. Note that most of the techniques roosed in the literature to realize reliable emergency message delivery allow to control the amount of consumed wireless resources, e.g. by changing the transmit ower level [12, 13], or by changing the number of MAC-layer reetitions of a message [14]. Since allocating resources to emergency message dissemination necessarily reduces the available resources for other VANET services (traffic reort, commercial services, and so on), the network designer is interested in finding the minimum value of that guarantees the required safety level on the road. To the best of our knowledge, the study reorted in this aer is the first attemt to carefully analyze this fundamental tradeoff in vehicular ad hoc networks. 3. THEORETICAL ANALYSIS In the following, we denote a set of n equally saced cars on a straight line road as a string S n of bits, where the least significant bit corresonds to the initiator, and the i-th bit to the i-th car in the cloud. In the following, we assume that cars are numbered from 0 (the initiator) to n 1, and are ordered accordingly. Without loss of generality, we assume that the initiator is the rightmost node in the cloud of cars. We also use notation i, j,... to denote both a generic node in the network and its osition in the cloud of cars. Bits encode whether a certain car has received the token (bit value is 1), or not (bit value is 0). In the following, we call a node holding the token a 1-node, and a node which has not yet received the token a 0-node. The value of S n evolves with time, as the token is roagated throughout the network. We exress this deendence with time with notation S n(t), where t denotes the round index. We assume that initially only the initiator holds the token, i.e., S n(0) = During the first round, the initiator transmits the message, and all nodes within distance r T (i.e., within r T ositions) from it become a 1-node with robability. At the generic round k, a certain number of transmitters is selected among the 1-nodes, and some of the 0-nodes (which are determined according to the radio channel and interference model) become a 1-node with robability. The goal of our analysis, re-stated in terms of bits and rounds, is to estimate the robability that a certain bit turns into a 1 within a certain number t of rounds. It is clear that this robability deends also on the strategy used to select transmitters at each round. In this aer, we are interested in characterizing the behavior of the emergency message roagation rocess under the best ossible conditions. This characterization is useful in understanding which is the best ossible achievable safety level for a certain value of the robability of correct message recetion, thus serving as a guideline to address the safety level / emergency message resource wastage tradeoff mentioned in the revious section. Studying life critical message roagation under the best ossible conditions requires identifying an otimal strategy for selecting the set of transmitters at each round, which is a very difficult roblem. To circumvent this roblem, in the following we define a simle multi-hase strategy to select transmitters at each round, and we formally rove that this strategy is within a constant factor from the otimal strategy. 3.1 Dissemination strategies Before resenting the strategy and roving the aroximation bound, we need the following definition: Definition 1 (Internal 0-nodes). Let i be the i-th car in the cloud, and assume i is a 0-node. Node i is said to be an internal 0-node at time t if and only if there exists a 1-node in S n(t) in osition j, with j > i. The strategy used to select the transmitters set at each round is reorted in Figure 2. We call this strategy Global, because it is a centralized strategy that uses global knowledge: the set of transmitters for round h and h + 1 (or h, h + 1, h + 2) is comuted by knowing the comlete status of S n(t) at time h 1. The strategy roceeds in stages, where every stage is comosed of a certain number of rounds, deending on the relative value of r I with resect to r T. A stage takes as inut string S n(t), and oututs sets T 1,..., T i of transmitters at round t + 1,..., t + i. We have i = 2 if r I = r T, and i = 3 if r T < r I 2r T. In the first round of the stage, the leftmost 1-node (call it node k) is selected to transmit, and a number of other 1-nodes are selected for concurrent transmission according to a greedy rule designed to avoid interference. At the end of the first round, nodes are marked as covered if they are within distance r T from one of the selected transmitters, and as uncovered otherwise. Then, grous G z of consecutive uncovered nodes of cardinality at most r T are formed by scanning all nodes starting from the right. The second round of the stage differs slightly deending on the value of the interference range. If r I = r T, all grous are scanned from right to left and, for every considered grou G z, a 1-node among the nodes in G z is scheduled for transmission and included in T 2. If all nodes in G z are 0-nodes, the closest 1-node j to the left of G z is scheduled for transmission and included in T 2. In case r T < r I 2r T, grous are alternately marked as 2 or 3 at the beginning of the round, and the above described rocedure is alied to the grous marked as 2. Round 3, which is executed only if r T < r I 2r T, is equal to round 2, excet that transmitters are selected only for grous marked as 3 in the revious round. We now rove that Global is within a constant factor from the otimal roagation strategy. Since we are dealing with a stochastic rocess, we need an aroriate definition of otimal strategy, and of aroximation bound for nonotimal strategies. We start observing that our goal is ensuring fast and reliable backward roagation of emergency messages in the cloud of cars: with fast we mean that the token should be roagated backward as quickly as ossible; with reliable, we mean that backward roagation should not come at the exense of leaving many uncovered nodes between the initiator and the front of the emergency message roagation area. Given this observation, consider string S n(t) at the generic round t, and denote by k the leftmost 1-node in S n(t). It is immediate that an otimal strategy cannot do better than the stochastic rocess that at every round i) turns every internal

4 Inut: Sn(t) Outut: set T h of transmitters for round t + h Bit(Sn(t), i) returns the i-th bit of Sn(t) Array T ag[ ] stores nodes tags Round 1 k = leftmost 1-node in Sn(t) for i = 1 to k do T ag[i] = uncovered T 1 = {k} i = k (r T + r I + 1) reeat while Bit(Sn(t), i)=0 do i = i 1 end while T 1 = T 1 {i} i = i (r T + r I + 1) until i > 0 for each i T 1 do for j = 0 to r T do T ag[i + j] = covered T ag[i j] = covered z = 1 Gz = i = 0 while i < k do if Tag[i] = uncovered then Gz = {i} j=1 while (T ag[i + j] = uncovered) (j < r T ) do Gz = Gz {i + j} j = j + 1 end while h = h + 1 Gz = i = i + j else i = i + 1 end while Round 2 (case r I = r T ) T 2 = for each Gz do if exists 1-node in Gz then j = leftmost 1-node in Gz T 2 = T 2 {j} else i = leftmost 0-node in Gz find closest 1-node j with j > i T 2 = T 2 {j} Round 2 (case r T < r I 2r T ) T 2 = Alternately mark grous Gz as 2 and 3 for each Gz do if Gz is marked 2 then if exists 1-node in Gz then j = leftmost 1-node in Gz T 2 = T 2 {j} else i = leftmost 0-node in Gz find closest 1-node j with j > i T 2 = T 2 {j} Round 3 (Needed only if r T < r I 2r T ) T 3 = for each Gz do if Gz is marked 3 then if exists 1-node in Gz then j = leftmost 1-node in Gz T 3 = T 3 {j} else i = leftmost 0-node in Gz find closest 1-node j with j > i T 3 = T 3 {j} Figure 2: The Global strategy. r t r t xx0x0xxx1... Figure 3: Ensuring both roerties i) and ii) of the Idealized strategy might be infeasible due to interference: the leftmost 1 must transmit at each round to ensure roerty ii); as a consequence, the internal 0-node highlighted in gray has robability 0 (rather than ) of turning into a 1 at the next round. 0-node into a 1-node, indeendently, with robability at each node (this is because in our channel model a node has robability at most of receiving a message), and ii) to every node i such that i > k and i k r t assigns robability of turning into a 1. At a slight abuse of terminology, we shall call this rocess the Idealized strategy, even though a transmission strategy satisfying i) and ii) may not exist for certain S(n) (see Figure 3). Definition 2 (Aroximation factor). Let E(k(t)) be the exected value of the random variable k(t) denoting the osition of the leftmost 1-node in S n(t) under the Idealized dissemination strategy. A certain message dissemination strategy S is said to be within a factor (α, β) from otimal if and only if: i) at each round gives chance at least /α of turning into a 1 to all internal 0-nodes, and ii) E S(k(t)) E(k(t)), where E β S(k(t)) is the analogue of E(k(t)) under strategy S. Before roving the aroximation bounds for Global we need the following technical lemmas: Lemma 1. At any time t, there exist at most r T 1 consecutive internal 0-nodes in S n(t). r i r t Proof. Consider any internal 0-node, and assume it is in osition i. By definition, there exists at least one 1-node in osition j, with j > i. Among such nodes, consider the one that became 1 (received a token) at the earliest time, t, and denote it by j. We must have t t 1, since i is an internal node at time t. But j must have received the token at time t from some node h, with h < j (because j is the earliest node to the left of i that received a token), and h j r t. At time t both h and j have a token. Since j < i < h, and h j r t, i is in a run of at most r t 1 0-nodes. Lemma 2. Consider two random rocesses P 1 and P 2. Define P 1 to be the rocess in which a stage is comosed of h rounds, with h 1, and the robability for a node to correctly receiving the token at each stage is. Define P 2 to be the rocess in which a node has robability q 1 (1 ) h 1 of correctly receiving the token at each round. Denote by E 1(i) the event a node has correctly received the token in at most i stages, and with E 2(hi) the event a node has correctly received the token in at most h i rounds. For every > 0, we have P rob(e 1(i)) P rob(e 2(hi)) for each i 1. Proof. In general, the robability that a node receives the token in t or fewer stes is 1 (1 ) t, where is the robability of correctly receiving the token at each ste. We then have to rove that which can be rewritten as 1 (1 ) i 1 (1 q) hi, (1 ) i (1 q) hi, which holds whenever q 1 (1 ) 1 h. The ratio between and q = 1 (1 ) h 1 is a decreasing function of, whose minimum in the (0, 1] interval is obtained when = 1 (see Figure 4). Note that the value of

5 this ratio corresonds to the value of the constant α in the aroximation bound for the Global strategy roved in the next theorem. It is interesting to observe that α decreases as communication becomes more and more reliable (increasing values of ), indicating that our roosed strategy is very close to otimal for high values of. k(t) 1 r t r i+1 S (t) n S (t) n 3 Aroximation bound Figure 5: Notation used in the definition of the Im- Global strategy. 2,75 bound 2,5 2,25 2 1,75 h = 2 h = 3 Inut: Sn(t) Outut: set T h of transmitters for round t + h k(t) = leftmost 1-node in Sn(t) T i = set of transmitters for round i comuted by Global executed on S n (t) 1,5 1, ,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Figure 4: Aroximation bound α for the Global strategy as a function of. Plot h = 2 refers to the case r I = r T, lot h = 3 to the case r T < r I 2r T. T 1 = {k(t)} T 1 erform all transmissions in T 1 k(t + 1) = leftmost 1-node in Sn(t + 1) T 2 = {k(t + 1)} T 2 erform all transmissions in T 2 (these instructions are needed only if r T < r I 2r T ) k(t + 2) = leftmost 1-node in Sn(t + 2) T 3 = {k(t + 2)} T 3 erform all transmissions in T 3 Figure 6: The ImGlobal strategy. Theorem 1. Strategy Global is within a factor ( 2) from otimal if r I = r T, and within a factor ( from otimal if r T < r I 2r T. 1 (1 ) (1 ) 1 3, 3) Proof. Due to lack of sace, the roof of this theorem is omitted. We have observed at the beginning of this section that dissemination of emergency messages in VANETs should be fast and reliable. We can interret these design goals under a stochastic ersective as follows. Fast backward roagation of emergency messages is related to the exected one-ho advancement of a message in the backward direction, and to the osition of the leftmost transmitter in the cloud of cars. For a given value of and r T (and, hence, a given value of the exected one-ho message advancement), backward message roagation is sed u as much as ossible by selecting the leftmost 1-node in the cloud at each round (this is art of the Idealized strategy). On the other hand, reliable dissemination is maximized by giving internal 0-nodes the maximum ossible chance of turning into a 1 at each round (this is also art of the Idealized strategy). These two asects are interrelated as follows. Consider a certain node i in the network. Its chances of correctly receiving the emergency message are 0 until a node within distance r T from it has received the message. From that oint on, node i has chance at most of turning into a 1 at each round. If we use a multihase strategy similar to Global, by Lemma 2 this is equivalent to starting a geometric rocess of arameter at node i, where the starting time t i of the geometric rocess deends on the distance of the node to the initiator, and on the message dissemination strategy. Hence, the robability of node i to turn into a 1 within a certain time t is maximized when t i is minimized, and =. Since achieving fastest ossible backward message roagation and best ossible reliability at the same time is (in the general case) imossible (recall Figure 3), our goal as network designers is to find the best ossible balance between reducing t i and increasing as much as ossible. It is our strong feeling that, considering that is likely to be relatively high in realistic VANET scenarios (say, 0.75), the dominating factor in the above described stochastic, rocess is the starting time t i of the geometric rocess. In other words, we believe erformance is otimized when t i is reduced as much as ossible, and the best ossible value of is obtained after minimizing t i. Strategy Global is not very effective in minimizing t i: the leftmost 1-node in the cloud is scheduled for transmission only once every 2 or 3 rounds, resulting in a backward message roagation which is a factor 2 or 3 away from otimal (see Theorem 1). In the following, we resent an imroved version of Global, which we call ImGlobal, aimed at seeding u backward message roagation as much as ossible, while roviding the same reliability erformance as Global on all but a negligible fraction of network nodes. Before introducing strategy ImGlobal we need some notation. Denote by k(t) the leftmost 1-node in S n(t), and with S n(t) the string comosed of the rightmost k(t) r T r I 1 bits in S n(t) (see Figure 5). Also, denote by T i the set of transmitters as comuted by round i of Global when comuted on S n(t). Strategy ImGlobal is reorted in Figure 6. The strategy is very simle. Essentially, the leftmost 1-node is scheduled for transmission at each round, thus ensuring otimal backward emergency message dissemination. Reliability within a 1 (1 ) h 1 factor from otimal (h is either 2 or 3 deending on r I) is ensured on all but a negligible fraction of 0-nodes by executing Global on S n(t). The following theorem can then easily roved: Theorem 2. Algorithm imglobal is within a factor (1 (1 ) 1 h, 1) from otimal, where h = 2 if ri = r T, and h = 3 if r T < r I 2r T. The bound on internal nodes is satisfied on all but at most c = 3r T +r I +1 nodes, which is asymtotically negligible as n grows to infinity. Proof. The straightforward roof of this theorem is omitted. Note that constant c = 3r T + r I + 1 above corresonds to the maximum ossible backward advancement of the leftmost 1- node k (3r T ), lus a term r I + 1 to avoid interference. Concerning time comlexity of our roosed dissemination strategies, the following theorem can be easily roved:

6 Theorem 3. Algorithms Global and ImGlobal have O(n) time comlexity. 3.2 Estimation of time-constrained recetion robability In this section, we rovide bounds to the robability P ( d, t, ) that a node at distance d from the initiator correctly receives the token within t rounds of communication. In our analysis, we consider the three dissemination strategies described in the revious section, namely Idealized, Global, and ImGlobal. As observed at the end of the revious section, we can rovide bounds to P ( d, t, ) by bounding: i) the robability that a geometric rocess of a certain arameter starts at osition d within a certain time t d t, and ii) the robability of success in the geometric rocess within t t d rounds. In the following, we denote by Start( d, t) the event geometric rocess starts at node d within time t with Idealized dissemination, and with Succ(, h) the event a geometric rocess of arameter succeeds within h rounds with Idealized dissemination. In order to estimate i), we start by comuting the exectation and variance of the 1-ho advancement of a message. Lemma 3. Suose node k(t) is the leftmost 1-node in S n(t), and node k(t) transmits at round t + 1. Denote by k(t + 1) the leftmost 1-node in S n(t + 1), and with X(t) the random variable X(t) = k(t + 1) k(t). In absence of interference, we have that and E(X(t)) = E(, r T ) = r T (1 )r T +1, V ar(x(t)) = V (, r T ) = = (1 ) ˆ1 P rob(start( d, t)) = P rob(k( t) d) = P rob(k( t) > d 1). (2r T + 1)(1 ) r T (1 ) 2r T +1. The latter term can be rewritten as 2 P rob(k( t) > t E(, r T ) ( t E(, r T ) d + 1). Proof. The robability of maximal advancement (r t ositions) in one ste is equal to the robability that the node The roof follows immediately by setting λ = t E(, r T ) d+1 at distance r t receives the message, i.e.,. In general, the in the inequality of Theorem 4, and by observing that t > d 1 robability that the one-ho advancement is h < r T, is equal E(,r T imlies λ > 0. ) to the robability that the node at distance h receives the message (), times the robability that none of the r T h The robability of event Succ(, h) can be comuted exactly, using elementary roerties of the geometric rocess: farther nodes receives the message, i.e., (1 ) (r T h). Alying the formulas for mean (E(x) = P h hx h ) and variance (V ar(x) = E(x 2 ) E(x) 2 P rob(succ(, h)) = 1 (1 ) h. ) of a discrete random variable, we obtain We now have the tools to derive lower bounds on robability P ( d, t, ) for the different dissemination strategies. We r X T E(, r T ) = h (1 ) r T h = r T (1 )r T +1. start with strategy Idealized, which is easier to analyze. h=1 Denote by EStart( d, t) the event geometric rocess starts and at node d exactly at time t. We can write P ( d, t, ) as follows: V (, r T ) = r T X h=1 h 2 (1 ) r T h E(, r T ) 2 = = (1 ) ˆ1 (2r T + 1)(1 ) r T (1 ) 2r T Since E(, r T ) and V (, r T ) do not deend on the secific round t, we can aly well-known theorems about exectation and variance of a sum of indeendent, discrete random variables to rove the following lemma. Lemma 4. Let k(t) be the random variable denoting the osition of the leftmost 1-node in S n(t). We have E(k(t)) = t E(, r T ) and V ar(k(t)) = t V ar(, r T ). However, Lemma 4 is scarcely useful in estimating P rob(start( d, t)), since it gives little information on how close random variable k(t) is to its exected value. To circumvent this roblem, we use the following concentration inequality, which is reorted in [4] Theorem 2.7, age 27: Theorem 4 ([4]). Let X 1, X 2,..., X k be non-negative, indeendent random variables. We have the following bounds for random variable X = P k i=1 Xi: for any λ > 0. P rob(x E(X) λ) e λ 2 2 Pk E(X 2 i=1 i ), In our framework, the above concentration inequality can be restated as follows: where P rob(k(t) t E(, r T ) λ) e λ 2 2 t E(X(t) 2 ), E(X(t) 2 ) = (r T + 1) rT + (2 )(1 )r T We can then rove the following lemma: Lemma 5. Let t be such that t > d 1 E(,r T ). Then, P rob(start( d, t)) 1 e ( te(,r T ) d+1) 2 2 te(x(t) 2 ). Proof. We have the following: P ( d, t, ) = t d rt X h=1 P rob(succ(, h)) P rob(estart( d, t h)). In fact, in the Idealized strategy an internal 0-node has robability of turning into a 1-node at each round, once the geometric rocess is started. On the other hand, the starting time of the geometric rocess cannot be lower than d r T, since the maximal backward advancement of the token at each round is r T. The above formula is quite difficult to handle. However, the following theorem can be used to determine, for a fixed target robability 0 < P < 1 and distance d, an uer bound to the minimum ossible value t min( d, ) of t such that P ( d, t, ) P.

7 Theorem 5. Given a target robability 0 < P ı < 1 and distance d to the initiator, define h ln(1 P ) =. Assume ln(1 ) emergency messages are disseminated according to strategy Idealized. Then, we have that min{ t : P ( d, t, ) P } t min( d, ), where the value of t min( d, ) is reorted in Figure 7. Proof. It is easy to see that we have P rob(succ(, h)) P for each h h. We can now lower bound P ( d, t, ) as follows: P ( d, t, ) = + = h 1 X P rob(succ(, h)) P rob(estart( d, t h)) + h=1 t d rt X h= h P P rob(succ(, h)) P rob(estart( d, t h)) t d rt X h= h P rob(estart( d, t h)) = P P rob(start( d, t h)). The above inequality imlies that t min( d, ) equals the minimum value of t such that P rob(start( d, t h)) P. By Lemma 5, this is equivalent to solving inequality 1 e (( t h) E(,rT ) d+1) 2 2( t h) E(X(t) 2 ) P. (1) Taking the logarithm, we can transform (1) into an inequality of degree 2 in t, whose minimal solution is reorted in Figure 7. Let us now consider the Global strategy. We start with the following lemma. Lemma 6. Let t be such that t >, where k = 2 k E(,r T ) if r I = r T and k = 3 if r T < r I 2r T. Then, d 1 t ( d+1) k E(,r T ) 2 P rob(start G ( d, t)) 1 e 2 t k E(X(t) 2 ). Proof. The roof is along the same lines as the roof of Lemma 5, given the observation that the random variable k G (t) denoting the osition of the the leftmost 1-node in S n(t) after t rounds of Global dissemination has the same distribution as the random variable k ` t k. Theorem 6. Given a target robability 0 < P < 1 and distance d to the initiator, define h G = k h, where k = 2 if r I = r T, k = 3 if r T < r I 2r T, and h is defined as in Theorem 5. Assume emergency messages are disseminated according to strategy Global. Then, we have that min{ t : P ( d, t, ) P } t G min( d, ), where t G min( d, ) is reorted in Figure 7-b). 1 (1 ) 1 k Proof. By Theorem 1, Global is at least as good as a strategy which is at most a factor from otimal for what concerns reliability. This means that, denoting with Succ G (, h) the same event as Succ(, h) when strategy Global is used for message dissemination, we have that P rob(succ G (, h)) = 1 (1 ) h k. To comute the minimum value h G such that P rob(succ G (, h G )) P, we observe that, in order for Lemma 2 to hold, h G must be a multile of k. Hence, we have & h G 1 = k ln(1 P ) 1 k = k h. ln(1 ) k By roceeding as in the roof of Theorem 5, we can write P ( d, t, ) P P rob(start G ( d, t h G )). The derivation of t min can then be done by alying Lemma 6 and roceedings along the same lines as in the roof of Theorem 5. Note that the value of t G min reorted in Figure 7-b) is obtained by omitting the floor oerator in the bound stated in Lemma 6, i.e. considering the t/k ratio as a real value. Hence, what reorted in Figure 7-b) is actually an aroximation of t G min. However, the error committed is very small (< k), and asymtotically negligible. Consider the ImGlobal strategy. In this case, backward message dissemination is otimal, and, similarly to strategy Global, the geometric rocess, once started, is at most a factor away from otimal (see Theorem 2) on all 1 (1 ) 1 k but a fraction at most c = 3r T + r I + 1 of internal nodes. These (at most) c nodes are those immediately to the right of the leftmost 1-node in the cloud of cars (see Figure 5). This imlies that ImGlobal is at least as good as a strategy with otimal backward message roagation, and whose starting time of the geometric rocess at node i is delayed such that the leftmost 1-node in the network is at least c ositions away from i. In other words, denoting with Start IG ( d, t) the same event as Start( d, t) when strategy ImGlobal is used for message dissemination, we can write: P rob(start IG ( d, t)) P rob(start( d + c, t)). By roceeding as in the roof of Theorem 6, we can then rove the following theorem: Theorem 7. Given a target robability 0 < P < 1 and distance d to the initiator, define h IG = k h, where k = 2 if r I = r T, k = 3 if r T < r I 2r T, and h is defined as in Theorem 5. Assume emergency messages are disseminated according to strategy ImGlobal. Then, we have that min{ t : P ( d, t, ) P } t IG min( d, ), where t IG min( d, ) is obtained from the formula reorted in Figure 7-a) by substituting d with ( d + c). 4. DISCUSSION 4.1 Distance-time vs. 1-ho reliability tradeoff Based on the bounds to t min derived in the revious section, we now analyze the tradeoff between the three arameters (distance d from the initiator, recetion time t, and 1- ho channel reliability ) influencing P ( d, t, ). This tradeoff is analyzed for the three dissemination strategies introduced in Section 3.1. Strategy Idealized is considered only for the sake of comarison. In investigating the above mentioned tradeoff, we identify three traffic scenarios (light, medium, and heavy traffic), which differ only on the inter-vehicle distance (60m, 30m, and 15m, resectively) 3. Indeendently of the traffic scenario, we 3 These inter-vehicle distances must be intended as the comosition of distances in a multi-lane road (e.g., a two lane road with 30m inter-vehicle distance in each lane corresonds to our high traffic scenario).

8 a) t min (d, ) = he(, r T ) 2 + ( d 1)E(, r T ) E((X(t) 2 r ) ln 1 P + E((X(t) 2 ) ln 1 P 2 de(, r T ) + 2E(, r T ) + E((X(t) 2 ) ln 1 P E(, r T ) 2 b) t G min ( d, ) = h + k (d 1)E(, r T ) E(X(t) 2 q )ln(1 P ) + k E(X(t) 2 )ln(1 P )( 2d E(, r T ) + 2E(, r T ) + E(X(t) 2 )ln(1 P )) E(, r T ) 2 Figure 7: Value of t min( d, ) in the statement of Theorem 5 (a)), and in the statement of Theorem 6 (b)). t_min as a function of distance - =0.5 t_min as a function of distance - = t_min 70 IDEAL =0.5 ImGLOBAL =0.5 GLOBAL =0.5 linear t_min IDEAL =0.9 ImGLOBAL =0.9 GLOBAL =0.9 linear d d Figure 8: Values of t min for fixed values of and varying values of d. Medium traffic scenario. P(t,d,) 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0 P(t,d,) at distance d=50 - GLOBAL vs ImGLOBAL t GLOBAL =0.9 ImGLOBAL =0.9 GLOBAL =0.5 ImGLOBAL =0.5 Figure 9: Values of P ( d, t, ) for d = 50 and varying values of t. Medium traffic scenario. make the following assumtions about the wireless channel model: r T = 120m, which corresonds to having a normalized value of the transmit range r T of 2, 4, and 8 under light, medium, and high traffic, resectively; and r I = 2r T = 240m. While larger values of the transmit range are in rincile ossible in VANETs, increase in 1-ho channel reliability usually comes at the exense of a reduced communication range. Recent research has indicated that a reliable transmit range in the order of one hundred meters is a realistic value [13]. Concerning the choice of the interference range value, assuming r I = 2r T corresonds to a worst-case situation for our roosed message dissemination strategies. Deendence on distance. We first investigate how (our bound on) P ( t, d, ) varies with distance. To this urose, we consider medium traffic conditions (r T = 4), and two different scenarios for 1-ho channel reliability: low reliability ( = 0.5), and high reliability ( = 0.9). In both scenarios, we evaluated the value of t min under the different dissemination strategies as a function of distance (see Figure 8). The target robability P is set to As seen from the figure, t min shows a sublinear increase with distance with Idealized and ImGlobal dissemination (both in case of low and high channel reliability), and a slightly suer-linear increase with Global dissemination. It is interesting to note that the sloe of t min exressed as a function of d dislayed under ImGlobal dissemination is very close to the one dislayed under the Idealized dissemination strategy. Regarding the effect of different reliability values of the wireless channel on the value of t min, we observe the following: a considerable increase in the channel reliability (from = 0.5 to = 0.9), results in a ercentage reduction of t min which decreases with distance, and varies from 37.8% ( d = 12) to 32.3% ( d = 55) with ImGlobal dissemination, and from 29.9% ( d = 12) to 25.1% ( d = 55) with Global dissemination. On the other hand, given the same channel reliability, the message dissemination strategy has a considerable influence on t min: at distance d = 55, using ImGlobal dissemination reduces t min of 50.39% with resect to the case of Global dissemination with low channel reliability, while the ercentage reduction becomes 54.9% in case of high channel reliability. The above results clearly indicates that designing a smart message dissemination strategy is fundamental in order to imlement fast and reliable multi-ho message roagation. With resect to this oint, we observe that while an increase in channel reliability tyically comes at the exense of some form of resource (bandwidth) wastage, designing a smart dissemination strategy does not entail any additional resource wastage. Actually, the oosite usually holds, i.e. smart message dissemination strategies (such as ImGlobal) tend to use the bandwidth in a more efficient way. Deendence on traffic conditions. The deendence of t min on the traffic conditions at different distances and under different channel reliability values is reorted in Table 1. The dissemination strategy is ImGlobal, and the value of the target recetion robability P is In the table, we

9 distance t min, = 0.5 t min, = 0.9 %reduction Heavy traffic 600m 411, 4ms 280, 43ms 31, 8% 1500m 525, 61ms 382, 76ms 27, 2% Medium traffic 600m 441, 93ms 281, 19ms 36, 4% 1500m 572, 23ms 385ms 32, 7% Light traffic 600m 502, 39ms 283, 77ms 43, 5% 1500m 662, 69ms 390, 77ms 41% Table 1: Values of t min under different traffic scenarios with target robability P = 0.95 and ImGlobal dissemination strategy. t_min t_min for varying 0,5 0,55 0,6 0,65 0,7 0,75 0,8 0,85 0,9 0,95 1 ImGLOBAL d=20 ImGLOBAL d=50 GLOBAL d=20 GLOBAL d=50 Figure 10: Values of P ( d, t, ) for fixed values of d and varying values of. Medium traffic scenario. assume that the size of an emergency message is 500Bytes, which is indicated as a reasonable acket size for VANETs in the security study reorted in [9]. Considering that the minimum (and more reliable) data rate available in DSRC is 3Mbs, we need about 1.33ms to send a acket. Given this, we assume a time slot duration of 10ms in Table 1, which gives adequate margins for overhead due to acket header, channel access, nodes coordination, and so on. The following observations can be made out of the data reorted in Table 1. First of all, fast and reliable message dissemination aears to be a realistic goal, since t min results always below 0.7s even at relatively long distances (1.5Km). Of course, what is reorted in Table 1 should not be considered as reliable quantities for a real imlementation, due to the many simlifying assumtions used in our analysis. Nevertheless, our findings seem to indicate that fast and reliable emergency message dissemination is indeed ossible. Another imortant observation regards the deendence of the relative benefits of increasing channel reliability on traffic conditions: more benefits can be observed under light traffic conditions, while the relative reduction of t min with = 0.9 over the case with = 0.5 tends to decrease under heavy traffic conditions. This behavior seems to suggest that the relative benefit of imlementing reliable 1-ho communication decreases with car density. Hence, network designers might design density-aware reliable 1-ho communication rimitives, where the desired value of reliability is tuned (e.g., by aroriately setting the transmit ower [12, 13], or by changing the number of MAC layer reetitions of a message [14]) deending on the observed car density 4. It is interesting to note that setting the desired channel reliability level deending on vehicle density has the additional advantage of overcoming the well known fact that achieving a certain level of 1-ho reliability incurs higher and higher bandwidth wastage as node density increases (in fact, channel reliability is inversely roortional to network congestion). Hence, by reducing the required channel reliability value in resence of high vehicle density we can considerably reduce resource wastage, while not significantly imacting fast and reliable 4 Techniques for estimating vehicle density have been roosed in the literature [1]. roagation of emergency messages. Deendence on time. The deendence of P ( d, t, ) on time, for a fixed value of d, and low and high channel reliability is reorted in Figure 9. From the figure, it is clearly seen the beneficial effect of ImGlobal dissemination on the correct message recetion time. It can also be observed that a higher 1-ho channel reliability induces a sharer transition (in the time domain) from near zero to close to 1 values of P ( d, t, ). Deendence on channel reliability. The deendence of P ( d, t, ) on channel reliability, for two different values of d is reorted in Figure 10. As seen from the figure, increasing 1-ho channel reliability above a certain value (about 0.9), which comes at the exense of considerable resource wastage, has only marginal effects on fast and reliable multi-ho message dissemination. 4.2 Generalizations We believe a main contribution of this aer is the definition of a methodology that can be used to analyze multi-ho emergency message roagation under more general models. The methodology consists in subdividing reliable multi-ho message roagation into two subrocesses: i) backward advancement of the emergency message coverage area, and ii) reliable message roagation to the nodes which are inside the coverage area, but they have not received the message yet. To study i), one needs to estimate 1-ho message advancement, and to use this estimation to rovide bounds on multi-ho message advancement. Process ii) can be modeled as a geometric rocess of a certain arameter, whose starting time is determined by i). We believe the methodology used in our analysis can be used in combination with more general network toologies and/or radio channel models. For instance, one could assume that cars are not equally saced, but saced according to some robability distribution. It is our belief that results very similar to the ones reorted in this aer hold also in a setting in which average node density is the same in all the network, but inter-vehicle distances can vary. In fact, 1-ho message advancement is determined by the exected number of nodes within transmitting range, and not by their relative distance; furthermore, Lemma 4 (which is used to bound multi-ho message advancement) is very general, and can be used also in this setting. Similarly, one can assume more general channel models, such as letting the robability of correct message recetion become a function of the transmitter/receiver distance. In this case, besides aroriately modeling i), to model ii) one should use a generalized notion of geometric rocess in which the robability of success in each exeriment can change. We are currently actively working on formally deriving bounds on P ( d, t, ) under the above mentioned generalizations of our model. 5. SIMULATIONS To assess the accuracy of the qualitative analysis reorted in the revious section, we have imlemented an ad hoc simulator. For the three traffic scenarios described in the revious section, the simulator disseminates emergency messages in synchronous rounds, according to the three dissem-