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On the convergence to stationarity of the many-server Poisson queue

Part of: Special processes Computer system organization

Published online by Cambridge University Press:  14 July 2016

Wolfgang Stadje  and
P. R. Parthasarathy
Wolfgang Stadje*
Affiliation:
University of Osnabrück
P. R. Parthasarathy*
Affiliation:
Indian Institute of Technology, Madras
*
∗Postal address: Fachbereich Mathematik/Informatik, University of Osnabrück, 49069 Osnabrück, Germany. Email address: wolfgang@mathematik.uni-osnabrueck.de.
∗∗Postal address: Department of Mathematics, Indian Institute of Technology, Madras, Chennai-600036, India.
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Abstract

We consider the many-server Poisson queue M/M/c with arrival intensity λ, mean service time 1 and λ/c < 1. Let X(t) be the number of customers in the system at time t and assume that the system is initially empty. Then pn(t) = P(X(t) = n) converges to the stationary probability πn = P(X = n). The integrals ∫0∞[E(X)-E(X(t))]dt and ∫0∞[P(X≤n) − P(X(t)≤n)]dt are a measure of the speed of convergence towards stationarity. We express these integrals in terms of λ and c.

Keywords

Many-server Poisson queuetransient probabilitiesstationarityspeed of convergence

MSC classification

Primary: 60K25: Queueing theory 68M20: Performance evaluation; queueing; scheduling
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 2 , June 1999 , pp. 546 - 557
Copyright
Copyright © Applied Probability Trust 1999 

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