Tail asymptotics of the stationary distribution of a two dimensional reflecting random walk with unbounded upward jumps

TL;DR

This paper analyzes the tail behavior of the stationary distribution of a two-dimensional reflecting random walk with unbounded upward jumps, providing explicit asymptotic results under light tail assumptions.

Contribution

It extends previous work by deriving tail asymptotics for a more general class of reflecting random walks with unbounded upward jumps.

Findings

Complete tail asymptotics for marginal stationary distributions.

Generalization of previous results for skip free walks.

Application to a two-node network with batch arrivals.

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 the upward jump size distributions have light tails, we completely 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) [Mathematics of Operations Research 34, 547-575]. We exemplify these results for a two node network with exogenous batch arrivals.