Phase-type distributions for failure times

Phase-type distributions describe the random time taken for a Markov process to reach an absorbing state. In the context of component failure, sequential movement through the transient states (phases) of such a system could describe the ageing process with movement out of these states (absorption) corresponding to failure. Thus, the lifetime of a component is the absorption time, and the probability distribution of these times can be written in terms of the solution of a system of differential equations for which there are many convenient computational algorithms. A variety of different distributions is possible by varying the parameters of the process, and hazard rates of various shapes can be constructed, allowing different patterns of variation in observed data to be modelled. These distributions are applied to some industrial data-sets and further features of the processes discussed.