The NonHomogeneous NonStationary

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The Non-Homogeneous (Non-Stationary) Poisson
Process
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The Non-Homogeneous (Non-Stationary) Poisson
Process

• in many applications, we would like the arrival
  process for a queue to incorporate time of day
  effects

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The axioms become

• P(two or more events in t,th)) o(h)

• number of events in non-overlapping intervals are
  independent

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If we define the mean-value function
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Also, if we define T(s) to be, at any instance s,
the random amount of time until the next arrival,
we can show that its pdf is
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One Method of Generation

• simulate a homogeneous Poisson process and
  rescale the time

Specifically,
Proof is homework!
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Another Method of Generation

• thin the process

Lewis and Schedler (1979)
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Why this thinning works (heuristics)
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Poisson Processes in Signal Encoding

• I want to send you a message (signal).

• While in transit, that signal gets corrupted by
  noise.

• You filter out the noise to retrieve the
  message.

Example Radio transmissions get corrupted by
electromagnetic signals.
Free book http//ee.stanford.edu/gray
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Poisson Processes in Signal Encoding
Think of the message as a function x(t), that
varies over time.
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Poisson Processes in Signal Encoding
and you receive this
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There are several ways to filter out the noise.
Example Kalman Filter

• gives an estimate whose expected value is the
  true signal

• gives an estimate with minimum variance

• pretty ugly diversion from our course

• take a time series course

• visit www.cs.unc.edu/welch/kalman/

• talk to Debbie!

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Poisson Processes in Signal Encoding

• transmit a signal

receive it
filter it
there is error

• encode a signal with a NHPP

transmit encoded signal
receive it
filter it
Increase system robustness against noise
un-encode it
there may be less error
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Poisson Processes in Signal Encoding
Let x(t) be the signal (real-valued function) to
be sent.
Assume max x(t) lt A.
and by marking each Poisson arrival with a
positive or negative sign I(t) sign(x(t)).
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Poisson Processes in Signal Encoding
Original signal x(t)
Encoded Signal P(t), a850 pulses
1
0
-1
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Specifics of the encoding

• suppose that x(t) is constant over an interval of
  length T

• distribute them uniformly over the interval

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Un-encoding

• fix a small time window of length T in which you
  will assume the signal x(t) is constant

• estimate the constant rate of the Poisson process
  in this window by

• counting the number of arrivals in the window

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Un-encoded signal
Decoded Signal T0.01
Decoded Signal T0.1
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Some Non-Homogeneous Poisson Processes
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Some Non-Homogeneous Poisson Processes
A convenient rate function
(type of Weibull)

• In this case, the time dependent component of the
  rate function enters multiplicatively.

Hence, we have the following interpretations

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A Non-Homogeneous Poisson Processes

• Suppose each customer stays in the store for a
  random time X with cdf F.

• Let N(t) be the number of customers in the store
  at time t. (Assume N(0)0.)