Bathtub curve

From Wikipedia, the free encyclopedia

The "bathtub" curve hazard function

The bathtub curve is widely used in reliability engineering, although the general concept is also applicable to humans. 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 name is derived from the cross-sectional shape of the eponymous device.

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.

In less technical terms, in the early life of a product adhering to the bathtub curve, the failure rate is high but quickly decreasing as defective products are identified and discarded, and early sources of potential failure such as handling and installation error are surmounted. In the mid-life of a product - generally, once it reaches consumers - the failure rate is low and constant. In the late life of the product, the failure rate increases, as age and wear take their toll on the product. Many consumer products strongly reflect the bathtub curve, such as computer processors.

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

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

While the bathtub curve is useful, not every product or system follows a bathtub curve hazard function. ^[1]

The term "Military Specification" is often used to describe systems in which the infant mortality section of the bathtub curve has been burned out or removed. This is done mainly for life critical or system critical applications as it severely reduces the possibility of the system failing early in its life. Manufacturers will do this at some cost generally by means similar to accelerated stress testing.