Measurement and performance of the strong stability method
Authors:
Louiza Bouallouche and Djamil Aïssani
Journal:
Theor. Probability and Math. Statist. 72 (2006), 1-9
MSC (2000):
Primary 60K25, 68M20, 90B22
DOI:
https://doi.org/10.1090/S0094-9000-06-00659-4
Published electronically:
August 10, 2006
MathSciNet review:
2168131
Full-text PDF Free Access
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Additional Information
Abstract:
The aim of this paper is to show how to use in practice the strong stability method and also to prove its efficiency. That is why we chose the $GI/M/1$ model for which there exist analytical results.
For this purpose, we first determine the approximation conditions of the characteristics of the $GI/M/1$ system. Under these conditions, we obtain the stability inequalities of the stationary distribution of the queue size.
We finally elaborate upon an algorithm for the approximation of the $GI/M/1$ system by the $M/M/1$ system, which calculates the approximation error with an exact computation. In order to give some idea about its application in practice, we give a numerical example.
The accuracy of the approach is evaluated by comparison with some known exact results.
References
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- D. Aĭssani and N. V. Kartashov, Strong stability of an imbedded Markov chain in an $M/G/1$ system, Teor. Veroyatnost. i Mat. Statist. 29 (1983), 3–7 (Russian). MR 727097
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References
- D. Aïssani and N. V. Kartashov, Ergodicity and stability of Markov chains with respect to operator topologies in a space of transition kernels, Doklady Akad. Nauk USSR, Ser. A 11 (1983), 3–5. MR 0728475 (85c:60110)
- D. Aïssani and N. V. Kartashov, Strong stability of an imbedded Markov chain in the $M/G/1$ queueing system, Teor. Veroyatnost. Mat. Statist. 29 (1983), 3–7; English transl. in Theor. Probab. Math. Statist. 29 (1984), 1–5. MR 0727097 (85d:60167)
- S. Fdida and G. Pujolle, Modèles de Systèmes et de Réseaux, Tome 1 et Tome 2, Eyrolle, 1989.
- E. Gelenbe and G. Pujolle, Introduction to Queueing Networks, Wiley, 1998. MR 0874339 (87m:60210)
- N. V. Kartashov, Strongly stable Markov chains, Stability Problems for Stochastic Models, Vsesoyuz. Nauchno-Issled. Inst. Sistem. Issled., Moscow, 1981, 54–59. MR 0874339 (87m:60210)
- N. V. Kartashov, Strong Stable Markov Chains, VSP/TBiMC, Utrecht/Kiev, 1996. MR 1451375 (99e:60150)
- R. Pedrono and J. M. Hellary, Recherche Opérationnelle, Hermann, Paris, 1983.
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Additional Information
Louiza Bouallouche
Affiliation:
L.A.M.O.S, Laboratory of Modelisation and Optimization of Systems, University of Béjaïa, 06000, Algeria
Email:
lamos_bejaia@hotmail.com
Djamil Aïssani
Affiliation:
L.A.M.O.S, Laboratory of Modelisation and Optimization of Systems, University of Béjaïa, 06000, Algeria
Keywords:
Queueing system,
Markov chain,
stability,
strong stability,
performance evaluation,
approximation
Received by editor(s):
July 30, 2003
Published electronically:
August 10, 2006
Article copyright:
© Copyright 2006
American Mathematical Society
