An Efficient and Numerically Stable Method for Computing Interval Availability Distribution Bounds

Abstract

The paper develops a method, called bounding regenerative transformation, for the computation with numerical stability and well-controlled error of bounds for the interval availability distribution of systems modeled by finite (homogeneous) continuous-time Markov chain models with a particular structure. The method requires the selection of a regenerative state and is targeted at a class of models, class C 1, with a “natural” selection for the regenerative state. For class C 1 models, bounds tightness can be traded-off with computational cost through a control parameter DC, with the option DC = 1 yielding the smallest computational cost. For large class C 1 models and the selection DC = 1, the method will often have a small computational cost relative to the model size and, with additional conditions, seems to yield tight bounds for any time interval or not small time intervals, depending on the initial probability distribution of the model. Class C 1 models with those additional conditions include both exact and bounding failure/repair models of coherent fault-tolerant systems with exponential failure and repair time distributions and repair in every state with failed components with failure rates much smaller than repair rates.