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A monotonicity result for a single-server queue subject to a Markov-modulated Poisson process

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Qing Du
Qing Du*
Affiliation:
Columbia University
*
∗ Postal address: Department of Industrial Engineering and Operations Research, Columbia University, New York, NY 10027, USA.
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Abstract

Consider a single-server queue with zero buffer. The arrival process is a three-level Markov modulated Poisson process with an arbitrary transition matrix. The time the system remains at level i (i = 1, 2, 3) is exponentially distributed with rate cα i . The arrival rate at level i is λ i and the service time is exponentially distributed with rate μ i . In this paper we first derive an explicit formula for the loss probability and then prove that it is decreasing in the parameter c. This proves a conjecture of Ross and Rolski's for a single-server queue with zero buffer.

Keywords

POISSON ARRIVALS PROCESSZERO BUFFER

MSC classification

Primary: 60K25: Queueing theory

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 4 , December 1995 , pp. 1103 - 1111
Copyright
Copyright © Applied Probability Trust 1995 

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References

Chang, C.-S. and Nelson, R. (1991) Perturbation Analysis of the M/M/1 queue in Markovian environment via the matrix geometric method. IBM Research Report #75593.Google Scholar
Chang, C.-S., 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
Fiedler, M. (1986) Special Matrices and their Applications in Numerical Mathematics. Martinus Nijhoff, Dordrecht.Google Scholar
Fond, S. and Ross, S. M. (1978) A heterogeneous arrival and service queueing loss model. Naval Res. Logist. Quart. 25, 483–488.Google Scholar
Heyman, D. P. (1982) On Ross's conjecture about queues with non-stationary arrivals. J. Appl. Prob. 19, 245–249.Google Scholar
Niu, S.-C. (1980) A single server queueing loss model with heterogeneous arrival and service. Operat. Res. 28, 584–593.Google Scholar
Rolski, T. (1984) Comparison theorems for queues with dependent interarrival times. In Proceedings of the International Seminar, Paris, France, 1983, ed. Baccelli, F. and Fayolle, G. Lecture Notes in Control and Information Sciences 60, 42–70. Springer-Verlag, Berlin.Google Scholar
Rolski, T. (1989) Queues with nonstationary inputs. Queueing Systems 5, 113–130.CrossRefGoogle Scholar
Ross, S. M. (1978) Average delay in queues with non-stationary arrivals. J Appl. Prob. 15, 602–609.Google Scholar
Svoronos, A. and Green, L. (1987) The N-seasons S-servers loss system. Naval Res. Logist. 34, 579–591.Google Scholar
Svoronos, A. and Green, L. (1988) A convexity result for single server exponential loss systems with non-stationary arrivals. J. Appl. Prob. 25, 224–227.Google Scholar