Parallel Implementation of the GTH Algorithm for Markov Chains

Extended Abstract

We are concerned with computing the steady-state distribution it of an finite irreducible Markov chain with transition matrix P. We will use the GTH [1]algorithm which has excellent numerical properties which have been demonstrated empirically[2] and mathematically [3] We have at our disposal workstations and a massively parallel computer; we want to see how execution times on the latter compare to execution times on the former. Embedded in this endeavor is an exploration of how to harness the massively parallel computer to work on the GTH algorithm. Our main conclusions are:

• Our massively parallel computer can solve a problem with one thousand states one hundred times as fast as a serial computer. Extrapolation of our experience using one-eighth of the available memory indicates that for a problem with eleven thousand states,the serial machine would require 10⁴ as much time as the parallel machine

• Having enough memory to store the transition matrix is the limiting factor for our parallel computer.

• When the transition matrix has the block tridiagonal form, our parallel computer can store many thousands of states (depending on the block size; 16 × 10⁶ states can be stored when the blocks are 2 × 2) and compute the steady-state distribution in a few hours. The 16 × 10⁶ state example can be done in 24 hours.