Far Eastern Mathematical Journal

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Asymptotic invariants in one channel queueing system $G|G|1|\infty$

G. Sh. Tsitsiashvili, N. V. Markova

2002, issue 1, Ñ. 52–57

This paper is devoted to construction and investigation of invariant characteristics of stationary distribution tails of waiting time in queueing systems $M|M|1|\infty$, $G|G|1|\infty$ defined by subexponential distributions. Tails of these distributions are defined with accuracy of slowly varying multipliers. Stationary characteristics invariant to these multipliers are searched. Idea of invariant characteristics construction is based on classification of subexponential distributions suggested by Goldie and Kluppelberg and Karamata theorem and Embrechts-Veraverbeke formula.


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[1] C. Kluppelberg, “On Subexponential Distributions and Integrated Tails”, J. Appl. Probab., 25 (1988), 132–141.
[2] S. Asmussen, V. Kalashnikov, D. Konstantinides, C. Kluppelberg, G. Tsitsiashvili, “A local Limit Theorem for Random Walk Maxima with Heavy Tails”, Statist. Probab. Lett., 56:4 (2002), 399–404.
[3] C. M. Goldie, C. Kluppelberg, Subexponential Distributions, 96(1), Johannes Guttenberg, Universitat Mainz, 1996, 20 pp.
[4] P. Emprechts, C. Kluppelberg, T. Mikosh, Extremal Events in Finance and Insurance, Springer, 1997.
[5] P. Embrechts and N. Veraverbeke, “Estimates for the Probability of Ruin with Special Emphasis on the Possibility of Large Claims”, Mathematics and Economics, 1 (1982), 55–72.
[6] S. Asmussen, Ruin Probabilities, World Scientific, Singapore, 2000.
[7] S. Foss, “Asymptotics for Distributions of Stationary Characteristics in Queueing Networks with Heavy Tails”, Modern Problems in Applied Probability (20–27 August), Novosibirsk, 2000, 9 pp.
[8] E. J. G. Pitman, “Subexponential Distribution Functions”, J. Austral. Math. Soc. Ser. A, 29 (1980), 337–347.

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