A New Finite Approximation Method for Evaluating Steady-State Performance of a Continuous-State Markov Chain with an Application to Queues with Customer Abandonment.

This article presents a novel deterministic finite approximation method for computing the stationary distribution and steady-state performance measures of continuous-state Markov chains (MCs) supported on the real line, with a particular application to GI/GI/1+GI queues featuring customer abandonment. The approach constructs finite-state proxy MCs by discretizing the transition kernel, providing explicit solutions with deterministic, nonasymptotic error bounds under the supremum norm, and demonstrating near-optimal convergence rates compared to other discrete approximation methods. The method does not rely on large market assumptions and accommodates general bounded patience time distributions, outperforming steady-state simulation, phase-type, diffusion, and fluid approximations especially in small- or medium-scale or overloaded queueing systems. Theoretical results include consistency and computable error bounds for both stationary distributions and performance measures, supported by numerical experiments and an extension framework for multidimensional MCs.