Takacs' asymptotic theorem and its applications: a survey

The book of Lajos Takacs Combinatorial Methods in the Theory of Stochastic Processes has been published in 1967. It discusses various problems associated with Pk,i=P{sup1≤n≤ρ(i) (N n-n)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 =∑ j=0 k 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 Takacs. In the present survey we discuss applications of Takacs’ 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 Takacs' theorem are discussed.