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 paper develops a new method for computing the stationary distribution and steady-state performance measures of stochastic systems that can be described as a continuous-state Markov chain supported on [Formula: see text]. The balance equations are solved by constructing a proxy Markov chain with finite states. We show the consistency of an approximate solution and provide deterministic nonasymptotic error bounds under the supremum norm. Our method is near optimal among all approximation methods using discrete distributions. We apply the developed method to compute the stationary distribution of virtual waiting time and associated performance measures for a GI/GI/1+GI queue in which the large market assumption may not hold and the patience time may follow any bounded distribution. Numerical experiments show that our method outperforms steady-state simulation, phase-type approximation, diffusion approximation, and fluid approximation, particularly for medium or small arrival intensities in overloaded or balanced loaded queues. Funding: S. Li and S. Mehrotra were supported by the National Science Foundation grant CMMI-1763035.