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On Queues with Interarrival Times Proportional to Service Times

Published online by Cambridge University Press:  27 July 2009

Israel Cidon ,
Roch Guréin ,
Asad Khamisy  and
Moshe Sidi
Israel Cidon
Affiliation:
Department of Electrical Engineering Technion, Haifa 32000, Israel
Roch Guréin
Affiliation:
IBM T. J. Watson Research Center, Yorktown Heights, New York 10598
Asad Khamisy
Affiliation:
Department of Electrical Engineering Technion, Haifa 32000, Israel
Moshe Sidi
Affiliation:
Department of Electrical Engineering Technion, Haifa 32000, Israel
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Abstract

We analyze a family of queueing systems where the interarrival time In+1 between customers n and n + 1 depends on the service time Bn of customer n. Specifically, we consider cases where the dependency between In+1 and Bn is a proportionality relation and Bn is an exponentially distributed random variable. Such dependencies arise in the context of packet-switched networks that use rate policing functions to regulate the amount of data that can arrive to a link within any given time interval. These controls result in significant dependencies between the amount of work brought in by customers/packets and the time between successive customers. The models developed in the paper and the associated solutions are, however, of independent interest and are potentially applicable to other environments.

Several scenarios that consist of adding an independent random variable to the interarrival time, allowing the proportionality to be random and the combination of the two are considered. In all cases, we provide expressions for the Laplace-Stieltjes Transform of the waiting time of a customer in the system. Numerical results are provided and compared to those of an equivalent system without dependencies.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 10 , Issue 1 , January 1996 , pp. 87 - 107
Copyright
Copyright © Cambridge University Press 1996

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References

1.Bala, K., Cidon, I., & Sohraby, K. (1990). Congestion control for high-speed packet switch. Proceedings of Infocom '90, San Francisco, 06: 520–526.Google Scholar
2.Boxma, O.J. (1979). On a tandem queueing model with identical service times at both counters, Pts. I and II. Advances in Applied Probability 11: 616–659.CrossRefGoogle Scholar
3.Cidon, I., Gopal, I., & Guérin, R. (1991). Bandwidth management and congestion control in Planet. IEEE Communications Magazine 29(10): 54–63.Google Scholar
4.Cidon, I., Guérin, R., Khamisy, A., & Sidi, M. (1993). Analysis of a correlated queue in a communication system. IEEE Transactions on Information Theory 39(2): 456–465.Google Scholar
5.Conolly, B.W. (1968). The waiting time process for a certain correlated queue. Operations Research 16: 1006–1015.CrossRefGoogle Scholar
6.Conolly, B.W. & Choo, Q.H. (1979). The waiting time process for a generalized correlated queue with exponential demand and service. SIAM Journal on Applied Mathematics 37(2): 263–275.CrossRefGoogle Scholar
7.Conolly, B.W. & Hadidi, N. (1969). A comparison of the operational features of conventional queues with a self-regulating system. Applied Statistics 18: 41–53.Google Scholar
8.Conolly, B.W. & Hadidi, N. (1969). A correlated queue. Journal of Applied Probability 6: 122–136.Google Scholar
9.Elwalid, A.I. & Mitra, D. (1991). Analysis and design of rate-based congestion control of highspeed networks, I: Stochastic fluid models, access regulation. Queueing Systems Theory and Applications (QUESTA) 9: 29–64.Google Scholar
10.Fendick, K.W., Saksena, V.R., & Whitt, W. (1989). Dependence in packet queues. IEEE Transactions on Communications COM-37(11): 1173–1183.CrossRefGoogle Scholar
11.Ghahramani, S. & Wolff, R.W. (1989). A new proof of finite moment conditions for GI/G/1 busy periods. Queueing Systems Theory and Applications (QUESTA) 4(2): 171–178.Google Scholar
12.Guérin, R. & Gün, L. (1992). A unified approach to bandwidth allocation and access control in fast packet-switched networks. Proceedings of INFOCOM '92, Florence, 05, pp. 1–12.Google Scholar
13.Hadidi, N. (1981). Queues with partial correlation. SIAM Journal on Applied Mathematics 40(3): 467–475.Google Scholar
14.Hadidi, N. (1985). Further results on queues with partial correlations. Operations Research 33: 203–209.Google Scholar
15.Kingman, J.F.C. (1962). On queues in heavy traffic. Journal of the Royal Statistical Society 24: 383–392.Google Scholar
16.Kingman, J.F.C. (1962). Some inequalities for the queue GI/G/1. Biometrika 49: 315–324.Google Scholar
17.Kleinrock, L. (1975). Queueing systems, Vol. I: Theory. New York: John Wiley & Sons.Google Scholar
18.Kleinrock, L. (1976). Queueing systems, Vol. 2: Computer applications. New York: John Wiley & Sons.Google Scholar
19.Langaris, C. (1986). A correlated queue with infinitely many servers. Journal of Applied Probability 23: 155–165.Google Scholar
20.Langaris, C. (1987). Busy-period analysis of a correlated queue with exponential demand and service. Journal of Applied Probability 24: 476–485.Google Scholar
21.Loynes, R.M. (1962). The stability of a queue with non-independent inter-arrival and service times. Proceedings of the Cambridge Philosophical Society 58(3): 497–520.Google Scholar