# CHAPTER 4

# Probability Distributions for Modeling Time to Failure

## 4.1 INTRODUCTION

In Chapter 1 we discussed a variety of reliability-related problems and mentioned the use of the systems approach to finding solutions to these problems. The systems approach involves the use of mathematical models. An important and critical factor in the building of these models is modeling the system (product) failures.

A system (product) is, in general, comprised of several parts and system failure is related to part failures. As such, the starting point is the modeling of part failures. For a nonrepairable part, we need to consider only the first failure. For repairable parts, it is necessary to differentiate first failure from subsequent failures, since the latter depend on the type of repair action taken. The failure of a part can be characterized in many different ways. These lead to the use of different mathematical formulations in modeling failures.

In this chapter, we focus our attention on the modeling of first failures based on the “black-box” characterization. Here, a part is characterized as being in one of two states—working or failed. We consider two cases—static and dynamic. In the static case, because of manufacturing defects a part produced can initially be in a failed state. When such a part is put into operation, its (failed) state is detected immediately. In the dynamic case, the part is in its working state to start with and fails after a certain length of time (called time ...