To illustrate the use of theorem 33 we consider

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Unformatted text preview: times of a Poisson process with rate . Each arriving particle that finds the counter free gets registered and locks the counter for an amount of time ⌧ . Particles arriving during the locked period have no e↵ect. If we assume the counter starts in the unlocked state, then the times Tn at which it becomes unlocked for the nth time form a renewal process. This is a special case of the previous example: ui = ⌧ , si = exponential with rate . In addition there will be several applications to queueing theory. The first important result about renewal processes is the following law of large numbers: Theorem 3.1. Let µ = Eti be mean interarrival time. If P (ti > 0) > 0 then with probability one, N (t)/t ! 1/µ as t ! 1 In words, this says that if our light bulb lasts µ years on the average then in t years we will use up about t/µ light bulbs. Since the interarrival times in a Poisson process are exponential with mean 1/ Theorem 3.1 implies that if N (t) is the number of arrivals up to time t in a Poisson process, the...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

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