# Variance reduction techniques for the simulation of Markov process - II. Matrix iterative methods

## Abstract

Let Xn, n≧0 be an irreducible, aperiodic, Markov chain with finite state space E, transition matrix P, and stationary distribution π. Let f be a real valued function on E and define r = πf. A method of reducing the variance of simulation estimates for r is presented. The method combines the techniques of numerical analysis and simulation by partially solving an appropriate system of linear equations using some matrix iterative procedure and then estimating the difference between the true and partial solutions via simulation. After k iterations of the iterative procedure, functions fν ν = 0, ..., k are defined so that r = πfgn for each ν. Let {Mathematical expression} and {Mathematical expression} where {Mathematical expression}. Then {Mathematical expression} a.s. as N → ∞ and β is chosen to minimize the asymptotic variance of {Mathematical expression}. Numerical results for a simple queueing model are presented. © 1980 Springer-Verlag.