Reliability theory

Reliability theory uses probability theory and statistical inference to describe whether a component or system continues to perform its specified function. For a nonnegative failure time $T$, its reliability at time $t$ is the survival function $R(t)=\mathbb{P}(T>t)$. If $T$ has probability density function $f$, the hazard function $h(t)=f(t)/R(t)$ describes the instantaneous failure rate conditional on survival, where $R(t)>0$.

System structure matters. For independent components with reliabilities $R_i(t)$, a series reliability system fails at the first component failure, so $R_{\mathrm{series}}(t)={\prod }_i{R}_i(t)$. A parallel reliability system works while at least one component works, so $R_{\mathrm{parallel}}(t)=1-{\prod }_i(1-R_i(t))$. Two independent components each having reliability $0.9$ therefore give $0.81$ in series and $0.99$ in parallel. These product formulas depend on independence; a common cause of failure can invalidate them.^[ref]

A parallel reliability system works whenever at least one of its components works. With component lifetimes $T_i$, its lifetime is $T={\max }_i{T}_i$. If component survival events at time $t$ are independent, their survival functions giveThe formula follows because system failure requires every component to have failed. For two independent units each reliable with probability $0.9$, the system reliability is $1-0.1^2=0.99$. Two backup power supplies illustrate the intended redundancy, provided either can carry the required load. A common power surge can make failures dependent and invalidate the product formula; parallel structure alone does not establish independence.^[ref]

A series reliability system works only when every required component works. With component lifetimes $T_i$, its lifetime is $T={\min }_i{T}_i$. For independent component survival events,Two independent components each reliable with probability $0.9$ therefore give system reliability $0.81$. A pump that requires both a motor and a controller is a functional series system even if its wiring is not literally in series. For independent exponential distributions of lifetimes with rates ${\lambda }_i$, $R(t)=e^{-t{\sum }_i{\lambda }_i}$: adding required components adds failure rates. Shared causes of failure require a joint probability distribution rather than this product.^[ref]

Life testing observes failure times under a specified test protocol to estimate a lifetime probability distribution or check a reliability theory model. A test can end before every unit fails. Under right censoring, one records $Y_i=\min (T_i,C_i)$ and ${\delta }_i=1_{\{T_i\le C_i\}}$, where $T_i$ is the failure time and $C_i$ is the observation limit.

For independent censoring, the part of the likelihood function involving a continuous lifetime model with probability density function $f$ and survival function $R$ isA unit that fails at 45 hours contributes $f(45)$; one still working when observation ends at 60 hours contributes $R(60)$. Treating the latter as a failure at 60 hours invents an event that was not observed. The test protocol and any dependence between failure and observation limits therefore affect what can be inferred.^[ref]