Accelerated Accuracy in the Simulation of Markov Chains

Abstract

This paper describes a method of obtaining results from the simulation of an n + 1 finite state positive recurrent aperiodic Markov chain at a cost considerably less than that required by pure random sampling to achieve the same accuracy. The method reorganizes k independent epochs (or tours) simulated serially into k replications simulated in parallel, and therefore produces cost savings by inducing selected joint distributions across replications. The joint distributions are derived by the use of rotation sampling, a special case of the antithetic variate method. The paper first describes the method as it applies to a finite state nearest neighbor chain. In particular, it shows that even for independent parallel replications, the cost of achieving a specified accuracy with serial simulation, relative to the cost for parallel simulation, has a lower bound O(k1/2/n) as k → ∞. When rotation sampling is used, this bound is O(k2/n(ln k)3). A simulation of the M/M/1 queueing model with finite capacity n illustrates the effectiveness of the technique for selected values of k, n and activity level ρ. The paper then describes how this variance reducing technique applies to the simulation of a more general finite state chain. In particular, the lower bound on relative cost is O(k2/ϕ(ln k)3), where ϕ is the total number of transitions that can occur from all n + 1 states. A computer program that implements the method for the general finite state case is briefly described.