Encyclopedia of Operations Research and Management Science

2001 Edition
| Editors: Saul I. Gass, Carl M. Harris

Renewal processes

  • Igor Ushakov
Reference work entry
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) = Ev[ N( t)], plays a fundamental role in all renewal problems:
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References

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

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Igor Ushakov
    • 1
  1. 1.QualcommSan DiegoUSA