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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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References

1. Anderson, D., Blom, J., Mandjes, M., Thorsdottir, H. & de Turck, K. (2014). A functional central limit theorem for a Markov-modulated infinite-server queue. Methodology and Computing in Applied Probability doi: 10.1007/s11009-014-9405-8.Google Scholar
2. Blom, J., Kella, O., Mandjes, M. & Thorsdottir, H. (2014). Markov-modulated infinite server queues with general service times. Queueing Systems 76: pp. 403–424.Google Scholar
3. Blom, J., de Turck, K. & Mandjes, M. (2013). A central limit theorem for Markov-modulated infinite-server queues. In Proceedings ASMTA 2013, Ghent, Belgium. Lecture Notes in Computer Science (LNCS) Series, 7984: pp. 81–95.Google Scholar
4. Blom, J., Mandjes, M. & Thorsdottir, H. (2013). Time-scaling limits for Markov-modulated infinite-server queues. Stochastic Models 29: pp. 112–127.Google Scholar
5. D'Auria, B. (2008). M/M/∞ queues in semi-Markovian random environment. Queueing Systems 58: pp. 221–237.Google Scholar
6. Coolen-Schrijner, P. & van Doorn, E. (2002). The deviation matrix of a continuous-time Markov chain. Probability in the Engineering and Informational Sciences 16: pp. 351–366.Google Scholar
7. Falin, G. (2008). The M/M/∞ queue in a random environment. Queueing Systems 58: pp. 65–76.Google Scholar
8. Fralix, B. & Adan, I. (2009). An infinite-server queue influenced by a semi-Markovian environment. Queueing Systems, 61, pp. 65–84.Google Scholar
9. Hellings, T., Mandjes, M. & Blom, J. (2012). Semi-Markov-modulated infinite-server queues: approximations by time-scaling. Stochastic Models 28: pp. 452–477.Google Scholar
10. Hilbert, D., Courant, R. (1924). Methoden der mathematischen physik. Vol II, Berlin: Springer.Google Scholar
11. Keilson, J. (1979). Markov chain models: rarity and exponentiality. New York: Springer.Google Scholar
12. Keilson, J. & Servi, L. (1993). The matrix M/M/∞ system: retrial models and Markov modulated sources. Advances in Applied Probability 25: pp. 453–471.Google Scholar
13. Kemeny, J. & Snell, J. (1961). Finite Markov chains. New York: Van Nostrand.Google Scholar
14. O'Cinneide, C. & Purdue, P. (1986). The M/M/∞ queue in a random environment. Journal of Applied Probability 23: pp. 175–184.Google Scholar
15. Schwabe, A., Dobrzyński, M., & Bruggeman, F. (2012). Transcription stochasticity of complex gene regulation models. Biophysical Journal, 103, 1152–1161.Google Scholar
16. Syski, R. (1978). Ergodic potential. Stochastic Processes and their Applications 7: pp. 311–336.Google Scholar
17. van Woensel, T. & Vandaele, N. (2007). Modeling traffic flows with queueing models: a review. Asia-Pacific Journal of Operational Research 24: 235–261.Google Scholar