James C. Fu

Distribution Theory of Runs: A Markov Chain Approach

Abstract The statistics of the number of success runs in a sequence of Bernoulli trials have been used in many statistical areas. For almost a century, even in the simplest case of independent and identically distributed Bernoulli trials, the exact distributions of many run statistics still remain unknown. Departing from the traditional combinatorial approach, in this article we present a simple unified approach for the distribution theory of runs based on a finite Markov chain imbedding technique. Our results cover not only the identical Bernoulli trials, but also the nonidentical Bernoulli trials. As a byproduct, our results also yield the exact distribution of the waiting time for the mth occurrence of a specific run.

Survey of reliability studies of consecutive-k-out-of-n:F and related systems

The consecutive-k-out-of-n:F and related systems have caught the attention of many researchers since the early 1980s. The studies of these systems lead to better understanding of the reliability of general series systems, In computation and structure. This paper is mainly a chronological survey of computing the reliability of these systems. >

Reliability of Consecutive-k-out-of-n:F Systems with (k-1)-step Markov Dependence

Reliabilities are studied for consecutive-k-out-of-n:F systems with component failures having (k-1)-step Markov dependence.

On Reliability of a Large Consecutive-k-out-of-n:F System with (k - 1)-step Markov Dependence

This paper studies the reliability of a large consecutive-k-out-of-n:F system when the component failure states have (k - 1)-step Markov dependence.

Reliability of a Large Consecutive-k-out-of-n:F System

The system reliability and some bounds are obtained for a large consecutive-k-out-of-n:F system with equal component failure probabilities.

A limit theorem of certain repairable systems

Many large engineering systems can be viewed (or imbedded) as a series system in time. In this paper, we introduce the structure of a repairable system and the reliabilities of these large systems are studied systematically by studying the ergodicities of certain non-homogeneous Markov chains. It shows that if the failure probabilities of components satisfy certain conditions, then the reliability of the large system is approximately exp (-β) for some β>0. In particular, we demonstrate how the repairable system can be used for studying the reliability of a large linearly connected system. Several practical examples of large consecutive-k-out-of-n:F systems are given to illustrate our results. The Weibull distribution is derived under our natural set-up.

On the average run lengths of quality control schemes using a Markov chain approach

Control schemes such as cumulative sum (CUSUM), exponentially weighted moving average (EWMA) and Shewhart charts have found widespread application in improving the quality of manufactured goods and services. The run length and the average run length (ARL) have become traditional measures of a control schemes performance. Determining the run length distribution and its average is frequently a difficult and tedious task. A simple unified method based on a finite Markov chain approach for finding the run length distribution and ARL of a control scheme is developed. In addition, the method yields the variance or standard deviation of the run length as a byproduct. Numerical results illustrating the results are given.

THE RELIABILITY OF A LARGE SERIES SYSTEM UNDER MARKOV STRUCTURE

Let Y, - -, Y,, be a finite Markov chain and let f be a binary value function defined over the state space of the Ys. We study the reliability of general series system having the structure function op(Y) = min {f(Y1), - -, f(Y,)} and show that, under certain regularity conditions, the reliability of the system tends to a constant c (1 > c > 0), where c often has the form c = exp {-,A}.

A Random-Discretization Based Monte Carlo Sampling Method and its Applications

Recently, several Monte Carlo methods, for example, Markov Chain Monte Carlo (MCMC), importance sampling and data-augmentation, have been developed for numerical sampling and integration in statistical inference, especially in Bayesian analysis. As dimension increases, problems of sampling and integration can become very difficult. In this manuscript, a simple numerical sampling based method is systematically developed, which is based on the concept of random discretization of the density function with respect to Lebesgue measure. This method requires the knowledge of the density function (up to a normalizing constant) only. In Bayesian context, this eliminates the “conjugate restriction” in choosing prior distributions, since functional forms of full conditionals of posterior distributions are not needed. Furthermore, this method is non-iterative, dimension-free, easy to implement and fast in computing time. Some benchmark examples in this area are used to check the efficiency and accuracy of the method. Numerical results demonstrate that this method performs well for all these examples, including an example of evaluating the small probability values of a high dimensional multivariate normal distribution. As a byproduct, this method also provides an easy way of computing maximum likelihood estimates and modes of posterior distributions.

Bounds for Reliability of Large Consecutive-K-out-of-N:F Systems with Unequal Component Reliability

This paper examines upper and lower bounds for the reliability of a large consecutive-k-out-of-n:F system with unequal failure probabilities of components. The reliability of a large consecutive-k-out-of-n:F system can be derived under certain conditions, from the upper and lower bounds. Examples are given.