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8. Assessing Product Reliability
8.1. Introduction
8.1.2. What are the basic terms and models used for reliability evaluation?

Failure (or hazard) rate

The failure rate is the rate at which the population survivors at any given instant are "falling over the cliff" The failure rate is defined for non repairable populations as the (instantaneous) rate of failure for the survivors to time \(t\) during the next instant of time. It is a rate per unit of time similar in meaning to reading a car speedometer at a particular instant and seeing 45 mph. The next instant the failure rate may change and the units that have already failed play no further role since only the survivors count. 

The failure rate (or hazard rate) is denoted by \(h(t)\) and is calculated from $$ h(t) = \frac{f(t)}{1 - F(t)} = \frac{f(t)}{R(t)} = \mbox{the instantaneous (conditional) failure rate.} $$ The failure rate is sometimes called a "conditional failure rate" since the denominator \(1 - F(t)\) (i.e., the population survivors) converts the expression into a conditional rate, given survival past time \(t\).

Since \(h(t)\) is also equal to the negative of the derivative of \(\mbox{ln}[R(t)]\), we have the useful identity: $$ F(t) = 1 - \mbox{exp} \left[ -\int_0^t h(t)dt \right] \,\, . $$ If we let $$ H(t) = \int_0^t h(t)dt $$ be the Cumulative Hazard Function, we then have \( F(t) = 1 - e^{H(t)}\). Two other useful identities that follow from these formulas are: $$ h(t) = - \frac{d \mbox{ln} R(t)}{dt} $$ $$ H(t) = - \mbox{ln} R(t) \,\, . $$ It is also sometimes useful to define an average failure rate over any interval \((T_1, T_2)\) that "averages" the failure rate over that interval. This rate, denoted by \( AFR(T_1, T_2)\), is a single number that can be used as a specification or target for the population failure rate over that interval. If \(T_1\) is 0, it is dropped from the expression. Thus, for example, \(AFR(40,000)\) would be the average failure rate for the population over the first 40,000 hours of operation. 

The formulas for calculating \(AFR\) values are: $$ AFR(T_2 - T_1) = \frac{\int_{T_1}^{T_2} h(t)dt}{T_2 - T_1} = \frac{H(T_2) - H(T_1)}{T_2 - T_1} = \frac{\mbox{ln}R(T_1) - \mbox{ln}R(T_2)}{T_2 - T_1} $$ and $$ AFR(0,T) = AFR(T) = \frac{H(T)}{T} = \frac{-\mbox{ln} R(T)}{T} \,\, . $$

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