Transient analysis of some rewarded Markov models using randomization with quasistationarity detection

Rewarded homogeneous continuous-time Markov chain (CTMC) models can be used to analyze performance, dependability and performability attributes of computer and telecommunication systems. In this paper, we consider rewarded CTMC models with a reward structure including reward rates associated with states and two measures summarizing the behavior in time of the resulting reward rate random variable: the expected transient reward rate at time t and the expected averaged reward rate in the time interval [0, t]. Computation of those measures can be performed using the randomization method, which is numerically stable and has good error control. However, for large stiff models, the method is very expensive. Exploiting the existence of a quasistationary distribution in the subset of transient states of discrete-time Markov chains with a certain structure, we develop a new variant of the (standard) randomization method, randomization with quasistationarity detection, covering finite CTMC models with state space S\cup {f_1, f_2, ..., f_A}, A\geq 1, where all states in S are transient and reachable among them and the states f_i are absorbing. The method has the same good properties as the standard randomization method and can be much more efficient. We also compare the performance of the method with that of regenerative randomization.Postprint (published version

Computationally efficient and numerically stable reliability bounds for repairable fault-tolerant systems

The transient analysis of large continuous time Markov reliability models of repairable fault-tolerant systems is computationally expensive due to model stiffness. In this paper, we develop and analyze a method to compute bounds for a measure defined on a particular, but quite wide, class of continuous time Markov models, encompassing both exact and bounding continuous time Markov unreliability models of fault-tolerant systems. The method is numerically stable and computes the bounds with well-controlled and specifiable-in-advance error. Computational effort can be traded off with bounds accuracy. For a class of continuous time Markov models, class C’’, including typical failure/repair reliability models with exponential failure and repair time distributions and repair in every state with failed components, the method can yield reasonably tight bounds ay a very small computational cost. The method builds upon a recently proposed method for the transient analysis of continuous-time Markov models called regenerative randomization.Postprint (published version

Computationally efficient and numerically stable reliability bounds for repairable fault-tolerant systems

The transient analysis of large continuous time Markov reliability models of repairable fault-tolerant systems is computationally expensive due to model stiffness. In this paper, we develop and analyze a method to compute bounds for a measure defined on a particular, but quite wide, class of continuous time Markov models, encompassing both exact and bounding continuous time Markov unreliability models of fault-tolerant systems. The method is numerically stable and computes the bounds with well-controlled and specifiable-in-advance error. Computational effort can be traded off with bounds accuracy. For a class of continuous time Markov models, class C’’, including typical failure/repair reliability models with exponential failure and repair time distributions and repair in every state with failed components, the method can yield reasonably tight bounds ay a very small computational cost. The method builds upon a recently proposed method for the transient analysis of continuous-time Markov models called regenerative randomization

An efficient and numerically stable method for computing interval availability distribution bounds

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 D_C, with the option D_C = 1 yielding the smallest computational cost. For large class C'_1 models and the selection D_C = 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.Preprin

A generalized method for the transient analysis of Markov models of fault-tolerant systems with deferred repair

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 paper, 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.Preprin