Emad El-Neweihi’s research while affiliated with University of Illinois Chicago and other places

In this paper we consider the problem of estimating the reliability of an exponential component based on a Ranked Set Sample (RSS) of size n. Given the first r observations of that sample, 1≤r≤n, we construct an unbiased estimator for this reliability and we show that these n unbiased estimators are the only ones in a certain class of estimators. The variances of some of these estimators are compared. By viewing the observations of the RSS of size n as the lifetimes of n independent k-out-of-n systems, 1≤k≤n, we are able to utilize known properties of these systems in conjunction with the powerful tools of majorization and Schur functions to derive our results.

The ‘minimal’ repair of a system can take several forms. Statistical or black box minimal repair at failure time t is equivalent to replacing the system with another functioning one of the same age, but without knowledge of precisely what went wrong with the system. Its major attribute is its mathematical tractability. In physical minimal repair, at system failure time t , we minimally repair the ‘component’ which brought the system down at time t . The work of Arjas and Norros, Finkelstein, and Natvig is reviewed. The concept of a rate function for minimal repairs of the statistical and physical types are discussed and developed. It is shown that the number of physical minimal repairs is stochastically larger than the number of statistical minimal repairs for k out of n systems with similar components. Some majorization results are given for physical minimal repair for two component parallel systems with exponential components.

The `minimal' repair of a system can take several forms. Statistical or black box minimal repair at failure time t is equivalent to replacing the system with another functioning one of the same age, but without knowledge of precisely what went wrong with the system. Its major attribute is its mathematical tractability. In physical minimal repair, at system failure time t, we minimally repair the `component' which brought the system down at time t. The work of Arjas and Norros, Finkelstein, and Natvig is reviewed. The concept of a rate function for minimal repairs of the statistical and physical types are discussed and developed. It is shown that the number of physical minimal repairs is stochastically larger than the number of statistical minimal repairs for k out of n systems with similar components. Some majorization results are given for physical minimal repair for two component parallel systems with exponential components.

Design engineers are well aware of the stochastic result which says that (under the appropriate assumptions) redundancy at the component level is superior to redundancy at the system level. Given the importance of the hazard rate in reliability and life testing, we investigate to what extent this principle holds for the stronger stochastic ordering, viz, hazard rate ordering. Surprisingly, this does not hold for even series systems if the spares do not match the original components in distribution. It is true for series systems however for matching spares, and we conjecture that this is the case in general for k-out-of-n:G systems. We also investigate this principle for cold-standby redundancy (as opposed to active or parallel redundancy)

An (n − r+ 1) out of n system is down (at time t0, say) due to the failure of at least r of its components. Each component is made up of t equally reliable parts assembled in parallel. All parts are assumed to function independently of one another. Three different policies are described for inspecting and repairing the parts of one component after another until all (or some) of the failed components are identified (and fixed). We assume that a less reliable part is cheaper to repair than a more reliable one. We show that in order to minimize the expected total cost of necessary repairs, for each of the three policies, we should inspect the weakest component first, then the second weakest and so on. We also show that the closer we follow the optimal order (in a specific sense described by a partial ordering ⩾b2 on the set of permutations of {1,…, n}) the less is the expected cost that we incur.

In this paper we survey the literature concerning the topic of measures of importance for components in binary coherent systems. The diverse measures discussed fall into one of the following three broad categories: structural measures, time dependent measures and time independent measures. Comparisons are made between various measures, and a framework for possible new measures is suggested. More research is needed in assessing the quantitative and qualitative properties of each of these measures as well as their interrelationships, and it is suggested that the most appropriate measure for a given situation is often highly dependent on the type of improvement for a system that is being envisaged.

The hazard rate ordering is an ordering for random variables which compares lifetimes with respect to their hazard rate functions. It is stronger than the usual stochastic order for random variables, yet is weaker than the likelihood ratio ordering. The hazard rate ordering is particularly useful in reliability theory and survival analysis, owing to the importance of the hazard rate function in these areas. In this paper earlier work on the hazard rate ordering is reviewed, and extensive new results related to coherent systems are derived. Initially we fix the form of a coherent structure and investigate the effect on the hazard rate function of the system when we switch the lifetimes of its components from the vector ( T 1 , · ··, T n ) to the vector ( T′ 1 , · ··, T′ n ), where the hazard rate functions of the two vectors are assumed to be comparable in some sense. Although the hazard rate ordering is closed under the formation of series systems, we see that this is not the case for parallel systems even when the system is a two-component parallel system with exponentially distributed lifetimes. A positive result shows that for two-component parallel systems with proportional hazards ( λ 1 r ( t ), λ 2 r ( t ))), the more diverse ( λ 1 , λ 2 ) is in the sense of majorization the stronger is the system in the hazard rate ordering. Unfortunately even this result does not extend to parallel systems with more than two components, demonstrating again the delicate nature of the hazard rate ordering. The principal result of the paper concerns the hazard rate ordering for the lifetime of a k -out-of- n system. It is shown that if τ k|n is the lifetime of a k -out-of- n system, then τ k|n is greater than τ k+ 1 |n in the hazard rate ordering for any k. This has an interesting interpretation in the language of order statistics. For independent (not necessarily identically distributed) lifetimes T 1 , · ··, T n , we let T k:n represent the k th order statistic (in increasing order). Then it is shown that T k + 1 :n is greater than T k:n in the hazard rate ordering for all k = 1, ···, n − 1. The result does not, however, extend to the stronger likelihood ratio order.

Convolutions of random variables which are either exponential or geometric are studied with respect to majorization of parameter vectors and the likelihood ratio ordering ([greater-or-equal, slanted]lr) of random variables. Let X[lambda], ..., X[lambda]n be independent exponential random variables with respective hazards [lambda]i (means 1/[lambda]i), i = 1 ..., n. Then if [lambda] = ([lambda]1, ..., [lambda]n) [greater-or-equal, slanted]m ([lambda]1', ..., [lambda]n') = [lambda]', it follows that [Sigma]i = 1n X[lambda] [greater-or-equal, slanted]lr [Sigma]i = 1n X[lambda]'1. Similarly if Xp1, ..., Xpn are independent geometric random variables with respective parameters p1, ..., pn, then p = (p1, ..., pn) [greater-or-equal, slanted]m(p'1, ..., p'n) = p' or log p = (log p1, ..., log pn) [greater-or-equal, slanted] m (log p1, ..., log pn) = log p' implies [Sigma]i = 1n Xpl [greater-or-equal, slanted] lr [Sigma]i = 1n XP'1. Applications of these results are given yielding convenient upper bounds for the hazard rate function of convolutions of exponential (geometric) random variables in terms of those of gamma (negative binomial) distributions. Other applications are also given for a server model, the range of a sample of i.i.d. exponential random variables, and the duration of a multistate component performing in excess of a given level.