Takács’ Asymptotic Theorem and Its Applications: A Survey

Abstract

The book of Lajos Takács Combinatorial Methods in the Theory of StochasticProcesses has been published in 1967. It discusses various problems associated withwhere N n = ν 1 + ν 2 + · · · + ν n is a sum of mutually independent, nonnegative integer and identically distributed random variables, π j = P{ν k = j }, j ≥ 0, π 0 > 0, and ρ(i) is the smallest n such that N n = n − i, 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 P k,i = Q k−i /Q k , where the sequence {Q k } k≥0 satisfies the recurrence relation of convolution type: Q 0 = 0 and Q k = k j =0 π j Q k−j +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 Q k as k → ∞. The theorem on asymptotic behavior of Q k 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.

Optimal control of a large dam with compound Poisson input and costs depending on water levels

“…A particular problem have been studied in [4]. A circle of problems associated with the results of [4] are discussed in a review paper [5].…”

“…25 should be used. For different applications of the aforementioned Takács' asymptotic theorems and Tauberian theorems of Postnikov see [5].…”

“…Along with previously mentioned paper [4], the method of asymptotic analysis closely related to subject matter of this paper have been considered by Abramov [2] and [3] and further reviewed in [5].…”

“…(1) L as L → ∞ that given by (2.1) is established on the basis of the known asymptotic behaviour of the sequence Q n as n → ∞ that given by (2.2) (see [31], p.22, [21] as well as recent paper [5]). To make the paper self-contained, the necessary results about the asymptotic behaviour of Q n as n → ∞ are given in Section 4.1.…”

Statistical Analysis of Single-server Loss Queueing Systems

“…In this section, we recall the known results for the main characteristics of the M/GI/1/n and GI/M/1/n queueing systems during their busy periods. For a more detailed information see [1], [9] or [10].…”