Markovian systems with state-dependent component sets and linear damage accumulation

A vehicle transmission box is modeled as a system, whose operating states (¸=speeds) perform a Markov-process. Every speed uses its own subset of components (¸=gears etc.). The failure rate of a component increases linearly with its accumulated operating time. The first failure of a gear causes system failure. Asymptotic results and bounds are given for the system survival function. Mean value and variance of the system lifetime tend towards their counterparts in case of a Rayleigh distribution, as component failure rates converge towards 0. The conditional posterior culprit probabilities, i.e. the probabilities to cause a system failure, given survival of a time span τ, converge asymptotically towards the stationary expected relative damage accumulation rates as τ increases. Estimators for the model parameters are given for a known stationary distribution of the Markov-process.