An Adaptive Strategy for Maximizing Throughput in MAC layer Wireless Multicast

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1 University of Pennsylvania ScholarlyCommons Departmental Papers (ESE) Department of Electrical & Systems Engineering May 24 An Adaptive Strategy for Maximizing Throghpt in MAC layer Wireless Mlticast Prasanna Chaporkar University of Pennsylvania Anita Bhat University of Pennsylvania Saswati Sarkar University of Pennsylvania, Follow this and additional works at: Recommended Citation Prasanna Chaporkar, Anita Bhat, and Saswati Sarkar, "An Adaptive Strategy for Maximizing Throghpt in MAC layer Wireless Mlticast",. May 24. Copyright ACM, 24. This is the athor's version of the work. It is posted here by permission of ACM for yor personal se. Not for redistribtion. The definitive version was pblished in Proceedings of the 5th ACM International Symposim on Mobile Ad Hoc Networking and Compting 24 (MobiHoc 24), pages Pblisher URL: This paper is posted at ScholarlyCommons. For more information, please contact

2 An Adaptive Strategy for Maximizing Throghpt in MAC layer Wireless Mlticast Abstract Bandwidth efficiency of wireless mlticast can be improved sbstantially by exploiting the fact that several receivers can be reached at the MAC layer by a single transmission. The mlticast natre of the transmissions, however, introdces several design challenges, and systematic design approaches that have been sed effectively in nicast and wireline mlticast do not apply in wireless mlticast. For example, a transmission policy that maximizes the stability region of the network need not maximize the network throghpt. Therefore, the objective is to design a policy that decides when a sender shold transmit in order to maximize the system throghpt sbject to maintaining the system stability. We present a sfficient condition that can be sed to establish the throghpt optimality of a stable transmission policy. We sbseqently design an adaptive stable policy that allows a sender to decide when to transmit sing simple comptations based only on limited information abot crrent transmissions in its neighborhood, and withot sing any information abot the network statistics. The proposed policy attains the same throghpt as the optimal offline stable policy that ses in its decision process past, present, and even ftre network states. We prove the throghpt optimality of this policy sing the sffi- cient condition and the large deviation reslts. We present a MAC protocol for acqiring the local information necessary for execting this policy, and implement it in ns-2. The performance evalations demonstrate that the optimal strategy significantly otperforms the existing approaches in adhoc networks consisting of several mlticast and nicast sessions. Keywords Compter-Commnication Networks, Network Architectre and Design, Algorithms, Design, Wireless mlticast, MAC layer schedling, throghpt optimal policy, stability Comments Copyright ACM, 24. This is the athor's version of the work. It is posted here by permission of ACM for yor personal se. Not for redistribtion. The definitive version was pblished in Proceedings of the 5th ACM International Symposim on Mobile Ad Hoc Networking and Compting 24 (MobiHoc 24), pages Pblisher URL: This conference paper is available at ScholarlyCommons:

3 An Adaptive Strategy for Maximizing Throghpt in MAC layer Wireless Mlticast Prasanna Chaporkar Department of Electrical and Systems Engineering University of Pennsylvania Anita Bhat Department of Electrical and Systems Engineering University of Pennsylvania Saswati Sarkar Department of Electrical and Systems Engineering University of Pennsylvania ABSTRACT Bandwidth efficiency of wireless mlticast can be improved sbstantially by exploiting the fact that several receivers can be reached at the MAC layer by a single transmission. The mlticast natre of the transmissions, however, introdces several design challenges, and systematic design approaches that have been sed effectively in nicast and wireline mlticast do not apply in wireless mlticast. For example, a transmission policy that maximizes the stability region of the network need not maximize the network throghpt. Therefore, the objective is to design a policy that decides when a sender shold transmit in order to maximize the system throghpt sbject to maintaining the system stability. We present a sfficient condition that can be sed to establish the throghpt optimality of a stable transmission policy. We sbseqently design an adaptive stable policy that allows a sender to decide when to transmit sing simple comptations based only on limited information abot crrent transmissions in its neighborhood, and withot sing any information abot the network statistics. The proposed policy attains the same throghpt as the optimal offline stable policy that ses in its decision process past, present, and even ftre network states. We prove the throghpt optimality of this policy sing the sfficient condition and the large deviation reslts. We present a MAC protocol for acqiring the local information necessary for execting this policy, and implement it in ns-2. The performance evalations demonstrate that the optimal strategy significantly otperforms the existing approaches in adhoc networks consisting of several mlticast and nicast sessions. Categories and Sbject Descriptors C.2 [Compter-Commnication Networks]: Network Architectre and Design This research was spported in part by NSF grants ANI and NCR Permission to make digital or hard copies of all or part of this work for personal or classroom se is granted withot fee provided that copies are not made or distribted for profit or commercial advantage and that copies bear this notice and the fll citation on the first page. To copy otherwise, to repblish, to post on servers or to redistribte to lists, reqires prior specific permission and/or a fee. MobiHoc 4, May 24 26, 24, Roppongi, Japan. Copyright 24 ACM /4/ General Terms Algorithms, Design Keywords Wireless mlticast, MAC layer schedling, throghpt optimal policy, stability 1. INTRODUCTION Many crrent day wireless applications need mlticast commnication, e.g., sensor networks and military operations. Wireless commnication is inherently broadcast in natre, i.e. all nodes in the transmission range of a sender can receive a transmission from the sender. Hence, at the MAC layer, it sffices to transmit each packet only once in order to reach all the intended receivers. We focs on designing an optimal MAC strategy for wireless mlticast that tilizes the broadcast natre of the wireless medim. Thogh the broadcast natre of wireless transmissions provides a possible approach to improve the efficiency of the mlticast commnication, it also imposes varios difficlties. A mlticast specific challenge is that some bt not all the receivers may be ready to receive. For example, in Figre 1, when is transmitting to, can not receive the transmission from as both the transmissions will collide at. However,, and can still receive the transmission. The readiness state of a receiver depends on the network load. The policy decision is whether shold transmit or it shold wait till all the receivers are ready. A transmission policy that does not transmit ntil a sfficient nmber of receivers are ready may lead to nstable systems that have nbonded qee lengths at the senders. On the other hand, if the senders transmit when only a few receivers are ready, then the transmitted packet will be lost at the receivers that were not ready, which may reslt in low system throghpt. Ths, there is a tradeoff between system stability and the throghpt. The system clearly needs to be stable. The challenge therefore is to design a MAC layer transmission strategy that maximizes the system throghpt, while maintaining system stability. Frthermore, the optimm transmission strategy shold be sch that each sender decides whether or not to transmit sing (a) simple comptations, (b) no information abot system statistics, (c) limited control message exchange and (d) limited information abot its neighbors. We propose a transmission strategy that attains the above objectives (Section 3). First step in this direction has been to develop a model that captres the essential featres of a wireless network

4 R4 S1 R 3 R1 Transmission Range of S 1 Transmission Range of S 2 R 2 Figre 1: An example to demonstrate the advantages and the challenges associated with wireless mlticast. The figre shows two senders, and 5 receivers to. to are s receivers, and is s receiver. Dashed circle indicates the commnication range of a sender. (Section 2). We sbseqently se this model to obtain a sfficient condition to establish the throghpt optimality of an arbitrary stable policy. Using the model, we show that the proposed policy maximizes throghpt among all policies that stabilize the system [5]. The transmission decisions of the proposed policy are based on the qee length at a sender and the nmber of ready receivers. The first qantity is easily available at a sender. We propose a few MAC protocols that allow a sender to estimate the nmber of ready receivers (Section 4). We evalate the performance of varios mlticast schemes sing ns-simlations in a wireless network consisting of several mlticast and nicast sessions (Section 5). Simlation reslts show that the proposed optimal policy provides significantly higher throghpt than existing approaches. We present discssion on the model and the analytical reslts in Section 6. The previos research in wireless mlticast have lead to the development of transport and network layer protocols. End-to-end error recovery protocols address the isse of reliable loss recovery with minimm cost of information exchange among nodes, e.g., [4, 17]. Several mlticast roting protocols have been proposed at the network layer, e.g., [8, 13, 16, 19, 23]. Protocols for energy efficient mlticast roting have been proposed in [22, 29, 31]. Zho et al. have investigated content based mlticast in adhoc networks [32]. Nagy et al. have investigated mlticast in celllar networks [15]. The proposed transport and network layer protocols can work with any nderlying MAC layer strategy. Thogh efficiency of these higher layer schemes depends on the efficiency of MAC layer strategy, MAC layer mlticast has not been adeqately explored. Singh et al. have proposed a MAC protocol for power aware broadcast [22]. Wang et al. have proposed a schedling and power control protocol to minimize the transmission power at the sender nodes [3]. Jaikaeo et al. have stdied mlticast with directional antennas [11]. Kri et al. have proposed a protocol for reliable packet delivery in wireless LANs [12]. The design of the protocol is based on many assmptions that hold in wireless LANs bt not in ad-hoc networks. For ad-hoc networks, Tang et al. have proposed a nicast based mlticast scheme that transmits a packet to each receiver separately in rond robin fashion [26]. Crrently poplar mltiple access standard IEEE implements mlticast by broadcasting a packet after disabling all control messages. Ths, second hop interference is ignored. Tang et al. have also proposed the threshold- mlticast scheme where a sender transmits a packet whenever at least one receiver is ready to receive [24, 25]. The S2 R 5 nicast based mlticast policy does not exploit the broadcast natre of wireless medim, and its mltiple transmissions of a packet wastes power and bandwidth. The broadcast based mlticast and threshold-1 mlticast policies case packet loss at receivers becase several receivers may not be ready at the time of transmission. The broadcast based mlticast also cases packet collision de to second hop interference. We have proposed a throghpt optimal mlticast strategy in [6]. The parameters of this policy can only be compted with knowledge of the network load and statistics of the arrival process, which sender may not know. We now present an adaptive policy that maximizes the throghpt withot reqiring any sch information. We believe that MAC layer mlticast presents several research challenges and design isses that wold be of interest to both theoreticians and practitioners. Considering all sch isses is beyond the scope of a single paper. We, however, otline open problems in this area in Section 7. We hope that or reslts wold lay a basis for developing a flly operational MAC layer mlticast protocol, and stimlate frther research to address the open problems. De to the space constraints proofs for all the reslts are presented in [5]. 2. SYSTEM MODEL We consider a wireless network with several MAC layer mlticast and nicast sessions. All the nodes in the network need not be in each other s transmission range. Each mlticast session comprises of a sender and a set of receivers (mlticast grop). At the MAC layer all the receivers are within the sender s transmission range. We consider transmission of data traffic. Time is slotted. We assme that each packet can be transmitted in a single slot; this assmption can be relaxed easily. A major design challenge in wireless mlticast is that several existing approaches for optimizing system performance do not apply. Consider the objective of maximizing system throghpt in a network with senders generating packets at rates respectively. Consider only the policies that ensre correct reception of every packet by at least one receiver. Now, throghpt is the sm of the nmber of packets received correctly per nit time over all the receivers. The stability region of a transmission policy is the set of arrival rates for which the senders have finite expected qee lengths. The stability region of the network (denoted as ) is the nion of that of all transmission policies. In nicast and wireline mlticast, a policy maximizes throghpt if and only if its stability region eqals. The latter happens if there exists a lyapnov fnction that has a negative drift for the policy in. This property has been sed to show that back-pressre based policies maximize the stability region and hence the system throghpt in packet radio and wireline mlticast networks [27, 2]. This systematic approach cannot be sed in wireless mlticast as a policy that attains need not maximize the throghpt and vice-versa. Example: Consider Figre 1. When transmits, receive the packet withot any error; receives the packet only if is not transmitting simltaneosly. When transmits, receives the packet withot any error. Consider two transmission policies and. Under each sender transmits whenever it has a packet. Under transmits whenever it has a packet, while transmits only when is not transmitting. We assme that knows s transmission decisions, and in each slot a packet arrives at ( ) with probability ( ). Policy s stability

5 ' Arrival Rate λ S R1... R G R 2 Rest of the Wireless Network Figre 2: Figre shows a MAC layer mlticast session and its interaction with the rest of the network. Node is a sender, where packets arrive at the rate. Sender transmits the packets to receivers to. R 7 S 2 R 6 S 3 I3 R8 I S 2 1 R I1 9 R1 R 5 R4 R3 R2 Figre 3: Figre shows a mlticast session from to receivers to and two nicast sessions from to and to in a mlti-hop wireless network. First we observe that a single network layer mlticast session corresponds to many MAC layer mlticast sessions, e.g., mlticasts to intermediate nodes to, mlticasts to the receivers to, etc. Consider a MAC layer mlticast session from sender MAC layer receivers, and. Observe that when is receiving data, is not ready to transmit, bt all the receivers are ready to receive. Frthermore, when is transmitting to, receivers and are not ready to receive, bt is ready to transmit and is is ready to receive. Ths, readiness states of and are correlated. This is also the region is network s stability region as a sender can transmit only one packet in each slot. The network throghpt nder for arrival rates is Now, s stability region is! " which is a strict sbset of the network s and s stability region. The throghpt nder for arrival rates # is Ths, the throghpt nder is strictly higher than that nder for in Ths, nlike attains the stability region of the network, bt for certain arrival rates its throghpt is less than s throghpt. The above observation has two conseqences. First, we mst maximize the throghpt sbject to stability. In other words, we mst design a stable transmission policy that maximizes the throghpt among all the stable policies. Second, the existing framework does not apply. Therefore, we need a new design techniqe to attain the objective. We develop a sfficient condition for a policy to attain the objective sing an abstraction of a network which we describe next. Figre 2 shows a single MAC layer mlticast session, and represents its interaction with the rest of the network. De to the broadcast natre of wireless medim, transmissions from other nodes in the network affect the performance of the mlticast session and vice versa. The effect of the rest of the network on the mlticast session to is that the receivers are not always ready to receive. A receiver will not be ready when there are transmissions in its neighborhood or the transmission condition is poor, or when it is in a sleep mode. For example, in Figre 3 the receivers and will not be ready when is transmitting to %. Frther, the readiness states of different receivers are correlated in the same slot. The correlation across slots is de to brsty channel errors. The impact of the session on the rest of the network is that the sender s transmission interferes with simltaneos transmissions in its neighborhood. The interference is controlled as follows. The sender does not transmit if any node in its neighborhood is receiving a packet. For example in Figre 3, does not transmit when is transmitting a packet to %. Also, the sender backs-off jst after transmitting a packet so that other senders can se the shared medim. Ths, a sender is not ready when it backs off or a node in its neighborhood is receiving a packet. Ths, the effect of the session on the rest of the network is controlled by reglating the sender s readiness states. The readiness states of the receivers may be correlated with the readiness state of the sender. We consider a single mlticast session with & receivers, and model its interaction with the rest of the network by considering stochastic readiness states of the sender and the receivers. For example, in Figre 2 we only consider the sender and the receivers to, and assme that the readiness states are governed by a stationary and ergodic stochastic process. The readiness process in a slot is described by a & dimensional vector '+*,' ' )( ',-, where the component '.* is 1 if the sender is ready and it is otherwise. Frther for all /1, '.2 is 1 if the /4345 receiver is ready and it is otherwise. The packet arrival process at the sender is an irredcible, aperiodic and time homogeneos Markov Chain (MC) of 6 states. A state of the MC indicates the nmber of arrivals in a slot. Here 6 denotes the maximm nmber of packets arriving in a slot, and denotes the expected nmber of arrivals in a slot nder the MC s stationary distribtion. Next we present some definitions that will be sed in the rest of the paper. DEFINITION 1. A transmission policy is an algorithm at a sender node that decides when to transmit a packet. A necessary condition for a sender to transmit a packet is that it is ready to transmit, and it has a packet to transmit The class of transmission policies inclde offline strategies that se in the decision process a prior knowledge of packet arrivals and readiness states at all (inclding ftre) slots. DEFINITION 2. A reward for a packet is the nmber of receivers that receive the packet sccessflly. DEFINITION 3. System throghpt is the expected reward per nit time. DEFINITION 4. The packet loss at a receiver is the fraction of transmitted packets that are either not received or received in error at the receiver. The system loss is the sm of the packet losses at all the receivers in the mlticast grop. DEFINITION 5. A system is said to be stable if the mean qee length is bonded. Frther, a transmission policy that stabilizes the system is called a stable policy. Note that for any stable policy the packet departre rate is eqal to the arrival rate.

6 ( 3 ' DEFINITION 6. A stable transmission policy is called -throghpt We show that is -throghpt optimal nder some additional assmptions on the readiness process. We assme that optimal if no other stable transmission policy can achieve throghpt more than pls the throghpt nder. the readiness process is an irredcible, aperiodic and time homogeneos Markov Chain (MC) with arbitrary transition probabilities. The state of the MC is the & dimensional vector DEFINITION 7. The bsy slots are the slots in which the sender s qee is non-empty. '+*' ' - that represents the readiness state. Note that the DEFINITION 8. A policy belongs to the class of generalized threshold policies, if it sets threshold & in every bsy slot based on arbitrary rles and then transmits a packet only when the sender and or more receivers are ready. The threshold may be selected based on past, present and ftre arrivals and readiness states. We note that for any transmission policy, there exists a generalized threshold policy that transmits in the same slots as. This can be seen as follows. Let, sing certain rles, select slots, in which it transmits. Consider a generalized threshold policy that comptes slots, sing the same rle as and sets threshold in these slots. In the remaining bsy slots, sets threshold &. Ths, and transmit in the same slots. Hence, it is sfficient to consider only generalized threshold policies. In the following lemma, we provide a sfficient condition for a generalized threshold policy to be -throghpt optimal. Let denote the nmber of bsy slots in which threshold is chosen till time nder a generalized threshold policy. Note that it is not necessary to select a threshold when qee length is zero, as a packet cannot be transmitted in this case. LEMMA 1. For any, a stable generalized threshold policy is -throghpt optimal w.p. 1 if the following condition holds for some threshold. &. 3 & w.p. 1 (1) Lemma 1 does not show how to design an -throghpt optimal policy. Nevertheless it is a sefl tool as it provides a sfficient condition to establish the -throghpt optimality of a stable generalized threshold policy. The tility of the lemma is similar to that of a lyapnov fnction. Recall that a sfficient condition for a policy to be stable is the existence of a lyapnov fnction with negative drift. Bt this sfficient condition does not in general show how to design a stable policy. In the next section, we design an adaptive transmission policy that satisfies condition (1) and hence is -throghpt optimal. 3. THROUGHPUT OPTIMAL TRANSMIS- SION POLICY ( ) We describe a parameterized transmission policy, that we prove to be -throghpt optimal. The policy selects a threshold vale based on the qee length at the sender in each slot. A packet is transmitted if (a) the sender is ready to transmit, (b) the nmber of ready receivers is greater than or eqal to the threshold and (c) the sender has a packet to transmit. The threshold vales are selected as follows. Let denote the qee length at the sender and let be some fixed positive integer. For )&, the threshold is if & & and threshold is if &. Ths, the threshold vale increases with decrease in qee length. The policy does not select a threshold when qee length is zero. MC has a finite nmber of states, since '.2 for every / &. Since we do not impose any restriction on the transition probabilities of the markov chain, the chain can captre the correlations of the sender s and the receiver s readiness states in the same and different time slots. Figre 3 shows how sch correlations arise in practice. Let denote the niqe steady state probability that the sender is ready to transmit and! receivers are ready to receive. Let denote the nmber of slots till time in which the sender and! receivers are ready. Now, by stationarity and ergodicity of the readiness process 3 " w.p. 1 (2) In general, from ergodicity we cannot conclde anything abot the to #&% rate of convergence of the empirical distribtion 3 3' the stationary distribtion. Bt for finite, aperiodic MC s, empirical distribtion converges to the steady state distribtion exponentially fast [3, 21]. We se this exponential convergence to prove the optimality of. The optimality of holds for any stationary and ergodic stochastic process that has the exponential convergence property. *) %+ '. Let the The throghpt of policy is denoted as ( maximm throghpt attained by a stable policy be (-,. 3 the following reslt in the appendix. THEOREM 1. If the arrival rate. We prove is less than the steady state probability that the sender is ready (/ * 1 ), then for any given there exists 2 sch that *) is -throghpt optimal for every 32. Formally, (4,. 3 5( %+ ' " w.p. 1. Frther, no policy is stable if /6 *. Note that the above reslt implies that any stable off-line policy that takes transmission decisions based on the knowledge of past, present and ftre arrivals and readiness states can not at- *) tain throghpt more than ( %+ '. This holds even thogh takes transmission decisions based on only the crrent packet availability and the crrent nmber of ready receivers. The intition behind the reslt is as follows. Consider a policy that selects the same threshold in every slot. The expected reward is a monotonically increasing fnction of the threshold. Hence a throghpt optimal policy shold select the largest threshold that stabilizes the system. This threshold mst satisfy 7 ) " 7 ) (3) The throghpt can be frther improved by appropriately randomizing between the threshold vales and. The randomization shold be sch that the system remains stable. This is the basic idea behind the design of the static optimm policy [6]. Intitively an adaptive optimm policy shold select the thresholds and most of the time. The difficlty, however, is that and s are not known, and ths cannot be compted. Now, we explain why will select the thresholds and most of the time. From (3), the rate at which slots with 8 or

7 more ready receivers arrive is more than the packet arrival rate, for every 8. On the other hand, for 8 the rate at which the samples with 8 or more ready receivers arrive is smaller than. This implies that for threshold vales greater than or eqal to, i.e., when &, the qee length process has a positive drift and as a reslt the qee length increases, and conseqently the threshold decreases. On the other hand for threshold vales less than or eqal to, i.e., when &, the qee length process has a negative drift and hence qee length decreases, and the threshold increases. Hence we observe that are selected when is large enogh the thresholds and! most of the time. Recall that a packet is lost at a receiver if the receiver is not ready at the time of transmission. The MAC protocol we propose may transmit a packet even when some of the receivers are not ready, and is therefore nreliable. Bt, wireless is an inherently nreliable medim. Ths, it is a standard practice to se a reliable transport layer strategy to retrieve the information lost at the MAC layer. Several existing MAC strategies for mlticast in adhoc networks, like broadcast based mlticast and threshold-1 mlticast are nreliable as well. Fortnately, several reliable transport layer schemes have been proposed specifically for wireless mlticast transmissions, which can be sed in conjnction with any MAC layer strategy [4, 17]. Bt, the efficiency of these schemes is severely impaired when the packet loss at the MAC layer is high. Or focs is to minimize the packet loss sbject to resorce limitations in the network. Now, there wold not be any loss if a packet is transmitted only when all the receivers are ready, bt then as discssed before, the system may become nstable. Note that stability is essential as otherwise the qee lengths at the sender wold be nbonded leading to nbonded delays. Ths or objective is to se a transmission policy that minimizes the packet loss among all stable policies. The next theorem shows that achieves this objective. THEOREM 2. If is ( )-throghpt optimal, then no sta- mi- ble policy can achieve loss smaller than the loss nder ns for any given. Assme that the system is stable. The system loss is &, where is the average reward received by the policy per packet. Ths, we need to maximize the mean reward in order to minimize the system loss. The throghpt of a transmission policy is. Ths, a throghpt optimal policy maximizes and therefore minimizes the system loss sbject to system stability. Since is throghpt optimal, it minimizes the loss. From Theorem 2, if the system loss for is more than that the system can tolerate, then the reqired loss constraint can not be garanteed by any stable transmission policy. Since stability is essential, the resorces available in this case are not enogh to deliver the reqired QoS, and other measres sch as admission control mst be resorted to. This is beyond the scope of this paper. Henceforth we do not consider loss explicitly. The vale of 2 depends on the system parameters. However, since optimality of that is similar to except that it trans- selects threshold if & &) for and threshold 1 if &. Now, s and s is garanteed for all large only a rogh estimate of 2 is necessary. Simlations show that the optimality is attained for modest vales of. If the qee length is greater than & then has threshold and can therefore transmit a packet even when no receiver is ready; bt, in this case, the transmission is seless. So, we consider a policy mits only when at least one receiver is ready. Now, transmission rles are the same once the thresholds are selected. DEFINITION 9. is a class of policies that transmit a packet only when the sender and at least one receiver is ready, and the sender has a packet to transmit. Note that the broadcast policy is not in there THEOREM 3. If!3/6, then for any given exists sch that is -throghpt optimal for every 2. Frthermore, if /6, then no policy in is stable. The intition for Theorem 3 is similar to that for Theorem 1. The policy and provide comptationally simple transmission rles. The sender only needs the nmber of ready receivers and need not know which particlar receivers are ready. This simplifies the protocol design problem. 4. MAC LAYER PROTOCOL The optimal decision rle is based on the sender s qee length, its readiness state, and the nmber of ready receivers. The sender is ready if it is not backing off, and none of its neighbors is receiving a packet. We explore varios possible approaches to inform the sender abot the nmber of ready receivers and transmissions in its neighborhood. First we present the system challenges experienced by the existing IEEE hand-shakes in case of mltiple receivers. Recall that IEEE ses RTS-CTS-DATA-ACK handshake for commnication. The handshake works well if the sender transmits to only one receiver. If all the nodes in a mlticast grop send CTS simltaneosly in response to a RTS from a sender, then these CTS messages will collide at the sender. Hence, the sender will not know whether the receivers are ready. If the sender knew that the colliding messages were CTS responses from the receivers, then the sender can infer the nmber of ready receivers by measring the power. Power is additive and power information is not destroyed even dring collision. Bt the power measrement does not solve the problem entirely becase of the following reasons. The CTS message conveys the dration of the data transfer, so that the neighboring nodes can defer their transmissions. This is no longer possible if CTS collides and hence data packets collide with other transmissions in the neighborhood. Similar problems exist for ACK transmission. We propose to se a bsy-tone based scheme [28]. The available bandwidth is divided into two channels (a) a message channel and (b) a bsy-tone channel. A sender initiates a data transfer by sending an RTS message on the message channel. After receiving RTS, all the receivers in the mlticast grop that are ready to receive send a bsy-tone. This bsy-tone acts as a CTS message. A sender estimates the nmber of ready receivers by measring the power of the bsy-tone signal. The total received power can be measred sing standard circits [18]. If this estimate is greater than a threshold vale, the sender transmits the data packet. Receivers transmit the bsy tone ntil the packet transmission is complete. Bsy-tone does not interfere with the data transmission as both are on different freqency bands. Neighboring nodes defer their transmission till the

8 Procedre MAC at Sender() begin /* refers to the crrent vale of contention window */ /* and refer to the minimm and the maximm vales for respectively*/ When a packet arrives from a higher layer, then do the following Enqee the packet in the MAC layer bffer; qee length = qee length + 1; Compte threshold based on the qee length as described in Section 3; if (qee length = 1) then /*if the sender s qee was empty before the arrival*/ ; Choose an integer niformly from ; Set back-off timer for time dration! #"!%'&("*) ; When the back-off timer expires do the following if (data or bsy tone channel is not idle) then /* the sender is not ready */,+.-/ ; Choose an integer niformly from ; Set back-off timer for time dration! #"!%'&("*) ; else /* the sender is ready */ Send RTS packet; Measre the power on bsy tone channel; Estimate the nmber of ready receivers based on the measred bsy tone power; if (The estimate 78 ) then 5+.-/12 43,9 ; Choose an integer niformly from ; Set back-off timer for time dration :;< #" <%'&("*) ; else Transmit data packet; end qee length = qee length - 1; Compte threshold based on the qee length as described in Section 3; if (qee length = ) then > ; Choose an integer niformly from 9? ; Set back-off timer for time #"!%A&("*) ; Figre 4: Psedo code describes the bsy tone based MAC layer protocol at the sender. bsy-tone stops. If the decision is not to transmit, then the sender transmits a release signal and backs-off for a random interval. The back-off intervals are independent and niformly distribted. The receivers stop emitting the bsy tone once they receive the release signal. Refer to Figres 4 and 5 for psedo code. The sender can infer the nmber of ready receivers from the received bsy-tone power only if the received power levels are the same for all the receivers. This is not possible withot any power reglation as the receivers are at different distances from the sender ( near-far effect ). If all the receivers know their distances from the sender, then they can reglate the power so that the bsy tones from all the receivers reach the sender at the same power level. The receivers can estimate their distances from the power level of the received RTS, assming that all the senders transmit at the same power. By providing nambigos and instantaneos feedback, the bsy tone scheme solves the problems originating from CTS and ACK Procedre MAC at Receiver() begin When RTS intended for the receiver is received do the following if (Bsy tone and data channels are idle) then Pt p a bsy tone and wait for data packet; Set RcvTimer for the dration eqals to for times maximm propagation delay; if (Data packet begins to arrive before RcvTimer expires) then Receive the data packet and pass it to the higher layer; Stop the bsy tone; else Stop the bsy tone; end Figre 5: Psedo code describes the bsy tone based MAC layer protocol at the receiver. collisions in IEEE based mlticast schemes. The limitations of the bsy tone scheme are the following. (1) The scheme is not totally compatible with IEEE 82.11, thogh it works on the same philosophy. Hence the implementation of this scheme will need protocol changes. Bt some other recent papers have advocated bsy tone based schemes as well [9, 14, 22]. (2) Accrate power control and power measrement are difficlt in practice. Hence, the sender s estimate of the nmber of ready receivers may not be accrate. Using simlations, we have verified that the proposed optimal policy is robst, i.e., its throghpt is close to optimal even in presence of the estimation errors (Figre 9). Frther, ftre extensions for IEEE are likely to spport power control and power measrement fnctionalities [1]. We have also proposed an agmentation of IEEE protocol to implement [7]. The MAC protocol in [7] does not need any power measrement and control fnctionalities. (3) The receivers power consmption increase de to the transmission of bsy tone throghot the data transmission. (4) The receivers need to transmit a bsy tone signal while receiving the data. Ths, two radios are reqired at the receivers. Most of the crrent wireless appliances have a single radio and hence continos transmission of the bsy tone while receiving data may not be feasible. The problems (3) and (4) can be addressed with an additional bsy tone channel [9]. On one bsy tone channel, receivers can send a short brst of signal indicating CTS message, and on the other bsy tone channel receivers can send a short brst indicating ACK. All the nodes in the neighborhood will not initiate any commnication till they receive ACK bsy tone for every CTS bsy tone they received. 5. SIMULATION RESULTS AND DISCUSSION We have proved that is -throghpt optimal, when the readiness states are Markovian. We examine the application layer throghpt in a wireless network with several nicast and mlticast sessions when every sender ses, and the readiness states are generated de to packet transmissions. The simlation reslts sbstantiate the claim that attains significantly higher throghpt than the other existing policies. Using ns-2, we simlate the performance of and the other existing mlticast policies like broadcast based mlticast, nicast based mlticast and threshold-1 mlticast. In [2], we describe how

9 7 U7 6 8 U6 m 7 U8 m m 6 8 U5 m5 M m 3 U 3 3 m U U 4 m m 2 4 U U7 U6 6 m6 m 7 7 m 8 8 U8 U5 5 U4 m 5 4 m4 m 3 M U m m1 2 1 U1 U2 (a) Topology 1 (b) Topology 2 Figre 6: We evalate the the performance of varios mlticast strategies in Topologies 1 and 2. Each topology has a mlticast session with sender and 8 receivers 8 to 8 and 8 nicast sessions. Unicast session has sender and receiver!,. The difference between the topologies 1 and 2 is that the nicast receivers! s are not in the transmission range of in topology 1, while they are in the transmission range of in topology 2. we implement these policies in ns-2. We have not compared the performance of with that of the MAC policies proposed for wireless LANs, e.g. [12], becase they can only be sed in the scenario where a single node is in the transmission range of all other nodes in the network. This assmption does not hold in adhoc networks. 5.1 Simlation Scenario We se UDP at the transport layer. We do not se TCP, as the interaction between TCP and wireless MAC is not well nderstood and hence the topic of research even for nicast sessions [1]. We measre the throghpt of a receiver as the nmber of packets it receives sccessflly per nit time, and the throghpt of a session as the sm of the throghpts of its receivers. We consider a time interval of 5 seconds. The choice of the channel capacity scales the throghpt of all the policies by a factor of. We select 1 Mbps. The RTS packet has 44 bytes. The length of a slot is 2 s and the maximm propagation delay is 2 s. We se the bsy tone based approach described in Section 4 (Figre 4 and 5) to implement and threshold-1 mlticast. We consider sample topologies shown in Figre 6. In both the topologies, we assme that the nicast sender generates packets at rate and the mlticast sender generates packets at rate. The packet arrival process is Poisson. The packets arriving at have length 552 bytes, while those arriving at have length 53 bytes. For larger packet sizes at nicast sessions the reslts differ only in magnitde, bt the trends remain the same. Frther, stability region of the mlticast session is small when the packet size for nicast sessions is large. We, however, stdy the impact of high network load on the mlticast session by increasing (Figres 1 and 11). Now we discss how the packet transmissions generate readiness states in topologies 1 and 2. In topologies 1 and 2, for every, is ready when it is not backing off and 8 is not generating bsy tone. Also, 8 is ready when is not transmitting a packet to!. In topology 1, is not ready when it backs off, while! is always ready. In topology 2, is not ready and when some! is generating bsy tone. Frther, none of the nicast receivers are ready when is transmitting. 5.2 Discssion on Simlation Reslts We have proved that any stable policy that selects any two consective thresholds and ( & ) most of the time maximizes throghpt as long as the readiness states are ergodic (Lemma 1). The readiness states are likely to be stationary and ergodic even when they are generated by packet transmissions. We needed the additional assmption that the readiness states are Markovian to show that chooses two consective thresholds most of the time. Figre 7(a) shows that selects two consective thresholds most of the time even when the readiness states are generated by packet transmissions. This validates the optimality reslt. Now, Theorem 1 garantees that wold select two consective thresholds most of the time only when is sfficiently large. The observation in Figre 7(b) that freqently selects more than three thresholds for low vales of frther validates the theorem. Also, Figre 8 shows that as increases the throghpt of the mlticast session nder converges to optimm vale. Now, we compare the throghpt of the mlticast session in topologies 1 and 2 nder with that nder broadcast based mlticast, threshold-1 mlticast and nicast based mlticast policies. We observe that achieves mch higher throghpt than the other existing policies in both the topologies (Figres 1 and 11). We next explain the performance difference. The broadcast based mlticast scheme does not exchange any hand shake messages, and ths cases freqent data packet collisions. Ths, the reward per packet is low reslting in low system throghpt. Threshold-1 policy exchanges hand-shake messages and avoids the data packet collisions. As a reslt, this policy provides mch better throghpt than broadcast based mlticast policy. The limitation of this scheme is that the threshold is always 1, and hence the policy may transmit even when only a few receivers are ready. The policy otperforms these policies as it prevents data packet collisions by exchanging hand-shake massages,

10 1 Histogram for the Fraction of Time Each Threshold is Chosen 1 Histogram for the Fraction of Time Each Threshold is Chosen 8 8 Fraction of time (%) 6 4 Fraction of time (%) Threshold Vale Threshold Vale (a) (b) Figre 7: Figres (a) and (b) plot the fraction of time each threshold is chosen by in topology 1 for eqal to 25 and 2, respectively. Here, 5 packets/sec and 5 packets/sec. and it also prevents transmission when only a few receivers are ready by choosing an appropriate threshold vale. The nicast based mlticast policy ses separate transmissions to reach different members of the mlticast grop even when they can be reached sing a single transmission. Hence, the total nmber of packets transmitted nder this policy is mch smaller than that nder other policies. This reslts in low throghpt. From Figres 1(b) and 11(b), we observe that the throghpt gain of over the nicast based mlticast policy increases with increase in. This is becase the stability region of the nicast policy is small, and hence the throghpt of the policy satrates even for small vales of (Figres 1(a) and 11(a)). On the other hand, has large stability region. Now, transmits packets per second in its stability region, and this nmber nmber increases with increase in. As increases the threshold chosen by may decrease in order to maintain the system stability. Hence the reward per packet nder may redce. Bt this decrease is compensated by the increase in the nmber of packets transmitted per nit time, which also explains increases the non-linear increase in throghpt for as (Figres 1(a) and 11(a)). Now, the throghpt gain of over threshold-1 and broadcast based mlticast policies decreases with the increase in. This is becase as increases the thresholds chosen by decreases, becomes closer to 1. Ths the difference between the throghpt of and threshold-1 and broadcast based mlticast policy decreases. Hence the gain also decreases. From Figres 1(e) and 11(e), we observe that as increases the throghpt gain of over all the existing policies increases. This is becase for small vales of, the fraction of slots in which the nicast senders transmit a packet is small. Hence the mlticast receivers are ready most of the time reslting in a large reward per transmission nder all the policies. Ths, the choice of threshold does not affect the throghpt. Bt for higher vales of, large nmber of mlticast receivers are ready only in a small fraction of slots. Unlike other policies transmits only in these slots de to an appropriate selection of threshold. Hence the s throghpt gain increases with increase in. We also stdy the throghpt of in topology 1, when makes errors in estimating the nmber of ready receivers. We consider binomial errors and plot s throghpt for varios vales of the variance in Figre 9. Simlation reslts show that the throghpt of mlticast session nder decreases only marginally when the estimate is erroneos. The difference between s throghpt withot estimation errors and with estimation errors increases as increases. Frthermore, the decreased throghpt is still significantly higher than that of the other existing policies (Figre 9). Figres 1(c) and 11(c) show that in both the topologies the throghpt of nicast sessions is similar to or higher than that nder any other the policy. Ths, increases the throghpt of the mlticast session by sending more packets when the nicast sessions are not transmitting and not by decreasing the throghpt of the nicast sessions. Now, we explain why the difference between the throghpts nder threshold-1 policy and is small for all vales of in topology 2 (Figre 11(a)). Recall that in topology 2, and 8 are not ready when! is generating a bsy tone. Ths, when is ready, all the mlticast receivers are also ready. Hence, the reward per packet shold be 8 irrespective of the choice of threshold. In other words, and threshold-1 mlticast policy shold achieve the same throghpt. The throghpt vales differ by a small amont becase of the collision of RTS messages from and at 8. In this case, 8 will not pt a bsy tone, and it will not receive the data packet even if decides to transmit. Since chooses an appropriate threshold vale, it may not transmit when 8" does not pt p a bsy tone, and ths nlike threshold-1 mlticast policy which always has threshold 1, avoids packet loss at 8!. We note that this is an extreme case, and in general provides a significant throghpt gain over threshold-1 mlticast, e.g. Figre 1(b).

11 DISCUSSION We have designed a policy which is throghpt opti- is adaptive, and mal constrained to stability. The policy takes transmission decisions based only on the qee length at the sender and the receiver readiness states in the crrent slot. Hence, can be implemented in distribted settings. The optimality of also holds nder the following more general scenarios than that considered here. The analytical framework can be generalized to allow three or more readiness states for each receiver. Optimality of holds even when the readiness process is an irredcible, aperiodic, time-homogeneos discrete time markov chain with ' 2 states where the / th state is ( ' 2 - ' 2 is the probability of error-free reception of a packet at the th receiver in the / th state. Recall that earlier ' 2 Ths, the optimality of holds nder a more general fading model. We have considered packets of nit lengths. The optimality of holds for iid packet lengths. Lemma 1 holds for any stationary and ergodic receiver readiness process (not necessarily markovian). Also, Theorems 1 and 3 hold for any stationary and ergodic readiness process for which the empirical distribtion converges to the stationary distribtion at exponential rate. Now, let s examine the featres that are not captred by the analytical model proposed here. In the analytical model, we assme that the receiver readiness process cannot be controlled, and then the throghpt for a mlticast session is maximized constrained to stability nder the given readiness process. This approach is jstified if the senders do not coordinate with each other explicitly to decide transmission schedle. The following example shows that by appropriately coordinating the transmissions, it is possible to control the receiver readiness process and thereby achieve better throghpt. Example: Consider topology 1 in Figre 6(a). Let arrival rate at a nicast sender be for every and for small. Also, let arrival rate at the mlticast sender be. Note that simltaneos transmissions from all the nicast senders can be received correctly at their respective receivers. In this scenario, the throghpt is maximized if s coordinate among each other so as to transmit only in odd slots, i.e., in slots and transmits only in even slots, i.e., in slots. Here, s throghpt is &. Observe that s throghpt may be less than & if s do not coordinate, e.g., if transmits in slots for each and, then in most of the slots only &# (and not & ) recievers will be ready. Ths, s throghpt will be & approximately. Designing a coordination among different senders typically leads to centralized schedling schemes. For example, Tassilas et al. have proposed a centralized algorithm for determining the optimal coordination for maximizing throghpt in nicast wireless networks [27]. Since we are interested in a distribted implementation, we do not consider explicit coordination among senders. Throghpt (packets/sec) Throghpt of the Mlticast Session as a Fnction of Qee Theshold Qee Threshold Figre 8: We plot the throghpt of the mlticast session nder as a fnction of in topology 1. Here, + packets/sec and packets/sec. The parameter is referred as qee threshold in the figre Topology 1: Throghpt of Mlticast Session nder Varios Policies Error Varience v = 1 Error Varience v = 3 Error Varience v = Arrival Rate at Mlticast Senders (packets/s) Figre 9: We plot the performance of when the mlticast sender makes random error in estimating the nmber of ready receivers at time. We assme that is binomial random variable with mean and variance, and is positive with probability 1/2. 7. CONCLUSION Maximizing the performance metrics in wireless mlticast presents challenges that are not encontered in wireless nicast or wireline mlticast networks. For example, a transmission policy that maximizes the stability region of the network need not maximize the network throghpt. The objective therefore is to maximize throghpt sbject to attaining system stability. We consider a single mlticast session, and model its interactions with the rest of the network sing stochastic readiness states. We present a sfficient condition that can be sed to establish the throghpt optimality of a stable transmission policy. We sbseqently design an adaptive stable policy that allows a sender to decide when to transmit sing simple comptations based only on its local information sch as its qee length, its readiness state and the nmber of ready receivers, and withot sing any information abot the network statistics. The proposed policy attains the same throghpt as the optimal offline stable policy that ses in its decision process past, present, and even