# Modeling Consecutive Failures of Repairable Systems, with Applications in Warranty Cost Analysis

Most engineered systems are inclined to fail sometime during their lifetime. Many of these systems are repairable and not necessarily discarded and replaced upon failure. Unlike replacements, where the failed system is replaced with a new and identical system, not all repairs have an equivalent effect on the working condition of the system. Describing the effect of repairs is a requirement in modeling consecutive failures of a repairable system–at the very least, it is assumed that a repair simply returns the failed system to an operational state without affecting its working condition (i.e. the repair is minimal). Although this assumption simplifies the modeling process, it is not the most accurate description of the effect of repair in real situations. Often, along with returning a failed system to an operational state, repairs can improve the working condition of the system, and thus, increase its reliability which impacts on the rate of future failures of the system. Repair models provide a generalized framework for realistic modeling of consecutive failures of engineered systems, and have broad applications in fields such as system reliability and warranty cost analysis. The overall goal of this research is to advance the state of the art in modeling the effect of general repairs, and hence, failures of repairable systems. Two specific types of system are considered: (i) a system whose working condition initially improves with time or usage, and whose lifetime is modeled as a univariate random variable with a non-monotonic failure rate function; (ii) a system whose working condition deteriorates with age and usage, and whose lifetime is modeled as a bivariate random variable with an increasing failure rate function. Most univariate lifetime distributions used to model system lifetimes are assumed to have increasing failure rate functions. In such cases, modeling the effect of general repairs is straightforward– the effect of a repair can bemodeled as a possible decrease, proportional to the effectiveness of the repair, in the conditional intensity function of the associated failure process. For instance, a general repair can be viewed as the replacement of the failed system with an identical system at a younger age, so that the conditional failure intensity following the repair is lower than the conditional failure intensity prior to the failure. When the failure rate function is initially decreasing, specifically bathtub-shaped, general repair models suggested for systems with increasing failure rate functions can only be applied when initial repairs are assumed to be minimal. In this study, we propose a new approach to modeling the effect of general repairs on systems with a bathtub-shaped failure rate function. The effect of a general repair is characterized as a modification in the conditional intensity function of the corresponding failure process, such that the system following a general repair is at least as reliable as a system that has not failed. We discuss applications of the results in the context of warranty cost analysis and provide numerical illustrations to demonstrate properties of the models. Sometimes the failures of a system may be attributed to changes in more than one measure of its working condition– for instance, the age and some measure of the usage of the system (such as, mileage). Then, the system lifetime is modeled as a bivariate random variable. Most general repair models for systems with bivariate lifetime distributions involve reducing the failure process to a one-dimensional process by, for instance, assuming a relationship between age and usage or by defining a composite scale. Then, univariate repair models are used to describe the effect of repairs. In this study, we propose a new approach to model the effect of general repairs performed on a system whose lifetime is modeled as a bivariate random variable, where the distributions of the bivariate inter-failure lifetimes depend on the effect of all previous repairs and following a general repair, the system is at least as reliable as a system that has not failed. The lifetime of the original system is assumed to have an increasing failure rate (specifically, hazard gradient vector) function. We discuss applications of the associated failure process in the context of two-dimensional warranty cost analysis and provide simulation studies to illustrate the results. This study is primarily theoretical, with most of the results being analytic. However, at times, due to the intractability of some of the mathematical expressions, simulation studies are used to illustrate the properties and applications of the proposed models and results.