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

Fast simulation of the leaky bucket algorithm

We use fast simulation methods, based on importance sampling, to efficiently estimate cell loss probability in queueing models of the Leaky Bucket algorithm. One of these models was introduced by Berger (1991), in which the rare event of a cell loss is related to the rare event of an empty finite buffer in an "overloaded" queue. In particular, we propose a heuristic change of measure for importance sampling to efficiently estimate the probability of the rare empty-buffer event in an asymptotically unstable GI/GI/1/k queue. This change of measure is, in a way, "dual" to that proposed by Parekh and Walrand (1989) to estimate the probability of a rare buffer overflow event. We present empirical results to demonstrate the effectiveness of our fast simulation method. Since we have not yet obtained a mathematical proof, we can only conjecture that our heuristic is asymptotically optimal, as k/spl rarr//spl infin/

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

New variance reduction methods in Monte Carlo rare event simulation

Para sistemas que proveen algún tipo de servicio mientras están operativos y dejan de proveerlo cuando fallan, es de interés determinar parámetros como, por ejemplo, la probabilidad de encontrar el sistema en falla en un instante cualquiera, el tiempo medio transcurrido entre fallas, o cualquier medida capaz de reflejar la capacidad del sistema para proveer servicio. Las determinaciones de estas medidas de seguridad de funcionamiento se ven afectadas por diversos factores, entre ellos, el tamaño del sistema y la rareza de las fallas. En esta tesis se estudian algunos métodos concebidos para determinar estas medidas sobre sistemas grandes y altamente confiables, es decir sistemas formados por gran cantidad de componentes, en los que las fallas del sistema son eventos raros. Ya sea en forma directa o indirecta, parte de las las expresiones que permiten determinar las medidas de interés corresponden a la probabilidad de que el sistema se encuentre en algún estado de falla. De un modo u otro, estas expresiones evaluan la fracción —ponderada por la distribución de probabilidad de las configuraciones del sistema—entre el número de configuraciones en las que el sistema falla y la totalidad de las configuraciones posibles. Si el sistema es grande el cálculo exacto de estas probabilidades, y consecuentemente de las medidas de interés, puede resultar inviable. Una solución alternativa es estimar estas probabilidades mediante simulación. Uno de los mecanismos para hacer estas estimaciones es la simulación de tipo Monte Carlo, cuya versión más simple es la simulación en crudo o estándar. El problema es que si las fallas son raras, el número de iteraciones necesario para estimar estas probabilidades mediante simulación estándar con una precisión aceptable, puede resultar desmesuradamente grande. En esta tesis se analizan algunos métodos existentes para mejorar la simulación estándar en el contexto de eventos raros, se hacen análisis de varianza y se prueban los métodos sobre una variedad de modelos. En todos los casos la mejora se consigue a costa de una reducción de la varianza del estimador con respecto a la varianza del estimador estándar. Gracias a la reducción de varianza es posible estimar la probabilidad de ocurrencia de eventos raros con una precisión aceptable, a partir de un número razonable de iteraciones. Como parte central del trabajo se proponen dos métodos nuevos, uno relacionado con Spliting y otro relacionado con Monte Carlo Condicional. Splitting es un método de probada eficiencia en entornos en los que se busca evaluar desempeño y confiabilidad combinados, escasamente utilizado en la simulación de sistemas altamente confiables sobre modelos estáticos (sin evolución temporal). En vi su formulación básica Splitting hace un seguimiento de las trayectorias de un proceso estocástico a través de su espacio de estados y multiplica su número ante cada cruce de umbral, para un conjunto dado de umbrales distribuidos entre los estados inicial y final. Una de las propuestas de esta tesis es una adaptación de Splitting a un modelo estático de confiabilidad de redes. En el método propuesto se construye un proceso estocástico a partir de un tiempo ficticio en el cual los enlaces van cambiando de estado y se aplica Splitting sobre ese proceso. El método exhibe elevados niveles de precisión y robustez. Monte Carlo Condicional es un método clásico de reducción de varianza cuyo uso no está muy extendido en el contexto de eventos raros. En su formulación básica Monte Carlo Condicional evalúa las probabilidades de los eventos de interés, condicionando las variables indicatrices a eventos no raros y simples de detectar. El problema es que parte de esa evaluación incluye el cálculo exacto de algunas probabilidades del modelo. Uno de los métodos propuestos en esta tesis es una adaptación de Monte Carlo Condicional al análisis de modelos Markovianos de sistemas altamente confiables. La propuesta consiste en estimar las probabilidades cuyo valor exacto se necesita, mediante una aplicación recursiva de Monte Carlo Condicional. Se estudian algunas características de este modelo y se verifica su eficiencia en forma experimental.For systems that provide some kind of service while they are operational and stop providing it when they fail, it is of interest to determine parameters such as, for example, the probability of finding the system failed at any moment, the mean time between failures, or any measure that reflects the capacity of the system to provide service. The determination of these measures —known as dependability measures— is affected by a variety of factors, including the size of the system and the rarity of failures. This thesis studies some methods designed to determine these measures on large and highly reliable systems, i.e. systems formed by a large number of components, such that systems’ failures are rare events. Either directly or indirectly, part of the expressions for determining the measures of interest correspond to the probability that the system is in some state of failure. Somehow, this expressions evaluate the ratio —weighted by the probability distribution of the systems’ configurations— between the number of configurations in which the system fails and all possible configurations. If the system is large, the exact calculation of these probabilities, and consequently of the measures of interest, may be unfeasible. An alternative solution is to estimate these probabilities by simulation. One mechanism to make such estimation is Monte Carlo simulation, whose simplest version is crude or standard simulation. The problem is that if failures are rare, the number of iterations required to estimate this probabilities by standard simulation, with acceptable accuracy, may be extremely large. In this thesis some existing methods to improve the standard simulation in the context of rare events are analyzed, some variance analyses are made and the methods are tested empirically over a variety of models. In all cases the improvement is achieved at the expense of reducing the variance of the estimator with respect to the standard estimator’s variance. Due to this variance reduction, the probability of the occurrence of rare events, with acceptable accuracy, can be achieved in a reasonable number of iterations. As a central part of this work, two new methods are proposed, one of them related to Splitting and the other one related to Conditional Monte Carlo. Splitting is a widely used method in performance and performability analysis, but scarcely applied for simulating highly reliable systems over static models (models with no temporal evolution). In its basic formulation Splitting keeps track of the trajectories of a stochastic process through its state space and it splits or multiplies the number of them at each threshold cross, for a given set of thresholds distributed between the initial and the final state. One of the proposals of this thesis is an adaptation of Splitting to a static network reliability model. In the proposed method, a fictitious time stochastic process in which the network links keep changing their state is built, and Splitting is applied to this process. The method shows to be highly accurate and robust. Conditional Monte Carlo is a classical variance reduction technique, whose use is not widespread in the field of rare events. In its basic formulation Conditional Monte Carlo evaluates the probabilities of the events of interest, conditioning the indicator variables to not rare and easy to detect events. The problem is that part of this assessment includes the exact calculation of some probabilities in the model. One of the methods proposed in this thesis is an adaptation of Conditional Monte Carlo to the analysis of highly reliable Markovian systems. The proposal consists in estimating the probabilities whose exact value is needed, by means of a recursive application of Conditional Monte Carlo. Some features of this model are discussed and its efficiency is verified experimentally

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 dynamic fault trees

Fault trees (FT) are a popular industrial method for reliability engineering, for which Monte Carlo simulation is an important technique to estimate common dependability metrics, such as the system reliability and availability. A severe drawback of Monte Carlo simulation is that the number of simulations required to obtain accurate estimations grows extremely large in the presence of rare events, i.e., events whose probability of occurrence is very low, which typically holds for failures in highly reliable systems. This paper presents a novel method for rare event simulation of dynamic fault trees with complex repairs that requires only a modest number of simulations, while retaining statistically justified confidence intervals. Our method exploits the importance sampling technique for rare event simulation, together with a compositional state space generation method for dynamic fault trees. We demonstrate our approach using two parameterized sets of case studies, showing that our method can handle fault trees that could not be evaluated with either existing analytical techniques, nor with standard simulation techniques

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

Cross-entropy optimisation of importance sampling parameters for statistical model checking

Statistical model checking avoids the exponential growth of states associated with probabilistic model checking by estimating properties from multiple executions of a system and by giving results within confidence bounds. Rare properties are often very important but pose a particular challenge for simulation-based approaches, hence a key objective under these circumstances is to reduce the number and length of simulations necessary to produce a given level of confidence. Importance sampling is a well-established technique that achieves this, however to maintain the advantages of statistical model checking it is necessary to find good importance sampling distributions without considering the entire state space. Motivated by the above, we present a simple algorithm that uses the notion of cross-entropy to find the optimal parameters for an importance sampling distribution. In contrast to previous work, our algorithm uses a low dimensional vector of parameters to define this distribution and thus avoids the often intractable explicit representation of a transition matrix. We show that our parametrisation leads to a unique optimum and can produce many orders of magnitude improvement in simulation efficiency. We demonstrate the efficacy of our methodology by applying it to models from reliability engineering and biochemistry.Comment: 16 pages, 8 figures, LNCS styl

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