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A monotonicity result for the workload in Markov-modulated queues

  • Nicole Bäuerle (a1) and Tomasz Rolski (a2)
Abstract

We consider a single server queue where the arrival process is a Markov-modulated Poisson process and service times are independent and identically distributed and independent from arrivals. The underlying intensity process is assumed ergodic with generator cQ, c > 0. We prove under some monotonicity assumptions on Q that the stationary workload W(c) is decreasing in c with respect to the increasing convex ordering.

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Corresponding author
Postal address: Department of Mathematics VII, University of Ulm, D-89069 Ulm, Germany. Email address: baeuerle@mathematik.uni-ulm.de.
∗∗ Postal address: Mathematical Institute, University of Wroclaw, 50384 Wroclaw, Poland.
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Work supported in part by KBN under grant 2 PO3A 04608(1995–97).

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References
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[1] Asmussen, S., Frey, A., Rolski, T., and Schmidt, V. (1995). Does Markov-modulation increase the risk? ASTIN Bull. 25, 4966.
[2] Bäuerle, N. (1997). Monotonicity results for M/GI/1 queues. J. Appl. Prob. 34, 514524.
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[6] Meester, L., and Shanthikumar, J. (1993). Regularity of stochastic processes. Prob. Eng. Inf. Sci. 7, 343360.
[7] Rolski, T. (1986). Upper bounds for single server queues with doubly stochastic Poisson arrivals. Math. Operat. Res. 11, 442450.
[8] Rolski, T. (1989). Queues with nonstationary inputs. Queueing Systems 5, 113130.
[9] Ross, S. (1978). Average delay in queues with non-stationary Poisson arrivals. J. Appl. Prob. 15, 602609.
[10] Shaked, M., and Shanthikumar, J. (1994). Stochastic Orders and Their Applications. Academic Press, New York.
[11] Stoyan, D. (1983). Comparison Methods for Queues and Other Stochastic Models. Wiley, Chichester.
[12] Szekli, R., Disney, R. L., and Hur, S. (1994). M/GI/1 queues with positively correlated arrival stream. J. Appl. Prob. 31, 497514.
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Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
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