Geometric Decay in a QBD Process with Countable Background States with Applications to a Join-the-Shortest-Queue Model

Abstract

A geometric tail decay of the stationary distribution has been recently studied for the GI /G /1 type Markov chain with both countable level and background states. This method is essentially the matrix analytic approach, and simplicity is an obvious advantage of this method. However, so far it can be only applied to the -positive case (or the jittered case, as referred to in the literature). In this paper, we specialize the GI /G /1 type to a quasi-birth-and-death process. This not only refines some expressions because of the matrix geometric form for the stationary distribution, but also allows us to extend the study, in terms of the matrix analytic method, to non--positive cases. We apply the result to a generalized join-the-shortest-queue model, which only requires elementary computations. The obtained results enable us to discuss when the two queues are balanced in the generalized join-the-shortest-queue model, and establish the geometric tail asymptotics along the direction of the difference between the two queues.

Exact asymptotics for the stationary distribution of a Markov chain: a production model

“…Theorem 2.2.1 of Li, Miyazawa and Zhao [15] is somewhat similar to our results though the Markov chain analyzed (QBD process) is different. The approach in [15] does not seem to give any information about the large deviation path.…”

“…If the r.h.s. of (15) converges to a finite positive constant as tends to infinity, then we have an exact asymptotic expression for π( , y). Now we investigate the convergence of the r.h.s.…”

“…The argument continues without changes until just after (15). Under the jitter conditions, the location ← − X (T ) where the time reversal hits Δ v converged to a proper distribution as → ∞.…”

The shorter queue polling model

“…There exists an extensive literature on dispatching policies and their optimality [9,26,27,35,36,37,42,46]. Among the dispatching policies, the join-the-shortest-queue (JSQ) policy has received considerable attention [5,6,10,19,20,23,24,28,30,44,45]. The JSQ policy in some scenarios has been proven to be the optimal policy; on the one hand it minimizes the customers mean waiting time [25] and on the other hand it stochastically maximizes the number of customers served by time t, t > 0 [43].…”

Analysis of Stationary Queue Length Distribution for Geo/T-IPH/1 Queue

“…For example, Takahashi [17], Haque [5], Motyer [11] and Li [7] considered the sufficient conditions when the stationary distribution of queue length has the characteristic of geometric decay along certain direction. As for numerical methods, one can see in Blanc [3] for power-series algorithm, in Rao [13] or Lian [8] for dimension-reduction method, or in Shi [15] for BRI algorithm.…”