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A monotonicity result for a single-server loss system

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Xiuli Chao  and
Liyi Dai
Xiuli Chao*
Affiliation:
New Jersey Institute of Technology
Liyi Dai*
Affiliation:
Washington University
*
∗ Postal address: Department of Industrial and Manufacturing Engineering, New Jersey Institute of Technology, Newark, NJ 07102, USA.
∗∗ Postal address: Department of Systems Science and Mathematics, Washington University, St. Louis, MO 63130, USA.
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Abstract

We consider a family of M(t)/M(t)/1/1 loss systems with arrival and service intensities (λt (c), μt (c)) = (λct , μct ), where (λt , μt ) are governed by an irreducible Markov process with infinitesimal generator Q = (qij )m × m such that (λt , μt ) = (λi , μi ) when the Markov process is in state i. Based on matrix analysis we show that the blocking probability is decreasing in c in the interval [0, c ∗], where c ∗ = 1/maxi Σj ≠iqij /(λi + μi ). Two special cases are studied for which the result can be extended to all c. These results support Ross's conjecture that a more regular arrival (and service) process leads to a smaller blocking probability.

Keywords

NON-STATIONARY QUEUEBLOCKING PROBABILITYMATRIX ANALYSISROSS'S CONJECTURE

MSC classification

Primary: 60K25: Queueing theory

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 4 , December 1995 , pp. 1112 - 1117
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, RC #75593, Yorktown Heights, NY.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
ÇlInlar, E. (1975) Introduction to Stochastic Processes. Prentice Hall, NJ.Google Scholar
Du, C. (1994) A monotonicity result for the single server queue subject to 3-level Markov modulated Poisson process. Technical Report, Dept. of IE/OR, Columbia University, NY.Google Scholar
Du, C. and Pinedo, M. (1994) A convexity property of a Markov modulated queueing loss system. Technical Report, Dept. of IE/OR, Columbia University, NY.Google Scholar
Fond, S. and Ross, S. (1978) A heterogeneous arrival and service queueing loss model. Naval Res. Logist. Quart. 25, 483–488.Google Scholar
Heyman, D. (1982) On Ross's conjecture about queues with non-stationary arrivals. J. Appl. Prob. 19, 245–249.Google Scholar
Horn, R. A. and Johnson, C. R. (1987) Matrix Analysis. Cambridge University Press.Google Scholar
Niu, S. C. (1980) A single server queueing loss system with heterogeneous arrival and service. Operat. Res. 28, 584–593.Google Scholar
Rolski, T. (1984) Comparison theorems for queues with dependent interarrival times. Modeling and Performance Evaluation Methodology, Proceedings of the International Seminar Paris. 42–70.Google Scholar
Rolski, T. (1990) Queues with nonstationary inputs. Queueing Systems 5, 113–130.CrossRefGoogle Scholar
Ross, S. (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-season S-server 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