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A monotonicity result for the workload in Markov-modulated queues

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Nicole Bäuerle  and
Tomasz Rolski
Nicole Bäuerle*
Affiliation:
University of Ulm
Tomasz Rolski*
Affiliation:
University of Wroclaw
*
∗Postal address: Department of Mathematics VII, University of Ulm, D-89069 Ulm, Germany. Email address: [email protected].
∗∗Postal address: Mathematical Institute, University of Wroclaw, 50384 Wroclaw, Poland.
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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.

Keywords

Markov-modulated queuestationary workloadincreasing convex orderingsupermodular function

MSC classification

Primary: 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 3 , September 1998 , pp. 741 - 747
Copyright
Copyright © Applied Probability Trust 1998 

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Footnotes

Work supported in part by KBN under grant 2 PO3A 04608(1995–97).

References

Asmussen, S., Frey, A., Rolski, T., and Schmidt, V. (1995). Does Markov-modulation increase the risk? ASTIN Bull. 25, 49–66.Google Scholar
Bäuerle, N. (1997). Monotonicity results for M/GI/1 queues. J. Appl. Prob. 34, 514–524.Google Scholar
Bäuerle, N. (1997). Inequalities for stochastic models via supermodular orderings. Commun. Statist.–Stochastic Models, 13, 181–201.Google Scholar
Borovkov, A. A. (1972). Random Processes in Queueing Theory. Nauka, Moskva.Google Scholar
Chang, C., Chao, X., and Pinedo, M. (1991). Monotonicity results for queues with doubly stochastic Poisson arrivals: Ross's conjecture. Adv. Appl. Prob. 23, 210–228.Google Scholar
Meester, L., and Shanthikumar, J. (1993). Regularity of stochastic processes. Prob. Eng. Inf. Sci. 7, 343–360.CrossRefGoogle Scholar
Rolski, T. (1986). Upper bounds for single server queues with doubly stochastic Poisson arrivals. Math. Operat. Res. 11, 442–450.CrossRefGoogle Scholar
Rolski, T. (1989). Queues with nonstationary inputs. Queueing Systems 5, 113–130.Google Scholar
Ross, S. (1978). Average delay in queues with non-stationary Poisson arrivals. J. Appl. Prob. 15, 602–609.Google Scholar
Shaked, M., and Shanthikumar, J. (1994). Stochastic Orders and Their Applications. Academic Press, New York.Google Scholar
Stoyan, D. (1983). Comparison Methods for Queues and Other Stochastic Models. Wiley, Chichester.Google Scholar
Szekli, R., Disney, R. L., and Hur, S. (1994). M/GI/1 queues with positively correlated arrival stream. J. Appl. Prob. 31, 497–514.CrossRefGoogle Scholar