Poisson Process Modeling

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Poisson Process Modeling
A First Look at Discrete Events Simulation
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The Poisson Process

• we consider a process in which events occur
  at random points in time

Examples

• arrivals of customers to a system

• photon emittance in radioactive decay

• neuron spike activity

• claims to an insurance company

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The Poisson Process
It turns out that with some basic uniformity and
independence assumptions, all such processes can
be modeled by a single, one-parameter probability
model.
where N(t) events occurring in 0,t).
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The Poisson Process
Poisson Axioms

• P(two or more events in t,th)) o(h)

• number of events in non-overlapping intervals are
  independent

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This works both ways
Suppose N(t) is a random process with independent
increments
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One way to get a Poisson process is to have
events arriving with independent exponential rate
interarrival times
Define N(t) as follows

• N(0) 0

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Then for kgt0,
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On the other hand, any Poisson process has
exponential interarrival times
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Q Why do we see the Poisson structure in so many
processes?
A Probably because there is less structure than
you think
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In general, we can show that if exactly n events
have occurred in the time interval 0,t,
the successive occurrence times have the same
distribution as the order statistics of n
unif(0,t) random variables!
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One more thing.
If we know that n Poisson events have occurred by
time t, then the number of events that have
occurred by time sltt has the _____________
distribution.