Reliability and Survival Analysis

The survival function S(t) = P(T > t) gives the probability that a component or subject survives beyond time t. The hazard function h(t) = f(t) / S(t) describes the instantaneous failure rate conditional on survival to time t.

Together, these two functions fully characterize the failure-time distribution and are the primary tools for reliability assessment.

The exponential distribution assumes a constant hazard rate, meaning the component is equally likely to fail in the next instant regardless of its age. This is the memoryless property.

It works well for electronic components during their useful life but poorly for mechanical parts subject to wear.

The Weibull distribution generalizes the exponential by adding a shape parameter that allows decreasing, constant, or increasing hazard rates. This makes it the default choice in industrial reliability.

Shape < 1 models infant mortality (early failures), shape = 1 recovers the exponential, and shape > 1 models wear-out. The scale parameter represents the characteristic life.

The gamma distribution models the time until the k-th failure and is useful when systems have built-in redundancy or when failures accumulate gradually.

The lognormal distribution arises when failure is driven by multiplicative degradation processes. It is common in semiconductor reliability and fatigue-crack growth models.

MTTF is the expected failure time for non-repairable items. MTBF applies to repairable systems and is the average time between successive failures.

These metrics are single-number summaries and can be misleading if the failure rate is not constant. Always pair them with survival curves or hazard plots for a complete picture.