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We will now consider
an operation

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that is, in some sense,
the opposite of merging.

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We have a node, and traffic
arrives to that node.

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And each time that we have
an arrival, we flip a coin.

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And with probability
q, we send that arrival

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to one particular stream.

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And with probability
1 minus q, we

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send the arrival
to another stream.

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So we get two streams
that are formed

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by taking the original
stream and splitting it

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into two pieces.

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And as an example,
these might be

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arrivals at a department store.

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And one stream
corresponds to the people

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who go to the clothes section
of the department store,

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whereas the other stream
corresponds to the people that

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go to all of the other
sections of the store.

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So let us now make our
model a little more precise.

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We have a Bernoulli
process, which

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is independent across time.

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We also use an
independent coin flip

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to deal with each
one of the arrivals.

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But we will also make one
additional assumption, namely

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that the Bernoulli
process is also

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independent from the
process of coin flips.

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With this assumption
in place, let us now

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continue, and let
us draw a picture.

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We have a Bernoulli
process with parameter p,

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and arrivals get recorded
at certain times.

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Each time that there is an
arrival, we will flip a coin.

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And with probability
q, the arrival

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will be sent to that stream.

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With probability 1
minus q, the arrival

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will be sent to
the other stream.

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So one possible outcome
of the experiment

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might be this one, where
these two arrivals were

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sent to this stream and
these two arrivals were

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sent to the top stream.

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And we have these
probabilities q and 1 minus q

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of sending the arrivals to
one or the other stream.

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What kind of
process is this one?

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We argue it is a
Bernoulli process.

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First, we need to
check independence.

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Here, the argument is more or
less the same as in the case

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when we studied the
merging of processes.

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For example, if we look at two
different slots and we ask,

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how is the event at that slot
and at that slot determined?

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Well, what happens in
this slot is determined

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by whether we had
an arrival here

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and what happened to the outcome
of the coin flip at that time.

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What happens in this
slot is determined

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by whether we had
an arrival here

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and what happened to the
coin flip at that time.

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Now, the coin flips
are independent

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from the original
Bernoulli process.

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And for either the coin flips
or the Bernoulli process,

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we have independence
across time.

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So all of the four random
variables involved here

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that determine what
happens in these two slots

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are independent of each other.

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So what happens in
this slot is a function

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of two random
variables here, which

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are independent from the
two random variables that

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determined what
happens in that slot.

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So what happens in these two
slots are independent events.

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And this argument goes
through more generally

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when we consider
multiple distinct slots.

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So this is the argument
for the independence

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of the different slots in
this particular process.

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And then during each
slot, what happens

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is that we will have
an arrival if and only

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if this process records
an arrival, which happens

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with probability p, and
the corresponding coin

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flip happens to send the
arrival in this direction, which

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happens with probability q.

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And so the conclusion
is that this process

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is a Bernoulli process
with parameter p times q.

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By a similar argument, the
other process that we obtain

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will also be Bernoulli but with
probability p times 1 minus q.

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And a final question--
are these two processes

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that we get after the splitting
independent of each other?

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This is a question
that we can answer

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by reasoning intuitively.

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If I tell you that there
was an arrival in this slot,

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what can you infer from this?

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Well, it tells me that
there was an arrival

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in the original stream,
which was sent here.

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But since it was sent
in this direction,

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it means that it was not
sent in the other direction.

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And so we do not have
an arrival in this slot.

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Knowing that we
have an arrival here

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means that we do not
have an arrival there.

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So information about
one of the streams

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gives us information about what
happened in the other stream.

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And therefore, we do
not have independence.

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So this is what happens when we
split two Bernoulli processes.

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And earlier we saw
what happens when

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we merge two independent
Bernoulli processes.

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These two operations of
merging and splitting

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are quite common in
constructing more complex models

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using Bernoulli processes as
the elements of those models.

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They are often useful models
either in transportation

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systems, where you
have streams of traffic

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that merge or split, also in
models of computer networks

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or any other kind
of queueing system.

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And these same operations
of merging and splitting

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will also show up when we
study the continuous time

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analog of the Bernoulli process,
namely the Poisson process.