Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-05-01T01:53:50.338Z Has data issue: false hasContentIssue false
  • English
  • Français

Superimposed non-stationary renewal processes

Published online by Cambridge University Press:  14 July 2016

S. Blumenthal ,
J. A. Greenwood  and
L. Herbach
S. Blumenthal
Affiliation:
New York University
J. A. Greenwood
Affiliation:
New York University
L. Herbach
Affiliation:
New York University
Get access Rights & Permissions [Opens in a new window]

Extract

For superposition of independent, stationary renewal processes, it is well known that the distribution of waiting time between events for the superimposed process is approximately exponential if the number of processes involved is sufficiently large, (see Khintchine (1960), Ososkov (1956)). We assume that all component processes have the same age t, and we generalize the classical result to show that even for t finite (non-stationary case), the limiting waiting time distribution (as the number of processes increases) is exponential with a scale parameter which depends on t through the average of the individual process renewal densities.

Type
Research Papers
Information
Journal of Applied Probability , Volume 8 , Issue 1 , March 1971 , pp. 184 - 192
Copyright
Copyright © Applied Probability Trust 1971 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barlow, R. and Proschan, F. (1965) Mathematical Theory of Reliability. John Wiley, New York.Google Scholar
Beutler, T. J. and Leneman, O. A. Z. (1966) The theory of stationary point processes. Acta Math. 116, 159–197.Google Scholar
Çinlar, E. (1968) On the superposition of m-dimensional point processes. J. Appl. Prob. 5, 169–176.Google Scholar
Cox, D. R. (1962) Renewal Theory. John Wiley, New York.Google Scholar
Doob, J. L. (1953) Stochastic Processes. John Wiley, New York.Google Scholar
Drenick, R. F. (1960) The failure law of complex equipment. J. Soc. Indust. Appl. Math. 8, 680–689.Google Scholar
Franken, P. (1963) A refinement of the limit theorem for the superposition of independent renewal processes. Theor. Probability. Appl. 8, 320–327.Google Scholar
Khintchine, A. Ya. (1960) Mathematical Methods in the Theory of Queueing. Griffin, London. See page 50.Google Scholar
Ososkov, G. A. (1956) A limit theorem for flows of similar events. Theor. Probability. Appl. 1, 248–255.Google Scholar
Ryll-Nardzewski, C. (1961) Remarks on processes of calls. Fourth Berkeley Symp. Math. Statist. Prob. Univ. California Press, Berkeley, 2, 455–465.Google Scholar