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On queues with periodic inputs

Published online by Cambridge University Press:  14 July 2016

Nicholas Bambos  and
Jean Walrand
Nicholas Bambos
Affiliation:
University of California, Berkeley
Jean Walrand*
Affiliation:
University of California, Berkeley
*
∗ Postal address for both authors: Department of Electrical Engineering and Computer Sciences, and Electronics Research Laboratory, University of California, Berkeley, CA 94720, USA.
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Abstract

We consider a single-server queue with a periodic and ergodic input. It is shown that if the traffic intensity is less than 1, then the waiting time process is asymptotically periodic. Limit theorems associated with the asymptotic behavior of the queue are also proven. The results are then extended to acyclic networks of queues with periodic inputs. Particular cases of these results had been previously obtained for a single queue with periodic Poisson arrival input process and with independent and identically distributed service times.

Keywords

PERIODIC RANDOM MARKED POINT PROCESSESERGODICITYSTABILITYASYMPTOTIC PERIODICITYLIMIT THEOREMS
Type
Research Papers
Information
Journal of Applied Probability , Volume 26 , Issue 2 , June 1989 , pp. 381 - 389
Copyright
Copyright © Applied Probability Trust 1989 

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Footnotes

Research supported in part by NSF Grant No. ECS-8421128.

References

Asmussen, S. (1987) Applied Probabilities and Queues. Wiley, New York.Google Scholar
Baccelli, F. and Bremaud, P. (1987) Palm Probabilities and Stationary Queues. Lecture Notes in Statistics 41, Springer-Verlag, New York.Google Scholar
Baccelli, F., Makowski, A. M. and Schwartz, A. (1987) The fork-join queue and related systems with synchronization constraints: Stochastic ordering, approximations and computable bounds. Technical Research Report, SRC, Univ. of Maryland.Google Scholar
Franken, P., Koenig, D., Arndt, U. and Schmidt, V. (1982) Queues and Point Processes. Akademie Verlag, Berlin.Google Scholar
Harrison, J. M. and Lemoine, A. J. (1977) Limit theorems for periodic queues. J. Appl. Prob. 14, 566–576.CrossRefGoogle Scholar
Heyman, D. P. and Whitt, W. (1984) The asymptotic behavior of queues with time-varying arrival rates. J. Appl. Prob. 21, 143–156.CrossRefGoogle Scholar
Konstantopoulos, T. and Walrand, J. (1989) Stationarity and stability of fork-join networks. J. Appl. Prob. 26.Google Scholar
Lemoine, A. J. (1981) On queues with periodic Poisson input. J. Appl. Prob. 18, 889–900.CrossRefGoogle Scholar
Loynes, R. M. (1962) The stability of a queue with non-independent inter-arrival and service times. Proc. Camb. Philos. Soc. 58, 497–520.CrossRefGoogle Scholar
Matthes, K., Kerstan, J. and Mecke, J. (1978) Infinitely Divisible Point Processes. Akademie Verlag, Berlin.Google Scholar
Rolski, T. (1981a) Stationary random processes associated with point processes. Lecture Notes in Statistics 5, Springer-Verlag, New York.CrossRefGoogle Scholar
Rolski, T. (1981b) Queues with non-stationary input stream: Ross's conjecture. J. Appl. Prob. 13, 603–618.CrossRefGoogle Scholar
Walrand, J. (1988) An Introduction to Queueing Networks. Prentice Hall, Englewood Cliffs. NJ.Google Scholar
Walters, P. (1982) An Introduction to Ergodic Theory. Springer-Verlag, New York.CrossRefGoogle Scholar