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ANALYSIS OF MARKOV-MODULATED INFINITE-SERVER QUEUES IN THE CENTRAL-LIMIT REGIME

Published online by Cambridge University Press:  30 March 2015

Joke Blom ,
Koen De Turck  and
Michel Mandjes
Joke Blom
Affiliation:
CWI, P.O. Box 94079, 1090 GB Amsterdam, the Netherlands E-mail: joke.blom@cwi.nl
Koen De Turck
Affiliation:
TELIN, Ghent University, St.-Pietersnieuwstraat 41, B9000 Gent, Belgium E-mail: kdeturck@telin.ugent.be
Michel Mandjes
Affiliation:
Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, the Netherlands, CWI, P.O. Box 94079, 1090 GB Amsterdam, the Netherlands and Eurandom Eindhoven University of Technology, Eindhoven, the Netherlands, and IBIS, Faculty of Economics and Business, University of Amsterdam, Amsterdam, the Netherlands E-mail: M.R.H.Mandjes@uva.nl
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Abstract

This paper focuses on an infinite-server queue modulated by an independently evolving finite-state Markovian background process, with transition rate matrix Q≡(qij)i,j=1d. Both arrival rates and service rates are depending on the state of the background process. The main contribution concerns the derivation of central limit theorems (CLTs) for the number of customers in the system at time t≥0, in the asymptotic regime in which the arrival rates λi are scaled by a factor N, and the transition rates qij by a factor Nα, with α∈ℝ+. The specific value of α has a crucial impact on the result: (i) for α>1 the system essentially behaves as an M/M/∞ queue, and in the CLT the centered process has to be normalized by √N; (ii) for α<1, the centered process has to be normalized by N1−α/2, with the deviation matrix appearing in the expression for the variance.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 29 , Issue 3 , July 2015 , pp. 433 - 459
Copyright
Copyright © Cambridge University Press 2015 

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