Solving large interval availability models using a model transformation approach

Abstract

Fault-tolerant systems are often modeled using (homogeneous) continuous time Markov chains (CTMCs).Computation of the distribution of the interval availability, i.e. of the distribution of the fraction of time in a time interval in which the system is operational, of a fault-tolerant system modeled by a CTMC is an important problem which has received attention recently. However, currently available methods to perform that computation are very expensive for large models and large time intervals. In this paper, we develop a new method to compute the distribution of the interval availability which, for large enough models and large enough time intervals, is signiÿcantly faster than previous methods. In the method, a truncated transformed model, which has with some arbitrarily small error the same interval availability distribution as the original model, is obtained from the original model and the truncated transformed model is solved using a previous state-of-the-art method. The method requires the selection of a "regenerative" state and its performance depends on that selection. For a class of models, including typical failure/repair models of coherent fault-tolerant systems with exponential failure and repair time distributions and repair in every state with failed components, a natural selection for the regenerative state exists and theoretical results are available assessing the performance of the method for that natural selection in terms of "visible" model characteristics. Those results can be used to anticipate when the method can be expected to be competitive for models in that class. Numerical results are presented showing that the new method can indeed be signiÿcantly faster than a previous state-of-the-art method and is able to deal with some large models and large time intervals in reasonable CPU times. ?

A New General-Purpose Method for the Computation of the Interval Availability Distribution

“…That construct was considered in Carrasco (2004a) and can be looked at as a particular case of a more general construct in which arbitrarily different randomization rates Λ i ≥ −a i,i , Λ i > 0, i ∈ Ω are associated with the states of a DTMC X. In that generalized construct, X has, of course, same state space and initial probability distribution as X and transition matrix…”

“…Most of the work has dealt with the case in which the behavior of the system is captured by an (homogeneous) continuous-time Markov chain (CTMC) model having up (operational) and down states. Computing the distribution of the interval availability of systems modeled by a CTMC has been proved to be a challenging problem (see Carrasco 2004aCarrasco , 2011Goyal and Tantawi 1988;Ross 1983; Rubino and Sericola 1992, 1993, 1995Sericola 1990; de Souza e Silva and Gail 1986; Takács 1957). The first effort is reported in Takács (1957), where a closed-form integral expression was obtained for a two-state CTMC model.…”

“…That second method was reviewed in Rubino and Sericola (1995) as Algorithm A, where it was taken as starting point to develop another method (Algorithm B) that is competitive when the number of operational states of the model is small and, furthermore, can deal with some class of CTMC models with denumerable state space. Recently, we have developed a method in Carrasco (2004a), which will be called here regenerative transformation, targeted at a class of CTMC models, class C 1 , including both exact and bounding failure/repair CTMC models of fault-tolerant systems with increasing structure function (Barlow and Proschan 1981), exponential failure and repair time distributions, and repair in every state with failed components, with failure rates much smaller than repair rates, which can be significantly less costly in terms of CPU time than Algorithm A if the interval availability has to be guaranteed to have a level close to one. The regenerative transformation method was taken as starting point in Carrasco (2011) to develop a method, bounding regenerative transformation, to compute bounds for the interval availability distribution.…”

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

“…labelled with 1 − p, so that IAVCD t p decreases as p increases, as it should happen. Being irreducible and finite, the CTMC is ergodic (see, for instance, Kulkarni 1995), and as predicted by renewal reward process and regenerative process theories (see, for instance, Ross 1983), for t → , IAVCD t p has an asymptotic shape with IAVCD t p = 1 for p < SSA and IAVCD t p = 0 for p ≥ SSA, but the convergence to that asymptotic shape is very slow, making meaningful the computation of the measure for very large values of t and stressing the need for methods that have a small computational cost for large t. Computing IAVD t p or IAVCD t p of a system modeled by a CTMC has been proved to be a challenging problem (Carrasco 2004;de Souza e Silva and Gail 1986;Goyal and Tantawi 1988;Ross 1983;Rubino and Sericola 1992, 1993, 1995Sericola 1990;Takács 1957). The first effort is reported in Takács (1957), where a closed-form integral expression for IAVCD t p is given for a two-state CTMC.…”

“…Finally, a method that we will call regenerative transformation, has been developed by Carrasco (2004). The method covers finite CTMC models with a particular structure.…”