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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
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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