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A limit theorem for point processes by applications

Published online by Cambridge University Press:  14 July 2016

Prem S. Puri
Prem S. Puri*
Affiliation:
Purdue University
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Abstract

Let 0 ≦ T1 ≦ T2 ≦ ·· · represent the epochs in time of occurrences of events of a point process N(t) with N(t) = sup{k : Tk ≦ t}, t ≧ 0. Besides certain mild conditions on the process N(t) (see Conditions (A1)– (A3) in the text) we assume that for every k ≧ 1, as t →∞, the vector (t – TN(t), t – TN(t)–1, · ··, t – TN(t)–k+1) converges in law to a k-dimensional distribution which coincides with that of a random vector ξ k = (ξ1, · ··, ξ k) necessarily satisfying P(0 ≦ ξ1 ≦ ξ2 ≦ ·· ·≦ ξk) = 1. Let R(t) be an arbitrary function defined for t ≧ 0, satisfying 0 ≦ R(t) ≦ 1, ∀0 ≦ t <∞, and certain mild conditions (see Conditions (B1)– (B4) in the text). Then among other results, it is shown that

The paper also deals with conditions under which the limit (∗) will be positive. The results are applied to several point processes and to the situations where the role of R(t) is taken over by an appropriate transform such as a probability generating function, where conditions are given under which the limit (∗) itself will be a transform of an honest distribution. Finally the results are applied to the study of certain characteristics of the GI/G/∞ queue apparently not studied before.

Keywords

POINT PROCESSESLIMIT THEOREMSRENEWAL PROCESSESNON-HOMOGENEOUS POISSON PROCESSESGI/G/∞QUEUEPROCESSES WITH IMMIGRATIONBRANCHING PROCESSES
Type
Research Papers
Information
Journal of Applied Probability , Volume 15 , Issue 4 , December 1978 , pp. 726 - 747
Copyright
Copyright © Applied Probability Trust 1978 

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