A Delay Analysis of Sender-Initiated and Receiver-Initiated Reliable Multicast Protocols

Transcription

1 A Delay Analysis of Sender-Initiated and Receiver-Initiated Reliable Multicast Protocols Miki Yamamoto y 1 James F. Kurose z 2 Donald F. Towsley z 2 Hiromasa Ikeda y y Department of Communications Engineering z Department of Computer Science Faculty of Engineering, Osaka University University of Massachusetts 2-1 Yamadaoka, Suita, Osaka 565 JAPAN Amherst, Massachusetts 13 U.S.A. Abstract A growing number of network applications require the use of a reliable multicast protocol to disseminate data from a source to a potentially large number of receivers. This paper presents an analytic performance analysis of the packet delay incurred under three generic sender- and receiver-initiated approaches towards reliable multicast. We focus on the host processing requirements of these protocols and derive expressions for average time between the initial arrival of a packet at a sender and its correct reception at a randomly chosen receiver. Our numerical results indicate that a NAK-based protocol that limits NAK generation by intentionally and randomly delaying NAK packets can achieve substantially higher throughput than the other two protocols examined, and can do so without suffering an appreciable higher delay over a range of system parameters. 1 Introduction A growing number of network applications require the use of a reliable multicast protocol to disseminate data from a source to a potentially large number of receivers. These applications include whiteboard teleconferencing [1], distribution of financial and billing data [4], and distributed interactive simulation. An earlier paper by Pingali et al.[6] identified two basic approaches towards reliable multicast sender-initiated ACK-based protocols and receiver-initiated NAK-based protocols and examined the maximum achievable sender and receiver throughputs for these approaches. It was shown that as the number of receivers grows large, receiver-initiated protocols provide substantially higher throughput than their sender-initiated counterparts when end system processing is the resource 1 This author was visiting the University of Massachusetts during the academic year, supported by a grant from the Ministry of Education, Science and Culture of Japanese Government. 2 This research was supported in part by the Defense Advanced Research Projects Agency under contract F C-146 and the National Science Foundation under grant NCR under consideration. This present paper builds on the work in [6] by presenting a delay analysis of the three generic sender- and receiver-initiated protocols similar to identified in [6]: A (a sender-initiated ACK-based protocol), (a receiverinitiated NAK-based protocol in which NAKs are returned to the sender via a point-to-point channel), and (a receiver-initiated NAK-based protocol in which receivers multicast NAKs to the sender). Our results indicate that the protocol, which limits NAK generation by intentionally and randomly delaying NAK packets, can achieve substantially higher throughput than the other two protocols examined, and can do so without suffering an appreciable higher delay over a range of system parameters. Our work differs from previous work in reliable multicast protocols in two important ways. First, our goal here is not to propose a specific reliable multicast protocol but rather to focus on the fundamental differences between sender- and receiver-initiated protocols and the sensitivity of their delay behavior to system parameters such as the number of receivers, network propagation delay, and timer values. Second, we focus on end-system protocol processing as the crucial resource. We expect the processing costs of end-system protocols to become increasingly important as network bandwidths continue to increase, particularly when there are many participants in a multicast group. The remainder of this paper is structured as follows. Section 2 presents a description of the A,, and protocols. Section 3 describes our network model and presents an analysis of the reliable multicast packet delay the time between the initial generation of a packet at the sender and its successful reception at a a randomly chosen receiver under the three generic protocols. Section 4 provides a numerical comparison of the protocols performance under various scenarios. Section 5 concludes this paper.

2 2 Three Generic Approaches Towards Reliable Multicast In this section we present three generic approaches [6] towards reliable multicast. We emphasize that our intent here is not to present specific new reliable multicast protocols, but rather to identify generic protocols that capture important general characteristics of ACK- and NAK-based protocols. We describe these protocols assuming a single sender and R receivers. However, the protocols below are equally applicable in the case where an end system can both send and receive data. 2.1 A Sender-Initiated Protocol: A In the sender-initiated protocol, denoted A, each time a receiver correctly receives a packet, it sends a positive acknowledgment (ACK) to the sender. The sender maintains a list of the receivers from which it has received an ACK for each packet. A sender timeout mechanism is used to recover from lost packets. Specifically, the sender starts a timer when a packet is transmitted and, if the timer expires prior to it having received ACKs for this packet from every receiver, the sender retransmits the packet and restarts the timer. Most traditional point-to-point reliable data transfer protocols (e.g., TCP, HDLC, TP4) and many early multicast/broadcast protocols [8, 7] follow this ACK-based approach. Rather than focus on a specific protocol, we will instead focus on a generic protocol which exhibits the following behavior: Whenever the sender transmits or retransmits a packet, it multicasts it to all receivers and starts a timer. Whenever a receiver receives a packet correctly, it transmits an ACK to the sender over a point-to-point connection. A timer is associated with each packet that has not been acknowledged by all receivers. Whenever a timer expires, the sender re-multicasts the corresponding packet (only) to the receivers. This protocol can be optimized in numerous ways, such as grouping ACKs for different packets into a single ACK and hierarchically grouping receivers [8, 5]. We will not consider such optimizations here. 2.2 A Receiver-Initiated Protocol: Receiver-initiated protocols place the responsibility for ensuring reliable packet delivery on the receivers. The sender transmits an initial copy of a packet and will only retransmit that packet if it receives a negative acknowledgment (NAK) from a receiver. When this occurs, the sender then re-multicasts the NAKed packet. Under protocol, NAKs are generated by a receiver and sent point-to-point to the sender when it (the receiver) discovers that the sender has transmitted a packet that it (the receiver) has not yet received. This occurs when the receiver first receives a packet with sequence number i +1 without having received packet i: As we will see, the time between the transmission of packet i and packet i +1at the sender thus plays an important role in determining the delay until a receiver realizes that it has lost packet i: A number of researchers have proposed having the sender multicast another copy of packet i (or its sequence number), even though no NAKs are receiver, whenever the delay between the arrival of packet i and i +1exceeds some threshold. In the following, we will ignore these forced retransmissions in our analysis although they can be accommodated rather easily. In order to recover from the of a NAK or the subsequently retransmitted packet, a receiver uses a timer, as discussed below. The first generic receiver-initiated protocol,, operates as follows: The sender multicasts all packets. Whenever a receiver detects a lost packet, it transmits a NAK for that packet to the sender over a point-topoint channel and starts a timer. The expiration of a timer without prior reception of the corresponding packet causes the receiver to generate another NAK, which is sent point-to-point to the sender. The timer is also restarted. Whenever the sender receives a NAK, it re-multicasts the packet to all receivers. 2.3 A Second Receiver-Initiated Protocol: The protocol presented below is a variation of the protocol discussed above. As suggested by Ramakrishnan and Jain [7] for a LAN and Jacobson and Floyd [1] for a WAN, it limits the number of NAKs generated using a NAK suppression mechanism. With NAK suppression, each receiver multicasts its NAK only after waiting a random amount of time after detecting packet. If a receiver receives a NAK from another receiver while waiting to transmit its own NAK for that packet, it cancels its pendingnak. thus has the following behavior: The sender multicasts all packets. Whenever a receiver detects a lost packet, it schedules a pending NAK transmission at a randomly chosen point of time in the future. In the meantime, if the receiver receives a NAK (generated by another receiver) for this packet, it cancels the transmission of its own pending NAK. This is called NAK suppression.

3 If, at the scheduled time for a pending NAK transmission, a NAK for this packet has not been received since the pending NAK transmission was first scheduled, the receiver multicasts a NAK to the sender and all other receivers, and starts a timer. Upon receipt of a NAK for a packet that the receiver has not yet received, but for which it has initiated the random delay prior to sending a NAK, the receiver sets a timer and behaves as if it had sent that NAK. The expiration of a timer without prior reception of the corresponding packet causes the receiver infer that a packet has been lost, and operate as in item 2 above. Whenever the sender receives a NAK, it re-multicasts the packet to all receivers. As with the sender-initiated protocols, there are numerous optimizations possible. For example, it is possible for a receiver that has correctly received a packet to itself retransmit a copy of the packet [1]. See [7, 1] for additional discussion of possible optimizations. 3 Delay Analysis of Reliable Multicast Protocols The goal of the analysis outlined in this section is to determine the delay between the initial arrival of a packet at the sender and its correct reception at a randomly chosen receiverunderthe A,, and protocols. As we will see, this delay will be influenced by many factors, including the processing delays experienced by a packet at the sender and receiver, the end-to-end network propagation delay, the packet probability, and per-protocol parameters (e.g., the interval over which NAKs are delayed in ). The delay analysis of each of the three protocols has two major steps: 1. In the first step, we first determine the packet arrival rates for the different types of requests (data packets plus ACK or NAK packets) at the sender and the receivers for a given protocol. Given this arrival rate, an M/G/1 queueing model of the sender and receiver gives the average delay of a request at that host: the time from the arrival/generation of a packet at a host to the completion of its processing at that host. Here, processing refers to the actions taken when a packet is sent and/or received. For example, the arrival of a NAK at the sender may result in the retransmission of a packet; NAK processing would thus include the host protocol processing needed to receive the incoming NAK and transmit the requested data packet. Packet arrival rates and sojourn times are derived in section 3.2 for the three protocols under study. 2. In the second step, we focus on a specific data packet when it first arrives at the sender for transmission to the senders, and analyze the sequence of message exchanges between the end systems that ultimately results in the data packet being successfully received at a randomly chosen receiver. The delays associated with these message exchanges together with the packet processing delays determined in step 1 together yield the overall delay between the initial arrival of the packet at the sender and its successful reception at the randomly chosen receiver. The delay analysis of the A, and protocols is detailed in sections 3.3, 3.4 and 3.5 respectively. 3.1 Network Model Our network model is quite simple. As noted above, we assume a single sender and R receivers. New packets to be sent to the receivers arrive at the sender according to a Poisson process with rate : We assume that a packet sent by the sender is lost by a receiver with probability p; which is the same for all receivers. In our ensuing analysis, this assumption can be relaxed without significant conceptual difficulties. We also assume that packet es are independent in both space and time. That is, of a particular packet is independent among receivers and independent of the of any previously transmitted packet. [9] discusses the spatial and temporal correlation observed in the MBone, an operational multicast network overlayed on the Internet, and observes that a simple model of spatial correlation is often appropriate. We also assume that ACKs and NAKs are never lost. This assumption can be relaxed by following the analysis given in [8]. Finally, we assume that all participants in the multicast are separated from each other by a delay of : 3.2 Processing Model We assume that the time required to service a data packet (either receive or send) is a random variable and that the time to process an ACK or NAK packet (either receive or send) is a random variable Y and that they have means and second moments E[], E[ 2 ], E[Y ] and E[Y 2 ]. In section 4, we discuss our measured values for these random variables. 3.3 Mean Delays at Sender and Receivers The flow of packets among the sender and receivers is shown in Figure 1 for the A and protocols and will be discussed shortly. The flows in are also similar to, except that receivers receive no NAKs via the network. Important quantities that arise throughout the analyses are - the number of times a packet must be transmitted before receiver i receives it correctly. When there is no confusion, we will drop the subscript i. M (r) has M (r) i

4 λ R S λ timeout λ λ S original data pkts a ACKs Receiver 1 ACKs data pkts (original and retrans) y multicast network A Sender λ R Receiver R ACKs λ 1 n,g λ 1 n,r λ S n data and NAKs Receiver 1 NAKs λ S r λ1 t NAKs λ data pkts (original and retrans) y multicast network original data pkts Sender Figure 1: Packet flows under protocols A and Receiver R distribution P [M (r) = m] =(1 p)p m 1, m1, mean E[M (r) ]=1=(1 p). M = max 1rR M r (r), the number of transmissions required until all receivers receive the packet, it has distribution P P (M m) =P(M (r) m) R =(1 p m ) R and mean 1 E[M ]= m=1 (1 (1 pm 1 ) R ) Mean Waiting Times Under Protocol A Consider first the sender under protocol A. Work performed by the sender comes from one of three sources. There are those packets that arrive for first time transmission at the sender. This flow, denoted S t ; has rate : There are also those packets that are retransmitted due to a timeout. Since each packet is transmitted on average E[M ] 1 times, the rate for this work flow, S timeout ; is (E[M ] 1): Packets in both of these flows have a service requirement of ; as noted above. Finally, the arrival rate of ACK packets to the sender, S a ; is RE[M ](1 p); with each request having a service requirement Y. Recall that a receiver always acknowledges a received packet, even it has already successfully received that packet. Given these flows arriving at the sender, the total load on the sender then is S A = E[M ]E[] +RE[M ](1 p)e[y ] If we assume that the arrival processes are Poisson and that service times are independent random variables, independent of the arrival process, then the system can be modeled as an M/G/1 queue. The mean waiting time for a request (data (re)transmission or ACK) waiting to be processed at the sender under protocol A, E[W S A ],isgiven (using the Pollaczek-Khinchine formula, [3, p. 191]) by: NAKs E[W S A ]= (S t +S timeout )E[ 2 ]+ S a E[Y2 ] 2(1 S A ) : Let us now focus on the receivers under A. At each receiver, there is only a single arriving flow, corresponding to the arrival of data packets, with rate R = E[M ](1 Servicing such a request corresponds to receiving the packet followed by the transmission of an ACK; hence the service time is + Y. The load on the receiver is then R A = E[M ](1 p)(e[]+e[y ]) and, under the assumption of Poisson arrivals, the mean waiting time, E[WA R ] is p): E[W R A ]=R (E[ 2 ]+E[Y 2 ]+2E[]E[Y]) 2(1 R A ) : Note that E[( + Y ) 2 ]=E[ 2 ]+E[Y 2 ]+2E[]E[Y] when and Y are independent random variables. Note that in deriving the equations for the flows at the sender, we have implicitly assumed that the timeout interval at the sender is always greater than the round trip propagation delay plus processing delay at the receivers, i.e., that there are no premature timeouts. We are currently investigating how to relax this assumption Waiting Times under Protocol In [6], the assumption was made that the sender would never retransmit a packet more than once in response to NAKs due to the of the previous retransmission, i.e., it was assumed that the sender could determine whether or not a received NAK was a duplicate NAK. We assume this to be true in this paper as well but note that this is an optimistic assumption, since in practice it is difficult for the sender to determine whether or not a just-received NAK will be satisfied by its most recent retransmission or whether another retransmission should be made. Techniques such as having a receiver increment a count field in a NAK packet each time it generates a NAK for a given data packet, or having estimates of the delays between the source and the receivers [1] can help a source avoid retransmitting in response to duplicate NAK s. In [1] we also treat a pessimistic scenario where every NAK triggers a retransmission. Under this scenario, the sender processes three flows. The first corresponds to the original transmissions. The second corresponds to the arrival of NAKs that trigger retransmissions and the third corresponds to the arrival of NAKs that are processed but do not generate a retransmission. The corresponding flow rates are S t = ;

5 S r = (E[M ] 1); a i timeout timeout S n = (RE[M (r) ] E[M] R +1): Here the subscripts t and r refer to the original transmissions and retransmissions respectively. The service times are, + Y and Y respectively. The load at the sender is then S = (E[]E[M ]+R(E[M(r) ] 1)E[Y ]) (1) WA S T S WA S sender receiver R S A W A S time τ WA R Y r i and the mean waiting time at the sender, E[W S ],is E[W S ] = f S t E[ 2 ]+ S r (E[2 ]+E[Y 2 ]+2E[]E[Y]) + S n E[Y 2 ]g=2(1 S ) (2) There are two work flows through the receiver under. One corresponds to the arrival of data packets from the sender and the other corresponds to the self-generation of NAK packets by the receiver. The respective arrival rates are R t = E[M ](1 p); R n = (E[M (r) ] 1): The respective service times are and Y. The receiver load is thus R = (E[]E[M ](1 p) +(E[M(r) ] 1)E[Y ]) and the mean wait time at a receiver, E[W R ] is E[W R ]= R t E[2 ]+ R n E[Y2 ] 2(1 R ) Waiting Times Under We next consider packet flows under protocol. Recall that has a more complicated NAK generation policy in which receivers needing to generate a NAK wait a random amount of time before doing so. The receipt of a NAK generated by another receiver causes the pending NAK transmission to be cancelled. As with our analysis of, we will again make an optimistic analysis of, assuming that a sender never retransmits a packet more than once in response to NAKs generated in response to the of the previous retransmission. Define the random variable N to be the total number of NAKs received for packet i by the sender under. Since NAKs are multicast, this is also the total number of NAKs of packet i either transmitted or received by a receiver under. In [1], we derive the value of E[N ]: Figure 2: Delay components of the A protocol Given this value and following similar reasoning as in the case of the optimistic protocol, we have the following expressions for the three flow rates into the sender, S t = ; S r = (E[M ] 1); S n = E[N ] S r : The service times are, + Y and Y respectively. The load and mean waiting time at the sender under, S and E[W S ]; are calculated using (1) and (2) (as in ) using the values of S t ;S r and S n immediately above. There are three flows at the receiver with rates R t = E[M ](1 p); R n;g = 1 R E[N ] R n;r = R 1 R E[N ]: R n;g and R n;r correspond to the rates at which the receiver generates NAKs and receives NAKs respectively. They each have service time Y: R t is the flow of data packets, with service time. Given these flows, the receiver load is R = (E[]E[M ](1 p) +E[N ]E[Y]) and the mean waiting time at the receiver under, under the assumption of Poisson flows, is E[W R ]=R t E[2 ]+( R n;g + R n;r )E[Y 2 ]) 2(1 R ) : 3.4 Overall Delay Analysis of Protocol A Having derived the rate of packet flows among the senders and receivers under the three protocols, we now consider the overall delay analysis of protocol A. Figure 2 shows the operation of protocol A. Packet i first arrives at the sender at time a i. If there are other packets (data or ACK) waiting for processing at the sender, packet

6 a i a i+1 S WS S W WS W data data data detection phase, D NAK S NAK R W Y R W Y R W T R r i recovery phase, R Figure 3: Delay components of the protocol i waits for processing under FCFS scheduling and is eventually sent. After being lost twice in this example (and having to wait for retransmission processing after each timeout) the packet is received correctly at the receiver at time r i. Recall that the sender timeout delay and propagation delay have fixed values T s and respectively. The probability that a randomly chosen receiver requires exactly j transmissions of the packets is p j 1 (1 p). Given these considerations, the mean waiting times (E[W S A ] and E[W R A ]) computed in section 3.3.1, and the service requirements and Y, the mean time between a packet s initial arrival to the sender and its successful receipt at a receiver under protocol A is given by E[S A ] 1 j=1 p j 1 (1 p)(j(e[w S A ]+E[]) + (j 1)T S) + + E[W R A ]+E[] = (E[WS A ]+E[]+pT S) (1 p) + + E[W R A ]+E[] 3.5 Delay Analysis of Protocol Figure 3 illustrates the operation of protocol. In this example, packet i first arrives at the sender at time a i ; awaits processing at the sender, and is eventually sent. Let us focus on a randomly chosen receiver, which (along with K other receivers) does not receive this initial copy of packet i: Upon the reception of the first packet, j,where j>iat any of these K +1receivers, a NAK is generated. We define the time from the initial arrival of packet i at the sender to the generation of this initial NAK by any of the K +1receivers as the detection phase. We present an approximate expression for the average length of this phase in section Once the is detected, those receivers that detect the will continue to generate NAKs periodically until they eventually receive packet i: We define the time between the end of the detection phase and the successful reception 1 A of the packet at the randomly chosen receiver as the recovery phase. We present an approximation for the length of the recovery phase in section In section we combine the results of sections and to derive the overall packet delay under protocol Loss Detection Phase Conditioning on the fact that our randomly chosen receiver has lost packet i, we further condition on the number of other receivers K that have also lost packet i. As noted above, the detection phase ends when any of the K +1 receivers successfully receives a packet with a higher sequence number. Given that K = k, the conditional probability distribution for L, the number of subsequent packets (i.e., packets with higher sequence number) that are lost by the randomly chosen receiver and the other K receivers is P (L = l j K = k) =p l(k+1) (1 p k+1 );l =;:::;k = ;:::;R 1;and E(Lj K = k) =p k+1 =(1 p k+1 ): Once the L +1st packet (i.e., the first packet after i that is successfully received by one of the K +1receivers) arrives, it will experience a processing delay at the sender of E[W S ]+E[]; a propagation delay to the receiver of and a processing delay at the receiver of E[W R ]+E[]: We define the detection phase to end after the NAK is sent. Conditioning on the value of K, we have E[D ] = R 1 k= R 1 k p k (1 p) R k 1 pk+1 = 1 p k+1 +E[W S ]+E[WR ]+ +2E[]+E[Y]: We note that one simplifying assumption we have made in computing E[D ] is that instead of using E[W R ],a more accurate expression would replace this quantity with the mean minimum processing time among the j receivers out of the k +1receivers that detect the of the L +1st packet Loss Recovery Phase The recovery phase of seen by our random receiver, denoted R and shown in Figure 3, is essentially a mirror image of protocol A. The receivers send one or more NAKs to the sender and the sender replies with a retransmitted packet, that is lost at the random receiver with probability p. From the random receiver s viewpoint, the recovery phase thus begins with a number of receiver timeout rounds, each of length T R + W R + E[Y ] where T R is the length of the timeout period and W R is the waiting time (see section 3.3.2). The recovery phase ends with

7 a final propagation delay of the NAK to the sender, sender processing/retransmission of the data packet, and receiver processing of the received packet. Following our derivation of the delay of the A protocol, the mean recovery delay E[R ]: E[R ] 1 1 p j 1 (1 p)(j 1)(T R + E[W R A ]+E[Y]) j=1 +2 +E[W S ]+E[]+E[WR ]+E[] = p(t R+E[W R ]+E[Y]) (1 p) +2(E[]+) Overall Delay Analysis + E[W S ]+E[WR ] Given the results from and for the lengths of the detection and recovery phases, respectively, for the protocol, we may now easily derive the mean overall delay. With probability (1 p) the randomly selected receiver receives the initial transmission of packet i and experiences a delay equal to the sum of the sender processing delay, propagation delay, and receiver processing delay. With probability p the receiver loses the initial transmission of packet i and then experiences a delay equal to the sum of the detection and recovery phases, as discussed in sections and Given these considerations, the overall mean delay between the initial arrival of packet i at the sender and its successful receptions at the randomly chosen receiver, E[S ] is: E[S ] = (1 p)(e[w S ]+E[WR ]+2E[]+) +p(e[d ]+E[R ]) = E[W S ]+E[WR ]+2E[]+ + p2 (T R +E[W R ]+E[Y]) (1 p) + p(e[d ]+) Note that a simplifying assumption is that the K +1 receivers proceed in rounds after detecting the of packet i. Such synchronization is clearly idealized. It also overlooks the possibility that some of the K +1receivers that lost the packets but were not among the subset of the K +1receivers that initially detected the, might themselves detect the while the rounds are ongoing. 3.6 Delay Analysis of Protocol Figure 4 illustrates the operation of protocol, using an example similar to that in Figure 3. The delay between the initial arrival of packet i at the sender and its reception at a randomly chosen receiver under protocol, S ; divides into three segments: a i a i+1 WS data WS data detection phase, D R W r i NAK Y random delay Phase B S WS WS T R NAK recovery phase, R data R W Y R W Figure 4: Delay components of the protocol As in the protocol, the overall delay S begins with a detection phase of length D.The derivation of the mean detection period length under, E[D ] is identical to the derivation of E[D ] in section 3.5. for the protocol, except that E[W S ] and E[W R ] are used instead of E[W S ] and E[W R ]. Also, here we do not include the NAK transmission time in E[D ]. As in the protocol, S ends with a recovery phase of length R. The derivation of the mean recovery period length is identical to the corresponding derivation in section for the protocol, except that E[W S ] and E[W R ] are used instead of E[W S ] and E[W R ]. Unlike the protocol, the receivers in wait a random amount of time until generating a NAK. This gives rise to the random delay phase showninfigure 4. We denote the length of the random delay phase B and derive its mean, E[B ] in [1]. Given the above considerations, the mean overall delay between the initial arrival of packet i at the sender and its successful receptions at the randomly chosen receiver under protocol is: E[S ] = E[W S ]+E[WR ]+2E[]+ +pe[b ]+ p2 (T R +E[W R ]+E[Y]) + 1 p R 1 R 1 p p k (1 p) R k 1 pk+1 = k 1 p k+1 + k= E[W S ]+E[WR ]+2(E[]+) 4 Numerical Results The analysis from sections 3.4, 3.5, and 3.6 give the expected delay between the initial arrival of a packet at the sender and its successful reception at the randomly chosen

8 receiver under protocols A,, and respectively. We will refer to this measure simply as the expected delay. In order to choose realistic values for the various processing times associated with sending and receiving a data or ACK/NAK packet, we use the measured values reported in [2]. Those measurements show that packet send and receive processing times for a Linux Ethernet/IP/UDP stack to be approximately 5 secs on a 133Mhz Pentium. ACK/NAK send/receive times are approximately 1 secs. In the following, we normalize the unit of time to be the data packet send and receive times, i.e., E[] =1;and E[Y ]=:2:We assume these per-packet processing costs are constant (i.e., have no variability) and choose a normalized propagation delay of =2:;corresponding to a 1 msec propagation delay. Recall that an important parameter of the protocol is the backoff timer for NAK suppression. Figure 5 shows the expected number of NAKs generated by the receivers as a function of the backoff interval length (measured in units of propagation delay). As expected, the expected number of NAKs generated decreases with an increasing NAK suppression interval length since the receivers are randomizing their NAK transmissions over an increasingly longer period of time. Note that after an initial decrease, the expected number of generated NAKs does not decrease significantly as the backoff interval length is increased. In the case of a small probability (p = :1), the expected number of receivers that lose a packet is small and hence the number of generated NAKs is small, regardless of the receivers timeout interval length. We note that a backoff interval of length 1 is a reasonable value for all of the probabilities shown, and so we use this value in subsequent computations for R = 1: Figures 6, 7 and 8 plot the expected delay under each of the three protocols as a function of the arrival rate for R = 1 and various p. Figure 6, with an x-axis plotted logarithmically, shows the expected delay under the NAK-based protocols is high at very low arrival rates. This results from the fact that a receiver can only detect a packet when a packet with a larger sequence number arrives. At low arrival rates, this wait can be significant. For such low arrival rates, we see that A outperforms both and. As the arrival rate increases, and perform essentially identically (although achieves a slightly higher throughput). Figure 7 shows that for a larger probability of p = :1; and have comparable delays, up to the point where becomes saturated. Similar behavior is evidenced in Figure 8 for the case p =:25: Figure 9 plots expected delay versus the arrival rate for the case of R = 5 receivers and p =:1:As in the case of Figure 7, and perform similarly up to the point where saturates. Also, the ACK overhead associated p=.25 p=.1 p= Backoff interval length Figure 5: NAKs generated under versus receiver timeout interval p=.1 R=1 ACK lambda Figure 6: Average delay versus arrival rate, p =:1 with A and the larger number of receivers again severely limits A s throughput. Comparing Figures 7 and 9, we note that with the increased number of receivers, the relative size of the gap between the maximum throughput of and has increased noticeably. In summary, we conclude that can achieve substantially higher throughput than both and A, and can do so without suffering an appreciable higher delay due to delays associated with its NAK suppression mechanism. 5 Conclusion In this paper we have derived analytic performance models of packet delay of three generic sender- and receiver-initiated protocols. For the range of parameter values studied, our results showed that can achieve substantially higher throughput than both and A, and do so

9 ACK p=.1 R= lambda Figure 7: Average delay versus arrival rate, p =: ACK p=.25 R= lambda Figure 8: Average delay versus arrival rate, p =: p=.1 R=5 ACK without suffering an appreciable higher delay due to delays associated with its NAK suppression mechanism. Our future work in this area is aimed at modeling packet correlation in the network (a problem we have examined experimentally in [9]), examining adaptive NAK generation policies under, and studying the distribution of the delay under the various protocols an important consideration for real-time reliable multicast protocols. References [1] S. Floyd, V. Jacobsen, S. McCanne, C.G. Liu, L. Zhang, A Reliable Multicast Framework for Light-weight Sessions and Application-Level Framing, Proc. ACM Sigcomm95, (Aug. 1995, Boston MA), pp [2] S. Kasera, J. Kurose, D. Towsley, Scalable Reliable Multicast Using Multiple Multicast Groups, to appear in Proc. ACM SIgmetrics 97. [3]. L. Kleinrock, Queueing Systems: Part 1: Theory, Wiley- Interscience, [4] J.C. Lin and S. Paul, RMTP: A Reliable Multicast Transport Protocol, Proc. IEEE Infocom96, (San Francisco, March 1996), pp [5] S. Paul, K. Sabnani, D. Kristol, Multicast Transport Protocols for High-Speed Networks, Proc. IEEE Int. Conference on Network Protocols, (Boston MA, 1994). [6] S. Pingali, J.F. Kurose, D. Towsley, A Comparison of Sender-initiated and Receiver-initiated Reliable Multicast Protocols, Proc ACM Sigmetrics Conf., May 1994, ftp://gaia.cs.umass.edu/pub/ping94:multicast.ps.z. [7] S. Ramakrishnan and B. N. Jain, A Negative Acknowledgment with Periodic Polling Protocol for Multicast over LANs, Proc. IEEE Infocom 87, pp , Mar-Apr [8] D. Towsley, An Analysis of a Point-to-Multipoint Channel Using a Go-Back-N Error Control Protocol, IEEE Trans. on Communications, 33: , March [9] M. Yajnik, J. Kurose, D. Towsley, Packet Loss Correlation in the MBone Multicast Network, submitted to IEEE Global Internet Conference. ftp://gaia.cs.umass.edu/pub/yajn96:loss.ps. [1] M. Yamamoto, J. Kurose, D. Towsley, H. Ikeda, A Delay Analysis of Sender- and Receiver-Initiated Reliable Multicast Protocols, Technical Report, Dept. of Computer Science, Univ. Massachusetts, June ftp://gaia.cs.umass.edu/pub/yamamoto96:delay.ps lambda Figure 9: Delay versus arrival rate, R = 5;p=:1