# To illustrate the use of theorem 33 we consider

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

Ask a homework question - tutors are online