We analyse the survival time of a general duplex system sustained by an auxiliary cold standby unit and subjected to priority rules. The duplex system is attended by two general repairmen Rp and Rh . Repairman Rp has priority in repairing failed units with regard to repairman Rh provided that both repairmen are jointly idle. Otherwise, the priority is overruled. The auxiliary unit has its own repair facility. The duplex system has overall, break-in priority (often called pre-emptive priority) in operation and in standby with regard to the auxiliary unit. The analysis of the survival time is based on advanced complex function theory (sectionally holomorphic functions). The main problem is to convert a functional equation into a (parameter dependent) Sokhotski–Plemelj problem.

A nanocomponent is a collection of atoms arranged to a specific design in order to achieve a desired function with an acceptable performance and reliability. The type of atoms, the manner in which they are arranged within the nanocomponent, and their interrelationship have a direct effect on the nanocomponent's reliability (survival) function. In this paper we propose models based on the notion of a copula that are used to describe the relationship between the atoms of a nanocomponent. Having defined these models, we go on to construct a ‘nanocomponent’ model in order to obtain the reliability function of a nanocomponent.

Lifetime distributions for multicomponent systems are developed through the interplay of ageing and stress shocks to the system. The ageing process is explicitly modeled by an exponential function with rate affected by the magnitude of stresses from a compound Poisson process shock model. Applications of these life distributions and associated failure rates towards the study of multicomponent system survival are discussed. In particular, we illustrate the behavior of these survival functions in relevant subsets of the parameter space.

In this paper we establish a characterization theorem for a general class of life-testing models based on a relationship between conditional expectation and the failure rate function. As a simple application of the theorem, we characterize the gamma, Weibull, and Gompertz distributions, since they have many probabilistic and statistical properties useful in both biometry and engineering reliability.

In this paper, we consider the life distribution H(t) of a device subject to shocks governed by a finite mixture of homogeneous Poisson processes. It will be shown that if (pk), the probabilities that the device fails on the kth shock, has a discrete phase-type (DPH) distribution, then H(t) is continuous phase-type (CPH). The relationship between the mean values of (pk) and H(t) is established.

Two arbitrary life distributions F and G can be ordered with respect to their Laplace transforms. We say is Laplace-smaller than for all s > 0. Interpretations of this ordering concept in reliability, operations research, and economics are described. General preservation properties are presented. Using these preservation results we derive useful inequalities and discuss their applications to M/G/1 queues, time series, coherent systems, shock models and cumulative damage models.

A model incorporating the effect of a common environment on several components (structurally independent) of a system is developed. A multivariate generalization of the Lomax (Pareto type 2) distribution is obtained by mixing exponential variables. Its relationship to other multivariate distributions is discussed. Several properties of this distribution are reported and their usefulness in reliability theory indicated. Finally, a further generalization of this multivariate Lomax distribution is presented.

If the survival function satisfies the functional equation

where e = (1, …, 1), and if the marginal distributions are exponential, then is the multivariate exponential distribution of Marshall and Olkin. The functional equation has many solutions if the requirement of exponential marginals is not imposed, but the class of possible marginals is somewhat limited (e.g., marginals must be absolutely continuous). The class of possible solutions of the equation is characterized in this paper, and several examples are obtained from models for dependence that may be of practical interest.

The class of life distributions for which , where , and , is studied. We prove that this class is larger than the HNBUE (HNWUE) class (consisting of those life distributions for which for x ≧ 0) and present results concerning closure properties under some usual reliability operations. We also study some shock models and a certain cumulative damage model. The class of discrete life distributions for which for 0 ≦ p ≦ 1, where , is also studied.

A vector X of patient prognostic variables is modeled as a linear diffusion process with time-dependent, non-random, continuous coefficients. The instantaneous force of mortality (hazard function) operating on the patient is assumed to be a time-dependent, continuous quadratic functional of the prognostic vector. Conditional on initial data X0, the probability of surviving T units of time is expressed in terms of the solution of a Riccati equation, which can be evaluated in closed form if the coefficients of the process and the hazard are constant. This conditional expectation does not preserve proportional hazards.

Suppose that a device is subjected to shocks and that denotes the probability of surviving k shocks. Then is the probability that the device will survive beyond t, where is the counting process which governs the arrival of shocks. A-Hameed and Proschan (1975) considered the survival function H(t) under what they called the pure birth shock model. In this paper we shall prove that is IFRA and DMRL under conditions which differ from those used by A-Hameed and Proschan (1975).