A quasi-renewal process is more realistic in modeling the behavior of a repairable system than traditional models such as perfect repair and minimal repair since it reflects the deterioration process of the system over time while traditional models do not. The quasi-renewal parameter is set to a value between 0 and 1 to indicate the rate of deterioration. Moreover, a quasi-renewal process can also be used to model the increasing time of maintenance actions due to the increasing difficulty of maintaining an aging system by setting the parameter to a value larger than 1. We construct a model where the operating times follow a quasi-renewal process and the corrective/preventive maintenance times follow another quasi-renewal process. A quasi-renewal function and two equivalent point availability expressions are developed for the model described by a quasi-renewal process with and age-dependent preventive maintenance policy. In addition, numerical results from various theoretical distributions are obtained to illustrate the behavior of the models. The two equivalent point availability functions each contains an infinite sum and must be truncated to obtain a numerical approximation. The two approximated point availability functions form upper and lower bounds on the real value. The bounds are useful for determining the result accuracy, which can be arbitrarily increased by adding more terms to the truncated summation. Our framework provides a new time-dependent availability model for a non-stationary process with a preventive maintenance policy without any cost structure or optimization problem. / Ph. D.

The behavior of repairable equipment is often modeled under assumptions such as perfect repair, minimal repair, or negligible repair. However the majority of equipment behavior does not fall into any of these categories. Rather, repair actions do take time and the condition of equipment following repair is not strictly "as good as new" or "as bad as it was" prior to repair. Non-homogeneous processes that reflect this type of behavior are not studied nearly as much as the minimal repair case, but they far more realistic in many situations. For this reason, the quasi-renewal process provides an appealing alternative to many existing models for describing a non-homogeneous process. A quasi-renewal process is characterized by a parameter that indicates process deterioration or improvement by falling in the interval [0,1) or (1,Infinity) respectively. This parameter is the amount by which subsequent operation or repair intervals are scaled in terms of the immediately previous operation or repair interval. Two equivalent expressions for the point availability of a system with operation intervals and repair intervals that deteriorate according to a quasi-renewal process are constructed. In addition to general expressions for the point availability, several theoretical distributions on the operation and repair intervals are considered and specific forms of the quasi-renewal and point availability functions are developed. The two point availability expressions are used to provide upper and lower bounds on the approximated point availability. Numerical results and general behavior of the point availability and quasi-renewal functions are examined. The framework provided here allows for the description and prediction of the time-dependent behavior of a non-homogeneous process without the assumption of limiting behavior, a specific cost structure, or minimal repair. / Ph. D.

This thesis explores methodologies of modeling and evaluating the operational performance of a typical aeroengine having field experience over two decades. Upon failure, the engine is repaired and restored to flight worthy condition and hence comes under the purview of repairable systems. Operational performance of the engine is being measured in terms of five functions of time, namely, M(t), which is the expected number of system failures in the time interval [0,t]; system failure rate m(t), which is an unconditional quantity and is simply the derivative of M(t); ρ(t), the conditional failure intensity given the history of a system Ht, which is nothing but limdt→1 Prob(System fails in [t,t + dt] |Ht); and M′(t) and m′(t), which are 0 dt conditional entities analogous to M(t) and m(t) defined in the same spirit as that of ρ(t), the details of which are given in the third chapter of the thesis. These functions are being estimated using field failure-repair data of 418 aeroengines, where the observations on time between failures are being measured in number of flying hours logged in between failures, and the corresponding repair duration is being measured in number of calendar days. To start with, using the superimposed renewal process model the above quantities M(t), m(t), m′(t), M′(t) and ρ(t) are estimated both in the frequentist as well as the Bayesian framework. Subsequently repair times have been incorporated into the model and analysed using both frequentist and Bayesian approaches. Next, the model of Lawless and Thiagarajah (1996) which incorporates both renewal and time trend, has been generalized to include repair time as well, and a comprehensive methodology of Bayesian model selection under this model has been developed. After introducing the research problem in the first chapter, the engineering system description leading to the identification of the failure modes, repair practice and the variables of interest is taken up in the following chapter at the outset, as a pre-requisite to the stochastic modeling and the statistical analysis that to follow in the remainder of the thesis. As the first stochastic model, the number of system failures in a given time interval is modeled as a superimposed renewal process with the constituent independent renewal processes running in different component sockets having Weibull inter failure times. This model is first empirically validated using the field failure data and then using this model, the five quantities of interest as mentioned above viz. M(t), m(t), ρ(t), M′(t) and m′(t) are analysed from a frequentist maximum likelihood perspective. A Bayesian analysis of the same follows in the subsequent chapter. Next, the repair effect is incorporated into the superimposed renewal process model by considering the Weibull parameters of inter failure times of the constituent renewal processes running in independent component sockets as a polynomial in the last repair time. The nature of this polynomial relationships are empirically deter-mined and the Weibull assumption is validated through a test of hypothesis. Different polynomial relationships lead to consideration of several models, with the correct ones chosen through a series of likelihood ratio tests. Next based on the appropriate models a maximum likelihood analysis of M(t), ρ(t) and M′(t) has been carried out. Like the simple superimposed renewal process model, Bayesian analysis of this model incorporating repair times is carried out in the following chapter. In the Bayesian setup however, the problem of model selection could be kept unrestricted to non-nested models as well (unlike the previous chapter, where only nested models could be considered), and a comprehensive model selection exercise has been carried out with the aid of intrinsic Bayes factors and training data sets. The last but one chapter presents a generalised model of Lawless and Thiagarajah (1996) for performance evaluation of aeroengines that incorporate renewals, time trends and the repair characteristics. Here also since the primary problem is one of model selection, the entire analysis like in the preceding chapter has been carried out under the Bayesian frame-work. The final chapter concludes the thesis by comparing the empirical results obtained in the previous five chapters, summarising the main contributions of the thesis and providing directions for future research.