A Generalized Method for the Transient Analysis of Markov Models of Fault-Tolerant Systems with Deferred Repair

Abstract

Randomization is an attractive alternative for the transient analysis of continuous time Markov models. The main advantages of the method are numerical stability, well-controlled computation error, and ability to specify the computation error in advance. However, the fact that the method can be computationally expensive limits its applicability. Recently, a variant of the (standard) randomization method, called split regenerative randomization has been proposed for the efficient analysis of reliability-like models of fault-tolerant systems with deferred repair. In this article, we generalize that method so that it covers more general reward measures: the expected transient reward rate and the expected averaged reward rate. The generalized method has the same good properties as the standard randomization method and, for large models and large values of the time t at which the measure has to be computed, can be significantly less expensive. The method requires the selection of a subset of states and a regenerative state satisfying some conditions. For a class of continuous time Markov models, class C'₂, including typical failure/repair reliability models with exponential failure and repair time distributions and deferred repair, natural selections for the subset of states and the regenerative state exist and results are available assessing approximately the computational cost of the method in terms of “visible” model characteristics. Using a large model class C’₂ example, we illustrate the performance of the method and show that it can be significantly faster than previously proposed randomization-based methods.

Keywords:

• Continuous-time Markov chains
• Deferred repair
• Fault-tolerant systems
• Randomization
• Transient analysis

Mathematics Subject Classification:

• 60J22

Notes

¹The case A = 0 was not previously considered because in that case the m(t) measure is identical to 0. The developments made in Temsamani and Carrasco (2004) for the case A ≥ 1 carry immediately to the more general case A ≥ 0 considered here.