= Parallel reliability system
{title2=$R_{\mathrm{parallel}}(t)$}
A <parallel reliability system> works whenever at least one of its components works. With component lifetimes $T_i$, its lifetime is $T=\max_iT_i$. If component survival events at time $t$ are <independent>, their <survival functions> give
$$
R(t)=\mathbb P(T>t)=1-\prod_i\bigl(1-R_i(t)\bigr).
$$
The 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>.<NIST and SEMATECH e-Handbook of Statistical Methods>[Section 8.1.8.3, parallel system model]{ref}
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