AN ANALYSIS OF A RELIABILITY MODEL FOR REPAIRABLE FAULT-TOLERANT SYSTEMS

The ARIES reliability model proposed by Ng and Avizienis [191 models a class of repairable and nonrepairable fault-tolerant systems by a Continuous Time Markov Chain. ARIES uses the Lagrange-Sylvester interpolation Formula to directly compute the exponential of the State Transition Rate Matrix (STRM) which appears in the solution of the Markov Chain. Following have been the main objections to this solution technique [9], [16]. First, that the method is prohibitively expensive in terms of computation; the computational complexity is O(n5) for a state transition rate matrix of size n. Second, that it is not clear that the solution technique is general enough as to handle all repairable fault-tolerant systems which ARIES models. Third, that the numerical stability of this method is unsatisfactory. In this paper, we analyze the properties of the STRM for ARIES repairable systems and, drawing from well established results in matrix theory, suggest an efficient solution for reliability computation when the eigenvalues of the STRM are distinct. Also, we identify a class of systems that ARIES models for which the solution technique is inapplicable. We employ several transformations which are known to be numerically stable in our solution method. Our solution method also offers a facility for incrementally computing reliability when the number of spares in the fault-tolerant system is increased by one.