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 -