Approximate solution of Markov renewal programs with finite time horizon

The present paper investigates the error committed by using an infinite tim e horizon Markov renewal program as an approximation of the (often more rea listic) Markov renewal program with a finite time horizon t(0). Under weak assumptions the error is shown to converge to zero exponentially fast when t(0) --> infinity. The convergence is based on explicit error bounds. Impro ved error bounds hold when the (transformed) transition law has a nontrivia l stochastic lower bound. Some bounds use the discounted renewal function. For the latter, monotone upper and lower bounds are obtained by an iterativ e method combined with an extrapolation. Several examples demonstrate the a pplicability of the results.