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On closed ring queueing networks

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Nicholas Bambos
Nicholas Bambos*
Affiliation:
University of California, Los Angeles
*
∗ Postal address: Electrical Engineering Department, School of Engineering and Applied Science, 405 Hilgard Avenue, Los Angeles, CA 90024–1594, USA.
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Abstract

In this paper we first study ring structured closed queueing networks with distinguishable jobs. Under assumptions of periodicity and ergodicity of the service times, essentially the most general, it is shown that the limits defining the average flows of the jobs exist almost surely, and methods for their estimation by simulation are given. However, it turns out that the values of the flows depend on the initial positions of the jobs, due to the emergence of distinct persistent blocking modes. The effect of these modes on the behavior of general networks with queueing loops is examined.

For independent and identically distributed service times, conditions are specified for the network to asymptotically approach a steady state at large times.

Finally, we study the special case of ring networks with indistinguishable items and stationary and ergodic service times. It is shown that as the number of jobs in the network increases towards infinity, the average circulation time converges to the maximum of the expectations of the service times.

Keywords

STATIONARY RING NETWORKSFLOW ERGODICITYSUBADDITIVE STOCHASTIC PROCESSESRESONANT BLOCKING MODESFLOW ASYMPTOTICS

MSC classification

Primary: 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 29 , Issue 4 , December 1992 , pp. 979 - 995
Copyright
Copyright © Applied Probability Trust 1992 

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References

[1] Asmussen, S. (1987) Applied Probabilities and Queues. Wiley, New York.Google Scholar
[2] Bambos, N. and Walrand, J. (1989) On queues with periodic inputs. J. Appl. Prob. 31, 381–389.Google Scholar
[3] Borovkov, A. (1987) Limit theorems for queueing networks I. Theory Prob. Appl. 31, 413–427.Google Scholar
[4] Borovkov, A. (1988) Limit theorems for queueing networks. II. Theory Prob. Appl. 32, 257–272.Google Scholar
[5] Chung, K. L. (1974) A Course in Probability Theory, 2nd edn. Academic Press, New York.Google Scholar
[6] Kelly, F. P. (1982) The throughput of a series of buffers. Adv. Appl. Prob. 14, 633–653.CrossRefGoogle Scholar
[7] Kingman, J. F. C. (1968) The ergodic theory of subadditive stochastic processes. J. R. Statist. Soc. B30, 499–510.Google Scholar
[8] Kingman, J. F. C. (1973) Subadditive ergodic theory. Ann. Prob. 1, 883–909.CrossRefGoogle Scholar
[9] König, D. and Schmidt, V. (1986) Limit theorems for single-server feedback queues controlled by a general class of marked point processes. Theory Prob. Appl. 30, 712–719.CrossRefGoogle Scholar
[10] Konstantopoulos, P. and Walrand, J. (1989) On the ergodicity of networks with/GI/1/N queues. Adv. Appl. Prob. 22, 263–267.Google Scholar
[11] Loynes, R. M. (1962) The stability of a queue with non-independent inter-arrival and service times. Proc. Camb. Phil. Soc. 58, 497–520.Google Scholar
[12] Shiryayev, A. N. (1984) Probability. Springer-Verlag, New York.Google Scholar
[13] Sigman, K. (1989) Notes on the stability of closed queueing networks. J. Appl. Prob. 26, 678–682.Google Scholar
[14] Sigman, K. (1990) The stability of open queueing networks. Stoch. Proc. Appl. 35, 11–25.Google Scholar
[15] Walrand, J. (1988) An Introduction to Queueing Networks. Prentice Hall, Englewood Cliffs, NJ.Google Scholar
[16] Walters, P. (1982) An Introduction to Ergodic Theory. Springer-Verlag, New York.Google Scholar