Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-22T19:14:57.479Z Has data issue: false hasContentIssue false
  • English
  • Français

Characterization problems in queueing and their stability

Published online by Cambridge University Press:  01 July 2016

V. V. Kalashnikov  and
S. T. Rachev
V. V. Kalashnikov*
Affiliation:
Institute for Systems Studies, Moscow
S. T. Rachev*
Affiliation:
Institute of Mathematics, Sofia
*
∗Postal address: Institute for Systems Studies, 9, Prospect 60 let Oktjabrja, 117312 Moscow, USSR.
∗∗Postal address: Institute of Mathematics, Bulgarian Academy of Sciences, P.O. Box 373, 1090 Sofia, Bulgaria.
Get access Rights & Permissions [Opens in a new window]

Abstract

The concept of a characterization and its stability for queueing models is introduced. The principle of two stages in the study of the stability property is formulated. A series of results concerning the G/G/1 model is obtained.

Keywords

QUEUEING MODELEXPONENTIAL DISTRIBUTIONPROBABILITY METRIC
Type
Research Article
Information
Advances in Applied Probability , Volume 17 , Issue 4 , December 1985 , pp. 868 - 886
Copyright
Copyright © Applied Probability Trust 1985 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barlow, R. E. and Proschan, F. (1975) Statistical Theory of Reliability and Life Testing: Probability Models , Holt. Rinehart and Winston, New York.Google Scholar
Barzilovich, E. Yu. Et Al. (1983) Problems of Reliability Theory (in Russian), Ed. Gnedenko, B. V. Radio i Sviaz, Moscow.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Borovkov, A. A. (1980) Asymptotic Methods in Queueing Theory (in Russian). Nauka, Moscow.Google Scholar
Dimitrov, B. N., Klebanov, L. B. and Rachev, S. T. (1982) Stability of the exponential law characterization (in Russian). In Stability of Stochastic Models , ed. Zolotarev, V. M. and Kalashnikov, V. V., VNIISI, Moscow, 39–46.Google Scholar
Dudley, R. M. (1976) Probability and Metrics. Aarhus Universitet Lecture Notes Series 45.Google Scholar
Fortet, R. and Mourier, E. (1953) Convergence de la répartition empirique vers la répartition théorique. Ann. Sci. Ecole Normale Sup. (3) 70, 266–285.Google Scholar
Galambos, J. and Kotz, S. (1978) Characterization of Probability Distributions. Lecture Notes in Mathematics 675, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Kalashnikov, V. V. (1978) Qualitative Analysis of the Behaviour of Complex Systems by Trial Functions Method (in Russian). Nauka, Moscow.Google Scholar
Kalashnikov, V. V. (1981) Estimates of convergence rate and stability for regenerative and renovative processes. In Coll. Math. Soc. J. Bolyai , 24; Point Processes and Queueing Problems, Keszthely, Hungary, 1978, North-Holland, Amsterdam, 163–180.Google Scholar
Kalashnikov, V. V. and Rachev, S. T. (1984) A characterization of queueing models and its stability (in Russian). In Stability of Stochastic Models , Ed. Zolotarev, V. M. and Kalashnikov, V. V., VNIISI, Moscow, 61–89.Google Scholar
Kalashnikov, V. V. and Zolotarev, V. M. (1983) Methoden zum Nachweis von Stetigkeitseigenschaften. In Handbuch der Bedienungstheorie , Vol. I, eds. Gnedenko, B. V. and König, D., Akademie-Verlag, Berlin, 337–351.Google Scholar
Kendall, D. G. and Lewis, T. (1965) On the structural information contained in the output of GI/G/8. Z. Wahrscheinlichkeitsth. 4, 144–148.Google Scholar
Kovalenko, I. N. (1965) On the recovery of the system characteristic using observations of output process (in Russian). Dokl. USSR Acad. Sci. 164, 979–981.Google Scholar
Obretenov, A. and Rachev, S. T. (1983) Characterizations of the bivariate exponential distribution and Marshall-Olkin distribution and stability. In Lecture Notes in Mathematics 982, ed. Zolotarev, V. M. and Kalashnikov, V. V., Springer-Verlag, Berlin, 136–150.Google Scholar
Obretenov, A. and Rachev, S. T. (1983a) Stability of the exponential law characterizations (in Russian). In Stability of Stochastic Models , ed. Zolotarev, V. M. and Kalashnikov, V. V., VNIISI, Moscow, 79–87.Google Scholar
Obretenov, A., Dimitrov, B. N. and Rachev, S. T. (1983) On the stability of the queueing system M/M/1/8 (in Russian). In Stability of Stochastic Models , ed Zolotarev, V. M. and Kalashnikov, V. V., VNIISI, Moscow, 71–79.Google Scholar
Yurinski, V. V. (1975) A smoothing inequality for estimates of the Levy-Prohorov distance. Theory Prob. Appl. XX, 1–10.CrossRefGoogle Scholar
Zolotarev, V. M. (1976) Metric distances in spaces of random variables and their distributions. Math. USSR Sb. 30, 373–401.CrossRefGoogle Scholar
Zolotarev, V. M. (1977) General problems of the stability of mathematical models. Bull. Internat. Statist. Inst. XLVII (2), 382–401.Google Scholar