Strong stability in a $G/M/1$ queueing system
Authors:
Mustapha Benaouicha and Djamil Aissani
Journal:
Theor. Probability and Math. Statist. 71 (2005), 25-36
MSC (2000):
Primary 60K25, 68M20, 90B22
DOI:
https://doi.org/10.1090/S0094-9000-05-00645-9
Published electronically:
December 28, 2005
MathSciNet review:
2144318
Full-text PDF Free Access
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Abstract: In this paper, we study the strong stability of the stationary distribution of the imbedded Markov chain in the $G/M/1$ queueing system, after perturbation of the service law (see Aissani, 1990, and Kartashov, 1981). We show that under some hypotheses, the characteristics of the $G/G/1$ queueing system can be approximated by the corresponding characteristics of the $G/M/1$ system. After clarifying the approximation conditions, we obtain the stability inequalities by exactly computing the constants.
References
- D. Aissani, Ergodicité uniforme et stabilité forte des chaines de Markov. Application aux systèmes de files d’attente, Séminaire Mathématique de Rouen 167 (1990), 115–121.
- Dzh. Aĭssani and N. V. Kartashov, Ergodicity and stability of Markov chains with respect to operator topologies in a space of transition kernels, Dokl. Akad. Nauk Ukrain. SSR Ser. A 11 (1983), 3–5 (Russian, with English summary). MR 728475
- 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
- J. Banks, J. S. Carson, and B. L. Nelson, Discrete-Event System Simulation, Prentice Hall, New Jersey, 1996.
- N. V. Kartashov, Strongly stable Markov chains, Problems of stability of stochastic models (Panevezhis, 1980) Vsesoyuz. Nauch.-Issled. Inst. Sistem. Issled., Moscow, 1981, pp. 54–59 (Russian). MR 668559
- N. V. Kartashov, Strong stable Markov chains, VSP, Utrecht; TBiMC Scientific Publishers, Kiev, 1996. MR 1451375
- L. Kleinrock, Queueing Systems, vols. 1 and 2, John Wiley and Sons, 1976.
References
- D. Aissani, Ergodicité uniforme et stabilité forte des chaines de Markov. Application aux systèmes de files d’attente, Séminaire Mathématique de Rouen 167 (1990), 115–121.
- D. Aissani and N. V. Kartashov, Ergodicity and stability of Markov chains with respect to operator topology in the space of transition kernels, Dokl. Akad. Nauk Ukr. SSR, ser. A 11 (1983), 3–5. MR 0728475 (85c:60110)
- D. Aissani and N. V. Kartashov, Strong stability of the imbedded Markov chain in an $M/G/1$ system, Theor. Probab. Math. Statist. 29 (1984), 1–5. MR 0727097 (85d:60167)
- J. Banks, J. S. Carson, and B. L. Nelson, Discrete-Event System Simulation, Prentice Hall, New Jersey, 1996.
- N. V. Kartashov, Strong stable Markov chains, Stability Problems for Stochastic Models, VNISSI, Moscow, 1981, pp. 54–59. MR 0668559 (84b:60089)
- N. V. Kartashov, Strong Stable Markov Chains, VSP/TBiMC, Ultrecht/Kiev, 1996. MR 1451375 (99e:60150)
- L. Kleinrock, Queueing Systems, vols. 1 and 2, John Wiley and Sons, 1976.
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Additional Information
Mustapha Benaouicha
Affiliation:
Laboratory of Modelization and Optimization of Systems, Faculty of Sciences and Engineer Sciences, University of Béjaia, 06000, Algeria
Djamil Aissani
Affiliation:
Laboratory of Modelization and Optimization of Systems, Faculty of Sciences and Engineer Sciences, University of Béjaia, 06000, Algeria
Email:
lamos_bejaia@hotmail.com
Keywords:
Queueing systems,
strong stability,
uniform ergodicity,
perturbations,
stability inequalities
Received by editor(s):
July 30, 2003
Published electronically:
December 28, 2005
Article copyright:
© Copyright 2005
American Mathematical Society
