Reliability of Semi-Markov Systems

Abstract

In many engineering problems, especially in dependability (reliability, availability, maintainability, safety, performability, …) analysis, semi-Markov processes are used. The main advantage of semi-Markov processes is to allow nonexponential distributions for transitions between states and to generalize several kinds of stochastic processes. Since in most real cases the lifetime and repair time are not exponential, this is very important. The numerical computation is unequally hard for the time-dependent and for the steady-state probabilities. The calculus of steady-state probabilities is very easy and requires only the computation of the mean time spent in each state and of the invariant distribution of the embedded Markov chain in the semi-Markov process. Concerning the time-dependent probabilities, the computation is more difficult than for Markov systems because of convolution products, i.e., matrices inversion in the convolution sense. The different methods for transient probabilities evaluation for semi-Markov systems are: transformation into Markov systems by state space expansion, direct Markov renewal equation solution, partial state space expansion; equivalent rate method; supplementary variables method; stochastic simulation methods; etc. In the standard Markov analysis, when we have several independent Markov processes, say X, F, Z,., W, the vector process (X, Y, Z,…, W) is also a Markov process on the product state space, whose generator is given by the direct Kronecker sum of the partial process generators. Unfortunately, this closure property is not still valid in the case of semi-Markov systems. This is the main inconvenience of semi-Markov systems in modeling dependability problems.