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Okay so previous lecture we looked at the
cross point complexity for a clos network
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but that is a very ideal scenario okay that
is a basically we know the value after which
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if the middle switches are grown up then it
will lead to a strictly non blocking property
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but as I actually have explained earlier in
dual light situation we do not want to build
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up strictly non blocking switches we actually
build up blocking switches in fact we do two
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things.
We either build only the blocking switches
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and use then to reduce the cost because certain
amount of blocking is acceptable or we do
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create a strictly non blocking switch and
then introduce blocking by using concerned
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data units that basically means.
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Unable to capture the image because lecture
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I am going to have a special kind of switch
where number of inputs are large number of
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outputs are small users strictly non blocking
switch and connect to the large one so I have
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introduced the blocking by use this is known
as concerned data unit okay and that is a
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switch this configuration also has been used
in lot of telecom switch implementations okay
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so in fact most of them actually where using
this so if I am going to actually do not satisfy
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that condition of 2xm-1 those many number
of middle state switches.
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Is going to be less than that what will be
blocking probability so that question actually
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remains so if you have a switch like here
so it is a three stage clos network now one
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important things is that I want to set up
a connection between two free incoming and
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outgoing port there other switches also here
which you should ask question what is going
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to happen with those how those will be incorporated
but now you take any pair you take this pair
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and this pair this is independent of when
you set up a connection between these input
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output port pairs because these links in these
are known as input links these are output
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links that is a totally different set when
I am considering these input and output ports.
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Okay so the blocking here will only be dependent
on the availability of a common link a free
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link has to be available here and a free link
has to be available here. Whether this link
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is available or not or this link is available
or not is immaterial, so those need not be
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accounted for and the situation you take any
pair the situation is same, so whatever blocking
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probability which have an estimate for these
two.
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Is going to be true for any other possible
pair, in fact that is the premise which we
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use to estimate the blocking probability.
So but we do know if my number of middle stage
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switches if this is this ports are m these
ports are m, the moment it is greater than
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2m-1 it should become 0, that’s what the
Clos theorem actually says we are actually
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we have seen this thing in the previous lecture.
Now, how to estimate it here? So there actually
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various ways so we will discuss two of them
so one is the lee’s approximation, okay.
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So this is very simple entry will exercises
this is not a great deal, so the caller arrival
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rate we define as A, the probability that
this link will be active or a call will be
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coming up, okay. So if this is m and these
are k number of links KR the middle stage
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switches.
And it is a Clos configuration, so what is
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the probability this link will be occupied
is a very simple calculation. So which actually
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means there are m number of ports each one
of them can be occupied with probability F
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this should be equal to the probability of
the any one of this link getting occupied
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that probability is PV call it, so P should
be equal to yeah, let us call it this way
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so P should be the probability that can any
one of this links will get occupied is MA/K.
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MK is larger than M so this is going to be
smaller, see I am not assuming that blocking
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is happening because of this switch this switch
the blocking will happen remember the M/N
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composite switch when N is M/N composite switch
was something like this, when this N is a
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smaller than M then blocking happens here
if N > = M there is no blocking, the condition
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here is K > = M, okay.
This even can be even larger in fact this
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lee’s approximation gives you an estimate,
okay. But this is still says the blocking
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is going to happen when it cannot happen when
your case going to be 2m – 1 the formula
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will is still blocking is going to happen
so this the probability that this link will
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be occupied okay, and similarly I can estimate
what is a blocking probability what is the
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probability that this link will get occupied
this will also be P this is them so call which
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is coming here is going to have an impact
connecting to anyone of the outgoing inputs.
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So affectively when I am looking at the probability
that call will coning here the arrival rate
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that is because of all these so this should
also be a I am looking at a complete symmetric
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condition any input is trying to connect to
any output with equal probability under those
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symmetric conditions both side it is a so
this occupancy probability will also the so
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this is here is k will also be given by ma/k
now the probability when the blocking will
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be happening when you cannot find the out
a path.
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Now how many possible paths are there you
can actually you can route through this that
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case such possible paths and when a path is
not available or blocked when this is been
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used to make a connection to somebody else
by some other busy port so this is occupied
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even if this link is free you cannot use this
path there is a possibility that some other
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output from a x this n connect here so this
is occupied even if this is available you
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cannot use this path there is a possibility
that some input here is connected some other
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output here.
And both of them are occupied you can always
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set up a connection between them only when
both input and output both links are available
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so essentially we have to find out when the
links will be available so when this a path
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will be busy either then when both of them
are busy of one of them is going to be busy
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or one man is that both of them are available
so the both of them are going to be available
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so this is the occupancy probability of a
link this link is not being occupied that
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probability is 1 – p.
This link is also not occupied is 1 – p
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so I will say 1- p2 is the probability that
this path is not occupied and can be used.
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Okay, so one this should means the part is
going to be occupied this essentially can
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happen because this is busy or both of this
links are busy so all three are being now
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taken here so remember this is not nothing
but equal to that both of them are busy which
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is going to be p2 +this is busy and this is
not busy and this is busy this is not busy
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which will be 2 x t x 1- p so remember this
both are same so I have just computed in the
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other way around.
So this is the probability that
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this part is busy now if all the possible
part of all of them all k of them are occupied
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then you cannot set up the communication between
this and this free input and free output port.
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So that is what is going to be give you a
probability of blocking remember this is the
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switch being in block testate this is a time
congestion this is not a call congestion this
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is the probability has which is in the blocking
state okay.
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So this is what is known as lees approximation
and of course I can replace p/ this so this
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will be come 1-, now the problem is if I actually
put k = 2m-1 this would become 0 but that
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does not happen you actually you can compute
this
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or any value which is higher than this
you take any value of m compute take some
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value of a which is the probability can take
value from 0 to 1 this is not going to be
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0 in this case.
So this is start deviating actually this start
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deviating for larger numbers but is a good
approximation if my case and if we are doing
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randomly the things than that will going to
be a good approximation okay but this is still
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in approximation is not by exact value of
probability of blocking and I think the mistake
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is that the probability that the call blocking
will happen that depends on the switches state
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we have not taken that so I think we have
to modify this approach so modified approach
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was given by Carlo
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So there is a Carlo approach which will give
you more exact blocking probability but this
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blocking probability will be call loss probability
so you will actually represent this switch
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by some state s and then there is going to
be caller arrival probability which is because
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of s okay.
So infect so arrival probability which cause
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s probability that you are switches in s state
and the probability that call will get lost
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when you in state s and you submit up over
all possible states and this is the all the
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cause which will be arriving this will give
you PL the corlos loss probability and interestingly
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this call loss probability and corresponding
the switch being in probability of switch
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being in blocked state ,time conjunction they
are also related by the same relation which
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we have derived earlier .
So next lecture we will be looking in to the
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Carlos approach carols approximation for finding
out the call loss probability.