The Effect of Limiting the Number of Retransmission Trials on the ...
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 49, NO. 4, JULY 2000
1449
The Effect of Limiting the Number of Retransmission Trials on the Stability of Slotted ALOHA Systems Katsumi Sakakibara, Member, IEEE, Hiroyuki Muta, and Yoshiharu Yuba, Member, IEEE
Abstract—We investigate the effect of limiting the number of retransmission trials on the stability of slotted ALOHA systems with no capture effect. Using an argument of the catastrophe theory, we prove that a slotted ALOHA system is mono-stable, if the number of retransmission trials is limited to, at most, eight. We also show how the bistable region is enlarged, as the number of retransmission trials grows over eight. Index Terms—Cusp catastrophe, limited retransmission trials, slotted ALOHA, stability.
I. INTRODUCTION
S
LOTTED ALOHA is one of the most widely implemented random access protocols, in particular, in wireless, mobile, or satellite communication systems. A number of researches have been done for bistable behavior of slotted ALOHA systems [1]. Among them bistable behavior can be successfully described in [2]–[4] by using the catastrophe theory [5]. It can be easily expected that the bistability can be mitigated, when we restrict the number of retransmission trials to a finite value and permit packet droppings [6], or when we can control the packet retransmission probability [7]. Furthermore, the capture effect can be also an effective countermeasure to the bistability in wireless or mobile environments [8]–[10]. In the literature [1]–[4], [7]–[10], however, it has been assumed that packets which failed in correct reception have to be retransmitted until correctly received. Although the effect of limiting the number of retransmission trials on throughput, transmission delay, and packet dropping was discussed in [6], taking into account the capture effect, the stability was not considered. In this work, we investigate the effect of limiting the number of retransmission trials on the stability of slotted ALOHA systems with no capture effect. It is proved that a slotted ALOHA system is mono-stable for any values of control parameters (the packet generation probability and the packet retransmission probability), if the number of retransmission trials is limited to, at most, eight. We also show how the bistable region is enlarged, as the number of retransmission trials grows over eight. Note that the effect of limiting the number of retransmission trials on the stability will be solely investigated, so that it can
Manuscript received November 8, 1998; revised December 2, 1999. This work was supported in part by the Ministry of Education, Science, Sport, and Culture in Japan under Grant-in-Aid for Encouragement of Young Scientists. The authors are with the Department of Communication Engineering, Okayama Prefectural University, Soja 719-1197, Japan (e-mail: [email protected]). Publisher Item Identifier S 0018-9545(00)04827-1.
Fig. 1. System model with the IFT protocol.
be distinguished from the corresponding effects of the capture effect and of controlling the packet retransmission probability. II. SYSTEM MODEL Consider a slotted ALOHA system of single-buffer users. We focus ourselves on the immediate first transmission (IFT) protocol. The system models employing the IFT protocol is shown in Fig. 1. A user with empty buffer (a user in the TH mode) generates a new packet with probability at the beginning of a slot and a user who generated a new packet immediately transmits the packet. A user in the RT mode retransmits its packet in the buffer with probability in each slot . We assume that . If only one user has access to a slotted ALOHA channel, the user succeeds in the (re)transmission and returns to the TH mode. On the contrary, if two or more users collide on the ALOHA channel, all the users involved in a collision fail in their (re)transmission. Then, the users in the TH mode enter the RT mode and the users in the mode . A packet RT mode enter the RT mode fails its redropping occurs when a user in the RT transmission, so that it moves back to the TH mode. Let be the number of users in the RT mode . Then, the slotted ALOHA system in Fig. 1 can be specified -dimensional Markov chain with state space of by an . The number of backlogged users can be . Clearly, . defined as III. BALANCE FUNCTION AND CUSP CATASTROPHE Let denote the offered load to the ALOHA channel. Then, that it follows from the assumption
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(1)
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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 49, NO. 4, JULY 2000
Using the Poisson approximation [2]–[4] for the number of (re)transmitted packets in a slot, we can formulate the average increment of the number of users in a slot for each mode as follows: • TH mode
• RT mode
• RT mode
The equilibrium points can be obtained by solving . Equivalently, we can obtain the following equation at the equilibrium points:
Fig. 2. Behavior of the function h(G;
L)
.
IV. MAIN RESULTS In this section, we will prove that no cusp catastrophe . From exists for slotted ALOHA systems with we obtain
(2) (5) , , and , (2) gives the number of backlogged For given users at the equilibrium points. It is easy to see that (2) has and at least one positive root, since . The slotted ALOHA system is mono. stable, if (2) has an unique positive root in the range of While the system is bistable, if (2) has three roots [1]. Hereafter, as the balance funcwe refer to the function tion. It follows from Appendices C and D in [2] that bistable behavior of slotted ALOHA systems can be described in terms of the catastrophe theory: Theorem 1: There may exists the cusp catastrophe in slotted ALOHA systems with at most retransmission trials, if the system of equations
and (6) where (7) and
(8) and (3) for given , where , has roots , and . When the cusp catastrophe exists, the cusp point is given by and , are defined the root of (3) and the bifurcation sets, as
(4)
In order for (5) and (6) to give the valid bifurcation sets, both and should be positive. It follows from (1) that the right . On the side of (5) is positive for any value of less than and should have the same signs other hand, to be positive. The following lemmas address in order for and . the signs of , then for any . Lemma 1: If Proof: We can easily check
, and . Hence, for any and . Q.E.D. , then for any . If Lemma 2: If , then there exists positive which satisfy . Proof: Let us assume that is the balance function for a certain dynamic system with system state parameter and control parameter . Then, it follows from the catastrophe
SAKAKIBARA et al.: EFFECT OF LIMITING THE NUMBER OF RETRANSMISSION TRIALS
Fig. 3.
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Bifurcation sets of slotted ALOHA systems with the IFT protocol for L = 9; 10; 15; 20; 30; 50 (solid lines) and for infinite L (dashed lines).
theory that there may exists the fold catastrophe [2], [5], if the system of equations and has roots
, where
and
We can solve numerically so that we . Substituting the values into obtain (10), we can easily confirm that
(9) . From (8) we have
and
(10)
which implies that there may exists the fold catastrophe and that . As a result, the fold point is given by satisfying , if , and there exists for any , if , as shown in that Fig. 2 This completes the proof. Q.E.D. From Lemmas 1 and 2 there may exist the cusp catastrophe, . In fact, we can successfully evaluate the cusp point if numerically from (3) and (4), and the bifurcation sets for
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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 49, NO. 4, JULY 2000
Fig. 5. System model with the DFT protocol.
Fig. 4. Asymptotic values of Np in the bifurcation sets.
respectively. Consequently, the main result of the present paper follows. Theorem 2: A slotted ALOHA system is mono-stable for any and , if the number of retransmission trials is values of limited at most eight. While there may exists the bistable region in a slotted ALOHA system when the number of retransmission trials is permitted more than eight.
, where . Then, the balance function for the DFT protocol can be evaluated as
Similarly to (5), we obtain V. DISCUSSIONS (12)
A. Bistable Region for The cusp point and the bifurcation sets, and , are iland 50. The solid lustrated in Fig. 3 for lines present the bifurcation sets for finite , while the dashed lines are those for infinite (conventional slotted ALOHA).1 For values of between and , slotted ALOHA systems are bistable. We can observe that the bistable region is stable with grows as goes larger. One bifurcation set respect to and it coincides with that for infinite , when is approximately larger than 30. The other bifurcation set moves to the left in Fig. 3 as grows. We can also observe from Fig. 3 that the bifurcation sets and converge to nonzero values of as goes large. From (5) we have (11) in the bifurcation sets can Then, the asymptotic values of be obtained by substituting the root of (8) into (11). Numerical results are given in Fig. 4 B. Expressions for the DFT Protocol Corresponding to Fig. 1, the system model with the DFT (delayed first transmission) protocol is shown in Fig. 5. We can specify the system by an -dimensional Markov chain with state and the offered load is given by space of
[the other equation, from (6), is unchanged]. It is straightforward to check that Lemmas 1 and 2 also hold for the DFT protocol. Consequently, Theorem 2 is also valid for the DFT protocol. However, the bifurcation sets in corresponding to Fig. 3 move slightly to the right, since the bifurcation sets is larger for the DFT protocol than for the IFT protocol [comparing (12) with (5)]. VI. CONLUSION We have considered a slotted ALOHA system without capture effect and proved that the slotted ALOHA system is mono, if the number of retransstable for any value of mission trials is limited to, at most, eight, and that the slotted ALOHA system exhibits bistable behavior, if more than eight retransmission trials are permitted. We have also illustrated how the bistable region grows, as the number of retransmission trials exceeds eight. It is also known that the capture effect can mitigate instability of a slotted ALOHA system [9], [10]. Thus, we can easily conjecture that a slotted ALOHA system will be mono-stable even though we restrict the number of retransmission trials more than eight, if the capture effect is taken into account. However, the mixed effect with the capture effect on the stability is left for further study. ACKNOWLEDGMENT
1Expressions for the conventional ALOHA systems with infinite L are presented in [3].
The authors would like to thank the anonymous reviewers for their constructive comments and suggestions.
SAKAKIBARA et al.: EFFECT OF LIMITING THE NUMBER OF RETRANSMISSION TRIALS
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] Y. Onozato and S. Noguchi, “On the thrashing cusp in slotted ALOHA systems,” IEEE Trans. Commun., vol. COM-33, pp. 1171–1182, Nov. 1985. [3] Y. Onozato, J. Liu, S. Shimamoto, and S. Noguchi, “Effect of propagation delays on ALOHA systems,” Comput. Net. ISDN Syst., vol. 12, pp. 329–337, 1986. [4] R. Nelson, “Stochastic catastrophe theory in computer performance modeling,” J. ACM, vol. 34, no. 3, pp. 661–685, July 1987. [5] M. Golubitsky, “An introduction to catastrophe theory and its applications,” SIAM Review, vol. 20, no. 2, pp. 352–387, Apr. 1978. [6] K. Sakakibara, “Performance approximation of a multi-base station slotted ALOHA for wireless LAN’s,” IEEE Trans. Veh. Technol., vol. 41, pp. 448–454, Nov. 1992. [7] 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. [8] 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. [9] C. van der Plas and J.-P. M. G. Linnartz, “Stability of mobile slotted ALOHA network with Rayleigh fading, shadowing, and near-far effect,” IEEE Trans. Veh. Technol., vol. 39, pp. 359–366, Nov. 1990. [10] 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, pp. 2092–2100, Oct. 1998.
Katsumi Sakakibara (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. From 1991 to 1995, he was with College of Industrial Technology, Amagasaki, 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).
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Hiroyuki Muta received the B.E. and M.E. degrees in communication engineering from Okayama Prefectural University, Soja, Japan, in 1997 and 1999, respectively. In 1999, he joined Matsushita Systems Engineering, Osaka, Japan.
Yoshiharu Yuba (M’70) received the B.E., M.E., and D.E. degrees in communication engineering from Osaka University, Suita, Japan, in 1954, 1956, and 1970, respectively. In 1956, he joined Osaka-Onkyo, Osaka, Japan. From 1967 to 1993, he was with the Department of Electronics and Information Science at Kyoto Institute of Technology, Kyoto, Japan. Since 1993, he has been with Okayama Prefectural University, Soja, Japan, where he is currently a Professor in the Department of Communication Engineering. His interests include millimeter-wave transmission and image processing. Dr. Yuba is a member of the Institute of Electronics, Information and Communication Engineers (IEICE) of Japan, and the Institute of Image Information and Television Engineers of Japan.
1449
The Effect of Limiting the Number of Retransmission Trials on the Stability of Slotted ALOHA Systems Katsumi Sakakibara, Member, IEEE, Hiroyuki Muta, and Yoshiharu Yuba, Member, IEEE
Abstract—We investigate the effect of limiting the number of retransmission trials on the stability of slotted ALOHA systems with no capture effect. Using an argument of the catastrophe theory, we prove that a slotted ALOHA system is mono-stable, if the number of retransmission trials is limited to, at most, eight. We also show how the bistable region is enlarged, as the number of retransmission trials grows over eight. Index Terms—Cusp catastrophe, limited retransmission trials, slotted ALOHA, stability.
I. INTRODUCTION
S
LOTTED ALOHA is one of the most widely implemented random access protocols, in particular, in wireless, mobile, or satellite communication systems. A number of researches have been done for bistable behavior of slotted ALOHA systems [1]. Among them bistable behavior can be successfully described in [2]–[4] by using the catastrophe theory [5]. It can be easily expected that the bistability can be mitigated, when we restrict the number of retransmission trials to a finite value and permit packet droppings [6], or when we can control the packet retransmission probability [7]. Furthermore, the capture effect can be also an effective countermeasure to the bistability in wireless or mobile environments [8]–[10]. In the literature [1]–[4], [7]–[10], however, it has been assumed that packets which failed in correct reception have to be retransmitted until correctly received. Although the effect of limiting the number of retransmission trials on throughput, transmission delay, and packet dropping was discussed in [6], taking into account the capture effect, the stability was not considered. In this work, we investigate the effect of limiting the number of retransmission trials on the stability of slotted ALOHA systems with no capture effect. It is proved that a slotted ALOHA system is mono-stable for any values of control parameters (the packet generation probability and the packet retransmission probability), if the number of retransmission trials is limited to, at most, eight. We also show how the bistable region is enlarged, as the number of retransmission trials grows over eight. Note that the effect of limiting the number of retransmission trials on the stability will be solely investigated, so that it can
Manuscript received November 8, 1998; revised December 2, 1999. This work was supported in part by the Ministry of Education, Science, Sport, and Culture in Japan under Grant-in-Aid for Encouragement of Young Scientists. The authors are with the Department of Communication Engineering, Okayama Prefectural University, Soja 719-1197, Japan (e-mail: [email protected]). Publisher Item Identifier S 0018-9545(00)04827-1.
Fig. 1. System model with the IFT protocol.
be distinguished from the corresponding effects of the capture effect and of controlling the packet retransmission probability. II. SYSTEM MODEL Consider a slotted ALOHA system of single-buffer users. We focus ourselves on the immediate first transmission (IFT) protocol. The system models employing the IFT protocol is shown in Fig. 1. A user with empty buffer (a user in the TH mode) generates a new packet with probability at the beginning of a slot and a user who generated a new packet immediately transmits the packet. A user in the RT mode retransmits its packet in the buffer with probability in each slot . We assume that . If only one user has access to a slotted ALOHA channel, the user succeeds in the (re)transmission and returns to the TH mode. On the contrary, if two or more users collide on the ALOHA channel, all the users involved in a collision fail in their (re)transmission. Then, the users in the TH mode enter the RT mode and the users in the mode . A packet RT mode enter the RT mode fails its redropping occurs when a user in the RT transmission, so that it moves back to the TH mode. Let be the number of users in the RT mode . Then, the slotted ALOHA system in Fig. 1 can be specified -dimensional Markov chain with state space of by an . The number of backlogged users can be . Clearly, . defined as III. BALANCE FUNCTION AND CUSP CATASTROPHE Let denote the offered load to the ALOHA channel. Then, that it follows from the assumption
0018–9545/00$10.00 © 2000 IEEE
(1)
1450
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 49, NO. 4, JULY 2000
Using the Poisson approximation [2]–[4] for the number of (re)transmitted packets in a slot, we can formulate the average increment of the number of users in a slot for each mode as follows: • TH mode
• RT mode
• RT mode
The equilibrium points can be obtained by solving . Equivalently, we can obtain the following equation at the equilibrium points:
Fig. 2. Behavior of the function h(G;
L)
.
IV. MAIN RESULTS In this section, we will prove that no cusp catastrophe . From exists for slotted ALOHA systems with we obtain
(2) (5) , , and , (2) gives the number of backlogged For given users at the equilibrium points. It is easy to see that (2) has and at least one positive root, since . The slotted ALOHA system is mono. stable, if (2) has an unique positive root in the range of While the system is bistable, if (2) has three roots [1]. Hereafter, as the balance funcwe refer to the function tion. It follows from Appendices C and D in [2] that bistable behavior of slotted ALOHA systems can be described in terms of the catastrophe theory: Theorem 1: There may exists the cusp catastrophe in slotted ALOHA systems with at most retransmission trials, if the system of equations
and (6) where (7) and
(8) and (3) for given , where , has roots , and . When the cusp catastrophe exists, the cusp point is given by and , are defined the root of (3) and the bifurcation sets, as
(4)
In order for (5) and (6) to give the valid bifurcation sets, both and should be positive. It follows from (1) that the right . On the side of (5) is positive for any value of less than and should have the same signs other hand, to be positive. The following lemmas address in order for and . the signs of , then for any . Lemma 1: If Proof: We can easily check
, and . Hence, for any and . Q.E.D. , then for any . If Lemma 2: If , then there exists positive which satisfy . Proof: Let us assume that is the balance function for a certain dynamic system with system state parameter and control parameter . Then, it follows from the catastrophe
SAKAKIBARA et al.: EFFECT OF LIMITING THE NUMBER OF RETRANSMISSION TRIALS
Fig. 3.
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Bifurcation sets of slotted ALOHA systems with the IFT protocol for L = 9; 10; 15; 20; 30; 50 (solid lines) and for infinite L (dashed lines).
theory that there may exists the fold catastrophe [2], [5], if the system of equations and has roots
, where
and
We can solve numerically so that we . Substituting the values into obtain (10), we can easily confirm that
(9) . From (8) we have
and
(10)
which implies that there may exists the fold catastrophe and that . As a result, the fold point is given by satisfying , if , and there exists for any , if , as shown in that Fig. 2 This completes the proof. Q.E.D. From Lemmas 1 and 2 there may exist the cusp catastrophe, . In fact, we can successfully evaluate the cusp point if numerically from (3) and (4), and the bifurcation sets for
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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 49, NO. 4, JULY 2000
Fig. 5. System model with the DFT protocol.
Fig. 4. Asymptotic values of Np in the bifurcation sets.
respectively. Consequently, the main result of the present paper follows. Theorem 2: A slotted ALOHA system is mono-stable for any and , if the number of retransmission trials is values of limited at most eight. While there may exists the bistable region in a slotted ALOHA system when the number of retransmission trials is permitted more than eight.
, where . Then, the balance function for the DFT protocol can be evaluated as
Similarly to (5), we obtain V. DISCUSSIONS (12)
A. Bistable Region for The cusp point and the bifurcation sets, and , are iland 50. The solid lustrated in Fig. 3 for lines present the bifurcation sets for finite , while the dashed lines are those for infinite (conventional slotted ALOHA).1 For values of between and , slotted ALOHA systems are bistable. We can observe that the bistable region is stable with grows as goes larger. One bifurcation set respect to and it coincides with that for infinite , when is approximately larger than 30. The other bifurcation set moves to the left in Fig. 3 as grows. We can also observe from Fig. 3 that the bifurcation sets and converge to nonzero values of as goes large. From (5) we have (11) in the bifurcation sets can Then, the asymptotic values of be obtained by substituting the root of (8) into (11). Numerical results are given in Fig. 4 B. Expressions for the DFT Protocol Corresponding to Fig. 1, the system model with the DFT (delayed first transmission) protocol is shown in Fig. 5. We can specify the system by an -dimensional Markov chain with state and the offered load is given by space of
[the other equation, from (6), is unchanged]. It is straightforward to check that Lemmas 1 and 2 also hold for the DFT protocol. Consequently, Theorem 2 is also valid for the DFT protocol. However, the bifurcation sets in corresponding to Fig. 3 move slightly to the right, since the bifurcation sets is larger for the DFT protocol than for the IFT protocol [comparing (12) with (5)]. VI. CONLUSION We have considered a slotted ALOHA system without capture effect and proved that the slotted ALOHA system is mono, if the number of retransstable for any value of mission trials is limited to, at most, eight, and that the slotted ALOHA system exhibits bistable behavior, if more than eight retransmission trials are permitted. We have also illustrated how the bistable region grows, as the number of retransmission trials exceeds eight. It is also known that the capture effect can mitigate instability of a slotted ALOHA system [9], [10]. Thus, we can easily conjecture that a slotted ALOHA system will be mono-stable even though we restrict the number of retransmission trials more than eight, if the capture effect is taken into account. However, the mixed effect with the capture effect on the stability is left for further study. ACKNOWLEDGMENT
1Expressions for the conventional ALOHA systems with infinite L are presented in [3].
The authors would like to thank the anonymous reviewers for their constructive comments and suggestions.
SAKAKIBARA et al.: EFFECT OF LIMITING THE NUMBER OF RETRANSMISSION TRIALS
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] Y. Onozato and S. Noguchi, “On the thrashing cusp in slotted ALOHA systems,” IEEE Trans. Commun., vol. COM-33, pp. 1171–1182, Nov. 1985. [3] Y. Onozato, J. Liu, S. Shimamoto, and S. Noguchi, “Effect of propagation delays on ALOHA systems,” Comput. Net. ISDN Syst., vol. 12, pp. 329–337, 1986. [4] R. Nelson, “Stochastic catastrophe theory in computer performance modeling,” J. ACM, vol. 34, no. 3, pp. 661–685, July 1987. [5] M. Golubitsky, “An introduction to catastrophe theory and its applications,” SIAM Review, vol. 20, no. 2, pp. 352–387, Apr. 1978. [6] K. Sakakibara, “Performance approximation of a multi-base station slotted ALOHA for wireless LAN’s,” IEEE Trans. Veh. Technol., vol. 41, pp. 448–454, Nov. 1992. [7] 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. [8] 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. [9] C. van der Plas and J.-P. M. G. Linnartz, “Stability of mobile slotted ALOHA network with Rayleigh fading, shadowing, and near-far effect,” IEEE Trans. Veh. Technol., vol. 39, pp. 359–366, Nov. 1990. [10] 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, pp. 2092–2100, Oct. 1998.
Katsumi Sakakibara (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. From 1991 to 1995, he was with College of Industrial Technology, Amagasaki, 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).
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Hiroyuki Muta received the B.E. and M.E. degrees in communication engineering from Okayama Prefectural University, Soja, Japan, in 1997 and 1999, respectively. In 1999, he joined Matsushita Systems Engineering, Osaka, Japan.
Yoshiharu Yuba (M’70) received the B.E., M.E., and D.E. degrees in communication engineering from Osaka University, Suita, Japan, in 1954, 1956, and 1970, respectively. In 1956, he joined Osaka-Onkyo, Osaka, Japan. From 1967 to 1993, he was with the Department of Electronics and Information Science at Kyoto Institute of Technology, Kyoto, Japan. Since 1993, he has been with Okayama Prefectural University, Soja, Japan, where he is currently a Professor in the Department of Communication Engineering. His interests include millimeter-wave transmission and image processing. Dr. Yuba is a member of the Institute of Electronics, Information and Communication Engineers (IEICE) of Japan, and the Institute of Image Information and Television Engineers of Japan.
