MESSAGE BROADCASTING IN WIRELESS VEHICULAR AD HOC NETWORKS

Transcription

1 MESSAGE BROADCASTING IN WIRELESS VEHICULAR AD HOC NETWORKS CARLA F. CHIASSERINI, ROSSANO GAETA, MICHELE GARETTO, MARCO GRIBAUDO, AND MATTEO SERENO Abstract. Message broadcasting is one of the fundamental services in multihop vehicular networks. In particular, many applications require to deliver the information to all vehicles traveling over a geographical area, with high reliability and low delay. We focus on road segments where there are not fixed nodes and message dissemination relies only on the ad hoc communication paradigm. We introduce an analytical framework to study the system performance, and derive several metrics relevant to message broadcasting. We first use a static analysis and derive the message blocking probability; then we study the transient network behavior and evaluate the delivery delay of broadcast messages. The standard message broadcasting used in based networks is enhanced through a novel application and channel access mechanism, which significantly improves the system performance. Analytical results are compared with the performance obtained through simulation using ns. Key words. Wireless vehicular networks, Message broadcasting, Performance analysis AMS subject classifications. 15A15, 15A09, 15A23 1. Introduction. Wireless vehicular communications, along with onboard driverassistance systems and fixed communication nodes located along the roads, enable a wide range of applications like road safety services, car-to-car audio/video communication, nomadic Internet access, and multimedia geocasting. For example, transportation safety can be provided by letting vehicles exchange warning messages that are generated by approaching emergency vehicles, or by vehicles stuck in a road tunnel because of an accident [1]. Multimedia geocasting, instead, can be performed by fixed nodes sending, either directly or relaying on intervehicular communications, advertisements and touristic information to the users traveling over a certain region. All these applications require broadcast messages to be disseminated to all vehicles within a geographical area, and need to be delivered in a reliable and timely manner. It is therefore important to develop protocol solutions that meet such requirements. In [2], an innovative protocol architecture for vehicular ad hoc networks is proposed, which enables high interaction among protocol layers and great flexibility in the control parameters setting. Self-organization and routing in vehicular networks supporting safety applications are addressed in [3], while solutions at the MAC (Medium Access Control) layer are studied in [4, 5, 6, 7, 8, 9]. In particular, the works in [5, 7, 8, 9] specifically deal with broadcast communications. In [8], an IEEE based scheme is proposed to address the broadcast storm and the hidden terminal problems in urban areas. The use of IEEE e EDCA scheme for priority access is investigated in [7], where the authors study through simulation the broadcast reception rate in presence of different channel propagation models. The performance of the optimum broadcast algorithm defined over the node minimum connected dominating set is studied in [9], in the case of a unidimensional ad hoc network. This work was supported by the PRIMO FIRB project Dipartimento di Elettronica Politecnico di Torino, Corso Duca degli Abruzzi 24, I Torino, Italy (chiasserini@polito.it). Dipartimento di Informatica, Università di Torino, Corso Svizzera 185, I Italy ({garetto,rossano,marcog,matteo}@di.unito.it). 1

2 2 C. F. CHIASSERINI, R. GAETA, M. GARETTO, M. GRIBAUDO AND M. SERENO Our contribution. We focus on a unidimensional vehicular network supporting message broadcasting and featuring ad hoc connectivity. Our objective is to study the dissemination of broadcast messages over the network, on road segments where there are not fixed nodes and dissemination relies on inter-vehicular communications only. We present a channel access scheme and an application mechanism that provide an efficient multihop broadcasting. We assume that vehicles can estimate the direction of arrival of a message, as well as their distance from the vehicle that is currently transmitting the message. We introduce a spatial differentiation approach at the MAC layer: vehicles that are about to rebroadcast the message access the channel with different priority, depending on their distance from the last vehicle that transmitted. The differentiation in accessing the channel is provided by using different values of contention window. This technique reduces the broadcast delay along the road, thus improving the message timeliness. Furthermore, to reduce the broadcast overhead, a vehicle refrains from rebroadcasting the information if it hears a vehicle located further ahead that retransmits the same message. We develop an analytical framework to study the system performance, and derive several metrics relevant to the dissemination of broadcast messages. We first use a static analysis and derive the message block probability. Then, we present an analytical framework to study the transient system behavior and evaluate the delivery delay of broadcast messages in the general case of inhomogeneous vehicle density. In the case where the vehicle density is constant along the road, we derive a simple but accurate Gaussian approximation that allows us to assess the message delivery delay with very low computational complexity. Analytical results are compared with the performance obtained through the network simulator ns. Chapter organization. The remainder of the chapter is organized as follows. Section 2 describes the network scenario under study. In this section we also reviews the main features of wireless networks following the IEEE and e standards, which are the basis of the proposed system. Section 3 presents the application and the channel access scheme that we propose. Section 4 presents our modeling techniques and discusses several issues related with the proposed framework. For all the presented techniques, in Section 5 we present some analytical results and compare them with the performance obtained through the ns simulator. Section 6 concludes the contribution and outlines some possible future directions. 2. System Description. We focus on a unidimensional inter-vehicular network, modeling a single- or multi-lane road segment. Vehicles are randomly distributed with spatial density that may vary along the road. Note that, the extension to the two-dimensional case is straightforward when the system behavior on different road segments can be assumed to be independent of each other. We look at a snapshot of the traffic stream; indeed, considering typical values for the vehicle speeds, the message length and the communication data rates, it results that the vehicle movement during a message broadcasting is negligible. As an example, consider the technology. By fixing the message length at 32 bytes, the data rate at 11 Mb/s and the contention window at 31, the average time to forward a broadcast message over one hop is about 522 µs. Assuming a vehicle speed of 100 km/h, during the message forwarding time vehicles move m. We therefore assume that vehicle positions remain constant during the message forwarding over the whole road. All vehicles have a common coverage radius, equal to R. Also, upon a message reception, a vehicle is able to detect whether the sender is located ahead or behind,

3 MESSAGE BROADCASTING IN WIRELESS VEHICULAR AD HOC NETWORKS 3 as well as its own distance from the sender. Several solutions can make this feasible. Vehicles may be equipped with a GPS device 1 and the vehicle position can be included in each transmitted message. Or, vehicles can use directional antennas to determine the signal direction of arrival, and the RSSI (Received Signal Strength Indicator) to estimate their distance from the sender. Finally, we focus on broadcasting applications and neglect other kinds of data traffic. This is justified by the fact that, if other applications are simultaneously supported by the network, typically broadcast messages have higher priority and their transmission on the wireless medium should not be affected by other types of traffic Background: The Standard. In this section we briefly describe the IEEE b and the IEEE e distributed channel access schemes, named, respectively Distributed Coordination Function (DCF) and Enhanced Distributed Channel Access (EDCA) [10, 11]. In networks time is divided into time intervals called time slots. The channel access is based on the Carrier Sensing Multiple Access mechanism, with Collision Avoidance (CSMA/CA). The IEEE DCF exploits both a physical and a virtual channel sensing, to determine whether the channel is idle or busy. Virtual sensing is implemented by including in all transmitted messages an indication of their duration so that the nondestination stations overhearing a transmission are aware of the time interval during which the channel will remain busy. A counter, called NAV (Network Allocation Vector), is set accordingly to keep track of the channel status. Once a station has set its NAV, it remains in overhearing state for the whole duration of the transmission. When a station wishes to access the channel, the physical and virtual carrier sense mechanisms are checked. If both of them detect the channel as idle for a time duration equal to the Distributed Inter Frame Space (DIFS), the node transmits. Otherwise, the station waits for the channel to become idle; then, within an interval of DIFS, it randomly selects a backoff value from the range called Contention Window ([0, W]) and sets its backoff counter to this value times the slot duration. The W value is doubled at every transmission attempt. The backoff counter is decremented at the end of each idle slot; as the backoff counter reaches zero, the node accesses the channel. Finally, after a node has sent its data, it draws a random value within the [0, W] range and starts a backoff procedure (the so-called post-backoff). With respect to the DCF scheme, the e EDCA introduces the following innovations: (1) a station that seizes the channel is entitled to transmit one or more messages till a maximum duration, called TXOP Limit, is reached; (2) various Access Categories (ACs) are defined, each of which corresponds to a different priority level and to a different set of parameters to be used for contending the channel. In particular, an e station includes up to four MAC queues; each queue corresponds to a different AC and represents a separate instance of the CSMA/CA protocol. A queue employs the following parameters to access the channel: (i) the Arbitration Inter Frame Spacing (AIFS[AC]), similar to the DIFS used in DCF, (ii) the Minimum and the Maximum Contention Window (W min [AC], W max [AC]), (iii) and the TXOP Limit[AC]. The higher the AC priority is, the smaller the AIFS[AC], W min [AC] and W max [AC] are; however, the values of W min [AC] and W max [AC] have to be care- 1 In the case of vehicles traversing a tunnel, one can simply imagine that positioning-aware communication devices are provided to the drivers at the tunnel entrance

4 4 C. F. CHIASSERINI, R. GAETA, M. GARETTO, M. GRIBAUDO AND M. SERENO fully chosen so as to avoid high collision probability among traffic flows belonging to the same AC, and the value of AIFS must be at least as long as the DIFS interval. Within every e station, a scheduler solves virtual collisions among the AC queues, i.e., among the various CSMA/CA instances, by always enabling the queue associated with the highest priority to transmit. Finally, we would like to highlight that, according to the IEEE standards, broadcast messages are never retransmitted at the MAC layer. 3. Dissemination of Broadcast Messages. In this section we propose a channel access scheme and an application that aim at reducing the delay and the overhead of broadcast messages. Our channel access scheme (Section 3.1) provides channel access priority by exploiting the concept of spatial differentiation, while the broadcasting application (3.2) is based on the vehicle ability to detect the message direction of arrival The Spatial Differentiation Approach. At the MAC layer, we envision an access scheme based on the CSMA/CA mechanism (e.g., based on the IEEE standard [10]). The binary exponential backoff procedure is employed, and the backoff time is a number b of time slot intervals of duration σ. We have that b is a random number uniformly distributed over [0, W], where W is the minimum contention window. Also, we consider that, whenever the MAC layer receives a message from the higher layers, it extracts a backoff value, so that a random time interval is waited before attempting to access the channel. Our key idea is to assign different access priorities to the vehicles that are currently in charge of forwarding the message, so that the advancement corresponding to a message hop is maximized. Let v be the last vehicle that (re)broadcasted the message. We define different forwarding zones within the coverage range of v, and assign to the vehicles belonging to each zone a different value of contention window W. The larger the distance from the sender v, the smaller the contention window. By doing so, vehicles belonging to the furthest zone have the highest priority in accessing the channel, and the probability that the message forwarding is performed by vehicles at distance close to the coverage radius of the previous sender is increased 2. Note that such differentiation mechanism could be implemented through the IEEE e technology [11]. Indeed, the traffic transmitted by vehicles belonging to different zones could be mapped onto different e Access Categories, based on the geographical position of the vehicle The Application. Consider a vehicle wishing to broadcast a message along the road; we define this vehicle as the broadcast message source. The application we devise allows for a smart broadcasting technique that, based on the vehicle ability to determine the direction of arrival of the message, reduces the communication overhead. The procedure followed by each vehicle application along the road (excluding the source) is presented below. (i) Upon the reception of the broadcast message, the application first checks whether the message is received for the first time and its direction of arrival. (ii) If the message has never been received before, the application passes the message to the MAC layer to rebroadcast the information. Recall that a random backoff time is waited at the MAC layer before accessing the channel. 2 Recall that vehicles are assumed to be able to estimate their distance from the sender

5 MESSAGE BROADCASTING IN WIRELESS VEHICULAR AD HOC NETWORKS 5 (iii) If it is a duplicated message, the application flushes out the previously received copy (which is buffered at the MAC layer waiting to be transmitted) and processes the newly arrived copy. The new copy will be either passed to the MAC layer or discarded, based on its direction of arrival. It will be passed to the MAC layer if its direction of arrival is the same as the one of the original message; it will be discarded otherwise. Note that, according to this procedure, a vehicle that detects the message being rebroadcasted further ahead, will abandon its transmission attempt, avoiding unnecessary message forwarding. Furthermore, as the message propagates along the road, a vehicle may receive multiple copies of the message from different vehicles located on its left side. In this case the distance between the vehicle and the last message sender decreases progressively. This implies that, at every message reception, a vehicle needs to start a new transmission attempt with an updated value of contention window. Our broadcast scheme meets this requirement and dynamically adapts to the message advancements, since the application always processes the newly arrived copy and discards the previous one. Section 4 presents our modeling techniques and discusses several issues related with the proposed framework. 4. The Modeling Framework. In this section we presents our modeling techniques and discusses several issues related with the proposed framework. In particular, he notations and the assumptions used to develop the system model are introduced in Section 4.1. We outline the developed model in Section 4.2. In Section 4.3 we derive the message block probability while Section 4.4 presents the computation of the message first reception probability. In Section 4.5 we introduce an approximate method of the transient system behavior in the case of homogeneous vehicle density Assumptions and Notations. We consider a single-lane road and we use the minimum distance between two vehicles to discretize space along the lane; it follows that vehicles can occupy only a discrete set of positions indexed by y, with y N. A vehicle at position y is at distance y from the origin. The normalized coverage radius r = R/ is the maximum number of vehicles receiving the message on either side of a transmitter. For the sake of simplicity, we assume that the source of the broadcast message is located at y = 0 and consider the message broadcasting only on one side of the source. The extension to the case where the source is located at y > 0, and the message broadcasting occurs on both sides of the source, is straightforward. We define the occupation probability ρ i as the probability that position i is occupied by a vehicle (with i > 0); note that the vehicle density can be expressed as ρ i /. We discretize time into slots of duration σ. The duration of a broadcast message is set to T slots, and the message originates from the source at time n = 0 (i.e., the first bit of the message is placed on the channel at time zero). The backoff time of a vehicle is uniformly distributed in [0, W j ], with j being the vehicle distance from the last message sender and W j the contention window expressed in time slots. At the physical layer, we make the following simplifying assumptions: (i) there is perfect capture, meaning that a vehicle simultaneously receiving more than one message at a time is able to lock on the strongest signal and receive the message correctly; (ii) the radio channel is error-free. It follows that a transmission is always correctly received by all vehicles within the sender s coverage radius. We will see in Section 5 what is the impact of such assumptions.

6 6 C. F. CHIASSERINI, R. GAETA, M. GARETTO, M. GRIBAUDO AND M. SERENO R r ρ y ρ i,y σ W j T P D P L P A [y] Table 4.1 Notations Coverage radius Normalized coverage radius Minimum distance between two vehicles Occupation probability at position y (single lane) Occupation probability on lane i at position y Duration of a backoff slot Contention window size (in time slots) of a vehicle at distance j from the last message sender Duration of a message (in time slots) Distribution of hop delay (in time slots) Distribution of hop length Distribution of the maximum distance reached by a message P R [y, n, h] Probability of first reception at position y, time n, in h hops P R [y, n] P R [y, h] P B [y] P T [y, n, h, l] P T [y, n, h] P(S = y, C = b, L = l) M(n) P M [y, n] Marginal probability of first reception at position y and time n Marginal probability of first reception at position y in h hops Message block probability at position y Probability that the message is transmitted at position y, time n, in h hops having been received from a predecessor at distance l Marginal probability that the message is transmitted at position y, time n, in h hops Probability that a vehicle in y extracts a backoff of b slots, it wins the contention and the message advancement is equal to l positions Maximum distance reached by the message at time n Normal distribution approximating the distribution of the maximum distance reached by the message at time n For the sake of clarity, Table 4.1 collects the parameters that we use and the functions that we define throughout the rest of the paper Model Outline. Message broadcasting (e.g., broadcast of safety messages) poses strict requirements in terms of reliability and message timeliness. Therefore, while evaluating the performance of our solution, we consider the following metrics:

7 MESSAGE BROADCASTING IN WIRELESS VEHICULAR AD HOC NETWORKS 7 message block probability P B [y], that is the probability that the message broadcast stops exactly at the vehicle occupying position y, i.e., the probability that the message will not be delivered to any other vehicle beyond y; first reception probability P R [y, n, h], that is the probability that a vehicle at position y receives for the first time the message at time n, in h hops. From P R [y, n, h] it is possible to derive the marginal probability that a vehicle in y receives the message for the first time at n, regardless of the number of hops (P R [y, n]), and the marginal probability of first reception in y, in h hops (P R [y, h]), regardless of the delay. The message block probability is computed in Section 4.3 by using a static analysis, that is without the need to explicitly considering the temporal dynamics of the system. The second metric, instead, requires a study of the temporal evolution of the message broadcast. An analysis of the transient behavior of the system, which provides P R [y, n, h] is presented in Section 4.4. To reduce the computation time of this important metric, in Section 4.5 we restrict ourself to the case of homogeneous occupation probability and introduce an approximated technique based on the central limit theorem. Such technique gives a good approximation of the maximum distance reached by the message at time n and of the marginal probability P R [y, n] Computation of the block probability. We first consider the case in which the road comprises a single lane. The extension to the case of multiple lanes is presented in Section Let us assume for now the case of homogeneous vehicle density, i.e., that the occupation probability is constant along the road (i.e., ρ y = ρ, y > 0). We first derive the probability P A [y] that the furthest spatial position reached by the message transmission as time goes to infinity is y, i.e., the probability that the transmission carrying the broadcast message has covered a physical distance up to position y. By definition, we have: P A [0] = 1. The meaning of P A [y] is highlighted in Fig The figure shows a snapshot of the system where the vehicle in x rebroadcasts the message: the upper illustration represents the case where the probability that the furthest distance reached by the message is equal to x+r differs from the reception probability at x+r. Indeed, since there is not a vehicle in x + r, the signal transmission reaches position x + r but the probability of reception by a vehicle at x + r is equal to 0. The lower figure instead presents the case where P A [y] and the probability of reception by a vehicle are the same. 0 x x+1 r x+r 0 x x+1 r x+r Fig Difference between P A [y] and the message reception probability

8 8 C. F. CHIASSERINI, R. GAETA, M. GARETTO, M. GRIBAUDO AND M. SERENO Probability P A [y] can be easily computed based on the following observations. Firstly, a message transmitted by a vehicle at position x can be received up to position x + r. Thus, the message broadcast stops at x only if there are no vehicles between x+1 and x+r. In other words, the message eventually arrives at the generic position y if between 0 and y there are no gaps in the vehicle distribution, of width greater than or equal to r. Secondly, computing the probability that there is connectivity between 0 and y maps onto the known problem of finding the probability that, in a sequence of y Bernoulli trials, there are no runs of length r [12]. (This is due to the assumption that each position i is independently occupied by a vehicle with given probability ρ i = ρ.) It follows that the most efficient way to calculate P A [y], with y > r + 1, is given by the following recursive equation: P A [y] = P A [y 1] ρ(1 ρ) r P A [y r 1] (4.1) starting from the initial values: P A [i] = 1, (0 i r) and P A [r + 1] = 1 (1 ρ) r. The above formula has the following intuitive explanation: a message transmission that has arrived up to position y 1 will reach also position y unless y 1 is the last position that completes a series of r empty positions (i.e., without vehicles). Figure 4.2 represents the case where the message does not reach the vehicle in y because of a gap of r positions, with r = 3. 0 y r 1 y 1 y r positions Fig Explanation of probability P A [y] It can be shown that in the case where the occupation probability varies along the road (i.e., inhomogeneous vehicle density), for any y > r + 1 (4.1) becomes: P A [y] = P A [y 1] ρ y r 1 y r j<y (1 ρ j )P A [y r 1] (4.2) starting from the initial values: P A [0]=1; P A [i] = ρ i, (1 i r); and P A [r + 1] = 1 1 j r (1 ρ j). The block probability P B [y] can be computed as the probability that the message reaches the vehicle in y and this is the last vehicle to receive the message. A vehicle in y will be the last one to receive the message if its location is followed by a connectivity gap, which happens with probability y+r j=y+1 (1 ρ j). Thus, we have: P B [y] = ρ y P A [y] y+1 j y+r (1 ρ j ) (4.3) The case of multiple lanes. Let us consider a road with n l lanes. The extension to the case of multiple lanes is straightforward if we assume that the total width of the road is small with respect to the radio range R. In this case, we can assume that a transmission from position y i on any given lane i (1 i n l ) reaches all locations x j on lane j (1 j n l ) if x j y i r, i, j. We allow the vehicle density to vary among lanes introducing ρ i,y, the occupation probability on lane i, 1 i n l, at position y. This can be used to model the fact that a faster lane has a

9 MESSAGE BROADCASTING IN WIRELESS VEHICULAR AD HOC NETWORKS 9 smaller occupation probability than a slower lane. The problem can be reduced to the single-lane case computing the effective occupation probability ρ y = 1 n l i=1 (1 ρ i,y), which is the probability that there is at least a vehicle at position y across all lanes. Then the probabilities P A [y] and P B [y] can be computed exactly in the same way as described in the single-lane case Computation of the probability of first reception. To evaluate the timeliness of the broadcast message delivery, we need to study the temporal dynamics of the message broadcasting along the road and to analyze the transient behavior of the system. The dynamics of the message broadcasting are illustrated in Figure 4.3 (in the figure the arrows indicate the direction of the message broadcast). Suppose that at time k vehicle x starts sending the message. All vehicles in range of x that were contending for the channel will suspend their transmission attempt for the duration T of the message. At time k + T the message is fully received by all vehicles up to position x + r. After that, the vehicles in [x + 1, x + r] contend among themselves to further forward the message, using a contention window that depends on their distance from x (see Section 3.1). The time spent in contention will be equal to the minimum backoff value among the contending vehicles. Notice that more than one vehicle can extract the minimum value of backoff, resulting in simultaneous transmissions at the end of the contention period; however the message will be successfully forwarded also in this case. Indeed, let y be the position of the furthest vehicle transmitting the message after the contention period: thanks to the assumption of perfect capture and ideal wireless channel, after time T the message sent by y will be successfully received by all vehicles up to position y + r. 0 x y x+r y+r Fig Dynamics of the message broadcasting In order to compute the probability of first reception P R [y, n, h], we introduce the following definitions: P T [y, n, h, l] is the transmission probability, i.e., the probability that a vehicle at position y transmits the message at time n in h hops, having received the message from its predecessor at distance l; P T [y, n, h] is the marginal transmission probability regardless of the predecessor: P T [y, n, h] = r l=1 P T[y, n, h, l] P(S = y, C = b, L = l) is the probability that a vehicle at position y extracts a backoff value of b slots, it wins the channel contention and the message advances by l positions on the road (i.e., the length of the hop between y and the location of the predecessor vehicle is equal to l). We first write the transmission probabilities P T [y, n, h, l] as: W l [ P T [y, n, h, l] = P T [y l, n T b, h 1]P(S = y, C = b, L = l) ] (4.4) b=0 for y > 0, n T, h 1, and 1 l r. The sum on the right hand side of (4.4) accounts for all possible values b of backoff time that the vehicle in y can extract. Also, note that the contention window used by the vehicle in y will depend on l, being l the distance between y and its predecessor.

10 10 C. F. CHIASSERINI, R. GAETA, M. GARETTO, M. GRIBAUDO AND M. SERENO By definition, the marginal probability P T [y, n, h], can computed from (4.4) through the following recursive equation: P T [y, n, h] = r W l { P T [y l, n T b, h 1]P(S = y, C = b, L = l) } (4.5) l=1 b=0 The two sums on the right hand side of (4.5) account, respectively, for all possible pairs (y l, b) representing the position of the previous transmitter with respect to y and the value b of backoff time that the vehicle in y can extract. The initial value of P T [y, n, h] is: P T [0, 0, 1] = 1, which accounts for the fact that the message source makes its transmission at time 0, performing the first hop. Next, we compute the joint probability P(S = y, C = b, L = l). For the sake of clarity, let us first present the case of occupation probability constant and equal to 1 (ρ i =ρ=1). We need to compute the probability that the contention period lasts for b slots and the message advances by l positions on the road. This event happens if and only if: (i) the vehicle at distance l from the vehicle transmitting the message in the previous hop extracts a backoff value equal to b ; (ii) all vehicles at distance j < l extract a backoff higher than or equal 3 to b ; (iii) all vehicles at distance j > l (within transmission range) extract a backoff value strictly higher than b. In general, a vehicle at distance j extracts a backoff value uniformly in the contention window [0, W j ], where W j depends on the distance j to allow for spatial differentiation. We obtain: P(S = y, C = b, L = l) = 1 l 1 W j b + 1 W l + 1 W j=1 j + 1 r j=l+1 W j b W j + 1 (4.6) Note that the three factors on the right hand side of (4.6) account for the events (i), (ii) and (iii) described above. We now consider that the occupation probability varies along the road (i.e., 0 < ρ i 1, with i > 0), and recompute P(S = y, C = b, L = l). In this case we have to take into account that there may be no vehicle at distance j : if so, that position does not contribute to the contention phase. The expression of P(S = y, C = b, L = l) therefore becomes: l 1 1 P(S = y, C = b, L = l) = ρ y P(B j b) W l + 1 j=1 r j=l+1 P(B j b + 1) (4.7) where P(B j b) = 1 ρ y l+j + ρ y l+j W j b + 1 W j + 1 By using (4.7) in (4.5), we obtain the marginal transmission probabilities P T [y, n, h], from which we compute P T [y, n, h, l] using (4.4). Finally, the probability of first reception P R (y, n, h) (with y > r, k 2T, h 1) can be written as, P R (y, n, h) = ρ y r m=1 r m {P T [y m, n T, h] P T (y m, n T, h, l)} (4.8) 3 Equality holds due to the perfect capture capability of the vehicles l=1

11 MESSAGE BROADCASTING IN WIRELESS VEHICULAR AD HOC NETWORKS 11 starting from the following initial values: P R (0, 0, 0) = 1, which accounts for the message source, and P R (i, T, 1) = ρ i for 0 < i r. Referring to (4.8), we have that: ρ y accounts for the probability that a vehicle is present at position y, while the sum over m considers all possible predecessors within distance r from y, i.e., all vehicles from which the one in y can receive the message. Given m, the probability that the message is received for the first time in y due to the transmission of a vehicle at distance m, is equal to the probability that the vehicle at y m transmits the message, and the message has not been heard by y before. That is, the vehicle in y m must have received the message from a predecessor whose distance from y is greater than r. Figure 4.4 illustrates the case where a transmission performed by the vehicle at position y m l reaches the vehicle in y m but not the one in y (Figure 4.4.(a)). The case where a transmission performed at y m l reaches both the vehicles (in y m and in y) is presented in Figure 4.4.(b). (a) r r y m r y m l y r y m y (b) r r y m r y r y m l y m y Fig Locations of the predecessor from which (a) a transmission is not received at position y, (b) a transmission is received in y By summing over the number of hops (h 1), we can derive the probability P R (y, n) that the vehicle at position y receives the message for the first time at time n. By summing over time, we derive the probability P R (y, h) that the message is received by a vehicle at position y in h hops. Note that it is also possible to derive the block probability similarly to 4.3: P B (y) = P R (y, n) (1 ρ i ) (4.9) n=0 y<i y+r 4.5. A Gaussian Approximation to the Transient System Behavior. The analysis presented in the previous section is general enough to deal with an inhomogeneous vehicle density, however it requires to calculate the three-variable function P T [y, n, h]. Here we restrict ourself to the case of homogeneous vehicle density (ρ y = ρ, y > 0) and derive an approximate analysis of the system transient behavior that provides the marginal probability P R [y, n] with low computational complexity.

12 12 C. F. CHIASSERINI, R. GAETA, M. GARETTO, M. GRIBAUDO AND M. SERENO We first consider the case with ρ y = ρ = 1, y > 0, i.e., the situation in which all positions along the road are occupied by vehicles; the analysis in the case of occupation probability smaller than one is briefly sketched at the end of the section. In Section 4.5.2, we consider the case of multiple lanes Transient analysis with ρ 1, single lane. Let us first consider the case ρ = 1. We start by observing that the message broadcasting along the road can be described by a sequence of independent hops having variable delay D and variable length L (with L expressed in number of spatial intervals of length ). The delay D is the sum of a deterministic term (the message duration T) and the variable term C given by the contention period. The maximum distance M(n) reached by the message at time n is therefore given by the sum of a random number H(n) (the number of hops) of random variables L (the hop length). By applying the central limit theorem, we can approximate the probability distribution of the furthest distance reached by the message at time n with a normal distribution P M [y, n]. To specify the normal distribution of M(n), we need to compute mean and variance of M(n). This can be done as follows. We observe that from the joint probability P(S = y, C = b, L = l) in (4.6), we can derive the marginal probability P D [d] that the total hop delay is equal to d, as well as the marginal probability P L [l] that the hop length is equal to l. Please note that the dependence on y drops due to the assumption of homogeneous occupation probability. We have: P D [d] = r P(S = y, C = d T, L = l) T d T +W l (4.10) l=1 W l P L [l] = P(S = y, C = b, L = l) 1 l r (4.11) b=0 From (4.10) we compute the mean hop delay E[D] and its variance Var[D]; from (4.11) we compute the first two moments of hop length E[L] and E 2 [L]. The average number of hops E[H(n)] done at time n is: E[H(n)] = n/e[d] [13]. The variance Var[H(n)] of the number of hops can be expressed as Var[H(n)] = Var[D] n (E[D]) 3 (4.12) Then, it is possible to calculate E[M(n)] and Var[M(n)] from the first two moments of H(n) and L as (see [13]): E[M(n)] = E[H(n)] E[L] (4.13) Var[M(n)] = E[H(n)] Var[L] + E 2 [L] Var[H(n)] (4.14) The Gaussian approximation is more and more accurate as n ; in practice, it already produces satisfactory results after the message has propagated for a few hops (namely, 10 hops). In the analysis above, we have neglected the fact that the first hop is special: it has a deterministic duration T and advances by a deterministic length r. However this fact can be easily taken into account considering the contribution of the first hop separated from the rest.

13 MESSAGE BROADCASTING IN WIRELESS VEHICULAR AD HOC NETWORKS 13 An approximated expression of P R [y, n] can be obtained by considering that a vehicle in y receives the message for the first time if its location falls within the spatial advancement of the last message hop, and such advancement occurs at time n. In particular, we can prove the following approximation, which turns out to be very good after the message has propagated for a few hops: P R [y, n] P M [y, n] E[L] E[D] (4.15) Proof. Let us define P Z [y, n] as the probability that a vehicle in y receives the broadcast message for the first time at time n, being y the furthest position reached by the message. Note that P Z [y, n] and P M [y, n] differ since P Z [y, n] is the probability that the furthest vehicle receives the message exactly at time n, whereas P M [y, n] returns the probability that, by time n, the furthest vehicle reached by the message is at position y. We can relate P M [y, n] with P Z [y, n] as follows: P M [y, n] = D d=1 P Z [y, n d] τ d P D [τ] Indeed, position y is still the furthest reached distance, if the message arrived here d time slots in the past, and the delay of the current hop is greater than or equal to d. Approximating P Z [y, n d] with P Z [y], and recalling than the mean of a discrete time variable can be expressed as the sum of its complementary cumulative distribution function, i.e., we obtain E[D] = D P D [τ] d=1 τ d P M [y, n] P Z [y, n] E[D] (4.16) Similarly, we can relate P Z [y, n] with P R [y, n], the probability of first reception at y at time n, being y a generic position (not necessarily the furthest reached one). We obtain: P R [y, n] = r l=1 D d=1 P Z [y l, n d]p D [d] x l P L [x] where we have considered all possible hop lengths l and all possible hop delays d for the last hop (the one that allows the message to advance and cover position y for the first time). In particular, y l is the furthest position reached by the message at the previous hop, and d is the delay associated with the last hop. Note that: (i) the expressions of the marginal probabilities P D [d] and P L [x] can be found in (4.10) and (4.11), respectively, (ii) values of hop length greater than or equal to l are considered because the message can be received even further than y (which is not necessarily the last reached position). Approximating P Z [y l, n d] with P Z [y, d] (such approximation is actually very good after the message has propagated for a few

14 14 C. F. CHIASSERINI, R. GAETA, M. GARETTO, M. GRIBAUDO AND M. SERENO hops, i.e., in the operational region we are considering), and exploiting again the fact that r E[L] = P L [x] we obtain l=1 x l P R [y, n] P Z [y, n]e[l] (4.17) Finally, by putting together approximations (4.16) and (4.17), we get: P R [y, n] P M [y, n] E[L] E[D] (4.18) The approximate analysis when ρ 1 presents two additional complications: (i) the number of vehicles contending for the channel at each hop is now randomly distributed; (ii) the message can be blocked at some point due to the lack of connectivity (see Section 4.3). We can first analyze the broadcast delay with ρ 1 assuming that the network is connected from the source up to position y, and derive P M [y, n connected ]. The probability P M [y, n connected ] can be easily obtained following the same approach as described above. Then, we decondition with respect to the assumption that there is connectivity from the source up to position y. To do so, we simply multiply probability P M [y, n connected ] by the probability P A [y] that the message indeed reaches position y. We therefore approximate P R [y, n] as: P R [y, n] ρp M [y, n connected ]P A [y] E[L] E[D] where ρ accounts for the probability that there is a vehicle at position y. (4.19) Transient analysis with ρ 1, multiple lanes. The more general case of multiple lanes (and ρ 1) can be addressed in a similar way, first computing the propagation delay under the assumption that the network is connected, and then considering the impact of connectivity. The conditional occupation probability on lane ρ i is ˆρ i = i 1 (1 ρ ). where ρ has been computed in Section The expression of r the joint probability P(S = y, C = b, L = l) becomes where l 1 P(S = y, C = b, L = l) = P(B j = b) P(B j b) i=1 j=1 r j=l+1 n l ( W ) j P(B j = b) = 1 1 ˆρ i + ˆρ i W j + 1 P(B j b + 1) is the probability that at least one vehicle at distance j extracts a backoff equal to b, whereas n l ( W j m + 1 ) P(B j m) = 1 ˆρ i + ˆρ i W j + 1 i=1 is the probability that all backoff values extracted at distance j are greater than or equal to m.

15 MESSAGE BROADCASTING IN WIRELESS VEHICULAR AD HOC NETWORKS Performance Evaluation. In this section we first assess the impact of the assumptions made while developing our analytical models; then we evaluate the performance of the mechanisms proposed in Section 3, using the analytical framework. We validate our models against detailed simulation experiments with ns [14]. We consider that the minimum distance between vehicles is = 5 m. The normalized coverage radius is fixed to r = 9. The MAC protocol considered in our experiments relies on the standard DCF as implemented in ns. The backoff slot duration is σ = 20 µs. The message payload is equal to 32 bytes. The total transmission time of a broadcast message, including all physical and MAC layer overhead, is set equal to 150 µs. Considering that each station must wait for a time DIFS = 50 µs before accessing the channel, the total transfer delay of a message (excluding the time spent in contention) is 200 µs, corresponding to T = 10 slots. Notice that broadcast messages are not acknowledged and are never retransmitted, thus they are lost in case of collision. The implementation of our access scheme in ns requires only the following minor modifications to the simulator code. (i) When a node receives the message for the first time, it extracts a backoff value that depends on the distance from the sender (to allow for spatial differentiation). Moreover, the node keeps track of the sequence number and of the direction of arrival. (ii) If the node receives another copy of the message, it checks the direction of arrival: if this is the same as the first copy, it extracts a new backoff based on the distance from the transmitter; otherwise, the node stops its transmission attempt The impact of power capture. Here we assess the impact of power capture effects on the delivery delay of the broadcast messages. Indeed, since our model is based on the simplifying hypothesis of perfect capture (which basically implies no collisions), it is important to assess the impact of this assumption 4. We denote by C th the power capture threshold (in db), i.e., the minimum ratio between the power of the strongest signal and that of the background noise plus interference that allows the receiver to lock on the strongest signal and correctly decode it. Perfect capture corresponds to C th = 0 db. As a more realistic value of this parameter, we choose C th = 6 db. We consider the case of occupation probability ρ = 1, the worst case interference scenario. We assume that the signal power decays exponentially with the distance as d α, where α is equal to either 2 or 4 (the transmission power is modified accordingly so as to have a constant coverage radius r). Furthermore, we assume two different values of contention window size, W = 31 and W = 7, assuming for now that they are fixed. Figure 5.1 reports the results of our simulation experiments. We observe that power capture effects have almost no impact when the contention window size is W = 31. This essentially occurs because in this case there are few collisions among contending nodes. When W = 7, the collision probability is higher, and so is the impact of power capture effects, which also depends on the power loss exponent α. In general, small values of W make the propagation of the message faster only under the hypothesis of good reception capabilities of the vehicles, because of the increased interference due to simultaneous transmissions. This is an additional motivation to introduce spatial differentiation so as to reduce the number of collisions. We conclude that, when the collision probability is low, the assumption of perfect capture is indeed acceptable. 4 We modified the ns code to properly simulate power capture effects

16 16 C. F. CHIASSERINI, R. GAETA, M. GARETTO, M. GRIBAUDO AND M. SERENO Maximum distance reached (m) C th = 0 db - CW = 7 C th = 6 db - α = 4 - CW = 7 C th = 6 db - α = 2 - CW = Time (ms) C th = 0 db - CW = 31 C th = 6 db - α = 4 - CW = 31 C th = 6 db - α = 2 - CW = 31 Fig The impact of power capture effects on broadcast delivery delay 5.2. The case of occupation probability ρ = 1. We now consider that all spatial positions along the road are occupied by vehicles, i.e., ρ = 1, and evaluate the performance of the proposed broadcast mechanism by using the Gaussian approximation presented in Section 4.5. In this case the message is never blocked because of lack of connectivity (P B [y] = 0, y), and eventually reaches all positions with probability 1. In this case the Gaussian approximation produces the best results, almost indistinguishable from simulation results as soon as the message has traveled a few hops. Maximum distance reached (m) ns - CW = 31/15/7 model - CW = 31/15/7 ns - CW = 7 model - CW = 7 ns - CW = 31 model - CW = Time (ms) Fig Average value of the maximum distance reached by the message as a function of time when ρ = 1, for three different contention window schemes In Figure 5.2 we compare the average value of the maximum distance (E[M(n)]

17 MESSAGE BROADCASTING IN WIRELESS VEHICULAR AD HOC NETWORKS 17 in (4.13)) reached by the message as a function of time in three different cases: (i) fixed contention window W = 31; (ii) fixed contention window W = 7; (iii) spatial differentiation according to the following rule, hereinafter called 31/15/7 scheme: W l = 31 if 1 l 3; W l = 15 if 4 l 6; W l = 7 if 7 l 9. In the plot, the curves representing analytical results are overlapped with the ones referring to simulation results (under the hypothesis of perfect capture), thus the two sets of curves cannot be distinguished. We observe that the spatial differentiation approach outperforms the others schemes, making the propagation of the broadcast message along the road faster ns model P M [y,n] P A Distance (m) Fig Distributions P M [y, n] sampled every 50 slots = 1 ms. Comparison between analysis and simulation Figure 5.3 reports instead the distribution P M [y, n] of the maximum distance reached by the message sampled every 50 slots (or, equivalently, 1 ms). In the plot, there are multiple curves, each of them corresponding to a different sampling time. We observe that the Gaussian approximation is very accurate already after 1 ms (first peak). As expected, the variance of the distribution increases with the passing of time, and the model captures this behavior perfectly. The approximation of the marginal probability P R [y, n] is instead shown in Figure 5.4, again sampled every 1 ms. Also for this performance metric, the prediction based on the Gaussian approximation produces an excellent match with simulation results The case of homogeneous occupation probability, ρ < 1. We now consider the more complex case of ρ < 1, under the assumption that the occupation probability is homogeneous along the road. We first study the block probability P B [y], which is a static metric that does not depend on the access scheme employed, but only on the occupation probability ρ. Figure 5.5 compares the block probability predicted by the model (Section 4.3) and the one measured on simulation, for four different values of ρ. The agreement is excellent for all values of y, since (4.3) is exact under the hypothesis that the message stops propagating only because of lack of connectivity, which is indeed the case. Except close to the source, we observe that under homogeneous occupation probability the

18 18 C. F. CHIASSERINI, R. GAETA, M. GARETTO, M. GRIBAUDO AND M. SERENO ns model P R [y,n] P R Distance (m) Fig Approximation of the marginal probability of first reception, P R [y, n], sampled every 50 slots = 1 ms block probability decays approximately geometrically with the distance ns model - ρ = 0.4 model - ρ = 0.3 model - ρ = 0.2 model - ρ = 0.1 Block Probability, P B e Position, y Fig Block probability P B [y] for four different values of occupation probability ρ Next we focus on the case ρ = 0.3, and compare the delivery delay of the broadcast message under different access schemes. Again, we use the Gaussian approximation as explained in Section 4.5. Figure 5.6 reports the average value of the maximum distance (E[M(n)] in (4.13)) reached by the message as a function of time, for the three access schemes already considered in Figure 5.2. Since the message soon or later stops propagating because of lack of connectivity, the maximum distance saturates to a maximum value independent of the access scheme, at about 350 m from the source. We also observe that