Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-22T20:14:49.151Z Has data issue: false hasContentIssue false
  • English
  • Français

Clustering of events in a stochastic process

Published online by Cambridge University Press:  14 July 2016

Joseph Glaz
Joseph Glaz*
Affiliation:
The University of Connecticut
*
∗Postal address: Department of Statistics, The University of Connecticut, Storrs, CT06268, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we derive bounds for the expected waiting time of clustering of at least n events of a stochastic process within a fixed interval of length p. Using this approach of clustering, we derive bounds for the expected duration of the period of time that at least n servers are busy in an ∞-server queue with constant service time. For the case of Poisson arrivals we derive the exact distribution of the duration of that period.

Keywords

CLUSTERINGSTOCHASTIC PROCESSLOSS SYSTEMGI/D/n/n QUEUEBUSY PERIODGI/D/∞ QUEUE
Type
Short Communications
Information
Journal of Applied Probability , Volume 18 , Issue 1 , March 1981 , pp. 268 - 275
Copyright
Copyright © Applied Probability Trust 

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

Cohen, J. W. (1976) On Regenerative Processes in Queueing Theory. Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, New York.CrossRefGoogle Scholar
Esary, J., Proschan, F. and Walkup, D. (1967) Association of random variables with applications. Ann. Math. Statist. 38, 1466–1474.CrossRefGoogle Scholar
Glaz, J. (1979) Expected waiting time for the visual response. Biol. Cybernetics 35, 39–41.Google Scholar
Grind, W. A. Van De, Koendrink, J. J., Heyde, G. L. Van Der, Landman, H. A. A. and Bouman, M. A. (1971) Adapting coincidence scalers and neural modeling studies of vision. Kybernetic 8, 85–105.Google Scholar
Huntington, R. J. and Naus, J. I. (1975) A simpler expression Kth nearest neighbor coincidence probabilities. Ann. Prob. 3, 894–896.Google Scholar
Huntington, R. J. (1978) Distribution of the minimum number of points in a scanning interval on the line. Stoch. Proc. Appl. 7, 73–78.CrossRefGoogle Scholar
Hwang, F. K. (1977) A generalization of the Karlin–McGregor theorem on coincidence probabilities and an application to clustering. Ann. Prob. 5, 814–817.CrossRefGoogle Scholar
Ikeda, S. (1965) On Bouman–Velden–Yammamoto's asymptotic evaluation formula for the probability of visual reponse in a certain experimental research in quantum biophysics of vision. Ann. Inst. Statist. Math. 17, 295–310.CrossRefGoogle Scholar
Lewis, P. A. W., (Ed.) (1962) Stochastic Point Processes: Statistical Analysis, Theory and Applications. Wiley, New York.Google Scholar
Neff, D. D. and Naus, J. I. (1980) The distribution of the size of the maximum cluster of points on a line. Selected Tables in Mathematical Statistcs. American Mathematical Society, Providence, RI.Google Scholar
Newell, G. F. (1963) Distribution for the smallest distance between any pair of kth nearest-neighbor random points on a line. Proc. Symp. Time Series, Brown University, Wiley, New York, 89–103.Google Scholar
Newell, G. F. (1973) Approximate Stochastic Behavior of n-Server Service Systems with Large n. Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, New York.Google Scholar
Runnenburg, J. Th. (1969) Limit theorems for stochastic processes occurring in studies of the light sensitivity of the human eye. Statist. Neerl, 23, 1–17.CrossRefGoogle Scholar
Saunders, I. W. (1978) Locating bright spots in a point process. Adv. Appl. Prob. 10, 587–612.Google Scholar
Solove'V, A. D. (1966) A combinatorial identity and its application to the problem concerning the first occurrence of a rare event. Theory Prob. Appl. 11, 276–285.Google Scholar
Syski, R. (1960) Introduction to Congestion Theory in Telephone Systems. Oliver and Boyd, Edinburgh and London.Google Scholar
Takács, L. (1958) On a coincidence problem concerning telephone traffic. Acta Math. Acad. Sci. Hung. 9, 47–81.CrossRefGoogle Scholar
Takács, L. (1969) On Erlang's formula. Ann. Math. Statist. 40, 71–78.Google Scholar
Velden, H. A. Van Der (1946) The number of quanta necessary for the perception of light of the human eye. Ophthalmologica 111, 321–331.Google Scholar