TY - JOUR

T1 - Takacs' asymptotic theorem and its applications: A survey

AU - Abramov, Vyacheslav

PY - 2010

Y1 - 2010

N2 - The book of Lajos Takacs Combinatorial Methods in the Theory of Stochastic Processes has been published in 1967. It discusses various problems associated with
P-k,P-i = P sup(1 = 0, pi(0) > 0, and rho(i) is the smallest n such that N-n = n - i, i >= 1. (If there is no such n, then rho(i) = infinity.)
Equation (*) is a discrete generalization of the classic ruin probability, and its value is represented as P-k,P-i = Q(k-i)/Q(k), where the sequence Q(k) (k >= 0) satisfies the recurrence relation of convolution type: Q(0) not equal 0 and Q(k) = Sigma(k)(j=0) pi(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 -> infinity. The theorem on asymptotic behavior of Q(k) as k -> infinity 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.

AB - The book of Lajos Takacs Combinatorial Methods in the Theory of Stochastic Processes has been published in 1967. It discusses various problems associated with
P-k,P-i = P sup(1 = 0, pi(0) > 0, and rho(i) is the smallest n such that N-n = n - i, i >= 1. (If there is no such n, then rho(i) = infinity.)
Equation (*) is a discrete generalization of the classic ruin probability, and its value is represented as P-k,P-i = Q(k-i)/Q(k), where the sequence Q(k) (k >= 0) satisfies the recurrence relation of convolution type: Q(0) not equal 0 and Q(k) = Sigma(k)(j=0) pi(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 -> infinity. The theorem on asymptotic behavior of Q(k) as k -> infinity 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.

UR - http://www.springerlink.com/content/n118341150vn6835/fulltext.pdf

U2 - 10.1007/s10440-008-9337-9

DO - 10.1007/s10440-008-9337-9

M3 - Article

VL - 109

SP - 609

EP - 651

JO - Acta Applicandae Mathematicae

JF - Acta Applicandae Mathematicae

SN - 0167-8019

IS - 2

ER -