ASYMPTOTIC PROPERTIES OF STATIONARY DISTRIBUTIONS IN TWO-STAGE TANDEM QUEUEING SYSTEMS

Abstract

This paper is concerned with geometric decay properties of the joint queue length distribution p(n_1,n_2) in two-stage tandem queueing system PH/PH/c_1 → /PH/c_2. We prove that, under some conditions, p(n_1,n_2) 〜 C(n_2)η^<n_1> as n_1 → ∞ and p(n_1,n_2) 〜 C^^-(n_1)η^^-^<n_2> as n_2 → ∞. We also obtain the asymptotic form of state probabilities when n_1 is large or when n_2 is large. These results prove a part of the conjecture of a previous paper [1]. The proof is a direct application of a theorem in [7] which proves geometric decay property of the stationary distribution in a quasi-birth-and-death process with a countable number of phases in each level.