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The purpose of this paper is to investigate the stochastic comparisons of the parallel system with independent heterogeneous Gumbel components and series and parallel systems with independent heterogeneous truncated Gumbel components in terms of various stochastic orderings.

The obtained results in this paper are obtained by using the vector majorization methods and results. First, the components of series and parallel systems are heterogeneous and having Gumbel or truncated Gumbel distributions. Second, multiple-outlier truncated Gumbel models are discussed for these systems. Then, the relationship between the systems having Gumbel components and Weibull components are considered. Finally, Monte Carlo simulations are performed to illustrate some obtained results.

The reversed hazard rate and likelihood ratio orderings are obtained for the parallel system of Gumbel components. Using these results, similar new results are derived for the series system of Weibull components. Stochastic comparisons for the series and parallel systems having truncated Gumbel components are established in terms of hazard rate, likelihood ratio and reversed hazard rate orderings. Some new results are also derived for the series and parallel systems of upper-truncated Weibull components.

To the best of our knowledge thus far, stochastic comparisons of series and parallel systems with Gumbel or truncated Gumble components have not been considered in the literature. Moreover, new results for Weibull and upper-truncated Weibull components are presented based on Gumbel case results.

Building on the steps to war model, this paper seeks to examine the impact that territorial Militarized Interstate Disputes (MID) have on the time it takes a dyad to go to war after it experiences its first MID.

A model common to epidemiological research, the hazard model, is employed to examine the dyadic relationship from the time of the first MID forward. This is an improvement to dyadic analysis, as most research examines the characteristics of individual MIDs in isolation.

Dyads with a history of territorial MIDs go to war much more quickly than dyads without a history of territorial MIDs. Future research should explore the relationship between territory, war, and power status to test the assertion that minor power states engage in power politics behavior less frequently.

Conflict resolution measures need to be employed more quickly when states have unresolved territorial issues. Mediation generally does not occur quickly, which may explain why territorial issues are less likely to be referred to mediators and less successfully mediated. The results presented herein highlight the need for flexible, quick responses to certain crises and the need to settle borders and other territorial disputes permanently to avoid war.

The paper tests a critical component of the steps to war model and examines the assertion that the historical relationship between states affects conflict decisions.

The purpose of this paper is to study likelihood ratio order for mixture and its components via their Glaser’s functions for weighted distributions. So, some theoretical examples using exponential family and their mixtures are presented.

First, Glaser’s functions of mixture and its components for weighted distributions in different scenarios are computed. Then by them the likelihood ratio order is investigated between mixture and its components.

The authors find conditions for weight functions under which the mixture random variable is between of its components in likelihood ratio order.

Results are obtained for weight function in general. It is well known that the some special weights are order statistics, up and down records, hazard rate, reversed hazard rate, moment generating function, etc. So, the results are valid for all of them.

In this paper, we consider a repair‐time limit replacement problem with imperfect repair and develop a graphical method to determine the optimal repair‐time limit which minimizes the expected total discounted cost over an infinite time horizon. The method proposed can be applied to an estimation problem of the optimal repair‐time limit from the empirical repair‐time data. Then, the modified scaled total time on test transform of the underlying repair‐time distribution function is used. Numerical examples are devoted to examine asymptotic properties of the nonparametric estimator for the optimal repair‐time limit.

In reliability studies, interests in discrete failure data came relatively late in comparison to its continuous analogue. Also, discrete failure data arise in several common situations. So, in this paper the authors try to study some reliability concepts such as reversed variance and reversed mean residual life functions based on discrete lifetime random variable.

Supposed T be a non‐negative discrete random variable, then based on reversed residual random variable T_k^*=(k−T|T≤k), some useful and applicable relations and bounds are achieved.

In this paper, the authors study the reversed variance residual life in discrete lifetime distributions, the results of which are not similar to the continuous case. Its relationship with reversed mean residual life and reversed residual coefficient of variation are obtained. Also, its monotonicity and the associated ageing classes of distributions are discussed. Some characterization results of the class of increasing reversed variance residual life, which is denoted by IRVR, are presented and the upper bound for reversed variance residual life under some conditions is obtained.

There are many situations where a continuous time is inappropriate for describing the lifetime of devices and other systems. For example, the lifetime of many devices in industry, such as switches and mechanical tools, depends essentially on the number of times that they are turned on and off or the number of shocks they receive. In such cases, the time to failure is often more appropriately represented by the number of times they are used before they fail, which is a discrete random variable.

All the results based on discrete reversed residual lifetime, such as the relationships among reversed mean, variance and coefficient of variation residual lifetime and also their monotonicity ageing classes, are new.

This study aims to focus on asset management of large‐scale information systems supporting infrastructures and especially seeks to address a methodology of their statistical deterioration prediction based on their historical inspection data. Information systems are composed of many devices. Deterioration process i.e. wear‐out failure generation process of those devices is formulated by a Weibull hazard model. Furthermore, in order to consider the heterogeneity of the hazard rate of each device, the random proportional Weibull hazard model, which expresses the heterogeneity of the hazard rate as random variables, is to be proposed.

Large‐scale information systems comprise many components, and different types of components might have different hazard rates. Therefore, when analyzing faults of information systems that comprise various types of devices and components, it is important to consider the heterogeneity of the hazard rates that exist between the different types of components. In this study, with this in consideration, the random proportional Weibull hazard model, whose heterogeneity of hazard rates is subject to a gamma distribution, is formulated and a methodology is proposed which estimates the failure rate of various components comprising an information system.

Through a case study using a traffic control system for expressways, the validity of the proposed model is empirically verified. Concretely, as for HDD, the service life at which the survival probability is 50 percent is estimated as 158 months. However, even for the same HDD, use environment differs according to usage. Actually, among the three different usages (PC, server, others), failures happen earliest in the case of PCs, which have the highest heterogeneity parameter and a survival probability of 50 percent after 135 months of usage. On the other hand, as for others, its survival probability is 50 percent at 303 months.

To operationally express the heterogeneity of failure rates, the Weibull hazard model is employed as a base, and a random proportional Weibull hazard model expressing the proportional heterogeneity of hazard rates with a standard gamma distribution is formulated. By estimating the parameter of the standard proportional Weibull hazard function and the parameter of the probability distribution that expresses the heterogeneity of the proportionality constant between the types, the random proportional Weibull hazard model can easily express the heterogeneity of the hazard rates between types and components.

The problem is to devise a life‐test acceptance procedure of an electrical item that has a Weibull failure distribution with an increasing hazard rate. The test‐bed facility has some constraints on the number of test samples and testing time.

The life‐test plan is obtained using censoring of experiments and the properties of order statistics. In this article, the author has derived expressions for order statistics and their moments for some commonly used hazard‐rate functions; for example, constant, linearly increasing, exponentially increasing, power function, etc. and the same is used in planning the life‐test acceptance procedure.

Results and findings are discussed in full. It is postulated that further research in this direction will definitely bring some fruitful results that have immense importance in the field of reliability analysis and life‐testing experiments.

The same methodology can be adopted for devising life‐test acceptance procedure using censoring of experiments.

In order to succeed in an action under the Equal Pay Act 1970, should the woman and the man be employed by the same employer on like work at the same time or would the woman still be covered by the Act if she were employed on like work in succession to the man? This is the question which had to be solved in Macarthys Ltd v. Smith. Unfortunately it was not. Their Lordships interpreted the relevant section in different ways and since Article 119 of the Treaty of Rome was also subject to different interpretations, the case has been referred to the European Court of Justice.

A distinction must be drawn between a dismissal on the one hand, and on the other a repudiation of a contract of employment as a result of a breach of a fundamental term of that contract. When such a repudiation has been accepted by the innocent party then a termination of employment takes place. Such termination does not constitute dismissal (see London v. James Laidlaw & Sons Ltd (1974) IRLR 136 and Gannon v. J. C. Firth (1976) IRLR 415 EAT).