Series reliability system ($R_{\mathrm{series}}(t)$)

A series reliability system works only when every required component works. With component lifetimes $T_i$, its lifetime is $T={\min }_i{T}_i$. 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]