Effect of exponential backoff scheme and

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Effect of Exponential Backoff Scheme and Retransmission Cutoff on the Stability of Frequency-Hopping Slotted ALOHA Systems Katsumi Sakakibara, Member, IEEE, Takehiko Seto, Daisuke Yoshimura, and Jiro Yamakita, Member, IEEE

Abstract—The combinatorial effect of an exponential backoff scheme and retransmission cutoff on the stability of frequencyhopping slotted ALOHA systems with finite population is investigated in terms of the catastrophe theory. In the systems, the packet retransmission probabilities are geometrically distributed with respect to the number of experienced unsuccessful transmissions and a packet will be discarded after a certain number of unsuccessful transmissions. Expressions which should be satisfied at equilibrium points are first derived. Then, the cusp point and the bifurcation sets are numerically evaluated. Finally, we visualize how the exponential backoff scheme and retransmission cutoff effect on the bistable region. Numerical results show that the exponential backoff scheme can mitigate bistable behavior of the system with finite population. However, it is also revealed that there is asymptotically no effect of the exponential backoff scheme on the stability of the system with infinite population. Index Terms—Catastrophe theory, exponential backoff scheme, frequency-hopping slotted ALOHA, retransmission cutoff, stability.

I. INTRODUCTION

S

INCE BISTABLE behavior of slotted ALOHA systems was first explicitly formulated in [1], intensive work has been done to devise medium access control protocols which can mitigate instability [2]. Among them an exponential backoff scheme [3], [4], [5], [6] and retransmission cutoff [7], [8] are of practical interest since they require neither to track feedback information nor to estimate the number of backlogged users. The exponential backoff scheme decreases the packet retransmission probabilities geometrically with respect to the number of unsuccessful transmissions experienced by a packet. In fact, the binary exponential backoff scheme has been already implemented in Ethernet [9]. On the other hand, in existing networks, retransmissions of a packet will be cut off and a packet will be discarded Manuscript received June 28, 2001; revised October 26, 2001 and December 13, 2001; accepted December 15, 2001.The editor coordinating the review of this paper and approving it for publication is M. Zorzi. This work was supported in part by the Okawa Foundation for Information and Telecommunications and in part by the Japan Society for the Promotion of Science under Grants-in-Aid for Scientific Research. K. Sakakibara and J. Yamakita are with the Department of Communication Engineering, Okayama Prefectural University, 719-1197 Soja, Japan (e-mail: [email protected]). T. Seto was with the Department of Communication Engineering, Okayama Prefectural University, 719-1197 Soja, Japan. He is now with TIS Inc., Tokyo, Japan. D. Yoshimura was with the Department of Communication Engineering, Okayama Prefectural University, 719-1197 Soja, Japan. He is now with Quest Co. Ltd., Tokyo, Japan. Digital Object Identifier 10.1109/TWC.2003.814345

after a certain number of unsuccessful transmissions in order to avoid excessive collisions among retransmitted packets and to satisfy delay constraints associated with a packet. The value of retransmission cutoff should be determined according to quality of service to be guaranteed [10]. For delay-sensitive traffic such as voice, it may be less than ten. In contrast, the value of retransmission cutoff may be much greater for loss-sensitive traffic such as data. The impact of the exponential backoff scheme has been discussed in [4]. Aldous [4] has proved that the binary exponential backoff scheme can not mitigate bistable behavior of slotted ALOHA systems with infinite population. Meanwhile, it has been proved that slotted ALOHA systems with finite population can be stabilized for any values of parameters, if retransmission cutoff is limited to not more than eight [8]. Stability of random access protocols in conjunction with frequency-hopped (FH) modulation techniques has been also extensively investigated [7], [11]–[13]. Kim has shown from the view point of the drift function that FH-slotted ALOHA systems with infinite population can be stabilized by adjusting the coding rate of Reed–Solomon codes employed [11] and the value of retransmission cutoff [7]. Murali and Hughes [12] also have shown that bistability of FH-slotted ALOHA systems can be eliminated by increasing the code length. Liu [13] has investigated dynamic behavior of FH-slotted ALOHA systems with retransmission cutoff and generalized backoff scheme. However, the analysis for finite population in [13] is limited to extremely small value (less than five) of retransmission cutoff due to computational complexity arising from a multidimensional Markov chain. It should be emphasized that analytical methods in [7], [11]–[13] are based on the drift analysis [1]. Although it enables us to determine whether the system exhibits bistable behavior or not for given parameters, the drift analysis requires greedy and heuristic approaches when we design the values of parameters in order for the system to operate with global stable equilibrium point. From the viewpoint of designing the system, an application of the catastrophe theory [14] has successfully supplied a view of the explicit bistable region in the space of control parameters for random access protocols such as classical slotted ALOHA [15], [16], [17], carrier-sense multiple-access with collision detection [18] and packet-reservation multiple access [10] (see Appendix A for an outline of the catastrophe theory). Utilizing the catastrophe theory, we can determine the values of control parameters of the system so as not to fall into the bistable region, but to operate with global stable equilibrium point. In this sense,

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SAKAKIBARA et al.: EFFECT OF EXPONENTIAL BACKOFF SCHEME AND RETRANSMISSION CUTOFF

Fig. 1. System model of slotted ALOHA with exponential backoff scheme and retransmission cutoff.

the catastrophe theory can provide a necessary and sufficient condition for the system to be monostable. Recently, the use of catastrophe theory has been extended to slotted ALOHA with the capture effect [19], [20] and with retransmission cutoff [8]. However, no backoff scheme is explicitly taken into account in the literature on the stability of random access protocols in terms of the catastrophe theory. The aim of the present paper is to investigate combinatorial effects of the exponential backoff scheme and retransmission cutoff on the stability of FH-slotted ALOHA systems with finite population in terms of the catastrophe theory. Applying the same technique in [8], we extend the results presented in [7] to a finite population scenario incorporating the exponential backoff scheme and we elaborate the results in [13] in order to enable us to evaluate the stability of the system with large value of retransmission cutoff. Taking into account the flow balance with respect to users’ state, we first formulate the balance function whose roots provide the offered traffic in equilibrium. Then, after indicating the existence of the cusp catastrophe, the cusp point and the bifurcation sets are numerically evaluated. Finally, we visualize dependency of the bistable region on the exponential backoff scheme and retransmission cutoff. II. SYSTEM DESCRIPTION A. System Model We consider an FH communication system consisting of single buffer users and a common receiver. The common receiver is assumed to be able to receive signals of all the users at a time. Each user transmits a unit-length packet to the common receiver according to a slotted ALOHA policy. Two strategies to resolve excessive packet collisions are incorporated in the system; an exponential backoff scheme and retransmission cutoff. The exponential backoff scheme decreases the packet retransmission probabilities geometrically by a factor according to the number of unsuccessful and transmissions experienced by a packet, where is referred to as the exponential backoff factor. By virtue of retransmission cutoff, a packet will be discarded after unsuccessful transmissions. The system model employing the exponential backoff scheme and retransmission cutoff is provided in Fig. 1 [5], [8], [23]. A user with empty buffer is said to be in the thinking (TH)

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mode, whereas a user with packet which has experienced unsuccessful transmissions is in the retransmission (RT ) mode. Due to the single-buffer assumption, every user belongs to either of the TH mode or the RT modes at the beginning of a slot. Each user in the TH mode independently generates a new packet of unit length with probability at the beginning of a slot and a user who has generated a new packet immediately transmits the packet in that slot. In this sense, packet arrivals are subject to the Bernoulli process. A user in the RT mode retransmits its packet in the buffer with probin each slot . Users who ability have succeeded in the (re)transmission return to the TH mode. On the contrary, mode transitions of unsuccessful users depend on the previous mode. Unsuccessful users in the TH mode enter mode the RT mode and those in the RT mode enter the RT . A packet dropping occurs when a user in mode fails its retransmission, so that it moves back the RT to the TH mode. B. FH Channels The available bandwidth is divided into frequency bins, each of which can convey an -ary signal (symbol). A user randomly changes frequency bins in a symbol-by-symbol manner. If two or more users simultaneously transmit their symbol in the identical frequency bin, then symbol errors occur. This event is referred to as a hit. In order to prevent symbol errors, each packet Reed–Solomon code over a finite field is encoded with an elements, where and is a power of a prime of [21]. Since Reed–Solomon codes are maximum-distance-separable, they can correct errors and erasures simultaneously holds. Background as long as the inequality noise is ignored, so that symbol errors/erasures are caused only by a hit. We also assume that all the users can obtain a positive (successful) or negative (failure) acknowledgment through error-free feedback channels immediately at the end of each slot (zero propagation delay). C. Packet Error Probability As shown in [7], [11], and [12], the conditional probability of hit, given that there are interfering packets, can be given by (1) for

, where for synchronous hopping (2) for asynchronous hopping

When the perfect side information is available at the common hits receiver, a Reed–Solomon code can correct up to with an erasure-only decoding algorithm [21]. Hence, the packet error probability with interfering packets is (3)

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where

represents the binomial distribution (4)

and for or , by for definition. On the other hand, if no side information is available, no more hits1 can be corrected with a bounded-distance than decoder of a Reed–Solomon code [21]. Then, the packet error probability with interfering packets can be evaluated as (5)

of discarded packets, and the last is the average number of newly generated and transmitted packets from the TH mode. • RT mode (10) • RT mode (11) The equilibrium points can be obtained by solving . First, we can get the recursive expression with respect to from (12)

III. BALANCE FUNCTION AND CUSP CATASTROPHE In this section, we first derive expressions which should be satisfied in equilibrium. be the number of users in the RT mode Let at the beginning of slot . Then, the slotted ALOHA system in Fig. 1 can be specified by a dis-dimensional Markov chain with state space crete-time . Since the -dimensional of Markov chain is irreducible, aperiodic, and homogeneous, there exists a steady-state [9]. Denoting a steady-state vector by , we can define the number of backlogged users as

. Next, substituting (12) into (6), the for number of backlogged users in equilibrium can be obtained by

(13)

Finally, substituting (12) and (13) into (9), we can obtain the following equation at the equilibrium points:

(6) . It is clear that denote the offered load to the FH-slotted ALOHA Let , , channel in Fig. 1. Then, it follows from that and (7) With the aid of the Poisson approximation2 for the number of colliding packets in a slot [8], [15]–[20], we can evaluate the by averaging (3) or (5) average packet error probability on the number of interfering packets (8) Then, we can evaluate the average increment of the number of users in a slot for each mode as follows: • TH mode (9) where the first term represents the average number of successful users per slot, the second is the average number 1bxc

is the maximum integer not greater than x. to [16] and empirical knowledge, the Poisson approximation is sufficiently accurate in analyzing the stability of slotted ALOHA systems for N > 50. 2According

(14)

, , , and , we can obtain the offered load at For given the equilibrium points as roots of (14). Hereafter, we refer to as the balance function.3 It the function . It follows from is clear that . Thus, the equation Appendix-B that has at least one positive root in the for given , , , and . range The slotted ALOHA system is monostable if (14) has a in the range of . The system unique positive root is bistable if (14) has three roots [1]. These three roots are categorized into: 1) a preferable stable equilibrium point (minimum root), which provides high throughput and small packet transmission delay; 2) an unstable equilibrium point (intermediate root), where the average sojourn time is indefinitely minimal; 3) an unpreferable stable equilibrium point (maximum root), which offers virtually zero throughput and huge packet transmission delay. According to [15, Appendixes C and D], bistable behavior of slotted ALOHA systems can be described in terms of the catastrophe theory. 3The

expression in [10], [15]–[19], corresponding to the balance function

A(G; Np; Nr; L; ) is described as a function of N , p, and r . However, it should be noticed that the balance function depends on Np and Nr .

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Theorem 1: There may exist the cusp catastrophe in slotted ALOHA systems with the exponential backoff factor and at most retransmission trials, if the system of equations

(15)

for given and , where has a solution , , and . When the cusp catastrophe exists, the cusp point is given by and , are defined the root of (15) and the bifurcation sets, as Fig. 2. Behavior of the denominator in (19) of FH-slotted ALOHA systems with asynchronous hopping, perfect side information, and [32, 20] Reed–Solomon code for q = 100.

(16)

(17) as shown in [8]. Notice that the results above are applicable to the classical slotted ALOHA systems if a [1, 1] Reed–Solomon , where two or more packet collisions code is employed for always destroy all the packets involved. IV. NUMERICAL EVALUATION Assuming the system with asynchronous hopping, perfect side information, and a [32, 20] Reed–Solomon code, we . Note that (3) is used present numerical results for to calculate the conditional packet error probability. These conditions are chosen in order to compare our results with those in [7]. A. Cusp Point and Bifurcation Sets , we obtain and as a funcSolving tion of as shown in (18) and (19), at the bottom of the page. , we can evaluate Substituting (18) and (19) into the cusp point numerically.4 It is necessary for the cusp point , and to be valid in the numerical results that all of , should be positive. From Appendix-C, (18) is positive for any . In the meantime, the sign of (19) depends on its denominator, since it can be easily verified that the numerator is 4An iterative method with the maximum tolerable error of 10 numerical evaluation.

is used in

positive for any . Furthermore, the denominator in (19) is independent of the exponential backoff factor . This results in the following lemma. Lemma 1: The minimum value of retransmission cutoff for which the cusp catastrophe may exist is independent of the exponential backoff factor . Let be the denominator of the right-hand side in as the balance function, that is, the (19). Regarding derivative of the potential function for a certain dynamic system with system parameter and control parameter , we can numerically obtain the fold point in terms of the fold catastrophe [15] by solving

(20)

It corresponds to [8, Lemma 2]. Under the conditions considered in this section, the fold point can be calculated as and the behavior of is shown in Fig. 2. for is negative. As a reFrom Fig. 2 and (19), , so that sult, the system has no cusp catastrophe for , , , and . On the contrary, it is monostable for any when retransmission cutoff is greater than two, there exists the cusp catastrophe. Note that this results in a striking contrast with [8, Th. 2]. Reference [8, Th. 2] addresses that there is no cusp point in the classical slotted ALOHA systems with no backoff schemes, when retransmission cutoff is limited to less than nine . and , are demonstrated The bifurcation sets, (solid lines), 0.70 (dashed in Fig. 3 for

(18)

(19)

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Fig. 3. Bifurcation sets of FH-slotted ALOHA systems with asynchronous hopping, perfect side information, and [32, 20] Reed–Solomon code for q = 100,  = 0:50 (solid lines), 0.70 (dashed lines), 0.90 (dashed–dotted lines), 1.00 (dotted lines), and L = 3; 4; 6; 10; 20; 30; 50; 100.

lines), 0.90 (dashed–dotted lines), 1.00 (dotted lines), and . The cusp point is provided and . In Fig. 3, for given , by the intersection of . In the the system is bistable in the range of bistable region there exist two stable equilibrium points in the system and the operating point of the system dynamically osin Fig. 3 is referred cillates between them. The area below to as the active region, in which the system is monostable and it operates at comparatively low offered load. The active region is preferable, since the system offers high throughput and

small transmission delay. The area lying to the right of in Fig. 3 is referred to as the saturated region, where the system is also monostable but its operating offered load is considerably high. In the saturated region, the system yields virtually zero throughput and enormous transmission delay. It can be observed in Fig. 3 that behavior of the bistable recontrasts with that for . The bistable gion for is monotonously spread according to an inregion for is almost crease of the value of ; that is, one bifurcation set moves to the left. On the stable for any , while the other

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transmissions are unlikely to retransmit their packets. This decreases the offered load and deletes the unpreferable stable equilibrium point, so that the system becomes monostable again. B. Bistable Region for Infinite Population

Fig. 4. Offered load at the stable equilibrium points for FH-slotted ALOHA systems with asynchronous hopping, perfect side information, and [32, 20] = 10, and = 1000. Reed–Solomon code for q = 100,

Np

Nr

contrary, for , increment of the value of first exmoves to the left). However, the pands the bistable region ( bistable region seems to turn to diminish in size by increasing moves to the right). For example, the value of further ( and . In this let us focus our attention on is monostable for case, the system with and bistable for . Then, for the system returns to be monostable. In order to elaborate on this, we depict in Fig. 4 the offered load at the stable equilibrium points, which can be obtained as the zeros of the balance function . In Fig. 4, we can find that the system is bistable only for . The ofwith fered load at the (preferable) stable equilibrium point is almost -invariant for . The offered load at the other (unpreferable) stable equilibrium point significantly depends on the value , however, the offered load at the unpreferof . For able stable equilibrium point is a narrow-sense increasing function with respect to . From Fig. 4, we can gain an insight on the nonmonotonicity . First, for , the of the bistable region for small value of retransmission cutoff results in frequent occurrence of packet dropping. In addition, users who discarded their packets have little opportunity to generate the next new packet, since the packet generation probability is considerably small . This can suppress the appearance of the unpreferfor , the unpreferable stable equilibrium point. Next, for suddenly appears because of able stable point with excessive collisions among retransmitted packets from the RT modes. An increment of decreases the probability of packet dropping so that the unpreferable stable point is likely to rise , the ofup. Although these excessive collisions last for fered load at the unpreferable stable equilibrium point turns to . Since the packet retransmission probability decrease for is decreasing by a factor , a user who experiences repeated unsuccessful transmissions tends to sojourn in the RT mode for large . This results in a decrease of the offered load at the unpreferable stable equilibrium point. When we increase further, the packet retransmission probabilities of the RT mode for large approach to zero. Thus, users with consecutive unsuccessful

Let us argue asymptotic behavior of the bistable region for infinite population and compare the result with that in [7]. Kim [7] has shown that FH-slotted ALOHA systems with infinite population, asynchronous hopping, perfect side information, and a [32, 20] Reed–Solomon code are bistable for and if the average input traffic is five packets per slot in [7]), as shown in [7, Fig. 2]. ( For infinite population, we can suppose that each (re)transmitted packet belongs to a different user. Hence, no packet retransmission needs to be considered and each newly generated packet can be always assigned to a user in the TH mode. This allows us to suppose that the infinite population model is indeed in some sense the limit of the finite population model if backlogged users are not distinguished from users in the TH mode and if the number of users is increased under the constraint that the total average packet arrival rate remains finite [22]. in our Therefore, the input traffic in [7] corresponds to system model and performance of infinite population is pro. In order to consider infinite , the devided for infinite nominator of the right-hand side in (19) should be zero

(21) has been shown in Fig. 2, Recalling that behavior of possesses two roots for . we can find that Then, for given , the asymptotic values of the bifurcation sets, and , for infinite population can be derived from (18)

(22) where is the roots of (21). Asymptotic behavior of the bifurcation sets is presented in Fig. 5 for the same values of parameters in [7]. It follows from intersects Fig. 5 that the asymptotic bifurcation sets at . Therefore, the system with infinite population is and bistable for . This coincides monostable for with the result in [7]. From Fig. 5, we can obtain perspective whether or not FH-slotted ALOHA systems with retransmission cutoff exhibit bistable behavior for given input traffic when the number of users can be considerably large. Since both (22) and (21) are independent of , the following lemma holds similarly to Lemma 1. Lemma 2: There is no effect of the exponential backoff scheme on the stability of FH-slotted ALOHA systems with infinite population. Lemma 2 agrees with [4, Th. 1] which states instability of the binary exponential backoff scheme of classical slotted ALOHA

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Fig. 5. Asymptotic behavior of the bifurcation sets of FH-slotted ALOHA systems with asynchronous hopping, perfect side information, and [32, 20] . Reed–Solomon code for q = 100 and

Nr ! 1

systems with infinite population and without retransmission cutoff. V. CONCLUSION Taking into consideration not only the exponential backoff scheme but also retransmission cutoff, we have investigated the stability of FH-slotted ALOHA systems with finite population in terms of the catastrophe theory. With the aid of the Poisson approximation of the number of colliding packets, we have first derived the balance function of the system, which should be satisfied in equilibrium. Then, the cusp point and the bifurcation sets have been numerically evaluated. Under the same conditions as discussed in [7], we have illustrated how the bistable region of the system fluctuates as the value of retransmission cutoff increases. The numerical results have shown that the bistable region of the system with the exponential backoff scheme is first expanded by increasing the value of retransmission cutoff, and then reduced in size. As an asymptotic result, it has been also revealed that the exponential backoff scheme does not affect the bistable region of the system with infinite population. Using the obtained results, we can design the values of parameters of FH-slotted ALOHA systems with the exponential backoff scheme and retransmission cutoff so that the system can operate with a global stable equilibrium point. Notice that the packet dropping probability is the inevitable performance measure when retransmission cutoff is employed. The packet dropping probability in the monostable region , where is the offered can be approximated by load at the stable equilibrium point obtained by solving , as in [23]. However, evaluation of the packet dropping probability is left for further study.

Suppose a dynamic system governed by a set of control be the potential parameters. Let , , and is the function of the system, where dimension of the state space of the system. The critical points are called equilibrium points. for which and An equilibrium point is stable if . An equilibrium point where unstable if is called a bifurcation point. The drift or , and its sign the balance function is defined as indicates the inverse direction of system evolution from point . The catastrophe theory is concerned with equilibrium points that arise when the potential function changes. The potential functions can be classified according to their shape, which are essentially determined by the number of control parameters. Thom’s list of seven elementary catastrophes is presented in Table I. In this paper, two elementary catastrophes are considered, the cusp and the fold catastrophes. For given and , in (14) can be viewed as the derivative of a certain potential function with two control parameters , and , and one-dimensional state space , . If , the system is likely to evolve so as to increase the offered load . On the other hand, if . the system tends to lessen Thus, there may exist the cusp or the fold catastrophe. In terms of the cusp catastrophe, the critical point for which is referred to as the cusp point and a set of bifurcation points, as in (21) can be the bifurcation set. On the other hand, regarded as the derivative of another potential function with one , , and one-dimensional state space control parameter , . Thus, there may exist the fold catastrophe. In terms of the fold catastrophe, a bifurcation point is referred to as the fold point. B. Proof of From (14) we have

(23)

Then, we obtain

APPENDIX A. Abstract of Catastrophe Theory We review here an abstract of the catastrophe theory established by Thom [14]. For more detailed description, see [14], [15], and [17].

(24)

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TABLE I SEVEN ELEMENTARY CATASTROPHES

since

, , and . This implies that

for .

C. Proof of It is evident that the numerator in (18) is positive for any . Substituting (12) and (13) into (7), we have

(25)

It implies that the relation (26)

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[8] K. Sakakibara, H. Muta, and Y. Yuba, “The effect of limiting the number of retransmission trials on the stability of slotted ALOHA systems,” IEEE Trans. Veh. Technol., vol. 49, pp. 1449–1453, July 2000. [9] M. E. Woodward, Communication and Computer Networks. Los Alamitos, CA: IEEE Comput. Soc. Press, 1994. [10] S. Nanda, “Stability evaluation and design of the PRMA joint voice data system,” IEEE Trans. Commun., vol. 42, pp. 2092–2104, May 1994. [11] S. W. Kim, “Stabilization of slotted ALOHA spread-spectrum communication networks,” IEEE J. Select. Areas Commun., vol. 8, pp. 555–561, May 1990. [12] R. Murali and B. L. Hughes, “Coding and stability in frequency-hop packet radio networks,” IEEE Trans. Commun., vol. 46, pp. 191–199, Feb. 1998. [13] Y.-S. Liu, “Performance analysis of frequency-hop packet radio networks with generalized retransmission backoff,” IEEE Trans. Wireless Commun., vol. 1, pp. 703–711, Oct. 2002. [14] R. Thom, Structural Stability and Morphogenesis. New York: W. A. Benjamin, 1975. [15] Y. Onozato and S. Noguchi, “On the thrashing cusp in slotted ALOHA systems,” IEEE Trans. Commun., vol. COM-33, pp. 1171–1182, Nov. 1985. [16] Y. Onozato, J. Liu, S. Shimamoto, and S. Noguchi, “Effect of propagation delays on ALOHA systems,” Comput. Netw. ISDN Syst., vol. 12, pp. 329–337, 1986. [17] R. Nelson, “Stochastic catastrophe theory in computer performance modeling,” J. ACM, vol. 34, no. 3, pp. 661–685, July 1987. [18] T. Yokohira, T. Nishida, and H. Miyahara, “Analysis of dynamic behavior in p-persistent CSMA/CD using cusp catastrophe,” Comput. Netw. ISDN Syst., vol. 12, pp. 277–289, 1986. [19] Y. Onozato, J. Liu, and S. Nogichi, “Stability of a slotted ALOHA system with capture effect,” IEEE Trans. Veh. Technol., vol. 38, pp. 31–36, Feb. 1989. [20] K. Sakakibara, M. Hanaoka, and Y. Yuba, “On the stability of five types of slotted ALOHA systems with capture and multiple packet reception,” IEICE Trans. Fundamentals, vol. E81-A, no. 10, pp. 2092–2100, Oct. 1998. [21] S. B. Wicker, Error Control Systems for Digital Communication and Storage. Engelwood Cliffs, NJ: Prentice-Hall, 1995. [22] R. Rom and M. Sidi, Multiple Access Protocols: Performance and Analysis. New York: Springer-Verlag, 1990. [23] K. Sakakibara, “Performance approximation of a multi-base station slotted ALOHA for wireless LANs,” IEEE Trans. Veh. Technol., vol. 41, pp. 448–454, Nov. 1992.

holds. Since both the numerator and the denominator in (18) are is completed. positive, the proof that ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their useful and constructive comments and suggestions. REFERENCES [1] A. B. Carleial and M. E. Hellman, “Bistable behavior of ALOHA-type systems,” IEEE Trans. Commun., vol. COM-23, pp. 401–410, Apr. 1975. [2] N. Abramson, Ed., Multiple Access Communications. New York: IEEE Press, 1993. [3] A. Fukuda, K. Mukumoto, and T. Hasegawa, “Adaptive retransmission randomization schemes for a packet switched random access broadcast channel,” in Proc. ICCC, Kyoto, Japan, Sept. 1978, pp. 543–548. [4] D. J. Aldous, “Ultimate instability of exponential back-off protocol for acknowledgment-based transmission control of random access communication channels,” IEEE Trans. Inform. Theory, vol. IT-33, pp. 219–223, Mar. 1987. [5] D. Raychaudhuri and K. Joseph, “Performance evaluation of slotted ALOHA with generalized retransmission backoff,” IEEE Trans. Commun., vol. 38, pp. 117–122, Jan. 1990. [6] D. G. Jeong and W. S. Jeon, “Performance of an exponential backoff scheme for slotted-ALOHA protocol in local wireless environment,” IEEE Trans. Veh. Technol., vol. 44, pp. 470–479, Aug. 1995. [7] S. W. Kim, “Frequency-hopped spread-spectrum random access with retransmission cutoff and code rate adjustment,” IEEE J. Select. Areas Commun., vol. 10, pp. 344–349, Feb. 1992.

Katsumi Sakakibara (S’86–M’90) received the B.E., M.E., and D.E. degrees in communication engineering from Osaka University, Suita, Japan, in 1985, 1987, and 1994, respectively. In 1987, he joined Toshiba R&D Center, Kawasaki, Japan. Since 1995, he has been with Okayama Prefectural University, Soja, Japan, where he is currently an Associate Professor in the Department of Communication Engineering. His interests include algebraic coding theory, error control schemes, multiple access protocols, and performance analysis of communication systems. Dr. Sakakibara is a Member of the Institute of Electronics, Information and Communication Engineers (IEICE) of Japan, the Society of Information Theory and Its Applications (SITA) of Japan, and the Japan Society of Industrial and Applied Mathematics (JSIAM).

Takehiko Seto received the B.E. and M.E. degrees in communication engineering from Okayama Prefectural University, Soja, Japan, in 2000 and 2002, respectively. In 2002, he joined TIS Inc., Tokyo, Japan, as a Software Engineer.

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Daisuke Yoshimura received the B.E. and M.E. degrees in communication engineering from Okayama Prefectural University, Soja, Japan, in 2001 and 2003, respectively. In 2003, he joined Quest Co. Ltd., Tokyo, Japan, as a Software Engineer.

Jiro Yamakita (M’75) received the B.E. and M.E. degrees in electrical engineering from Kyoto Institute of Technology, Kyoto, Japan, in 1969 and 1971, respectively, and the D.E. degree in electrical engineering from Osaka Prefectural University, Sakai, Japan, in 1979. He is currently a Professor in the Department of Communication Engineering at Okayama Prefectural University, Soja, Japan. His interests include scattering of waves and analysis of waveguides. Dr. Yamakita is a member of the Institute of Electronics, Information and Communication Engineers (IEICE) of Japan, the Information Processing Society (IPS) of Japan, and the Optical Society of America (OSA).