Tail Decay Rates in Double QBD Processes and Related Reflected Random Walks

Abstract

A double quasi-birth-and-death (QBD) process is the QBD process whose background process is a homogeneous birth-and-death process, which is a synonym of a skip-free random walk in the two-dimensional positive quadrant with homogeneous reflecting transitions at each boundary face. It is also a special case of a 0-partially homogenous chain introduced by Borovkov and Mogul'skii. Our main interest is in the tail decay behavior of the stationary distribution of the double QBD process in the coordinate directions and that of its marginal distributions. In particular, our problem is to get their rough and exact asymptotics from primitive modeling data. We first solve this problem using the matrix analytic method. We then revisit the problem for the 0-partially homogenous chain, refining existing results. We exemplify the decay rates for Jackson networks and their modifications.

Rare event asymptotics for a random walk in the quarter plane

“…Geometric decay requires the dominant singularity to be a pole, whereas it could be a singularity of a different nature like a branch point. The recent work of Miyazawa [24] greatly enlarges the scope of applicability of the matrix-analytic techniques, because it is no longer restricted to the boundary condition. For the general class of skip-free random walks in the quarter plane, Miyazawa characterizes both the rough and exact asymptotics of the tail decay rates, for coordinate directions and marginal distributions.…”

Wiener-Hopf Factorizations for a Multidimensional Markov Additive Process and their Applications to Reflected Processes

“…However, there are few results for computing the decay rates from primitive modeling parameters. Moreover, most of them are limited to the two dimensional case without background state although exact asymptotics are also obtained (e.g., see [5,21,22]). We are interested to find the rough asymptotics for the multidimensional reflected MAP which may have background states, where reflection at the boundary may be arbitrarily given.…”

“…The upper bound (5.15) may not be easy to use since it involves the unknown factor µ 0 . There have been some studies to find bounds for µ 0 * (θ) for the two dimensional reflecting process without background state (e.g., see [7,21,22,23]). 3.…”

Decay rates for some quasi-birth-and-death processes with phase-dependent transition rates

“…Sufficient conditions under which the stationary distribution for the level process of a QBD process with countably many phases has a geometric tail were obtained by Takahashi et al [14] and Haque et al [5]. Miyazawa [9] essentially provided a complete study of the exact decay behaviour of such models in the direction of the axes. Similar conditions were obtained for specific examples by Foley and McDonald [4], Miyazawa [8], and Adan et al [1].…”

“…Since (2.2) may be satisfied by altering onlyQ 1 , Theorem 2.4 of [6] suggests that, by changing the transition structure of the QBD process at level 0, the stationary distribution may be forced to possess any decay rate from the range of z values. Furthermore, even when the process does not possess the level-phase independence property, there are many situations where the asymptotic decay rate is one of the values of z for which R has a z −1 -invariant measure; see, for example, [9]. In order to find a vector w and scalar z satisfying (2.1), we apply Theorem 5.4 of [12], which states that if w ∈ 1 and z ∈ (0, 1) satisfy…”