Inhomogeneous perturbation and error bounds for the stationary performance of random walks in the quarter plane

TL;DR

This paper develops a method to approximate the stationary performance of random walks in the quarter plane using inhomogeneous perturbations, providing explicit error bounds and demonstrating improved accuracy over homogeneous approaches.

Contribution

It introduces inhomogeneous transition rates for perturbed random walks with geometric stationary distributions, enabling tighter error bounds and more flexible modeling.

Findings

Inhomogeneous perturbations can produce smaller error bounds than homogeneous ones.

Explicit error bounds are derived using the Markov reward approach.

Numerical experiments confirm the effectiveness of inhomogeneous perturbations.

Abstract

A continuous-time random walk in the quarter plane with homogeneous transition rates is considered. Given a non-negative reward function on the state space, we are interested in the expected stationary performance. Since a direct derivation of the stationary probability distribution is not available in general, the performance is approximated by a perturbed random walk, whose transition rates on the boundaries are changed such that its stationary probability distribution is known in closed form. A perturbed random walk for which the stationary distribution is a sum of geometric terms is considered and the perturbed transition rates are allowed to be inhomogeneous. It is demonstrated that such rates can be constructed for any sum of geometric terms that satisfies the balance equations in the interior of the state space. The inhomogeneous transitions relax the pairwise-coupled structure on these geometric terms that would be imposed if only homogeneous transitions are used. An explicit expression for the approximation error bound is obtained using the Markov reward approach, which does not depend on the values of the inhomogeneous rates but only on the parameters of the geometric terms. Numerical experiments indicate that inhomogeneous perturbation can give smaller error bounds than homogeneous perturbation.