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Approximations of general discrete time queues by discrete time queues with arrivals modulated by finite chains

Part of: Stochastic processes Operations research and management science Markov processes Special processes

Published online by Cambridge University Press:  01 July 2016

Vinod Sharma
Vinod Sharma*
Affiliation:
Indian Institute of Science
*
*Postal address: Department of Electrical Engineering, Indian Institute of Science, Bangalore 560012, India.
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Abstract

Recently, Asmussen and Koole (Journal of Applied Probability30, pp. 365–372) showed that any discrete or continuous time marked point process can be approximated by a sequence of arrival streams modulated by finite state continuous time Markov chains. If the original process is customer (time) stationary then so are the approximating processes. Also, the moments in the stationary case converge. For discrete marked point processes we construct a sequence of discrete processes modulated by discrete time finite state Markov chains. All the above features of approximating sequences of Asmussen and Koole continue to hold. For discrete arrival sequences (to a queue) which are modulated by a countable state Markov chain we form a different sequence of approximating arrival streams by which, unlike in the Asmussen and Koole case, even the stationary moments of waiting times can be approximated. Explicit constructions for the output process of a queue and the total input process of a discrete time Jackson network with these characteristics are obtained.

Keywords

MARKED POINT PROCESSESDISCRETE TIME QUEUESAPPROXIMATIONS OF POINT PROCESSESCONVERGENCE OF STATIONARY MOMENTS

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60G55: Point processes 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 90B22: Queues and service
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 29 , Issue 4 , December 1997 , pp. 1039 - 1059
Copyright
Copyright © Probability Trust 1997 

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