A stochastic process concerned with the times of replacement of components that are replaced as soon as they break down. Suppose a system has a single component, which is replaced immediately when it breaks down. Suppose the jth component has a lifetime Xj. The time, Tj, at which this component breaks down is given by ![renewal process](Images/oree/doc/10.1093/acref/9780199679188.001.0001/acref-9780199679188-math-0575-full.gif)
![renewal process](Images/oree/doc/10.1093/acref/9780199679188.001.0001/acref-9780199679188-graphic-104-full.gif)
The sequence T1, T2,…is a renewal process. A useful result is that, if N(t) is the number of breakdowns by time t then![renewal process](Images/oree/doc/10.1093/acref/9780199679188.001.0001/acref-9780199679188-math-0576-full.gif)