Bathtub Hazard Rate Curve

During the useful-life period, the hazard rate remains constant. Some of the reasons for the occurrence of failures in this region are higher random stress than expected, low safety factors, undetectable defects, human errors, abuse, and natural failures. 

Finally, during the wear-out period, the hazard rate increases with time t due to reasons such as follows [6,22]  

X(f) is the time-dependent failure rate (i.e., hazard rate). 

The following equation can be used to represent the bathtub hazard rate curve mathematically [8,9]  

X t) = hazard rate or time-dependent failure rate X and P = scale parameters m and n = shape parameters 

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Write down probability density function for the bathtub hazard rate curve distribution. 

Generally, bathtub hazard rate curve is used to describe the failure rate of engineering items/systems and is shown in Figure 3.5. 

The following equation obtained from a probability distribution presented in Chapter 2 can be used to represent the bathtub hazard rate curve, shown in Figure 3.5, mathematically [23] ... 

At P = 0.5, Equation 3.1 gives the shape of the bathtub hazard rate curve shown in Figure 3.1. 

This probability distribution can be used to represent the entire bathtub hazard rate curve presented in Chapter 3. The probability density function of the distribution is defined by [13,14]... 

For a = 0.5, this probability distribution gives the bathtub-shape hazard rate curve, and for a = 1, it gives the extreme value probability distribution. In other words, the extreme value probability distribution is the special case of this probability distribution at a = 1. 

Figure 1. A bathtub curve—hazard rate function over time.

For nonrepairable systems (a system composed of many subassemblies and components), the instantaneous failure rate, termed the hazard rate h t), follows a pattern that changes with time. It is usually represented by a bathtub-shaped failure curve over time, as shown in Fig. 1 [1,1a]. It begins with an initial decreasing hazard rate attributed to premature failure due to defects. This is followed by a useful life period with an almost constant hazard rate due to intrinsic failures and, finally, a wearout period where the hazard rate increases rapidly with time. 

The hazard rate concept is very useful in elucidating the typical life pattern associated with electronic components the bathtub-shaped failure curve over a function of time is a good example. 

In the most general case, the hazard function follows the so-called bathtub curve (Fig. 1, case 1), which shows three distinct phases in the life of a comptMient the first phase correspraids to a failure rate decreasing with time, and it is characteristic of the irfant mortality or bum-in period whereupon the more the stracture, equipment, or system lives with no failures, the lower its probability of failure itself becomes (this period is... 

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