Numerical methods for reliability evaluation of Markov closed fault-tolerant systems

Abstract

This paper compares three numerical methods for reliability calculation of Markov, closed, fault-tolerant systems which give rise to continuous-time, time-homogeneous, finite-state, acyclic Markov chains. The authors consider a modified version of Jensen's method (a probabilistic method, also known as uniformization or randomization), a new version of ACE (acyclic Markov chain evaluator) algorithm with several enhancements, and a third-order implicit Runge-Kutta method (an ordinary-differential-equation solution method). Modifications to Jensen's method include incorporating stable calculation of Poisson probabilities and steady-state detection of the underlying discrete-time Markov chain. The new version of Jensen's method is not only more efficient but yields more accurate results. Modifications to ACE algorithm are proposed which incorporate scaling and other refinements to make it more stable and accurate. However, the new version no longer yields solution symbolic with respect to time variable. Implicit Runge-Kutta method can exploit the acyclic structure of the Markov chain and therefore becomes more efficient. All three methods are implemented. Several reliability models are numerically solved using these methods and the results are compared on the basis of accuracy and computation cost.