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A BAYESIAN APPROACH TO FIND RANDOM-TIME PROBABILITIES FROM EMBEDDED MARKOV CHAIN PROBABILITIES

Published online by Cambridge University Press:  22 October 2007

Winfried K. Grassmann  and
Javad Tavakoli
Winfried K. Grassmann
Affiliation:
Department of Computer ScienceUniversity of SaskatchewanSaskatoon, S7N 5C9Canada E-mail: [email protected]
Javad Tavakoli
Affiliation:
Department of Mathematics, Statistics and PhysicsUniversity of British Columbia OkanaganKelowna, V1V 1V7Canada E-mail: [email protected]
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Abstract

The embedded Markov chain approach is widely used in queuing theory, in particular in M/G/1 and GI/M/c queues. In these cases, one has to relate the embedded equilibrium probablities to the corresponding random-time probabilities. The classical method to do this is based on Markov renewal theory, a rather complex approach, especially if the population is finite or if there is balking. In this article we present a much simpler method to derive the random-time probabilities from the embedded Markov chain probabilities. The method is based on conditional probability. Our approach might also be applicable in such situations.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 21 , Issue 4 , October 2007 , pp. 551 - 556
Copyright
Copyright © Cambridge University Press 2007

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References

1.Chen, X. (1993). The use of derivatives for optimizing steady state queues. Master's thesis, University of Saskatchewan.Google Scholar
2.Ferreira, F.& Pacheco, A. (2006). Analysis of GI/M/s/c queues using uniformization. Computers and Mathematics with Applications 51: 291–304.Google Scholar
3.Grassmann, W.K. (1982). The GI/PH/1 queue: A method to find the transition matrix. INFOR 20: 144–156.Google Scholar
4.Grassmann, W.K., Chen, X., & Kashyap, B.B.K. (2001). Optimal service rates of the state-dependent M/G/1 queues in steady state.Operations Research Letters 29: 57–63.Google Scholar
5.Grassmann, W.K. & Zhao, Y.Q. (1996). Heterogeneous multiserver queues with general input. INFOR 35: 208–224.Google Scholar
6.Gross, D. & Harris, C.M. (1971). On one-for-one ordering inventory policies with state-dependent lead times. Operations Research 19: 735–760.CrossRefGoogle Scholar
7.Gross, D. & Harris, C.M. (1998). Fundamentals of queueing theory, 2nd ed.New York: Wiley.Google Scholar
8.Jain, J.L., Mohanty, S.G., & Böhm, W. (2007). A course on queueing models. London: Chapman & Hall/CRC.Google Scholar
9.Kleinrock, L. (1975). Queueing systems, Vol. 1. New York: Wiley.Google Scholar
10.Prabhu, N. (1965). Queues and inventories: A study of their basic stochastic properties. New York: Wiley.Google Scholar
11.Ross, S.M. (1970). Applied probability models with optimization applications. San Francisco: Holden Day.Google Scholar