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Computational schemes for two exponential servers where the first has a finite buffer

Published online by Cambridge University Press:  10 May 2011

Moshe Haviv  and
Rita Zlotnikov
Moshe Haviv
Affiliation:
Department of Statistics, The Hebrew University of Jerusalem, 91905 Jerusalem, Israel. moshe.haviv@gmail.com
Rita Zlotnikov
Affiliation:
Department of Statistics, The Hebrew University of Jerusalem, 91905 Jerusalem, Israel. moshe.haviv@gmail.com
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Abstract

We consider a system consisting of two not necessarily identical exponential servers having a common Poisson arrival process. Upon arrival, customers inspect the first queue and join it if it is shorter than some threshold n. Otherwise, they join the second queue. This model was dealt with, among others, by Altman et al. [Stochastic Models20 (2004) 149–172]. We first derive an explicit expression for the Laplace-Stieltjes transform of the distribution underlying the arrival (renewal) process to the second queue. Second, we observe that given that the second server is busy, the two queue lengths are independent. Third, we develop two computational schemes for the stationary distribution of the two-dimensional Markov process underlying this model, one with a complexity of $O(n \log\delta^{-1})$, the other with a complexity of $O(\log n \log^2\delta^{-1})$, where δ is the tolerance criterion.

Keywords

Memoryless queuesquasi birth and death processesmatrix geometric
Type
Research Article
Information
RAIRO - Operations Research , Volume 45 , Issue 1 , January 2011 , pp. 17 - 36
Copyright
© EDP Sciences, ROADEF, SMAI, 2011

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References

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