Mission Time Analysis of Large Dependable Systems

The mission time analysis of large dependable systems that can be described by Markov Reward Models (MRM) with phase type distributed impulse and constant rate rewards is considered in the paper. A single Laplace transform domain description of the distribution of completion time is provided through a new analysis approach. Based on this description an effective numerical method is introduced which allows the evaluation of models with large state space ( 106 states). The applied analysis approach makes the use of an expanded Markov chain, but the state space expansion is much less than for common “phase type expansion”, because the expanded state space is composed by the union (instead of product) of the original state space and the state space of the phase type structure of non-zero impulse rewards. (Roughly speaking, the applied state space expansion is additive in contrast with the multiplicative state space expansion used for phase type modeling.) The proposed method, which is a counterpart of the analysis method of accumulated reward of MRMs with rate and impulse rewards, provides the moments of reward measures approximately on the same computational cost and memory requirement as the transient analysis of the expanded Continuous Time Markov Chain. Numerical example demonstrates the abilities of the proposed method.