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Theory of Probability and Mathematical Statistics

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Measurement and performance of the strong stability method


Authors: Louiza Bouallouche and Djamil Aïssani
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 72 (2005).
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
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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 [Enhancements On Off] (What's this?)

  • 1. 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)
  • 2. 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)
  • 3. S. Fdida and G. Pujolle, Modèles de Systèmes et de Réseaux, Tome 1 et Tome 2, Eyrolle, 1989.
  • 4. E. Gelenbe and G. Pujolle, Introduction to Queueing Networks, Wiley, 1998. MR 0874339 (87m:60210)
  • 5. 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)
  • 6. N. V. Kartashov, Strong Stable Markov Chains, VSP/TBiMC, Utrecht/Kiev, 1996. MR 1451375 (99e:60150)
  • 7. R. Pedrono and J. M. Hellary, Recherche Opérationnelle, Hermann, Paris, 1983.
  • 8. S. T. Rachev, The problem of stability in queueing theory, Queueing Systems 4 (1989), 287-318. MR 1018523 (91c:60132)

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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

DOI: https://doi.org/10.1090/S0094-9000-06-00659-4
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

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