A comparison of numerical splitting-based methods for Markovian dependability and performability models

Iterative numerical methods are an important ingredient for the solution of continuous time Markov dependability models of fault-tolerant systems. In this paper we make a numerical comparison of several splitting-based iterative methods. We consider the computation of steady-state reward rate on rewarded models. This measure requires the solution of a singular linear system. We consider two classes of models. The first class includes failure/repair models. The second class is more general and includes the modeling of periodic preventive test of spare components to reduce the probability of latent failures in inactive components. The periodic preventive test is approximated by an Erlang distribution with enough number of stages. We show that for each class of model there is a splitting-based method which is significantly more efficient than the other methods.Postprint (published version

Bounding steady-state availability models with group repair and phase type repair distributions

We propose an algorithm to obtain bounds for the steady-state availability using Markov models in which only a small portion of the state space is generated. The algorithm is applicable to models with group repair and phase type repair distributions and involves the solution of only four linear systems of the size of the generated state space, independently on the number of “return” states. Numerical examples are presented to illustrate the algorithm and compare it with a previous bounding algorithm.Postprint (published version

Tight steady-state availability bounds using the failure distance concept

Continuous-time Markov chains are commonly used for dependability modeling of repairable fault-tolerant computer systems. Realistic models of non-trivial fault-tolerant systems often have very large state spaces. An attractive approach for dealing with the largeness problem is the use of pruningmethods with error bounds. Several such methods for computing steady-state availability bounds have been proposed recently. This paper presents a new method which exploits the failure distance concept to bound more efficiently the behavior in the non-generated state space. It is proved that the bounding method gives tighter bounds than previous methods. Numerical analysis shows that the new bounds can be significantly tighter.Postprint (published version

Accelerating mean time to failure computations

In this paper we consider the problem of numerical computation of the mean time to failure (MTTF) in Markovian dependability and/or performance models. The problem can be cast as a system of linear equations which is solved using an iterative method preserving sparsity of the Markov chain matrix. For highly dependable systems, system failure is a rare event and the above system solution can take an extremely large number of iterations. We propose to solve the problem by dividing the computation in two parts. First, by making some of the high probability states absorbing, we compute the MTTF of the modified Markov chain. In a subsequent step, by solving another system of linear equations, we are able to compute the MTTF of the original model. We prove that for a class of highly dependable systems, the resulting method can speed up computation of the MTTF by orders of magnitude. Experimental results supporting this claim are presented. We also obtain bounds on the convergence rate for computing the mean entrance time of a rare set of states in a class of queueing models