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A Convexity Property of a Markov-Modulated Queueing Loss System

Published online by Cambridge University Press:  27 July 2009

Michael Pinedo
Affiliation:
Department of Industrial Engineering and Operations Research, Columbia University, New York, New York 10027

Abstract

In this note we consider a single-server queueing loss system with zero buffer. The arrival process is a nonstationary Markov-modulated Poisson process. The arrival process in state i is Poisson with rate λi. The process remains in state i for a time that is exponentially distributed with rate Cαi, with c being a control or speed parameter. The service rate in state i is exponentially distributed with rate μi. The process moves from state i to state j with transition probability qij. We are interested in the loss probability as a function of c. In this note we show that, under certain conditions, the loss probability decreases when the c increases. As such, this result generalizes a result obtained earlier by Fond and Ross.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1995

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References

1.Chang, C.-S., Chao, X., & Pinedo, M. (1991). Monotonicity results for queues with doubly stochastic Poisson arrivals: Ross's conjecture. Advances in Applied Probability 23: 210228.CrossRefGoogle Scholar
2.Chang, C.-S. & Nelson, R. (1991). Perturbation analysis of the M/M/l queue in Markovian environment via the matrix geometric method. Research Report 75593, IBM. T.J. Watson Research Center, Yorktown Heights, NY.Google Scholar
3.Du, C. (1994). A monotonicity result for a single server queue subject to a Markov modulated Poisson process. Journal of Applied Probability, forthcoming.Google Scholar
4.Fiedler, M. (1986). Special matrices and their applications in numerical mathematics. Dordrecht, The Netherlands: Martinus Nijhoff, p. 57.CrossRefGoogle Scholar
5.Fond, S. & Ross, S.M. (1978). A heterogenous arrival and service queueing loss model. Naval Research Logistics Quarterly 25: 483488.CrossRefGoogle Scholar
6.Heyman, D.P. (1982). On Ross's conjecture about queues with non-stationary arrivals. Journal of Applied Probability 19: 245249.CrossRefGoogle Scholar
7.Niu, S.-C. (1980). A single server queueing loss model with heterogeneous arrival and service. Operations Research 28: 584593.CrossRefGoogle Scholar
8.Rolski, T. (1989). Queues with nonstationary inputs. Queueing Systems 5: 113130.CrossRefGoogle Scholar
9.Ross, S.M. (1978). Average delay in queues with non-stationary arrivals. Journal of Applied Probability 15: 602609.CrossRefGoogle Scholar
10.Svoronos, A. & Green, L. (1987). The N-seasons S-servers loss system. Naval Research Logistics 34: 579591.3.0.CO;2-K>CrossRefGoogle Scholar
11.Svoronos, A. & Green, L. (1988). A convexity result for single server exponential loss systems with non-stationary arrivals. Journal of Applied Probability 25: 224227.CrossRefGoogle Scholar