Tail Asymptotics of the Stationary Distribution of a Two-Dimensional Reflecting Random Walk with Unbounded Upward Jumps

Abstract

We consider a two-dimensional reflecting random walk on the nonnegative integer quadrant. This random walk is assumed to be skip free in the direction to the boundary of the quadrant, but may have unbounded jumps in the opposite direction, which are referred to as upward jumps. We are interested in the tail asymptotic behavior of its stationary distribution, provided it exists. Assuming that the upward jump size distributions have light tails, we find the rough tail asymptotics of the marginal stationary distributions in all directions. This generalizes the corresponding results for the skip-free reflecting random walk in Miyazawa (2009). We exemplify these results for a two-node queueing network with exogenous batch arrivals.

Wiener-Hopf factorizations for a multidimensional Markov additive process and their applications to reflected processes

“…This allows us to consider precise asymptotics of a system with autoregressive input. For the multi-dimensional case, we give some preliminary results which, together with the other results in this paper, are further developed in [16]. We have not considered other applications, but one may think of them.…”

A unified approach for large queue asymptotics in a heterogeneous multiserver queue

“…We may consider (2.22) and (2.23) as stationary equations. For studying the asymptotic problems related to the stationary distribution of L(•) ≡ {L(t); t ≥ 0}, one may consider to apply the techniques developed in [18], [22], which are based on the stationary equation similar to (2.23) and its variant called a stationary inequality. However, one must be careful about it.…”

Simple error bounds for the QBD approximation of a special class of two dimensional reflecting random walks

“…It is known that, even if µ E = 0, the 2D-RRW {Z(ℓ)} can be stable though its stationary distribution must be heavy-tailed. For details, see [6,11].…”

“…Unfortunately, it is, in general, difficult to obtain a closed-form expression of the stationary distribution of 2D-RRWs. This is primarily why the tail asymptotics of 2D-RRWs and their generalizations have been extensively studied (see [2,8,10,11,16,15,19,20,21,22] and the references therein). These studies focus on identifying the decay rate of the stationary distribution, though they make a limited contribution to the performance evaluation of the queueing systems mentioned above.…”