Dynamic and Steady-State Solutions for a General Availability Model

Exact dynamic and steady-state solutions are given for the state equations of a fairly general availability model with known, constant, failure and repair rates for each component of the system. The states of all components are mutually statistically independent. Rules for writing the transition matrix and its eigenvalues and eigenvectors are given, and explicit formulas for the solutions are given in terms of the latter. Three special cases are derived from this model. A dynamic solution is given for a reliability model with no repair. Steady-state solutions are found for a model in which all components have the same failure and repair rates and for a model in which repair is required before a second failure can occur. For all models, approximate upper bounds are developed for the time required for the systems to arrive at the steady state; the longest time constant in the solution is given in terms of the failure and repair rates.