Some large availability models: computation and bounds

Abstract

A dynamic analytic solution is described for a 2/sup N/ state general availability model with N components having constant failure and repair rates. From this model, a family of models is developed using truncation and/or attenuation of transition rates. Expressions are derived for steady-state solutions. Then spread-sheet programs are: (1) given for obtaining these solutions, and (2) compared with BASIC programs yielding the same results. State probabilities of these truncation and level-attenuation models are either greater than or less than comparable states in the general model. Thus the states of the general model become either lower bounds or upper bounds for states in these two model types. Other bounds can be constructed from single exponentials based on steady-state probabilities. From this family of models, bounds should exist on state probabilities in models of similar structure but different constraints on failure and repair rates. A specific model is pursued where failures are restricted to any 2 components; and the failure rate of one component is assumed to change on second level of failure. Under these conditions, dynamic bounds on state-probabilities of the initial-state and some, but not all, steady state bounds on the other state probabilities can be found. Examples illustrate various bounds.