= Series reliability system
{title2=$R_{\mathrm{series}}(t)$}
A <series reliability system> works only when every required component works. With component lifetimes $T_i$, its lifetime is $T=\min_iT_i$. For <independent> component survival events,
$$
R(t)=\mathbb P(T>t)=\prod_iR_i(t).
$$
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.<NIST and SEMATECH e-Handbook of Statistical Methods>[Section 8.1.8.2, series system model]{ref}
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