On Sensitivity of Systems Reliability Characteristics to the Shape of Their Elements Life and Repair Time Distributions

The paper deals with the problem of the systems ‎‹M2/GI/1›‎ and ‎‹GI2/M/1› reliability characteristics sensitivity to the shape of their elements life and repair times distributions under restrictions on the availability of recovery. Partial differential equation for the time dependent and usual differential equations for the stationary micro-state probabilities of these systems are proposed. Explicit expressions for the micro-and macro-state stationary probabilities of these systems are given and they show their strong dependability on the shape of their elements life and repair times distributions. This dependence represents in terms of moment generation functions non-exponential distribution in the point of the exponential distribution parameters. Special software tool based on the MATLAB computer system has been developed for the numerical analysis of the system failure probability sensitivity to the shape of its elements life and recovery distributions and its comparison with the simplest Markov system. The numerical analysis shows that this dependence becomes negligible and vanishes for “fast” recovery (with recovery rate increasing). In particular, it has been shown that the failure probabilities of the systems ‎‹M2/GI/1›‎ and ‹GI2/M/1› with Gamma and Weibull-Gnedenko distributions instead of the general ones quickly converge to zero with increasing recovery rate and coincide with the simplest Markov system ‹M2/M/1› for special value of the particular value of the parameter c =1.0.

reliability systems, failure probabilities, sensitivity to the shape elements life and recovery time distributions, micro-and macro-states, stationary and non-stationary probabilities