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ON PROBABILITIES ASSOCIATED WITH THE MINIMUM DISTANCE BETWEEN EVENTS OF A POISSON PROCESS IN A FINITE INTERVAL

Published online by Cambridge University Press:  23 April 2010

Shai Covo
Shai Covo
Affiliation:
Department of Mathematics, Bar Ilan University, 52900 Ramat-Gan, IsraelE-mail: [email protected]
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Abstract

We revisit the probability that any two consecutive events in a Poisson process N on [0, t] are separated by a time interval that is greater than s (<t) (a particular scan statistic probability) and the closely related probability (recently introduced by Todinov [8], who denotes it as pMFFOP) that before any event of N in [0, t], there exists an event-free interval greater than s. Both probabilities admit simple explicit expressions, which, however, become intractable for very large values of t/s. Our main objective is to demonstrate that these probabilities can be approximated extremely well for large values of t/s by some very tractable and attractive expressions (actually, already for t larger than a few multiples of s).

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 24 , Issue 3 , July 2010 , pp. 423 - 439
Copyright
Copyright © Cambridge University Press 2010

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References

1.Bingham, N.H., Goldie, C.M. & Teugels, J.L. (1987). Regular variation. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
2.Comtet, L. (1974). Advanced combinatorics. Dordrecht: Reidel.Google Scholar
3.Gates, D.J. & Westcott, M. (1985). Accurate and asymptotic results for distributions of scan statistics. Journal of Applied Probability 22: 531–542.Google Scholar
4.Korevaar, J. (2002). A century of complex Tauberian theory. Bulletin of the American Mathematical Society 39: 475–531.CrossRefGoogle Scholar
5.Naus, J.I. (1982). Approximations for distributions of scan statistics. Journal of the American Statistical Association 77: 177–183.Google Scholar
6.Penrose, O. & Elvey, J.S.N. (1968). The Yang–Lee distribution of zeros for a classical one-dimensional fluid. Journal of Physics A 1: 661–674.Google Scholar
7.Todinov, M.T. (2002). Statistics of defects in one-dimensional components. Computational Materials Science 24: 430–442.Google Scholar
8.Todinov, M.T. (2003). Minimum failure-free operating intervals associated with random failures of non-repairable components. Computers & Industrial Engineering 45: 475–491.Google Scholar
9.Todinov, M.T. (2004). A new reliability measure based on specified minimum distances before the locations of random variables in a finite interval. Reliability Engineering & System Safety 86: 95–103.Google Scholar
10.Todinov, M.T. (2005). Reliability and risk models: Setting reliability requirements. West Sussex, UK: Wiley.Google Scholar