1 Introduction. Abstract CH2702-9/S9/0000/05S4$ IEEE

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1 Delay Minimization of the Adaptive GeBack-N ARQ Protocole for Point-to-Multipoint Communication Jonathan L. Wang Bell Communications Research 3 Corporate Place Piscataway, New Jersey John A. Silvester Communications Sciences Institute University of Southern California Los Angeles, California Abstract This paper studies some data link layer error control go-back-n ARQ protocols suitable for point-to-multipoint communication over broadcast channels where data are delivered to the destinations in the order they are sent. We study a series of protocols differing in the way that the sender usea the outcomes of the previous transmissions. The protocols have been described in detail in a previous paper where we also performed analysis of the protocols to determine the optimal number of copies the sender should transmit to maximize the system throughput. In this paper, the system delay instead of the throughput is the o p timization measure. We determine the optimal number of copies the sender should transmit to minimize the time between when the sender first transmits a data frame and when the data frame is accepted by all the receivers. The results show that by sending the optimum number of copies of a data frame instead of just a single copy, the delay performance is significantly improved. 1 Introduction Real world data communication is subject to errors caused by various sources. Protocols for achieving efficient, reliable communication between a transmitter and the corresponding receiver(s) are thus very important in order to ensure communication quality. One of the widely used techniques for handling transmission error at the data link layer in data communication systems is error detection incorporated with automatic-repeat-request (ARB). The protocols fall into three basic categories: (1) Stopand-wait; (2) Go-back-N; and (3) Selective-repeat. Much work has been done to analyze these basic protocols [MMN], (RSYGl], (Stu631, [BAHFM], [BS72]. Generally speaking, the stopand-wait protocol suffers from inefficiency due to the fact that the channel is idle between the transmission of the message and the reception of the ACK/NACK from the receiver. This inefficiency is particularly serious when the round trip delay between the transmitter and the receiver is long compared to the transmission time of a message. On the other hand, the selective repeat protocol offers the best performance in terms of throughput among the three, but has an increased requirement for buffers at the receiver end. These buffers are needed to store those messages which are received out of sequence. In an environment where the error rate is high and the round trip delay is long, the number of these resequencing buffers required may be considerable. Therefore, in this paper, we concentrate on the go-back-n ARQ scheme. Variations to the three basic protocols have also been extensively studied [Chu74], [Sas75], [Mor78], [Mor79], TOW^^], [SirSl], [YLSO], [YL81], [EJW82], [CL84], [BM86]. One main idea of these variations is to send multiple copies of a message instead of just sending a single copy. Recent work done by Bruneel and Moeneclaey [BM86] found the optimum value of the number of copies to be transmitted as a function of the round trip delay and the block error probability. Another important conclusion of the work is that the optimum value does not depend on the num ber of times the message has been sent. That is, the optimum number of copies of the message that should be sent in the first transmission, first retransmission, second retransmission, etc. is the same. This is intuitively correct since the channel condition remains the same no matter how many times the message has been sent. All of the work mentioned above concentrated on single destination type communication. For multidestination type communication, Calo and Easton in [CEXl] described a broadcast protocol for large file transfer in a satellite network. The protocol basically is a stopand-wait multidestination ARQ scheme. Mase et al. in [MTYS83] described a go-back-n ARQ scheme suitable for point-to-multipoint satellite communication. They also presented a tandem error control scheme", where the uplink and the down-link use separate go-back-n ARQ's. The idea is to decrease the round trip delay so that the system throughput can be increased. However, the performance analysis is done by simulation except for the single receiver case. Gopal and Jaffe in [GJ84] presented three different go-back-n ARQ schemes suitable for point-to-multipoint communication. Analytic expressions of the throughput for the three schemes is given. For the scheme that can achieve the highest throughput, the Full Memory Scheme, an embedded Markov chain is solved to get the throughput expres sion. Exact solution is obtained only for the case of two receivers. An approximation solution is presented to solve the case where the number of receivers is larger than two. As to selective-repeat schemes, Chandran and Lin in [CL861 described and evaluated one such scheme that is suitable for point-to-multipoint communication. Recently, Towsley and Mithal [TM87] also studied a finite buffer multidestination selective-repeat protocol which is based on the point-to-point protocol first presented by Weldon [E J W 82). Throughput performance is the main concern for all the papers mentioned above, much less work has been done to analyze delay performance of the ARQ error control mechanism. Towsley and Wolf (TW791 analyzed the statistical behavior of the multiplexer using the stopand-wait and go-back-n protocols. In [Sha87b,Sha87a,ST88], Shacham and Towsley studied the resequencing delay and buffer occupancy in a receiver(s) of a selective-repeat ARQ scheme with different system configurations, such as multiple receivers and multiple channels. Hayashida et al. in [HFT88] studied the delay performance of CH2702-9/S9/0000/05S4$ IEEE 584

2 a go-back-n scheme using multiple copy transmissions in a single receiver system. In this paper, we study a series of optimal g-back-n ARQ protocols suitable for point-to-multipoint communication. The protocols have been described in detail in aprevious paper [WSSS] where we also analyzed the protocols to determine the optimal number of copies the sender should transmit in order to maximize the system throughput. In this paper, we concentrate on the delay analysis of the protocols and determine the optimal number of copies that the sender should transmit to minimize the delay. The optimal number of copies depends on the transmission error probability, the round trip propagation delay, the arrival process, and the number of receivers. 2 The System Environment The environment we consider consists of a sender and K receivers, where the communication between the sender and receivers is over a broadcast channel such as a satellite or broadcast radio channel. Data messages are sent in fixed length data frames and time is segmented into fixed length slots whose duration is equal to the transmission time of a data frame. Each data frame includes a cyclic redundancy code at the end to enable the receivers to detect errors. A receiver that receives the data frame first checks the CRC code, and sends an acknowledgement packet (ACK) back to the sender (on a separate channel) if the frame is successfully decoded or a negative acknowledgement packet (NACK) if the receiver fails to decode the data frame. The feedback channel is assumed to be error free. We also assume that there is always a data frame waiting to be transmitted at the sender. We wume that the nodal processing time is negligible and that the timeout period is set to be exactly equal to the round trip propagation delay. During the round trip delay, N data frames can be sent. As to the error processes, we make the simple assumption that two kinds of errors can happen in the forward transmission channel: "share" error and "individual" error. A share error occurring in the data frame causes a data frame be received in error by all the receivers. This kind of error can model the error that occurs in the uplink. As in [GJ84], we let the probability that this share error does not occur be p.. The second kind of error, "individual" error, causes an error in a data frame that is received by a particular receiver. The individual error process can reflect the error in the down link. We let the probability that an "individual" error does not occur be pi. We further assume that the "individual" error processes are independent and identical among all receivers. Therefore, the probability that a receiver correctly receives a data frame is p.. pi. 3 The Protocols We now describe a series of protocols that are suitable for pointto-multipoint communication over a broadcast channel. The sender sends (possibly multiple copies) data frames to the receivers and starts the timeout clock. The number of copies sent depends on the number of receivers that have not yet success- fully responded, K, the "success" probability, p., pi, the arrival process A with mean p~ and variance UA', and the round trip delay N. The optimum number of copies ie denoted by "*(K,p#,pi,pA,pA',N). For given p,, pi, A, and N, the optimum number of copies is abbreviated as n*(k). Receivers that receive the data frame and decode it successfully send back ACKs while those that decode it in error send back NACKs. After a round trip delay, the sender can check whether all the receivers have received the data frame successfully. If not, the sender retransmits the data frame. The sender maintains a list for each outstanding data frame called the frame-outstanding list to keep track of those receivers that have not yet ACKed. The protocols differ in the way the sender maintains and updates the frame-outetanding list. Let M be the parameter that distinguishes the protocols, that is, the MB protocol denotes that the sender does not use the outcome of previous transmiasions; in the M-1 protocol the sender use the outcomes of the first outstanding data frame (FODF), i.e., the sender "remembers" the framcoutstanding list for the FODF; in M2 protocol the sender "remembers" the framcoutstanding lists for the first two outstanding data frames, and 80 on. The ultimate protocol is one that the sender " remembers" the frame-outstanding lists of all the data frames transmitted in a round trip delay. This protocol is called the FullX protocol. That is, full utilization is made of the outcomes of previous transmissions, not just of the FODF as in M-1. The performance of other protocols should fall between that of protocols M-1 and Full-M. A data frame becomes the FODF if at the time of its transmission, all the data frames transmitted before it will never be retransmitted again. Thus, a data frame must become a FODF before it can be successfully received by all the receivers and a given data frame may stay as FODF for multiple attempts before it is received by all the receivers. To be more precise about M-0 protocol, the sender in this case, before retransmitting a data frame, reinitializes the frame-outstanding list. This might correspond to the case where the sender has limited memory so that it cannot afford to maintain the frame-outstanding lit. Note that for the single receiver case (K = l), all the protocols are the same. For the detailed operation of the protocols, please refer to [WSSS]. In this paper, we consider the delay optimization analysis for these protocols and determine the optimum number of copies of a data frame that the sender should transmit for a given N (round trip delay); K (number of receivers that have not yet received the data frame); p,, pi (success probability); and p ~, ga2 (arrival process). First let qj(k) be the probability that all K receivers receive the data frame in i copies, then qj(~) = [I- (1- ifp. = 1 (2) For simplicity, the analysis is done for the case p, = 1 in this paper. 585

3 4 Delay Minimization Analysis For an ARQ scheme, three delay measures can be considered: (1) Delay from the time a data frame is first transmitted until it is correctly accepted by all the receivers. We call this delay measure the reception delay, denoted by TI. For the g-back-n scheme, this delay includes the retransmissions of the data frame due to error in a data frame with lower sequence number. (2) Delay from the time a data frame is first transmitted until a specific receiver can pass it on to the destination host. For the eelectivarepeat scheme, this delay includes the reception delay and the resequencing delay [ST88]. (3) Delay from the time a data frame is available for transmission until it is passed on by a receiver. Thie includes the delay described in (2) and the queueing delay, denoted by T,, introduced by the transmitter not being available due to backlogged frames awaiting retransmission. We call this delay the system delay, denoted by TI. In this paper, we focus on the system delay. Note that, for a g-back-n ARQ scheme, the system delay includes only queueing delay and reception delay, since no resequencing delay is incurred at the receiver. We assume that the sender queue has infinite capacity. We let the random variable A denote- the number of data frames arriving in a time slot and we further assume this arrival process is independent and identical in each slot. GA(z) is the generating function, is the mean, and U A is ~ the variance of A, respectively. To get the system delay, standard tools of queueing analysis are adopted which is similar to the approach used in [TW79] [HFT88]. Distribution of the queueing delay (T,) The queueing delay can be further decomposed into two parts: (1) TI, the time the data frame must wait due to data frames that are already in the sender queue before its arrival, (2) Tz, the time it must wait due to data frames within its own group which are served before it. A group includes all the data frames that arrive in the same time slot. (Note, if a data frame arrives to find the queue empty, then Tz = Ti = Let T: be the random variable that denotes the number of time slots required for all the receivers to accept the data frame, see figure 1. GT,(z) is the generating function, PT, is the mean, and o~~~ is the variance of Tt, respectively. The generating functions of Ti and T2 can be found with the approach used by Towsley and Wolf [TW79]: (3) Tq = Ti f Tz, we thus have with mean Distribution of the reception delay (T,) lay can also be decomposed into two parts: The reception de- (1) Ts, the number of time slots elapsed between when the sender first transmits the data frame and the time Tend. All the receivers have received at least one copy of the data frame transmitted before Tend. See figure 1. (2) TI, the number of time slots elapsed from Tend to the time when all the receivers have received the data frame, which equals to + - I. with T, = Ts+T4 and G,(z) = GT~(Z).GT,(Z). Since T4 = $-1, we have G,(t) = GT=(z)z(T-'). 4.1 M-0 Protocol In M-0 protocol none of the results from previous transmissions has been used, thus the number of time slots required, i, when a data frame is received by all the receivers in the jth (re)transmission cycle is i = j(n(k) + N - 1) + n(k) with probability Pr[T: = i] = (1 - z)jz which is probability that all K receivers successfully receive at least one out of n(k) copies of the data frame. Thus, we can find the generating function GT, (t), the mean PT,, and the variance qtz as follows: (let /3 = n(k)+n-1) where z = [l - (1 - UTt' = q22 1-2) (9) As for T,, we have Ts equal to the number of time slots elapsed from the slot the data frame is first transmitted to the time Tend. For the M-0 protocol, Ts equals to i = ja + 1 if all the receivers receive the data frame before the lth slot of the jth (re)transmission cycle. The probability that TJ = i is Pr[Ts = i] = (1-2)' [ql(k) - q~-l(k)] (4) 586

4 Thus, we have the generating function GT, (2) and the mean p ~, of the random variable T, GT,(z) = id+-') 00 xpr[ts = i]zi i=o j=o I=1 n(k) = 1.. Finally, we have the system delay T, = Tq + T, and the generating function GT, and the mean p~. are PT, = Nil + a;i>:;~za GT.(z) = GT, (~)GT, (2) (12) n(k) = 1 E:::)-l[l-(l-pi)']K 2.n K $ aaaptl +CA GI a WA(~-#APT~ For a Poisson arrival process with rate A, we have CA(%) = n(k) 1 2 (13) PA = OAz = Substituting into equation (13), we then have I n(k) = 1 2piK(piK-X(piK+(1-piK)B)) PT, = n(k) ; Ft-, [l-(l-pi)']k + mn(k)+(l-z)p 22 + za.n(k]+(1-z)z&a(l-z),p Zz(z-A(rm(K)+ (l-c)@]) n(k) 1 2 (14) As p ~ +, 00 in equation (14), the arrival rate X becomes 2 An, = z *?a( K) + (1 -.)a which is in concordance with equation (8) in [WSSS]. For system delay minimization, we can just minimize equation (14) over n(k). Figure 2, 3, and 4 shows the system delay versus various system parameters for different copy sizes. The delay of the M1) protocol, which is the minimum, is just the envelope of the different fixed copy curves. 4.2 M-1 Protocol We here consider the analysis of the M1 protocol without adap tivity, that is, the sender does not adjust the number of copies of a data frame he transmits according to the number of nodes that have not yet received the data frame. The sender, instead, selects a fixed copy size for a certain K and does not change it throughout the transmission(s) of the data frame. The difference between the M-0 and M-1 protocols is that in the M1 protocol, the sender memorizes the outcomes of previous transmissions. For example, if receiver 1 sends an ACK for the first transmission of a data frame, then the sender memorizes this fact and can neglect the response (whether it is an ACK or NACK) from receiver 1 in the following retransmissions of the data frame. Delay performance of the adaptive M-1 protocol is done by simulation. If all the receivers receive at least one copy of the data frame within the jth (re)transmission cycle, then the number of slots elapsed is i = jb + n( K) with probability Pr[Tt ;I = qj.n(k) - q(j-l).n(k) The generating function of Tt is From this equation, we can find the mean and the variance of T: by noting that PTr = GTt'(1) The generating function and the mean value of Tp can then be evaluated from equations (5) and (6). To find T,, we note that when all the receivers receive at least one copy of a data frame at the Ith slot of the j (re)transmiesion cycle, the number of slots required is i = j@ + I with probability Therefore, the generating function of T, is which corresponds to the maximum throughput that can be achieved by this go-back-n M-0 protocol. Substituting in x and,9, we can get the optimum number of copies that the sender should transmit for this scheme as We also can find the mean of T, by differentiating G,(z). Combining equation (17) and (la), we can obtain the generating function of T, and subsequently the mean of T,, p~, ~

5 Figure 5, 6, and 7 show the system delay versus various system parameters for different copy sizes of the M-1 fixed copy protocol. Table 1 shows some typical optimum copy size of the MD and M-1 protocols. 4.3 FullM Protocol The analysis of the FullX protocol is complex due to the fact that the system delay for each data frame is not independent. For example, it is more likely that data frame 2 will have a shorter system delay if data frame 1 has a long one. We thus resort to simulation. Two steps of simulation is done. The first step is to determine the optimum number of copies the sender should transmit in terms of the number of receivers that have not yet received the data frame. Then, based on the optimum number of copies we obtained, another simulation is performed to determine the system delay for the adaptive FulLM protocol. Delay comparisons between the MD, fixed M-1, adaptive MA, fixed F du, and adaptive Full-A4 protocols are shown in figure 8, 9, and 10. The results show that (1) by sending the optimum number of copies of a data frame, the delay performance is improved significantly; (2) as the sender uses more of the outcomes of the previous transmissions, the delay performance becomes better, however, the performance difference decreases with the sender transmitting the optimal number copies of a data frame. (3) as the error probability increases, it is more advantageous to use more of the outcomes of the previous transmissions; (4) as the arrival rate increases, it is more advantageous to use more of the outcomes of the previous transmissions. (5) as the number of receivers increases, the advantage of using more of the outcomes of the previous transmissions is less phenomenal. Finally, table 2 and 3 show delay versus throughput for different number of copies of a data frame the sender transmits'. For the throughput analysis, please refer to [WS88]. These tables can be used for system design, for instance, if the number of receivers is K = 20 and the error probability is 0.10 (pi = 0.90) then to achieve the system delay T, 5 5 time unitsz and the system throughput , the sender should send 3 copies of each data frame (using the MB or M-1 protocol). 5 Conclusion This paper discusses a series of optimal go-back-n ARQ schemes for point-to-multipoint communication. The protocols differ in the way that the sender utilizes the outcomes of previous transmissions. We determine analytically the optimal number of copies 'In the tables, the arrival process is assumed to be Poisson with rate A = 0.1 for the delay analysis; while the heavy traffic assumption (the sender always has a data frame in its transmitting buffer waiting to be transmitted) is adopted for the throughput analysis. sone time unit is equal to the transmitting time of a single data frame. of a data frame the sender should send to minimize the system delay for both M-0 and MA protocols. These optimal values can be stored in memory for further selection. The results show that as the sender uses more of the outcomes of the previous transmission, the delay performance becomes better, however, the performance difference decreases when the sender transmits the optimal number of copies of a data frame rather than just a single copy. The advantage of using more of the outcomes of the previous transmissions is more sensitive to the error probability and the arrival rate than to the number of receivers. We also conclude that by sending optimum number of copies of a data frame, the system delay can decrease substantially. References [BAHF64] R. J. Benice and Jr. A. H. hey. An analysis of retransmission system. IEEE Trans. Commun. Technol., , Dec [BirSl] [BM86] [BS72] [CE81] [Chu74] [CL841 [CL861 [EJW82] [GJ84] N. D. Birrel. Pre-emptive retransmission for communication over noisy channels. IEE Proc., Part F, , Nov H. Bruneel and M. Moeneclaey. On the throughput performance of some continuous ARQ strategies with repeated transmissions. IEEE Trans. Commun., , Mar H. 0. Burton and D. D. Sullivan. Errors and error control. Proc. IEEE, , Nov S. B. Calo and M. C. Easton. A broadcast protocol for file transfers to multiple sites. IEEE Trans. Commun., , W. W. Chu. Optimal message block size for computer communications with error detection and retransmission strategies. IEEE Trans. Commun., 151C1525, Oct Y. Chang and C. Leung. On Weldon's ARQ strategy. IEEE Trans. Commun., , Mar S. R. Chandran and S. Lin. A selective-repeat ARQ scheme for point-to-multipoint communications and it's throughput analysis. In Proc. ACM SIGCOM Conference, pages , Stowe, VT, Aug Jr. E. J. Weldon. An improved selectiverepeat ARQ strategy. IEEE Trans. Commun., , Mar I. S. Gopal and J. M. Jaffe. Point-to-multipoint communication over broadcast links. IEEE Trans. Commun., COM-32: , Sep

6 [HFT88] Y. Hayashida, S. Fujii, and Y. Tezuka. Delay performance of a continuous ARQ system with copytransmissions. In IEEE INFOCOM, pages , New Orleans, LA, [WS88] [MM601 J. J. Metzner and K. C. Morgan. Coded bi- [YL80] nary decision-feedback communication system. IRE. Trans. Commun. Syst., CS-8:lOl-113, June [Mor781 J. M. Morris. On another go-back-n ARQ technique [YL81] for high error rate conditions. IEEE Tranu. Commun., , Jan [Mor791 J. M. Morris. Optimal blocklengths for ARQ error control schemes. IEEE Trans. Commun., , Feb [MTYS83] K. Mase, T. Takenaka, H. Yamamoto, and M. Shinohara. Go-back-N ARQ schemes for point-tomultipoint satellite communications. IEEE Tram. Commun., COM-31: , Apr [RSYGl] [Sas75] [Sha87a] [Sha87b] [ST881 (Stu631 [TM87] [Tow791 [TW79] B. Reiffen, W. G. Schmidt, and H. L. Yudkin. The design of an error-free data transmission systems for telephone circuits. AIEE Trans. Commun. Electron., , A. R. K. Sastry. Improving automatic repeatrequest (ARQ) performance on satellite channels under high error rate conditions. IEEE Trans. Commun., , Apr N. Shacham. Packet resequencing in reliable trans mission over parallel channels. In IEEE ICC, pages , N. Shacham. Queueing analysis of a selectiverepeat ARQ receiver. In IEEE INFOCOM, pages , N. Shacham and D. Towsley. Reaequencing delay and buffer occupancy in selective repeat ARQ with multiple receivers. In IEEE INFOCOM, pages , New Orleans, LA, R. D. Stuart. An insert system for use with feedback communication links. IEEE Trans. Commun. Syst., , Mar D. Towsley and S. Mithal. A selective repeat ARQ protocol for a point to multipoint channel. In Proc. INFOCOM, pages , San Franscisco, CA, Mar D. Towsley. The stutter go-back-n ARQ protocol. IEEE Trans. Commun., , D. Towsley and J. Wolf. On the statistical analysis of queue length and waiting times for statistical multiplexers with ARQ retransmission schemes. IEEE Trans. Commun., COM-27: , Apr J. L. Wang and J. A. Silvester. Optimal adaptive ARQ protocols for point-t-multipoint communication. In IEEE INFOCOM, pages , New Orleans, LA, P. S. Yu and S. Lin. An efficient error control scheme for satellite communications. IEEE Trans. Commun., , Mar P. S. Yu and S. Lin. An efficient selective repeat ARQ scheme for satellite channels and ita throughput analysis. IEEE Trans. Commun., , Mar

7 pi = 0.90 x = 0.1 x = 0.01 N=5 N=20 N=5 N=20 n*= n*= _ - n*=3 n*=4 n* = 5 pi = 0.99 x = 0.1 x = 0.01 N=5 N=20 N=5 N= n*=2 n*=3 n*=4,* = ii M1 1. Computation is only performed up to K = (1 - b means from K = o to K = b the optimal number of copies is n., e.g., for the g-back-n MD protocol with pi = 0.90, X = 0.1, N = 6, n*(6) = n'(7) =... = n'(67) = 9. Table 1: Some typical optimal n*(k) values I GBN M-0 Protocol Delay r. pi = 0.85 pi = 0.80 pi = 0.75 I GBN MB Protocol pi = pi = ; = ~~ ~~~ ThroughDut Table 2: Delay vs. throughput for the go-back-n scheme with p. = 1, N = 5, pi = 0.90 Table 3: Delay vs. throughput for the go-back-n scheme with pi = 1, N = 5, K =

8 111 loow 10 I arrival of frame 1 I 'end arrival of All receivers frame 2 receive frame 1 Figure 1: A typical data frame (re)transmission cycle. t 1.I.001.o 1 Arrival itate (M Figure 3: System delay vs. arrival rate (A) for the M-0 protocol with N = 5, K = 20, pi = 0.9. B c1 0 Y $. m 7.5 CI F 8 ri 0 m I o Number of Nodes (ZQ Figure 2: System delay vs. number of nodes (K) for the M-0 protocol with N = 5, X = 0.1, pi = 0.9. Robability of Success (pi) Figure 4: System delay vs. probability of succw (pi) for the MD protocol with N = 5, X = 0.1, K =

9 I Wumber of Nodes (Kl Figure 5: System delay vs. number of nodes (K) for the M-1 protocol with N = 5, X = 0.1, pi = I.o Probability of Success (pi) Figure 7: System delay vs. probability of success (pi) for the M1 protocol with N = 5, X = 0.1, K = 20. B s Bo E cn ;,,,.,...I, 1 I....I. v.... q 00 I 01 I I Arrival Rate (A) Figure 6: System delay vs. arrival rate (A) for the M-1 protocol with N = 5, K = 20, pi = : Number of Nodes (M Figure 8: Comparison of the system delay vs. number of nodes (K) among various protocols with N = 5, X = 0.1, pi =

10 I I ;: I!I B d 1 1,,,,,,,,,,,,,,,, *,,,,,,.oo 1.o 1 I I Arrival. Rate (h) Figure 9: Comparison of the system delay vs. arrival rate (A) among various protocols with N = 5, K = 20, pi = I * I ' 1 ' 1 ' ) o Probability of Success (pi) Figure 10: Comparison of the system delay vs. probability of success (pi) among various protocols with N = 5, X = 0.1, K =