# Renewal processes

Reference work entry

First Online:

**DOI:**https://doi.org/10.1007/1-4020-0611-X_880

A *renewal process* is a stochastic point process {*N*(*t*) = number of occurrences by time *t*} which describes the appearance of a sequence of instant random events where the times between occurrences (e.g., called interarrival times in queueing theory) are a sequence of independent and identically distributed (i.i.d.) random variables. It is common to write the interoccurrence distribution function as *F*(*t*) and its density (if it exists) as *f*(*t*), with expected value 1/μ. The Poisson process represents a particularly important renewal process in which the intervals between occurrences are identically exponentially distributed (Cox, 1960; Cox and Isham, 1980; Feller, 1966; Smith, 1955).

The so-called

*renewal equation*for the process expectation (or renewal function)*H*(*t*) =*v[***E***N*(*t*)], plays a fundamental role in all renewal problems:This is a preview of subscription content, log in to check access.

## References

- [1]Cox, D.R. (1960). Renewal Theory, Methuen, New York.Google Scholar
- [2]Cox, D.R. and Isham, V. (1980). Point Processes, Chapman and Hall, New York.Google Scholar
- [3]Feller, W. (1966). Introduction to Probability Theory and Its Applications, vol. II, John Wiley, New York.Google Scholar
- [4]Smith, W.L. (1955). “Regenerative Stochastic Processes,” Proc. Royal Society, Ser. A, 232, 6–31. Google Scholar

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