A semicrystalline polymer fiber is a composite material consisting of difficult-to-deform crystals joined by more easily deformed and more easily broken amorphous materials. The failure process begins at the atomic level in the amorphous regions where random thermal fluctuations cause, at some time, a molecule to slip relative to other molecules or to rupture at one of its atomic bonds. The frequency of such random events is greatly enhanced by small increases in stress. As molecules slip or rupture, neighboring molecules become overloaded, thus increasing their failure rates. Such molecule failures accumulate locally and give rise to growing microcracks, although the exact kinetic mechanisms are not well understood. These growing minute cracks are the irreversible changes in the microstructure of the material that ultimately lead to macroscopic failure of the fiber.

A renewal process [N(t), t ≥ 0] with interarrival times Xi, for i ≥ 1 and renewal function m (t) is considered. Let Gn, λ denote the gamma distribution with parameters n and λ–that is, dGn, λ(x) = λε–λx(λx)n-1/(n – 1)

In Section 1 we show how m(t) can be approximated by f m(s) dGn, n/t(s). In addition, we show that these approximations constitute an increasing sequence of lower bounds when the interarrival distribution has the decreasing failure rate property. In Section 2 we show how the integrated renewal function can be approximated in a similar fashion by a decreasing sequence of upper bounds. In Section 3 we consider the problem of approximating the residual life (also called excess life) and the renewal age distribution and their means, and in Section 4 we consider the distribution of N(t). Finally, in Section 5 we remark on the relationship between our approximations and the Feller technique for inverting a Laplace transform.

Diversity of bugs or faults in a software system is a factor contributing to software unreliability which has not yet been appropriately emphasized. This paper is written with the intention of demonstrating the impact of fault diversity on the time to detection of software bugs. A new discrete software reliability model based on the multinomial distribution is introduced. It is shown that for models of this type, the more diverse the fault probabilities are, the longer (stochastically) it takes to detect or eliminate any n faults, while the smaller (stochastically) will be the number of faults detected or eliminated during a given amount of time (or during a given number of inputs to the system). The impact of fault diversity is also demonstrated for the Jelinski–Moranda model.

Resources are to be allocated sequentially to activities to maximize the total expected return, where the return from an allocation is the product of the value of the resource and the value of the activity. The set of activities and their values are given ahead of time, but the resources arrive according to a Poisson process and their values are independent random variables that are observed upon arrival. It is assumed that either there is a single random deadline for all activities, which is the same as discounting the returns, or the activities have independent random deadlines. The model has applications machine scheduling, packet switching, and kidney allocation for transplant. It is known that the optimal policy in the discounted case has a very simple form that does not depend on the activity values. We show that this is also true when the deadlines are independent and in this case the solution can expressed in terms of solutions to single activity models. These results also hold when there are batch arrivals of resources. The effects of pooling separate identical systems with a single activity into a combined system is investigated for both models. When activities have independent deadlines it is optimal to reject a resource in the combined system if and only if it is optimal to reject it in the single activity system. However, when returns are discounted, it is sometimes optimal to accept a resource in the combined system that would be rejected in the single activity system.

An inventory whose stock decreases linearly with time is considered. The inventory may be replenished at the instants at which a deliveryman arrives provided that the level of the inventory does not exceed a certain threshold; deliveries are made according to a Poisson process. A partial differential equation for the distribution function of the level of the inventory is solved to yield a formula for the corresponding Laplace–Stieltjes transform. The evaluation of the transform is discussed and explicit results are obtained for the stationary case.

This paper deals with a two-unit reliability model having one operating unit and one cold standby unit and with ample repair facilities. An approximate method is given for the calculation of the probability distribution of the proportion of time that the system is available in a given time period. The approximate method first computes the mean-time-to-failure and the meantime-to-repair and next approximates the up-and-down process of the two-unit reliability system by an alternating renewal process having exponentially distributed on and off periods. Numerical investigations show a satisfactory performance of the approximation. Also, a sensitivity analysis is given.

Using data obtained by the general capture/recapture sequential sampling process, an exact analytical expression for the maximum likelihood (ML) estimate of the population size, N, is introduced. As a consequence, it is shown that bounded likelihood functions have at most two maxima. For the simple one-by-one case the ML estimate is unique.

A large lot is to be sampled by a number of inspectors for the purpose of ascertaining its number of defective units. However, instead of a defective unit always being detected if it is sampled, there are unknown probabilities, depending on the inspectors, that the unit will not be identified as a defective. We offer an approach to estimating the number of defective units in the lot.