The Invariant Measure of Random Walks in the Quarter-plane: Representation in Geometric Terms

Invariant measures and error bounds for random walks in the quarter-plane based on sums of geometric terms

We consider homogeneous random walks in the quarter-plane. The necessary conditions which characterize random walks of which the invariant measure is a sum of geometric terms are provided in Chen et al. (arXiv:1304.3316, 2013, Probab Eng Informational Sci 29(02):233–251, 2015). Based on these results, we first develop an algorithm to check whether the invariant measure of a given random walk is...

The Invariant Measure of Homogeneous Markov Processes in The Quarter-Plane: Representation in Geometric Terms

We consider the invariant measure of a homogeneous continuoustime Markov process in the quarter-plane. The basic solutions of the global balance equation are the geometric distributions. We first show that the invariant measure can not be a finite linear combination of basic geometric distributions, unless it consists of a single basic geometric distribution. Second, we show that a countable li...

Random Walks in the Quarter-Plane: Invariant Measures and Performance Bounds

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

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 o...

Two non-holonomic lattice walks in the quarter plane

We present two classes of random walks restricted to the quarter plane whose generating function is not holonomic. The non-holonomy is established using the iterated kernel method, a recent variant of the kernel method. This adds evidence to a recent conjecture on combinatorial properties of walks with holonomic generating functions. The method also yields an asymptotic expression for the numbe...