In this study the Maximum Likelihood Estimator is utilized to identify the characteristics of failure of class-H insulation by considering accelerated life test data under censored situations from Nelson. The hazard rate function is considered in terms of the reliability, h(R), so-called AE model. The AE model is used to model the failures which are expressed as the serial connection between three modes, namely the turn, phase, and ground. This is the so-called competing failure. The main concern in the present investigation relates to the characteristic of changes in cumulative damage with temperature. The characteristic of the damage process basically change, with less capability of cumulation. The failure tends to be unpredictable in a constant hazard rate situation in much higher temperature environments. The parameters of the model are related to the temperature and follow the Arrhenius law. The numerical results indicate that the AE model is well fitted to the data and gives more information to identify the failure modes with fewer parameters. This is better than the using Weibull distribution with both parameters varied with temperature. According to the predicted lifetime, the turn needs to be rearranged primarily, followed by the phase. The ground mode only has influence on the failure at much higher temperatures.

In this paper different failure mechanisms which yield cumulative damage are investigated through two types of hazard rate functions. They have been studied during the past two decades. Type A was developed early by assuming the hazard rate as a function of reliability. There are two kinds of trend, one follows the negative logistic decay model, the other the negative Gompertz decay. Some modifications are suggested according to the failure tendency and convenience of fittings. Type B is developed recently by assuming the hazard rate as a function of the expected operation time, T, which is defined as the integration of reliability over the time, normalized by the mean-time-between-failure. In both types the proposed hazard rates grow with the time monotonically. Typical examples are taken to examine these models, meanwhile the comparisons with the Weibull-typed distribution are also made. The results show that the most of proposed relations are successful in the expression of cumulative damage phenomenon, especially the Type B is a better choice even compared with the Weibull-typed description in some respects. The advantages of the models are discussed based on the physical meanings involved in the parameters.

A system is discussed which is subject to a conditional Poisson stream of failures, whose intensity function representing the cumulative damage level is a Markov chain.

A system is subject to shocks; each shock at time t increases the cumulative damage λ (t) by a constant amount, while the system is subject to repair in between the shocks which brings down λ (t) at a constant rate. The shock arrival process is an inhomogeneous Poisson process with intensity function λ (t) and each shock weakens the system making it more expensive to run. The long-run expected cost per unit time of running the system is obtained as well as the variance of the cost which are used to get optimal times of replacement of the system.

A single device shock model is studied. The device is subject to some damage process. Under the assumption that as the cumulative damage increases, the probability that any additional damage will cause failure increases, we find sufficient conditions on the shocking process so that the life distribution will be increasing failure rate.