Abstract
Recently developed methods of qualitative analysis for regenerative processes arising in queueing are presented. These methods are essentially qualitative and use notions such as coupling, probability metrics, etc. They are developed for studying various properties of regenerative models, including convergence rate to a stationary regime, continuity of their characteristics with respect to some parameters and first-occurrence time of an event such as queue overflowing. In spite of their qualitative nature they lead to good quantitative estimates of underlying properties with computer methods available to calculate them.
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References
E. Nummelin,General Irreducible Markov Chains and Non-negative Operators (Cambridge Univ. Press, 1984).
W.L. Smith, Renewal theory and its ramifications, J. Roy. Stat. Soc. B20, no. 2 (1958).
H. Thorisson, The coupling of regenerative processes, Adv. Appl. Prob. 15 (1983) 531–561.
V.M. Zolotarev,A Modern Theory of Summarizing Independent Random Variables (Nauka, Moscow, 1986) (in Russian).
V.V. Kalashnikov and S.T. Rachev,Mathematical Methods of Queueing Models Construction (Nauka, Moscow, 1988) (in Russian).
L.V. Kantorovich and G.Sh. Rubinstein, On a space of completely additive functions, Vestnik Leningrad. Univ., Ser. Math., Mech., Astronom. 2, no. 7 (1958) 52–59 (in Russian).
V.M. Zolotarev, Metric distances in spaces of random variables and their distributions, Math. Sbornik 101 (143), no. 3 (1976) 416–454 (in Russian).
V. Strassen, The existence of probability measures with given marginals, Ann. Math. Stat. 36 no. 2 (1965) 423–439.
V.V. Kalashnikov,Qualitative analysis of complex systems behaviour by Trial Function Method (Nauka, Moscow, 1978) (in Russian).
T. Lindvall, On coupling of renewal processes with the use of failure rates, Stoch. Proc. Appl. 22 (1986) 1–15.
A.A. Borovkov,Asymptotic Methods in Queueing (Nauka, Moscow 1978), (in Russian).
G.Sh. Tzitziashvili, Piece-wise linear Markov chains and the study of their stability, Theory Prob. Appl. 20 (1975) 345–358.
D.L. Iglehart and G.S. Shedler,Regenerative Simulation of Response Times in Networks of Queues (Springer, Berlin, 1980).
V.V. Kalashnikov and S. Yu. Vsekhsviatskii, Metric estimates of the first occurrence time in regenerative processes,Stability Problems for Stochastic Models, Lecture Notes in Mathematics, no. 1155 (Springer, Berlin, 1985) pp. 102–130.
V.V. Kalashnikov, Estimates of convergence rate in Renyi's theorem,Stability Problems for Stochastic Models (VNIISI, Moscow, 1983) (in Russian).
V.V. Kalashnikov and S.Yu. Vsekhsviatskii, Estimates in Renyi's theorem in terms of renewal theory, Theory Appl. 33 (1988) 369–373.
V.V. Kalashnikov and S.Yu. Vsekhsviatskii, Estimates of the first-occurrence times and their connection with renewal theory,Stability Problems for Stochastic Models, Lecture Notes in Mathematics (Springer, Berlin, 1989) (to appear).
V.V. Kalashnikov, Analytical and simulation estimates of reliability for regenerative models, System Analysis, Modelling, Simulation (1989) (to appear).
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Kalashnikov, V.V. Regenerative queueing processes and their qualitative and quantitative analysis. Queueing Syst 6, 113–136 (1990). https://doi.org/10.1007/BF02411469
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DOI: https://doi.org/10.1007/BF02411469