Bathtub curve

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The "bathtub" curve hazard function

The bathtub curve is widely used in reliability engineering, although the general concept is also applicable to humans (for example in actuarial work). It describes a particular form of the hazard function which comprises three parts:

The first part is a decreasing failure rate, known as early failures or infant mortality.
The second part is a constant failure rate, known as random failures.
The third part is an increasing failure rate, known as wear-out failures.

The bathtub curve is generated by mapping the rate of early "infant mortality" failures when first introduced, the rate of random failures with constant failure rate during its "useful life", and finally the rate of "wear out" failures as the product exceeds its design lifetime.

The bathtub curve is often modeled by a piecewise set of three hazard functions,

$y(t) = \begin{cases} c_0-c_1t+\lambda, & 0\le t \le c_0/c_1 \\ \lambda, & c_0/c_1 < t \le t_0 \\c_2(t-t_0)+\lambda, & t_0 < t \end{cases} \!$