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On renewal theory, counter problems, and quasi-Poisson processes†

Published online by Cambridge University Press:  24 October 2008

Walter L. Smith
Walter L. Smith
Affiliation:
University of North Carolina
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Extract

The power and appropriateness of renewal theory as a tool for the solution of general problems concerning counters has been amply demonstrated by Feller (7), who considered a variety of counter problems and reduced them to special renewal processes. The use of what may be called renewal-type arguments had certainly been made by authors other than Feller (e.g. in § 3 of Domb (3)), but it was only in (7) that the simplicity of the renewal approach to counter problems was recognized and systematically applied. More recently, Hammersley (8) was concerned with the generalization of a counter problem previously studied by Domb (2). This problem may be introduced, mathematically, as follows. Let {xi}, {yi} be two independent sequences of independent non-negative random variables which are non-zero with probability one (i.e. two independent renewal processes). The {xi}, are distributed in a negative-exponential distribution with mean λ-1, and we write Eλ for their distribution function and say ≡ {xi} is a Poisson process to imply this special property of ; the {yi} have a distribution function ‡ B(x) with mean b1 ≤ ∞. Form the partial sums and define ni to be the greatest integer k such that Xk ≥ t, taking X0 0 and nt = 0 if x1 > t. Then define the stochastic process

Hammersley'sx counter problem concerns the stochastic process

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 53 , Issue 1 , January 1957 , pp. 175 - 193
Copyright
Copyright © Cambridge Philosophical Society 1957

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References

REFERENCES

(1)Cox, D. R. and Smith, W. L.On the superposition of renewal processes. Biometrika, 41 (1954), 91–9.Google Scholar
(2)Domb, C.The problem of random intervals on a line. Proc. Camb. Phil. Soc. 43 (1947), 329–41.Google Scholar
(3)Domb, C.The statistics of correlated events. I. Phil. Mag. (7) 41 (1950), 969–82.Google Scholar
(4)Doob, J. L.Renewal theory from the point of view of the theory of probability. Trans. Amer. Math. Soc. 63 (1948), 422–38.CrossRefGoogle Scholar
(5)Feller, W.On the integral equation of renewal theory. Ann. Math. Statist. 12 (1941), 243–67.CrossRefGoogle Scholar
(6)Feller, W.Fluctuation theory of recurrent events. Trans. Amer. Math. Soc. 67 (1949), 98–119.CrossRefGoogle Scholar
(7)Feller, W.On probability problems in the theory of counters. Courant Anniversary Volume (1948), pp. 105–15.Google Scholar
(8)Hammersley, J. M.On counters with random dead time. I. Proc. Carrib. Phil. Soc. 49 (1953), 623–37.CrossRefGoogle Scholar
(9)Pollaczek, F.Sur la théorie stochastique des compteurs électroniques. C.R. Acad. Sci., Paris, 238 (1954), 766–8.Google Scholar
(10)Smith, W. L.Asymptotic renewal theorems. Proc. Roy. Soc. Edinb. A, 64 (1954), 9–48.Google Scholar
(11)Smith, W. L.Extensions of a renewal theorem. Proc. Camb. Phil. Soc. 51 (1955), 629–38.Google Scholar
(12)Takács, L.On secondary processes generated by a Poisson process and their applications in physics. Acta Math. Hung. 5 (1954), 203–35.Google Scholar
(13)Widder, D. V.The Laplace transform (Princeton, 1941).Google Scholar