Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-29T19:04:07.145Z Has data issue: false hasContentIssue false
  • English
  • Français

An application of the theory of regenerative phenomena

Published online by Cambridge University Press:  24 October 2008

J. F. C. Kingman
J. F. C. Kingman
Affiliation:
Mathematical Institute, University of Oxford
Get access Rights & Permissions [Opens in a new window]

Extract

One of the classical models of applied probability is that which may be described as the covering of a line with a random collection of intervals of random length. It appears as such in ((9), pp. 23–25), but also as the problem of Type II counters with random dead time (1, 4, 10), as a model for a pedestrian crossing a busy road(5), and as the queue M/G/∞. If an excuse is required for returning to a problem which has been the object of so much research in the past, it is to be found in the fact, noted by Kendall (5), that the model gives rise to the general stable infinitely divisible p-function, and that the theory of the semigroup P of standard p-functions (6) requires deeper information than previous, practically motivated, studies have provided.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 68 , Issue 3 , November 1970 , pp. 697 - 701
Copyright
Copyright © Cambridge Philosophical Society 1970

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

REFERENCES

(1)Feller, W. On probability problems in the theory of counters. Courant Anniversary Volume, pp. 105–115 (Interscience; New York, 1948).Google Scholar
(2)Feller, W.An introduction to probability theory and its applications, vol. II (Wiley; New York, 1966).Google Scholar
(3)Fortet, R.Remarques finales. Queueing theory, pp. 215–218 (English Universities Press; London, 1967).Google Scholar
(4)Hammersley, J. M.On counters with random dead time I. Proc. Cambridge Philos. Soc. 49 (1953), 623–637.Google Scholar
(5)Kendall, D. G.Delphic semi-groups, infinitely divisible regenerative phenomena, and the arithmetic of p-functions. Z. Wahrscheinlichkeitstheorie und Verw.Gebiete 9 (1968), 163–195.Google Scholar
(6)Kingman, J. F. C.The stochastic theory of regenerative events. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1964), 180–224.CrossRefGoogle Scholar
(7)Kingman, J. F. C.Markov population processes. J. Appl. Probability 6 (1969), 1–18.Google Scholar
(8)Kingman, J. F. C. Infinitely divisible p-functions (to appear).Google Scholar
(9)Roach, S. A.The theory of random clumping (Methuen; London, 1968).Google Scholar
(10)Takacs, L.On a probability problem arising in the theory of counters. Proc. Cambridge Philos. Soc. 52 (1956), 488–498.CrossRefGoogle Scholar