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 α-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-α-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.

Keywords:

• α-positivity
• Countable background states
• Decay rate
• Generalized join-the-shortest-queue model
• QBD process
• Stationary distribution

Mathematics Subject Classification:

• 60K25
• 60K20
• 60F10
• 68M20

ACKNOWLEDGMENTS

This research was supported in part by JSPS under grant No. 18510135 and by NSERC Discovery Research grants.