Calculation of a system reliability function is one of the principal problems in reliability theory. It is well known that even for the simplest double redundant repairable system this function has not been analytically calculated yet in case when reliability and recovery functions of system elements have general distributions.
In papers of V.V.Rykov [1] multi-dimensional alternative processes have been used for studying the complex reliability systems. A case has been considered of a system with no limitations on the number of repair facilities. When there is enough repair units, the components of the process describing the system behavior become independent which allows to calculate the system characteristics. However, in the case of a limited number of repair facilities the problem of calculation of reliability function has not been solved yet, and it is considered in the current paper for the case of a heterogeneous hot standby system with only one repair unit. On the other hand in series of classical papers of B.V. Gnedenko, A.D. Solov'ev and others it was shown that under “quick” restoration the system life time distribution becomes asymptotically insensitive to the shapes of its elements' life and repair time distributions and in scale of the system mean life time it tends to the exponential one. In other works by V.V.Rykov et al. [2-4] the problem of system's steady state reliability characteristics sensitivity to the shape of life and repair time distributions of its elements has been considered for the simple case of a cold standby double redundant system when one of the input distributions (either of life or repair time) is exponential. For these models explicit expressions for stationary probabilities have been obtained which show their evident dependence on the non-exponential distributions in the form of their Laplace-Stiltjes transforms. At that, the numerical investigations, performed by D.V.Kozyrev [5], show that this dependence becomes vanishingly small under “quick” restoration.
In current paper we generalize the previous investigations in two directions: firstly, we generalize the previous results for heterogeneous systems, and, secondly, the problem of reliability function calculation is solved for a special case of reliability model with a limited amount of repair facilities.