A Semi Markovian model allowing for inhomogenities with respect to process time

摘要

A non-homogeneous SemiMarkov process is considered as an approach to model reliability characteristics of components or small systems with complex test resp. maintenance strategies. This approach generalizes previous results achieved for ordinary inhomogenous Markov processes. This paper focusses on the following topics to make application of Semi Markovian models feasable: Rather than transition probabilities Q_ij (t), which are used in normal mathematical text books to define Semi Markov processes, transition rates #lambda#_ij () are used, as is usual for ordinary Markov processes. These transition rates may depend on two types of time in general: on process time and on sojourn time in state i. Such transition rates can be followed from failure and repair rates of the underlying technical components, in much the same way, as this is known for ordinary Markov processes. Rather than immediately starting to solve the Kolmogorov equations, which would result in N~2 integral equations, a system of N itnegral equations for frequency densities of reaching states is considered. Once this system is solved, the initial value problem for state probabilities can be solved by straight forward integration. An example involving 14 states has been solved as an illustration using the approach.