Techniques for the Fast Simulation of Models of Highly dependable Systems

With the ever-increasing complexity and requirements of highly dependable systems, their evaluation during design and operation is becoming more crucial. Realistic models of such systems are often not amenable to analysis using conventional analytic or numerical methods. Therefore, analysts and designers turn to simulation to evaluate these models. However, accurate estimation of dependability measures of these models requires that the simulation frequently observes system failures, which are rare events in highly dependable systems. This renders ordinary Simulation impractical for evaluating such systems. To overcome this problem, simulation techniques based on importance sampling have been developed, and are very effective in certain settings. When importance sampling works well, simulation run lengths can be reduced by several orders of magnitude when estimating transient as well as steady-state dependability measures. This paper reviews some of the importance-sampling techniques that have been developed in recent years to estimate dependability measures efficiently in Markov and nonMarkov models of highly dependable system

Importance Sampling Simulations of Markovian Reliability Systems using Cross Entropy

This paper reports simulation experiments, applying the cross entropy method suchas the importance sampling algorithm for efficient estimation of rare event probabilities in Markovian reliability systems. The method is compared to various failurebiasing schemes that have been proved to give estimators with bounded relativeerrors. The results from the experiments indicate a considerable improvement ofthe performance of the importance sampling estimators, where performance is mea-sured by the relative error of the estimate, by the relative error of the estimator,and by the gain of the importance sampling simulation to the normal simulation

Rare event simulation for highly dependable systems with fast repairs

Stochastic model checking has been used recently to assess, among others, dependability measures for a variety of systems. However, the employed numerical methods, as, e.g., supported by model checking tools such as PRISM and MRMC, suffer from the state-space explosion problem. The main alternative is statistical model checking, which uses standard simulation, but this performs poorly when small probabilities need to be estimated. Therefore, we propose a method based on importance sampling to speed up the simulation process in cases where the failure probabilities are small due to the high speed of the system's repair units. This setting arises naturally in Markovian models of highly dependable systems. We show that our method compares favourably to standard simulation, to existing importance sampling techniques and to the numerical techniques of PRISM

Path-ZVA: general, efficient and automated importance sampling for highly reliable Markovian systems

We introduce Path-ZVA: an efficient simulation technique for estimating the probability of reaching a rare goal state before a regeneration state in a (discrete-time) Markov chain. Standard Monte Carlo simulation techniques do not work well for rare events, so we use importance sampling; i.e., we change the probability measure governing the Markov chain such that transitions `towards' the goal state become more likely. To do this we need an idea of distance to the goal state, so some level of knowledge of the Markov chain is required. In this paper, we use graph analysis to obtain this knowledge. In particular, we focus on knowledge of the shortest paths (in terms of `rare' transitions) to the goal state. We show that only a subset of the (possibly huge) state space needs to be considered. This is effective when the high dependability of the system is primarily due to high component reliability, but less so when it is due to high redundancies. For several models we compare our results to well-known importance sampling methods from the literature and demonstrate the large potential gains of our method.Comment: 25 pages, accepted for publication in ACM TOMAC

Adapted importance sampling schemes for the simulation of dependability models of Fault-tolerant systems with deferred repair

This paper targets the simulation of continuous-time Markov chain models of fault-tolerant systems with deferred repair. We start by stating sufficient conditions for a given importance sampling scheme to satisfy the bounded relative error property. Using those sufficient conditions, it is noted that many previously proposed importance sampling techniques such as failure biasing and balanced failure biasing satisfy that property. Then, we adapt the importance sampling schemes failure transition distance biasing and balanced failure transition distance biasing so as to develop new importance sampling schemes which can be implemented with moderate effort and at the same time can be proved to be more efficient for balanced systems than the simpler failure biasing and balanced failure biasing schemes. The increased efficiency for both balanced and unbalanced systems of the new adapted importance sampling schemes is illustrated using examples.Postprint (published version

Simulation of steady-state availability models of fault-tolerant systems with deferred repair

This paper targets the simulation of continuous-time Markov chain models of fault-tolerant systems with deferred repair. We start by stating sufficient conditions for a given importance sampling scheme to satisfy the bounded relative error property. Using those sufficient conditions, it is noted that many previously proposed importance sampling schemes such as failure biasing and balanced failure biasing satisfy that property. Then, we adapt the importance sampling schemes failure transition distance biasing and balanced failure transition distance biasing so as to develop new importance sampling schemes which can be implemented with moderate effort and at the same time can be proved to be more efficient for balanced systems than the simpler failure biasing and balanced failure biasing schemes. The increased efficiency for balanced and unbalanced systems of the new adapted importance sampling schemes is illustrated using examples.Preprin

Hypothesis testing for rare-event simulation: limitations and possibilities

One of the main applications of probabilistic model checking is to decide whether the probability of a property of interest is above or below a threshold. Using statistical model checking (SMC), this is done using a combination of stochastic simulation and statistical hypothesis testing. When the probability of interest is very small, one may need to resort to rare-event simulation techniques, in particular importance sampling (IS). However, IS simulation does not yield 0/1-outcomes, as assumed by the hypothesis tests commonly used in SMC, but likelihood ratios that are typically close to zero, but which may also take large values.\ud In this paper we consider two possible ways of combining IS and SMC. One involves a classical IS-scheme from the rare-event simulation literature that yields likelihood ratios with bounded support when applied to a certain (nontrivial) class of models. The other involves a particular hypothesis testing scheme that does not require a-priori knowledge about the samples, only that their variance is estimated well

Rare event simulation for highly dependable systems with fast repairs

Probabilistic model checking has been used recently to assess, among others, dependability measures for a variety of systems. However, the numerical methods employed, such as those supported by model checking tools such as PRISM and MRMC, suffer from the state-space explosion problem. The main alternative is statistical model checking, which uses standard Monte Carlo simulation, but this performs poorly when small probabilities need to be estimated. Therefore, we propose a method based on importance sampling to speed up the simulation process in cases where the failure probabilities are small due to the high speed of the system’s repair units. This setting arises naturally in Markovian models of highly dependable systems. We show that our method compares favourably to standard simulation, to existing importance sampling techniques, and to the numerical techniques of PRISM