Acta Applicandae Mathematicae

, Volume 109, Issue 2, pp 609–651

Takács’ Asymptotic Theorem and Its Applications: A Survey

Article

DOI: 10.1007/s10440-008-9337-9

Cite this article as:
Abramov, V.M. Acta Appl Math (2010) 109: 609. doi:10.1007/s10440-008-9337-9

Abstract

The book of Lajos Takács Combinatorial Methods in the Theory of Stochastic Processes has been published in 1967. It discusses various problems associated with
$$P_{k,i}=\mathrm{P}\left\{\sup_{1\leq n\leq\rho(i)}(N_{n}-n)<k-i\right\},$$
(*)
where Nn=ν1+ν2+⋅⋅⋅+νn is a sum of mutually independent, nonnegative integer and identically distributed random variables, πj=P{νk=j}, j≥0, π0>0, and ρ(i) is the smallest n such that Nn=ni, i≥1. (If there is no such n, then ρ(i)=∞.)

Equation (*) is a discrete generalization of the classic ruin probability, and its value is represented as Pk,i=Qki/Qk, where the sequence {Qk}k≥0 satisfies the recurrence relation of convolution type: Q0≠0 and Qk=∑j=0kπjQkj+1.

Since 1967 there have been many papers related to applications of the generalized classic ruin probability. The present survey is concerned only with one of the areas of application associated with asymptotic behavior of Qk as k→∞. The theorem on asymptotic behavior of Qk as k→∞ and further properties of that limiting sequence are given on pp. 22–23 of the aforementioned book by Takács. In the present survey we discuss applications of Takács’ asymptotic theorem and other related results in queueing theory, telecommunication systems and dams. Many of the results presented in this survey have appeared recently, and some of them are new. In addition, further applications of Takács’ theorem are discussed.

Keywords

Asymptotic analysis Tauberian theory Ballot problems Queueing theory Applications of queueing theory 

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.School of Mathematical SciencesMonash UniversityClaytonAustralia

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