Importance Functions for RESTART Simulation of General Jackson Networks

RESTART is an accelerated simulation technique that allows the evaluation of extremely low probabilities. In this method a number of simulation retrials are performed when the process enters regions of the state space where the chance of occurrence of the rare event is higher. These regions are defined by means of a function of the system state called the importance function. Guidelines for obtaining suitable importance functions and formulas for the importance function of two-stage networks were provided in previous papers. In this paper, we obtain effective importance functions for RESTART simulation of Jackson networks where the rare set is defined as the number of customers in a particular (‘target’) node exceeding a predefined threshold. Although some rough approximations and assumptions are used to derive the formulas of the importance functions, they are good enough to estimate accurately very low probabilities for different network topologies within short computational time

Dependability Estimation For Non-Markov Cosecutive-K-Out-Of-N: F Repairable Systems By Restart Simulation

The reliability of consecutive-k-out-of-n: F system (or C (k, n: F) system) has aroused great interest since it was first studied by Kontoleon in 1980 [1]. The system consists of a sequence of n ordered components along a line such that the system fails if and only if at least k consecutive components in the system have failed. A list of typical applications of C (k, n: F) system was given by Yam et al. [2]. A research book by Chang et al. [3] provide rich information about C (k,n: F) system

RESTART Simulation of Non-Markov Consecutive-K-Out-of-N: F Repairable Systems

The reliability of consecutive-k-out-of-n: F repairable systems and (k−1)-step Markov dependence is studied. The model analyzed in this paper is more general than those of previous studies given that repair time and component lifetimes are random variables that follow a general distribution. The system has one repair service which adopts a priority repair rule based on system failure risk. Since crude simulation has proved to be inefficient for highly dependable systems, the RESTART method was used for the estimation of steady-state unavailability, MTBF and unreliability. Probabilities up to the order of 10−16 have been accurately estimated with little computational effort. In this method, a number of simulation retrials are performed when the process enters regions of the state space where the chance of occurrence of a rare event (e.g., a system failure) is higher. The main difficulty for the application of this method is to find a suitable function, called the importance function, to define the regions. Given the simplicity involved in changing some model assumptions with RESTART, the importance function used in this paper could be useful for dependability estimation of many systems

Estimadores bayesianos de la fiabilidad con muestreo censurado

El estudio de la fiabilidad de componentes y sistemas tiene gran importancia en diversos campos de la ingenieria, y muy concretamente en el de la informatica. Al analizar la duracion de los elementos de la muestra hay que tener en cuenta los elementos que no fallan en el tiempo que dure el experimento, o bien los que fallen por causas distintas a la que es objeto de estudio. Por ello surgen nuevos tipos de muestreo que contemplan estos casos. El mas general de ellos, el muestreo censurado, es el que consideramos en nuestro trabajo. En este muestreo tanto el tiempo hasta que falla el componente como el tiempo de censura son variables aleatorias. Con la hipotesis de que ambos tiempos se distribuyen exponencialmente, el profesor Hurt estudio el comportamiento asintotico del estimador de maxima verosimilitud de la funcion de fiabilidad. En principio parece interesante utilizar metodos Bayesianos en el estudio de la fiabilidad porque incorporan al analisis la informacion a priori de la que se dispone normalmente en problemas reales. Por ello hemos considerado dos estimadores Bayesianos de la fiabilidad de una distribucion exponencial que son la media y la moda de la distribucion a posteriori. Hemos calculado la expansion asint6tica de la media, varianza y error cuadratico medio de ambos estimadores cuando la distribuci6n de censura es exponencial. Hemos obtenido tambien la distribucion asintotica de los estimadores para el caso m3s general de que la distribucion de censura sea de Weibull. Dos tipos de intervalos de confianza para muestras grandes se han propuesto para cada estimador. Los resultados se han comparado con los del estimador de maxima verosimilitud, y con los de dos estimadores no parametricos: limite producto y Bayesiano, resultando un comportamiento superior por parte de uno de nuestros estimadores. Finalmente nemos comprobado mediante simulacion que nuestros estimadores son robustos frente a la supuesta distribuci6n de censura, y que uno de los intervalos de confianza propuestos es valido con muestras pequenas. Este estudio ha servido tambien para confirmar el mejor comportamiento de uno de nuestros estimadores. SETTING OUT AND SUMMARY OF THE THESIS When we study the lifetime of components it's necessary to take into account the elements that don't fail during the experiment, or those that fail by reasons which are desirable to exclude from consideration. The model of random censorship is very usefull for analysing these data. In this model the time to failure and the time censor are random variables. We obtain two Bayes estimators of the reliability function of an exponential distribution based on randomly censored data. We have calculated the asymptotic expansion of the mean, variance and mean square error of both estimators, when the censor's distribution is exponential. We have obtained also the asymptotic distribution of the estimators for the more general case of censor's Weibull distribution. Two large-sample confidence bands have been proposed for each estimator. The results have been compared with those of the maximum likelihood estimator, and with those of two non parametric estimators: Product-limit and Bayesian. One of our estimators has the best behaviour. Finally we have shown by simulation, that our estimators are robust against the assumed censor's distribution, and that one of our intervals does well in small sample situation

Asymptotic optimality of RESTART estimators in highly dependable systems

Abstract We consider a wide class of models that includes the highly reliable Markovian systems (HRMS) often used to represent the evolution of multi-component systems in reliability settings. Repair times and component lifetimes are random variables that follow a general distribution, and the repair service adopts a priority repair rule based on system failure risk. Since crude simulation has proved to be inefficient for highly-dependable systems, the RESTART method is used for the estimation of steady-state unavailability and other reliability measures. In this method, a number of simulation retrials are performed when the process enters regions of the state space where the chance of occurrence of a rare event (e.g., a system failure) is higher. The main difficulty involved in applying this method is finding a suitable function, called the importance function, to define the regions. In this paper we introduce an importance function which, for unbalanced systems, represents a great improvement over the importance function used in previous papers. We also demonstrate the asymptotic optimality of RESTART estimators in these models. Several examples are presented to show the effectiveness of the new approach, and probabilities up to the order of 10-42 are accurately estimated with little computational effort

Transformación de estimadores bayesianos de la fiabilidad de una distribución exponencial con muestreo censurado

Supposing that the hazard rate is distributed according to the Gamma a priori distribution, two Bayes estimators of the reliability function of an exponential distribution under random censorship are derived. We make transformations of them that are necessary for obtaining the moments of such estimators and also for studying their asymptotic distribution. The mean squared errors of the Bayesian estimators are compared by simulation with that of the maximum likelihood estimator for several censoring modelsSuponiendo que la tasa de fallo sigue una distribución a priori Gamma, se obtienen dos estimadores Bayesianos de la función de fiabilidad de una distribución exponencial con muestreo censurado. Se realizan unas transformaciones de los mismos necesarias para poder calcular los momentos de dichos estimadores, así como sus distribuciones asintóticas. Mediante simulación se compara el error cuadrático medio de los estimadores Bayesianos con el de Máxima Verosimilitud para diversos modelos de censur

Importance functions for restart simulation of general Jackson networks

RESTART is an accelerated simulation technique that allows the evaluation of extremely low probabilities. In this method a number of simulation retrials are performed when the process enters regions of the state space where the chance of occurrence of the rare event is higher. These regions are defined by means of a function of the system state called the importance function. Guidelines for obtaining suitable importance functions and formulas for the importance function of two-stage networks were provided in previous papers. In this paper, we obtain effective importance functions for RESTART simulation of Jackson networks where the rare set is defined as the number of customers in a particular ('target') node exceeding a predefined threshold. Although some rough approximations and assumptions are used to derive the formulas of the importance functions, they are good enough to estimate accurately very low probabilities for different network topologies within short computational time.Simulation Queueing Rare event probabilities RESTART