Asymptotic Properties of Stationary Distributions in Two-Stage Tandem Queueing Systems

Abstract

Abstruet This paper is concerned with geometric decay properties of the joint queue length distributiori p(ni,n2) in two-stage tandem queueing system Pll'fPff/ci-)v IPfflc2. We prove that, under some conditions, p(ni,n2) N C(n2)n"i as nico and p(ni,n2)-C(ni)ii'i' as n2oo. We also obtain the asymptoeic form of state probabilities whell ni is large or when n2 is large. These results prove a part of the conjectllre of a previolls paper [1]. The proof is a direct application of a theorem i] [7] which proves geometric decay property of the st,ationary distribution in a quasi-birth-and-death process with a countable number of phases in each level.

Upper bound for the decay rate of the joint queue-length distribution in a two-node Markovian queueing system

“…2 ) c 2 from Lemma 5.3, and this coincides with the exact decay rate given in [2]. For the bound using η * k , among the eight types of Theorem 6.2, five types, (a-0), (a-1), (a-2), (b) and (d), may occur under the condition b K(1) 2 < 0.…”

“…, the decay rate is given by (η h 1 1 ) c 1 (η h 2 2 ) c 2 when −c 1 /c 2 is sufficiently close to 0 (Theorem 3.2 of [2]). …”

“…In one of them, η(c 1 , c 2 ) is given by (η h 1 1 ) c 1 (η h 2 2 ) c 2 (see (6.13)). In another type, η(c 1 , c 2 ) is given by (η h 2 1 ) c 1 (6.11)). These types seem to correspond to the two cases above conjectured in [1].…”

Matrix Product-Form Solutions of Stationary Probabilities in Tandem Queues

“…In Fajirnoto et al [7] that O < epT2 < Now we check (3.4) for m, > 1 and n > 1, Insertioll of (3,14) in (3.4) by n7'ep: yields (uo x ui X u2)(T G S/t o S2) == (3,17) in right hand sicle will balances (3,4 Observe that (3,8), (3,9), (3.11) and (3,I2) (3,20). In addition, 7r.o and To.…”

“…Note that h(x) is convex as shown in [7]. (3,21), Similarly, starting with ri,i, Ti,2, T2,] and 7T`i,3, we can fincl 7ro,., n }l 3, by (3,22 (8), (9), (11) and (12) …”