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We often think of
a Bernoulli process

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as a stream of arriving traffic.

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What happens if we
merge two streams?

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For example, consider a
server that receives traffic

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from two independent sources.

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How do we describe
the total traffic

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that arrives to this server?

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Here's a precise model.

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We have two streams
that correspond

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to Bernoulli processes with
some parameters each, p and q,

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respectively.

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And each one of these
processes receives arrivals

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at certain times that
we indicate by crosses;

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and similarly, for
the second process.

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We assume that these two
processes are independent.

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And what we mean by this
is that any collection

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of random variables associated
with the first process

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will be independent from any
collection of random variables

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associated with
the second process.

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We now merge the two
processes as follows.

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Whenever there's an arrival in
any of the original processes,

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we record an arrival
in the merged process,

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as in this picture.

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Notice that we do not to make a
distinction between those slots

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at which there was
an arrival in only

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one of the original
[processes] versus those slots

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in which there was an arrival in
both of the original processes.

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In both cases, we
just say that there

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was an arrival in
the merged process,

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and so collisions--
arrivals in both

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of the original
processes-- are counted

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as only one arrival
in the merged process.

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Now, what can we say
about the merged process?

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We will argue that
it is a Bernoulli

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process with a certain
parameter that we will compute.

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To check the Bernoulli property
for the merged process,

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the first thing
we need to ensure

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is the independence assumption,
independence across slots.

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Let us look at
two typical slots.

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And to do this, it helps
to define some notation

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that Xt and Yt be the
original processes,

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and let Zt be the
merged process.

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The random variable Zt
is determined in some way

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by the random
variables Xt and Yt.

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If I tell you there was
an arrival in the first

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and to the second
process, you can

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tell whether there was an
arrival in the merged process.

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And similarly, the
random variable Zt plus 1

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is determined in some way
from Xt plus 1 and Yt plus 1.

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What this is saying
is that whether we

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have an arrival at
this slot is determined

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by what happens in
these two slots.

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And whether we have an
arrival in this slot

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is determined by whatever
happens in these two slots.

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Now, we have assumed that the
two processes are independent.

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So these two random
variables are

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independent from those
two random variables.

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And furthermore, across
time, this random variable

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will be independent from
that random variable.

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And this random variable
will be independent

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from that random variable.

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So these four random variables
are independent of each other.

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Because of this, we
have Zt, a function

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of two random variables that
are independent from the two

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random variables that
determine Zt plus 1.

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And for this reason, Zt and
Zt plus 1 will be independent.

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This proves a pairwise
independence property

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for the merged process, but
we can extend this argument

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to argue that the collection of
random variables, Z1 up to Zt,

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is a collection of
independent random variables.

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So we have the
independence property.

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Now, let us calculate the
probability of an arrival

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during a typical slot.

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During a typical
time slot, there

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are four possibilities
for what may occur.

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And these possibilities
have to do with

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whether in the X process,
we have an arrival

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or not; and in the Y process,
whether we have an arrival

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or not.

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The probability that we have
an arrival in both processes,

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because of independence, is
the product of the probability

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that we have an arrival
in the first process

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with the probability
that we have

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an arrival in the
second process.

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Similarly, there's
a probability p

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of an arrival in
the first process

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and no arrival in the second.

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There's a probability
1 minus p of no arrival

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in the first process
and [a probability q of]

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an arrival in the
second process.

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And finally, there is
probability 1 minus p times 1

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minus q of no arrival in
either of the two processes.

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The probability that we have an
arrival in the merged process

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is the probability
of this green event.

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These are the cases in
which an arrival gets

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recorded in the merged process.

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So the probability of an
arrival in the merged process

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is the sum of those
three probabilities.

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Or another way to calculate
it is 1 minus this probability

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here; namely, 1 minus 1
minus p times 1 minus q.

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And after you
expand this product

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and do some
cancellations, you end up

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with this expression, which is
the probability of an arrival

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during a slot in
the merged process.

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Of course, this probability
is constant across time.

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And this, together with
the independence property,

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establishes that the
merged process is actually

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a Bernoulli process.

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Now, let us end by
answering one more question.

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If I tell you that at
a certain time slot,

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there was at least one arrival
in the two processes, which

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means that there was an
arrival in the merged process,

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what is the
probability that there

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was an arrival in
the first process?

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Now, the event that there was
an arrival in the first process

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is this event here.

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So we're trying to calculate
the conditional probability

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of the blue event given that
the green event has occurred.

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We use the definition of
conditional probabilities.

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A conditional probability
is equal to the probability

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that both events happen,
which in this case

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is the intersection of the
blue and the green event,

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which coincides
with the blue event.

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And the probability
of the blue event

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is the sum of these two
numbers, is equal to p.

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And then we divide by the
probability of the conditioning

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event, this is the
probability of an arrival.

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But this is what we
have just calculated,

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which is p plus q
minus p times q.