  Applications of stochastic models often involve the evaluation of
steady-state performance, which requires solving a set of balance equations. In
most cases of interest, the number of equations is infinite or even
uncountable. As a result, numerical or analytical solutions are unavailable.
This is true even when the system state is one-dimensional. This paper develops
a general method for computing stationary distributions and steady-state
performance measures of stochastic systems that can be described as
continuous-state Markov chains supported on R. The balance equations are
numerically solved by properly constructing a proxy Markov chain with finite
states. We show the consistency of the approximate solution and provide
deterministic non-asymptotic error bounds under the supremum norm. Our finite
approximation method is near-optimal among all approximation methods using
discrete distributions, including the empirical distributions generated by a
simulation approach. We apply the developed method to compute the stationary
distribution of virtual waiting time in a G/G/1+G queue and associated
performance measures under certain mild but general differentiability and
boundedness assumptions on the inter-arrival, service, and patience time
distributions. Numerical experiments validate the accuracy and efficiency of
our method, and show it outperforms a standard Markov chain Monte Carlo method
by several orders of magnitude. The developed method is also significantly more
accurate than the available fluid approximations for this queue.
