Abstract
Consider an M/G/c queue with homogeneous servers and service time distribution F. It is shown that an approximation of the service time distribution F by stochastically smaller distributions, say F n , leads to an approximation of the stationary distribution π of the original M/G/c queue by the stationary distributions π n of the M/G/c queues with service time distributions F n . Here all approximations are in weak convergence. The argument is based on a representation of M/G/c queues in terms of piecewise deterministic Markov processes as well as some coupling methods.
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Asmussen, S.: Applied Probability and Queues. Springer, New York (2003)
Asmussen, S., Johansen, H.: Über eine Stetigkeitsfrage betreffend das Bedienungssystem GI/GI/s. Elektron. Inf.verarb. Kybern. 22(10/11), 565–570 (1986)
Asmussen, S., Møller, J.: Calculation of the steady state waiting time distribution in GI/PH/c and MAP/PH/c queues. Queueing Syst. 37, 9–29 (2001)
Asmussen, S., Nerman, O., Olsson, M.: Fitting phase-type distributions via the EM algorithm. Scand. J. Stat. 23(4), 419–441 (1996)
Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968)
Breiman, L.: Probability. SIAM, Philadelphia (1968)
Breuer, L.: Transient and stationary distributions for the GI/G/k queue with Lebesgue-dominated inter-arrival time distribution. Queueing Syst. 45, 47–57 (2003)
Costa, O.: Stationary distributions for piecewise-deterministic Markov processes. J. Appl. Probab. 27(1), 60–73 (1990)
Davis, M.: Piecewise-deterministic Markov processes: A general class of non- diffusion stochastic models. J.R. Stat. Soc. Ser. B 46, 353–388 (1984)
Davis, M.: Markov Models and Optimization. Chapman & Hall, London (1993)
Dufour, F., Costa, O.L.: Stability of piecewise-deterministic Markov processes. SIAM J. Control Optim. 37(5), 1483–1502 (1999)
Kiefer, J., Wolfowitz, J.: On the theory of queues with many servers. Trans. Am. Math. Soc. 78, 1–18 (1955)
Kimura, T.: Approximations for multi–server queues: system interpolations. Queueing Syst. 17, 347–382 (1994)
Lindvall, T.: Lectures on the Coupling Method. Wiley, Chichester (1992)
Lucantoni, D.M., Ramaswami, V.: Algorithms for the multi-server queue with phase type service. Stoch. Models 1(3), 393–417 (1985)
Neuts, M.F.: Matrix-Geometric Solutions in Stochastic Models. Johns Hopkins University Press, Baltimore (1981)
Orey, S.: Lecture Notes on Limit Theorems for Markov Chain Transition Probabilities. Van Nostrand, London (1971)
Rachev, S.: Probability Metrics and the Stability of Stochastic Models. Wiley, Chichester (1991)
Schassberger, R.: Warteschlangen. Springer, Wien (1973)
Stoyan, D.: Comparison Methods for Queues and Other Stochastic Models. Wiley, Chichester (1983)
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Breuer, L. Continuity of the M/G/c queue. Queueing Syst 58, 321–331 (2008). https://doi.org/10.1007/s11134-008-9073-x
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DOI: https://doi.org/10.1007/s11134-008-9073-x