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Introduction to Queuing Theory - Industrial Engineering & Operational Research | GATE Mechanical - YouTube
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Before, I start discussing anything about
this unit, have a look at this very interesting
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and a funny video.
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So, you might think that why we are looking
at this video.
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And how this video is related to what we are
about to study.
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Look, the name of this unit is queuing theory
or at some places, you will also find the
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name to be queuing models.
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So, the basic term basics, the basic phrase
in this whole name is a Queue.
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Generally, in India we call it as a line.
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Whenever you go to a counter or anywhere where
you have to make a line for the service, maybe
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when you鈥檙e booking a ticket, maybe when
you鈥檙e ordering some food.
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So, we call it a line.
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But, line is not the proper term for it.
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The proper term for it is Queue.
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So, in the video you saw that there were two
different queues and there was a person who
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was confused between which of the queue to
join.
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Both of these queues were being served by
2 person.
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You might not be able to see those two person
in the video.
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But, in the end 2 person exited from the counter.
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So, those two were serving the other people
in the queue.
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There was this one person who was confused
whether to join this or to join the other.
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So, what were the common things present in
such a model, in such a system?
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There were more than one queues.
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There were some people who were serving the
people in the queue.
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The people in the queue were served by someone.
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The speed with which they were serving was
changing.
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Sometimes one queue was moving faster.
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Sometimes the other queue was moving faster
and a person was constantly shifting from
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1 queue to the other.
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These were some of the things that we observe
from that funny video clip.
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How is this going to be relevant for us?
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In this unit, what we are going to study is
the mathematical behavior of queues.
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How queues behave mathematically, as a mathematical
entity?
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You have to look at queue, not just as a collection
of people, as a bunch of people standing one
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after the other.
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But, look it, the queue from mathematical
point of you.
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Consider it like a mathematical entity whose
length is increasing and decreasing and the
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rate at which it is increasing is different
the rate at which it is decreasing is also
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different.
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And, when there are more than one queues,
then you can also compare this mathematical
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entities with each other.
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Their length you can compare.
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The time in which they are getting finished,
the time in which these queues are getting
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0 that you can compare.
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The speed with which they are getting increase
or decrease that you can compare.
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So, basically for this unit you need to look
at these queues from mathematical of view
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as if they are some mathematical entity.
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Their mathematical behavior we need to understand.
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And, not only their mathematical behavior,
but as a result of that what other parameters
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come into action, whenever someone is standing
in a queue, whenever someone is entering into
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this mathematical model you can say then what
other factors come into picture, those we
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will also consider here.
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And, don't think that this particular queuing
theory or queuing model unit is irrelevant
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from the practical point of view.
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You may think that why as an engineer you
are supposed to study about queues.
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You are not going to be in such a place where
you have to count the number of people in
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the queue and things like that.
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You might not able to relate it, but in queues
it is not necessary that only people will
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be there.
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In queues, there can be products as well.
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Some products waiting to go to a machining
center to getting service, to getting machines.
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Consider this thing from industrial point
of view where there is a big industry, there
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are different machines, different assembly
lines in which products are moving, just like
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people are moving.
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And just like people were getting served.
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Those products will also get served at different
machining center at different machining heads.
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And then even after finalizing, when the product
is ready then again in packaging and distributing
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everywhere, this thing is coming into the
picture.
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So, I hope you got an idea of how this queuing
theory or queuing model unit is going to be
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very relevant in different industries, when
mass production is going on and different
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products are moving in different lines either
for machining, for the manufacturing or for
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packaging or ordering or for delivering.
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So, this unit is again important from practical
point of view.
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From the point of view of GATE also, it's
pretty much decent.
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It鈥檚 not very important or it is not very
less important.
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It is moderately important.
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Again, I am telling you that the units in
general what we are covering in industrial
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engineering and operational research, they
are going to be very scoring and there is
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not going to be highly conceptual or innovative
question because the syllabus is defined and
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what is expected from you as an undergraduate,
that is also defined.
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So, again this unit is very simple, very easy
and very scoring.
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So, let's start the discussion about queues
and the very first thing that you need to
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understand in queues is that whatever queuing
system you have, wherever a queue is being
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made a queue is served, there are three basic
behaviors.
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First behavior is arrival in a queue.
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You have a queue, so people arrive in a queue.
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This is the basic behavior.
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The Second behavior is that after arriving
in the queue and after waiting for their turn,
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they are getting serviced.
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There is a counter.
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Obviously, there is a counter for which the
queue is there, maybe that counter having
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a person or may not be having a person.
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Maybe it鈥檚 an ATM machine.
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So, that is also a server.
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That is also a place where people are getting
service.
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They make a queue and one by one they are
getting serviced.
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So, arrival then service happens at the station,
at the server and then they depart from the
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queue.
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They leave from the queue.
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So, arrival, service and departure, these
are the 3 basic elements of any queue.
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And, there is some mathematical pattern into
this that we obviously have to study.
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Now, you may think that how can we study such
a random thing.
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Arriving in a queue and you can say that servicing
maybe have some mathematical connection.
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How much time it is taking and how fast it
can service depending upon the people out
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on the machine.
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But, you may think that arrival, how is it
even a mathematical thing?
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Anyone can arrive anytime, but it is not like
that.
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Recall, what I have told you in the last unit.
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Even random data like height, mass, blood
pressure, such a random thing that also follows
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a normal distribution.
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So, you may think that something as a laymen,
may look very random to you.
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But, when you observe it mathematically, there
will be some pattern there.
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There will be some distribution, normal distribution
and beta distribution, different types of
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distribution are there.
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So, there will be some mathematical pattern
that you can figure out.
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And on the basis of that you can study that
behavior.
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So, let's start this unit.
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As, I have mentioned before, the place where
the queue is getting serviced is called as
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a server, where maybe people maybe standing
in the queue to withdraw money from the ATM.
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So, ATM is this server.
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People standing in a queue to order their
food, so, the person standing at the food
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counter doing the billing that is the part
of the server.
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Maybe, they might be going to some garage
to get their cars serviced, to get their cars
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washed.
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So, that place where the car is getting washed
and serviced that is the server.
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So, basically the place where ultimately the
queue, the member of the queue are getting
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the service for which they were waiting is
called as a server.
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The server can be a single one or there can
be multiple servers also.
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For example, in the video that we saw, there
were two servers but they can be more than
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two servers as well.
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This is the example of Server 1, Server 2
and Server 3.
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This is the top fuel, let鈥檚 say.
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And these are the people who are standing
in a queue.
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You might have seen different queues, more
than one queues in banks or in railway stations
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or in airports.
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So, this is the example of multi-server or
multiple server queue.
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There are more than one server.
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Similarly, there can be a single server queue
as well.
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For example, if there is a place where a single
ATM machine is there.
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So, only one queue can be formed there.
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So, that's only one server is there basically
and in India you can form three queue for
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a single server which is not an ideal situation.
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But, generally if you have a single server,
so a single server queue will be formed.
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So, basically when you have a single server,
so for that server there will be a queue.
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There will be people in the queue who are
waiting for the service.
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And they approach the server and they get
served and after getting the service they
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get happy and then they leave.
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So, this is a single server queue system.
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This is a multiple server queue system.
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Now, whoever is joining the queue, there can
be some situations where there are definite
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number of people who can join the queue.
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In some cases there can be such situations
where the number of people or the number of
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members of the queue can be limited or they
can be unlimited that is called as queue length.
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That how many people or how many members,
in flow I may called it as people but it is
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not necessary that only people are joining
the queue.
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They can be, as I told you, objects also,
machine also, they can be any living or non-living
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thing in the queue.
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So, what I was telling you is that in different
systems or for a different queuing models,
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there can be some situations where only limited
number of objects or people or member can
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join the queue.
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Or there can be a situation where theoretically
infinite people can join the queue.
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For example, if you have a station, garage
station where cars are coming.
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They are getting washed.
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They are getting serviced and then they leave.
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And in the garage, there are only 10 slots
to park the car.
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The owners always brings their car and there
are only 10 slots there.
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So, what will happen?
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The length of the queue has been restricted.
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Only 10 cars can be there in the system to
get served in a queue one after the other.
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If the 11th car comes, when all the slots
are filled.
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What will happen?
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That car either it cannot join the system
or it will have to leave the system.
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Don't think that it will stand just outside
of the slot and you will call this also as
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a queue.
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We are again talking about a very organized
system.
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Don't consider as any garage next-door in
our country, where you can form any number
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of queues and you can break the queue and
you can just make the chaos of a queue.
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Don鈥檛 consider those situation.
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Consider it to be an organized situation,
where if there are 10 slots, so only 10 cars
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can be parked.
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If 11th car comes either it will stay out
of the system or it will have to leave the
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system.
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So, in such a case, the queue length is finite,
only 10 members, 10 cars, 10 people can join
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the queue.
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So, such a system is called as a finite queue
system.
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But, there can be other systems where the
queue can be theoretically infinite.
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For example, in any religious place, in any
temple let鈥檚 say, in any bank let鈥檚 say,
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for any ATM machine.
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Theoretically, it can be infinite.
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Any number of people can join the queue.
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There is no restriction of how many people
can join it.
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The queues can extend out of the boundaries
of that particular bank or that particular
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temple as well.
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The queues are sometimes extended way long,
I have seen them at some places.
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So, practically the length of those queues
are not finite.
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There, not limited people who can join the
queue and any number of people can join that
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queue.
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So, such queue system is called as infinite
queue system.
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In generally, the type of system that we will
study, generally is going to be infinite queue
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system where there is no limit on how much
the length of queue is?
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The next thing about queue is who can join
the queue?
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If you have a population of a lot of people,
then who among those can join the queue?
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Again, I am telling you, don't think that
I am only talking about people.
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When I am saying that we have a lot of population,
it does not mean that I am only talking about
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people.
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I can also talk about the machine or any object
or any non-living thing here.
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So, depending upon who can join the cube or
what members of the population of any living
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or non-living population can join the queue.
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You can further have 2 categorization and
that is - infinite population model and finite
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population model.
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Look, in infinite population model, basically
anyone can join the queue.
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Generally, the queue at doctors.
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The queue at reservation center, and at booking
counters, at food stalls, anyone and everyone
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is free to join the queue.
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That's an example of infinite population model.
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Again, don't confuse between these two.
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This tells the number of people who can join
the queue.
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But, this tells the type of people who can
join the queue.
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In infinite population model, any type of
person can join the queue.
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But, if talk about finite population model,
then there are only certain people who can
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join the queue.
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For example, if you have been to the airport.
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You will see that there is a different line
you can say for the business class people
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or for any privileged person who is some member
of that particular airline.
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Similarly, there can be some queue dedicated
for women or if I leave aside the people and
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I come to the example of non-living thing.
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Then, suppose you have a factory and in that
factory there is a maintenance office where
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whatever parts of a machine (in that factory)
are not working properly, so they can come
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at the maintenance office and get it repaired.
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And in that factory, let's say there are 30
machines.
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So, for that server, for that maintenance
server, the population is finite.
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It is not going to service any random machine
coming from anywhere.
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Even within the factory also, you cannot call
the maintenance office to fix the fan or to
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fix the light.
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There is a different definite type of machine
that they are going to service, that is going
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to be served.
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And those machines are present within the
factory, not any machine can randomly come
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forward for the service.
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So, that is an example of finite population
model.
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So, there were three categorization which
we discussed here.
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The first was the categorization of server-
single server or multiple server.
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The second was the categorization on the basis
of number of members in the queue whether
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that number is fixed or finite or it can theoretically
be infinite.
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And third was the type of member in the queue,
whether the type of member in the queue can
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be any given member that belongs to infinite
population or they were a definite set of
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people or definite set of objects that can
join the queue.
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These were the three categorizations.
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Theoretical questions can be asked from this.
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The general type of discussion or the general
type of queue in system that we will study,
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will be having infinite queue system or infinite
queue length system.
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So, Infinite queue length system with infinite
population model.
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So, these are the general types of systems
that we study in our discussion of this unit.
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