Theory concerned with determining the probability that a system (with n components) is working. Let xj=1 if the jth component is working and let xj=0 if it has failed. The vector x=(x1 x2...xn) is called the state vector. The function ϕ(x), which takes the value 1 when the system is working and 0 when it has failed, is called the structure function. For n components in series,ϕ(x)=min{x1, x2,..., xn}=x1×x2×...×xn.For n components in parallel,ϕ(x)=max{x1, x2,..., xn}=1-(1-x1)(1-x2)...(1-xn).If ϕ(x)=1 then x is a path vector: it traces a set of connected working components. If failure of any of its working components results in system failure, the vector is a minimal path vector. Correspondingly, if ϕ(x)=0 then x is a cut vector and, if it is the case that repair of any of the failed components in x leads to the system working, then the vector is a minimal cut vector.
If pj denotes the probability that the jth component continues to work during the next unit of time, then the probability that a structure consisting of n components in series continues to work is p1×p2×···×pn with the corresponding probability for n components in parallel being
Reliability theory. Complicated systems can always be subdivided into combinations of components arranged either in series or in parallel.
