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

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 (QBD) 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.