The failure rate of a mixture of even the most standard distributions used in reliability can have a complicated shape. However, failure rates of mixtures of two carefully selected distributions will have the well-known bathtub shape. Here we show that mixtures of whole families of distribtions can have a bathtub-shaped failure rate.

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.

Suppose that there is a sequence of programs or jobs that are scheduled to be executed one after another on a computer. A program may terminate its execution because of the failure of the computer, which will obliterate all work the computer has accomplished, and the program has to be run all over again. Hence, it is common to save the work just completed after the computer has been working for a certain amount of time, say y units. It is assumed that it takes a certain time to perform a save. During the saving process, the computer is still subject to random failure. No matter when the computer failure occurs, it is assumed that the computer will be repaired completely and the repair time will be negligible. If saving is successful, then the computer will continue working from the end of the last saved work; if the computer fails during the saving process, then only unsaved work needs to be repeated. This paper discusses the optimal work size y under which the long-run average amount of work saved is maximized. In particular, the case of an exponential failure time distribution is studied in detail. The properties of the optimal age-replacement policy are also derived when the work size y is fixed.

In this paper two burn-in procedures for a general failure model are considered. There are two types of failure in the general failure model. One is Type I failure (minor failure) which can be removed by a minimal repair or a complete repair and the other is Type II failure (catastrophic failure) which can be removed only by a complete repair. During a burn-in process, with burn-in Procedure I, the failed component is repaired completely regardless of the type of failure, whereas, with burn-in Procedure II, only minimal repair is done for the Type I failure and a complete repair is performed for the Type II failure. In field use, the component is replaced by a new burned-in component at the ‘field use age’ T or at the time of the first Type II failure, whichever occurs first. Under the model, the problems of determining optimal burn-in time and optimal replacement policy are considered. The two burn-in procedures are compared in cases when both the procedures are applicable.

When a mission is assigned, it often is the case that the component used to perform the task is required to work properly during the period of the mission time. In other words, the probability of the event that this component does not fail within the allowable mission time should be as large as possible. This problem is considered for the case when the lifetime of a component has a bathtub-shaped failure rate function, and it is found that burn-in procedure is beneficial. An application of this result to the problem of minimizing the mean number of failures in a given period of mission time is also considered.

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.