Parallel reliability system ($R_{\mathrm{parallel}}(t)$)

A parallel reliability system works whenever at least one of its components works. With component lifetimes $T_i$, its lifetime is $T={\max }_i{T}_i$. 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]