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Unformatted text preview: ory property of the exponential
distribution is crucial for many of the special properties of the Poisson process derived in this chapter. However, in many situations the assumption of
exponential interarrival times is not justiﬁed. In this section we will consider
a generalization of Poisson processes called renewal processes in which the
times t1 , t2 , . . . between events are independent and have distribution F .
In order to have a simple metaphor with which to discuss renewal processes,
we will think of a single light bulb maintained by a very diligent janitor, who
replaces the light bulb immediately after it burns out. Let ti be the lifetime
of the ith light bulb. We assume that the light bulbs are bought from one
manufacturer, so we suppose
P (ti t) = F (t)
where F is a distribution function with F (0) = P (ti 0) = 0.
If we start with a new bulb (numbered 1) at time 0 and each light bulb is
replaced when it burns out, then Tn = t1 + · · · + tn gives the time that the nth
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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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