In this paper, a new point process is introduced. It combines the nonhomogeneous Poisson process with the generalized Polya process (GPP) studied in recent literature. In reliability interpretation, each event (failure) from this process is minimally repaired with a given probability and GPP-repaired with the complementary probability. Characterization of the new process via the corresponding bivariate point process is presented. The mean numbers of events for marginal processes are obtained via the corresponding rates, which are used for considering an optimal replacement problem as an application.

In this paper we study the allocation problem of relevations in coherent systems. The optimal allocation strategies are obtained by implementing stochastic comparisons of different policies according to the usual stochastic order and the hazard rate order. As special cases of relevations, the load-sharing and minimal repair policies are further investigated. Sufficient (and necessary) conditions are established for various stochastic orderings. Numerical examples are also presented as illustrations.

Inaccuracy and information measures based on cumulative residual entropy are quite useful and have attracted considerable attention in many fields including reliability theory. Using a point process martingale approach and a compensator version of Kumar and Taneja's generalized inaccuracy measure of two nonnegative continuous random variables, we define here an inaccuracy measure between two coherent systems when the lifetimes of their common components are observed. We then extend the results to the situation when the components in the systems are subject to failure according to a double stochastic Poisson process.

Providing optimal strategies for maintaining technical systems in good working condition is an important goal in reliability engineering. The main aim of this paper is to propose some optimal maintenance policies for coherent systems based on some partial information about the status of components in the system. For this purpose, in the first part of the paper, we propose two criteria under which we compute the probability of the number of failed components in a coherent system with independent and identically distributed components. The first proposed criterion utilizes partial information about the status of the components with a single inspection of the system, and the second one uses partial information about the status of component failure under double monitoring of the system. In the computation of both criteria, we use the notion of the signature vector associated with the system. Some stochastic comparisons between two coherent systems have been made based on the proposed concepts. Then, by imposing some cost functions, we introduce new approaches to the optimal corrective and preventive maintenance of coherent systems. To illustrate the results, some examples are examined numerically and graphically.

In this note we revisit the discussion on minimal repair in heterogeneous populations in Finkelstein (2004). We consider the corresponding stochastic intensities (intensity processes) for items in heterogeneous populations given available information on their operational history, i.e. the failure (repair) times and the time since the last failure (repair). Based on the improved definitions, the setup of Finkelstein (2004) is modified and the main results are corrected in accordance with the updating procedure for the conditional frailty distribution.

In this paper we study the moment generating function order and the new better than used in the moment generating function order (NBUMG) life distributions. A closure property of this order under an independent random sum is deduced, and stochastic comparisons among the block replacement policy, the age replacement policy, the complete repair policy, and the minimal repair policy of an NBUMG component are investigated.

The process of deterioration of repairable systems with each repair is modeled using converging geometric-type processes. It is proved that the expectation of the number of repairs in each interval of time is infinite. A new regularization procedure is suggested and the corresponding optimization problem is discussed.

The notion of minimal repair is generalized to the case when the lifetime distribution function is a continuous or a discrete mixture of distributions (heterogeneous population). The statistical (black box) minimal repair and the minimal repair based on information just before the failure of an object are considered. The corresponding stochastic intensities are defined and analyzed for the point processes generated by both types of minimal repair. Some generalizations are discussed. Several simple examples are considered.

In this paper, an inspection–repair–replacement (IRR) model for a deteriorating system with unobservable state is studied. Assume that the system state can only be diagnosed by inspection and an inspection is imperfect. After inspection, if the system is diagnosed as being in a down state, a minimal repair will be undertaken, otherwise we do nothing. Assume further that the system lifetime is a random variable having increasing failure rate. A feasible IRR policy is studied. An algorithm is then suggested for determining an optimal feasible IRR policy for minimizing the long-run average cost per unit time after a finite-step search.

Often in the study of reliability and its applications, the goal is to maximize or minimize certain reliability characteristics or some cost functions. For example, burn-in is a procedure used to improve the quality of products before they are used in the field. A natural question which arises is how long the burn-in procedure should last in order to maximize the mean residual life or the conditional survival probability. In the literature, an upper bound for the optimal burn-in time is obtained by assuming that the underlying distribution of the products has a bathtub-shaped failure rate function; however, no lower bound is available. A similar question arises in studying replacement policy, warranty policy, and inspection models. This article gives a lower bound for the optimal burn-in time, and lower and upper bounds for the optimal replacement and warranty policies, under the same bathtub-shape assumption.

A new burn-in procedure for a repairable component is proposed. During a burn-in period, the failed component is only minimally repaired rather than being completely repaired. This procedure is shown to be economical and efficient when the minimal repair method is applicable during a burn-in process. The properties of the optimal burn-in time b∗ and block replacement policy T∗ are also given.

An optimal repair/replacement problem for a single-unit repairable system with minimal repair and random repair cost is considered. The existence of the optimal policy is established using results of the optimal stopping theory, and it is shown that the optimal policy is a ‘repair-cost-limit’ policy, that is, there is a series of repair-cost-limit functions gn(t), n = 1, 2,…, such that a unit of age t is replaced at the nth failure if and only if the repair cost C(n, t) ≥ gn(t); otherwise it is minimally repaired. If the repair cost does not depend on n, then there is a single repair cost limit function g(t), which is uniquely determined by a first-order differential equation with a boundary condition.

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.

Stochastic comparison results for replacement policies are shown in this paper using the formalism of point processes theory. At each failure moment a repair is allowed. It is performed with a random degree of repair including (as special cases) perfect, minimal and imperfect repair models. Results for such repairable systems with schemes of planned replacements are also shown. The results are obtained by coupling methods for point processes.

First, some basic concepts from the theory of point processes are recalled and expanded. Then some notions of stochastic comparisons, which compare whole processes, are introduced. The use of these notions is illustrated by stochastically comparing renewal and related processes. Finally, applications of the different notions of stochastic ordering of point processes to many replacement policies are given.

Burn-in is a widely used method to improve quality of products after they have been produced. For a repairable component there are two common types of repair, complete repair and minimal repair. Preventive maintenance policies such as age replacement and block replacement are often employed in field operation. The present paper takes burn-in, maintenance and repair into consideration at the same time and considers related cost structures. The properties of the corresponding optimal burn-in times and optimal maintenance policies are discussed.

In a counting process considered at time t the focus is often on the length of the current interarrival time, whereas points in the past may be said to constitute information about the process. The paper introduces new concepts on how to quantify predictability of the future behavior of counting processes based on the past information and considers then situations in which the future points become more (or less) predictable. Various properties of our proposed concepts are studied and applications relevant to the reliability of repairable systems are given.

In this study we examine repairable systems with random lifetime. Upon failure, a maintenance action, specifying the degree of repair, is taken by a controller. The objective is to determine an age-dependent maintenance strategy which minimizes the total expected discounted cost over an infinite planning horizon. Using several properties of the optimal policy which are derived in this study, we propose analytical and numerical methods for determining the optimal maintenance strategy. In order to obtain a better insight regarding the structure and nature of the optimal policy and to illustrate computational procedures, a numerical example is analysed. The proposed maintenance model outlines a new research channel in the area of reliability with interesting theoretical issues and a wide range of potential applications in various fields such as product design, inventory systems for spare parts, and management of maintenance crews.

Complete repair and minimal repair models with a block maintenance policy are considered. Each of these models gives rise to a counting process, and these processes are compared stochastically. This contrasts with most previous work on maintenance policies where only univariate marginal comparisons were made. Also a more general block schedule is considered than is customary.

In this paper, we develop general repair models for a repairable system by using the idea of the virtual age process of the system. If the system has the virtual age Vn –1 = y immediately after the (n – l)th repair, the nth failure-time Xn is assumed to have the survival function where is the survival function of the failure-time of a new system. A general repair is represented as a sequence of random variables An taking a value between 0 and 1, where An denotes the degree of the nth repair. For the extremal values 0 and 1, An = 1 means a minimal repair and An= 0 a perfect repair. Two models are constructed depending on how the repair affects the virtual age process: Vn = Vn– 1+ AnXn as Model 1 and Vn = An(Vn– 1 + Xn) as Model II. Various monotonicity properties of the process with respect to stochastic orderings of general repairs are obtained. Using a result, an upper bound for E[Sn] when a general repair is used is derived.