WEBVTT
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This segment is probably
the most critical one
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for the purpose of understanding
what the Poisson process really
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is and how it behaves.
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There will be almost no
mathematical formulas.
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But the segment
will be quite dense
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in terms of
conceptual reasoning.
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So pay a lot of attention.
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In a nutshell, we will argue
that the Poisson process has
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memorylessness properties that
are entirely similar to those
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that we have seen for
the Bernoulli process.
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This should not be surprising,
since the Poisson process can
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be thought of as a limiting
case of the Bernoulli process.
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We will reason through
these properties,
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not in the style of a
formal mathematical proof,
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but with an intuitive argument.
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But I would like to assure you
that the intuitive argument can
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be translated into
a rigorous proof.
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The first property
is the following.
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The process starts at time 0.
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You come in and start
watching at let's say time 7.
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Or more generally,
instead of time 7,
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suppose that you
come in and start
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watching at some time, little t.
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The important thing here is
that little t is a constant.
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It's a deterministic number.
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Starting at that time,
what will you see?
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Well, the original
process was Poisson.
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This means that
disjoint intervals
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in the original process
are independent.
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Therefore, disjoint
intervals in the process
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that you will be seeing
will also be independent.
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Furthermore, during any
little interval of length
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delta in the
process that you see
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will still have
probability lambda times
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delta, approximately,
of seeing and arrival.
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Therefore, what you see also
satisfies the properties
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of a Poisson process, and
is itself a Poisson process.
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Second, the original
process was Poisson.
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So different intervals
are independent.
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So whatever happens
in this interval
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is independent from whatever
happens in that interval.
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But that interval corresponds
to the future of the process,
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and therefore, the future of the
process, what you get to see,
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is independent from
the past history.
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And so the conclusion is that
the process that you get to see
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is a Poisson process, which
is independent of the history
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until the time that
you started watching.
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And we say, therefore,
that what you see
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is a fresh starting process.
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The Poisson process
starts fresh at time t.
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We have the fresh
start property.
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And similar to the language we
use for the Bernoulli process,
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the fresh start property means
that you see a process that's
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independent of
the past and which
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has the same statistical
properties as if this was time
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0, as if the process was
just starting right now.
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One consequence of this
fresh start property
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is the following.
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You start watching at time t.
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And you're interested
in the time
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it takes until the next arrival.
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What are the properties
of this random variable?
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Well, since you have a fresh
starting Poisson process
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at this time, this is the
time until the first arrival
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in this fresh starting
Poisson process.
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And the time until
the first arrival
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in a process that
is just starting,
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we know that it has an
exponential distribution.
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So this is going to be an
exponential random variable
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with the same parameter, lambda.
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Furthermore, because the
process starts fresh,
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whatever happens in the future
is independent from the past.
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And so this random variable,
the remaining time,
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is independent of whatever
happened in the past until time
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t.
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Now let us look at a
somewhat different situation.
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You start watching the
process at time T1.
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Time T1 is the time
of the first arrival.
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And you start
watching from here on.
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What is it that
you're going to see?
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Suppose that the
first arrival happens,
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let's say, at time equal to 3.
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So we're conditioning
on this event.
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In that case, you start
watching the process at time 3.
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And you also know that the first
arrival happened at time 3.
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But this fact about the first
arrival happening at time 3
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belongs to the history of
the process until time 3.
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This is information
about the past,
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and does not affect what is
going to happen after time 3.
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The process after time 3 will
be independent from the history
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until time 3 and whatever
happened until that time.
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So starting at that particular
time 3, what you see
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is a Poisson process that is
independent from the past.
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Now, this argument
is valid even if I
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were to use here a
3.5 or 3.4 or 3.7.
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No matter when this
first arrival occurred,
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what I see starting
from this time
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is a Poisson process which
is independent from the past.
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At the time of
the first arrival,
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the process just starts fresh.
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As a consequence of this, and by
repeating the argument that we
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carried out for the remaining
time until the next arrival
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up here, we can
repeat this argument
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and argue that the time
until the next arrival
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in this fresh
starting process, this
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will also be an exponential
random variable.
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Now, this time until
the next arrival
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is the difference between
the second arrival
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time and the first arrival time.
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And we denote it by T2.
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What we just argued is that
this time until the next arrival
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is going to be an
exponential random variable.
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And also, it is
independent from the past.
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And in particular, it
is independent from T1.
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So the time until
the second arrival,
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starting from the first arrival,
the second inter-arrival time
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is a random variable that has
an exponential distribution that
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is the same distribution
as that of T1,
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and is independent from T1.
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Now we can extend this
argument and look at the kth
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inter-arrival time.
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For example, if the
arrival numbered k minus 1
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occurred here, and the
k arrival occurs here,
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this difference, here
we denote it by Tk,
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and by arguing in a similar
way that the process starts
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fresh at this particular time,
the time until the next arrival
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will also be an
exponential random variable
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with the same distribution.
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And furthermore,
will be independent
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from the past history,
and therefore,
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independent from the
earlier inter-arrival times.
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And this has lots of
important implications.
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For example, the time
until the kth arrival,
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which is the sum of the
first k inter-arrival times,
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is the sum of independent,
identically distributed,
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exponential random variables.
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In particular, this means
that we can find the PDF of Yk
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by convolving the
exponential PDF
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of these inter-arrival times,
convolving this exponential PDF
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with itself k times.
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And this is indeed one
way to find the PDF of Yk.
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But fortunately for us,
we were able to find it
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with a much simpler argument.
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And we already know what it is.
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But this property here is
also useful for finding
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the mean and the variance of Yk.
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The mean of the sum is
the sum of the means.
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And since the random
variables are independent,
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the variance of the sum is
the sum of the variances.
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We know what is the mean and
the variance of an exponential.
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And so by multiplying
that by k, we
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obtain the mean of
the kth arrival time
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and the variance of
the kth arrival time.
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And so we now know
the mean and variance
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of the Erlang PDF of order k.
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A second implication
of this property
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is more theoretical,
more conceptual.
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Recall that we defined
the Poisson process
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in terms of an
independence assumption
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and an assumption on the
probability of arrivals
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during a small interval.
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But we could have defined the
Poisson process as follows.
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Consider a sequence of
independent, identically
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distributed exponentials.
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Call them Tk.
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And use these to define
the arrival times.
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This is a way of
constructing a process.
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What we argued in
this segment is
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that a Poisson process under
the original definition
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satisfies this new definition.
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One can complete
the argument to show
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that the two definitions
are equivalent.
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It is possible to argue
that if we define an arrival
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process in this manner, this
arrival process will also
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satisfy the basic properties
of the Poisson process.
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This argument can
indeed be carried out,
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but we will not go through it.
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A final implication, which
is a little more practical.
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If you want to simulate
the Poisson process,
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how would you do it?
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Given what we now know, the most
natural way is the following.
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We generate independent,
identically distributed,
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exponential random
variables, using
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for example a random
number generator.
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And then use these
exponential random variables
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to construct the values of
the inter arrival times.
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And this way, construct
a complete time history
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of the Poisson process.