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On the input-output map of a G/G/1 queue

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Cheng-Shang Chang
Cheng-Shang Chang*
Affiliation:
National Tsing Hua University
*
∗Postal address: Department of Electrical Engineering, National Tsing Hua University, Hsinchu, Taiwan 30043, R.O.C.
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Abstract

In this note, we consider G/G/1 queues with stationary and ergodic inputs. We show that if the service times are independent and identically distributed with unbounded supports, then for a given mean of interarrival times, the number of sequences (distributions) of interarrival times that induce identical distributions on interdeparture times is at most 1. As a direct consequence, among all the G/M/1 queues with stationary and ergodic inputs, the M/M/1 queue is the only queue whose departure process is identically distributed as the input process.

Keywords

STATIONARY AND ERGODIC INPUTSDEPARTURE PROCESS

MSC classification

Primary: 60K25: Queueing theory
Type
Research Article
Information
Journal of Applied Probability , Volume 31 , Issue 4 , December 1994 , pp. 1128 - 1133
Copyright
Copyright © Applied Probability Trust 1994 

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References

[1] Anantharam, V. (1993) The input-output map of a monotone discrete time quasireversible node. IEEE Trans. Inf. Theory. 39, 543–552.CrossRefGoogle Scholar
[2] Anantharam, V. (1992) Uniqueness of stationary ergodic fixed point for a /M/K node. Ann. Appl. Prob. To appear.CrossRefGoogle Scholar
[3] Baccelli, F. and Bremaud, P. (1994) Elements of Queueing Theory. Springer-Verlag, New York.Google Scholar
[4] Bambos, N. and Walrand, J. (1990) An invariant distribution for the G/G/1 queueing operator. Adv. Appl. Prob. 22, 254–256.Google Scholar
[5] Berman, M. and Westcott, M. (1983) On queueing systems with renewal departure processes. Adv. Appl. Prob. 15, 657–673.CrossRefGoogle Scholar
[6] Daley, D. J. (1968) The correlation structure of the output process of single server queueing systems. Ann. Math. Statist. 39, 1007–1019.CrossRefGoogle Scholar
[7] Gray, R. (1988) Probability, Random Processes and Ergodic Properties. Springer-Verlag, New York.Google Scholar
[8] Kamae, T., Krengel, U. and O'Brien, G. L. (1977) Stochastic inequalities on partially ordered spaces. Ann. Prob. 5, 899–912.CrossRefGoogle Scholar
[9] Ross, S. M. (1983) Stochastic Processes. Wiley, New York.Google Scholar
[10] Stoyan, D. (1983) Comparison Methods for Queues and Other Stochastic Models. Wiley, Berlin.Google Scholar
[11] Tchen, A. H. T. (1980) Inequalities for distributions with given marginals. Ann. Prob. 8, 814–827.Google Scholar
[12] Whitt, W. (1981) Comparing counting processes and queues. Adv. Appl. Prob. 13, 207–220.CrossRefGoogle Scholar