Adv. Appl. Prob. 37, 1075–1093 (2005) Printed in Northern Ireland  Applied Probability Trust 2005 LIGHT-TAILED ASYMPTOTICS OF STATIONARY PROBABILITY VECTORS OF MARKOV CHAINS OF GI / G/1 TYPE QUAN-LIN LI, ∗ Tsinghua University YIQIANG Q. ZHAO, ∗∗ Carleton University Abstract In this paper, we consider the asymptotic behavior of stationary probability vectors of Markov chains of GI/G/1 type. The generating function of the stationary probability vector is explicitly expressed by the R-measure. This expression of the generating function is more convenient for the asymptotic analysis than those in the literature. The RG-factorization of both the repeating row and the Wiener–Hopf equations for the boundary row are used to provide necessary spectral properties. The stationary probability vector of a Markov chain of GI/G/1 type is shown to be light tailed if the blocks of the repeating row and the blocks of the boundary row are light tailed. We derive two classes of explicit expression for the asymptotic behavior, the geometric tail, and the semigeometric tail, based on the repeating row, the boundary row, or the minimal positive solution of a crucial equation involved in the generating function, and discuss the singularity classes of the stationary probability vector. Keywords: Markov chain of GI/G/1 type; Markov chain of GI/M/1 type; Markov chain of M/G/1 type; asymptotic analysis; light tail; censoring method; RG-factorization; batch Markov arrival process 2000 Mathematics Subject Classification: Primary 60K25; 60K15 Secondary 60J22 1. Introduction Consider a Markov chain whose transition probability matrix is given by P = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ D 0 D 1 D 2 D 3 ··· D -1 A 0 A 1 A 2 ··· D -2 A -1 A 0 A 1 ··· D -3 A -2 A -1 A 0 ··· . . . . . . . . . . . . . . . ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , (1) where the matrices A i , -∞ <i< ∞, D 0 , and D j and D -j ,j ≥ 1, are of sizes m × m, m 0 ×m 0 , m 0 ×m, and m×m 0 , respectively. This Markov chain is referred to as being of GI/G/1 type. Throughout the paper, the Markov chain of GI/G/1 type is assumed to be irreducible and positive recurrent, and its stationary probability vector π is partitioned accordingly into vectors (π 0 ,π 1 ,π 2 ,...). Received 17 September 2002; revision received 15 August 2005. ∗ Postal address: Department of Industrial Engineering,Tsinghua University, Beijing, 100084, P. R. China. ∗∗ Postal address: School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada K1S 5B6. Email address: zhao@math.carleton.ca 1075 at https://www.cambridge.org/core/terms. https://doi.org/10.1239/aap/1134587754 Downloaded from https://www.cambridge.org/core. IP address: 181.215.101.49, on 12 Sep 2019 at 10:32:25, subject to the Cambridge Core terms of use, available