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In this lecture, we introduce
the Poisson process, which is
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a continuous time analog of
the Bernoulli process.
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One way of thinking about
it is as follows.
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Time is continuous, but
conceptually we divide it into
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a very large number of slots.
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And during each slot,
we have a tiny
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probability of an arrival.
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This probability is proportional
to the
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length of the slot.
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Furthermore, we have an
independence assumption for
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the different slots.
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The Poisson process is a very
elegant model of arrival
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processes in continuous time.
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It models many real-world
phenomena.
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And it also has a very clean
mathematical structure that
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allows us to calculate
practically
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every quantity of interest.
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Our development will parallel
our analysis of
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the Bernoulli process.
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For example, we will find the
PMF of the number of arrivals
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during a time interval
and the PDF of the
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time of the kth arrival.
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We will discuss the
memorylessness properties of
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the Poisson process.
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Similar to the case of the
Bernoulli process, this is
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just a consequence of the
independence assumptions that
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we are making.
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We will then exploit these
independence properties to
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argue that the interarrival
times are independent
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exponential random variables.
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And we will conclude with
a comprehensive example.