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Waiting Lines and Queueing Theory Models-2 | Models with Solved Example with QM for Windows - YouTube
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problem from the book 14-12 and it says that
from historical data Harry's car wash estimated
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that dirty cars arrive @ 10/hr so that means
with the crew working the wash line Harry
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figures that cars can be cleaned @ 1 every
5 minutes the thing is lambda if we are saying
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that this is in terms of per hr so we have
to convert service times also in terms of
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per hr so if you have 1 every 5 minutes so
obviously in 1 hr they can clean 12 assuming
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Poisson arrivals and ex[exponential service
times basically it says that this is model-1
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and we need to find various things like avg
number of cars in the line avg time a car
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waits before it is washed and so on so we
will go to QM for Windows to do all those
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calculations and answer the question let me
open that I'll just say ok and automatically
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it lists under module all the different tools
which are available and which are related
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to this book and we are interested in waiting
lines so we will click on waiting lines then
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we can say new and it has different options
1, 2, 3, 4 because we are dealing with single-channel
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situation I'll say we use 1 so it opens up
this window here and you have like no-cost
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situation and use-cost situation in this example
we don't have any cost so we are fine with
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just going and clicking ok arrival rate we
can enter here 10 and service rate mu 12 so
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please note that we have converted mu to 12
book says every 5 minutes 1 so that we have
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to convert to per hr before we can use this
and then you have number of servers obviously
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that is 1 we cannot change that and as soon
as we do this it has done all the calculations
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so it tells us avg service utilization is
0.83 83% that's very high so server utilization
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means how often server is going to be busy
so 0.83 means 83% of the time the server is
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busy only 17% of the time it is idle then
you have avg number in the queue 4.17 so if
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you go back and look at the questions the
1st question was avg number of cars in line
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so basically that is avg number in the queue
so the answer is 4.17 part-b was avg time
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a car waits before it is washed avg time in
queue here is 0.42 and please note that this
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is in hrs so in minutes it is 25 minutes so
they have to wait about 25 minutes before
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being washed avg time this is c avg time a
car spends in the service system so this one
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avg time in the queue is different from avg
time in the system avg time in the system
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means time spent in queue plus time spent
getting the service like getting the car washed
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so obviously avg time in the system which
includes avg time in the queue so this is
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a bigger number so you can see that about
30 minutes so it is always going to be a bigger
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number part-d is utilization rate of the car
wash so 83% we have seen that and then probability
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that no car are in the system so you can also
get an idea from server utilization so 1-0.83
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you get 17% so if you read the question you
will be able to find that lambda in this case
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is 12/hr mu is 15/hr and the difference now
is that we also have cost figures so service
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cost $10 per hr waiting cost is $50 now we
can use this data and go to QM for Windows
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once again I can click module and then waiting
lines I don't want to save it let me open
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1st one so now we can choose use-cost and
say ok and you will see that there are more
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options now so lambda is 12 mu is 15 and then
you have number of servers we don't want to
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change that because that is 1 so now we can
solve this problem so part-a is what is the
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avg time that catalog customers must wait
before their calls are transferred to order
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clerk so the avg waiting time in this case
we are talking about waiting time in the queue
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which is 0.27 or 16 minutes part-b is what
is the avg number of callers waiting to place
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an order so that is basically avg number in
the queue which is about 3.2 similarly you
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can also modify the information to answer
part-c which is Ashley is considering adding
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2nd clerk to take calls the store would pay
that person the same $10 per hr let's look
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at 2nd model because this model deals with
only 1 server so let me write down model-2
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here and then we can answer part-c so model-2
is M/M/m so we are dealing with m=2 and we
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want to find out what will be savings if we
can go for 2 so let's go back and see what
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was the total cost with one server now if
you look at here cost labor + number waiting
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cost which is 170 so we can make a small table
so total cost per hr and this is with M/M/1
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model and then we compare with M/M/2 model
so with M/M/1 we have $170 now let's go back
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and see what will be the cost when we have
2 instead of 1 so I'm going to edit data see
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if we can do it this will not allow because
we used a single server so we will close this
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and open a new file and select multi-channel
system we are also using cost so select cost
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so once again lambda is 12 mu is 15 number
of servers now we are saying 2 server cost
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was 10 and waiting cost was 15 so if we click
on solve it will solve and give this answer
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so the number we want to look at is this always
when we are comparing we should look at cost
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labor plus waiting time and waiting cost 27.62
is the cost so if you subtract we get the
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savings equals so this way we can compare
and see what happens if we add one more server
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now let's look at model-3 so we are looking
in this chapter at 4 models and this is the
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3rd one so this model is M/D/1 now when we
use D it means we are talking about constant
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service times like coffee machine so every
time it is going to take that much time for
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one cup of coffee and 1 once again means single
sever so doe this let's look at problem 27
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so 27 says customers arrive at an automated
coffee wending machine so that example indicates
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constant service time @ 4 per minute so that's
your lambda and it also says following a Poisson
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distribution so we are using a situation where
arrivals are modeled by Poisson distribution
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but please remember arrivals could be something
else also so most of the time Poisson will
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work but in many cases it could be something
different so we should use the model only
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when it is confirmed that arrivals are Poisson
coffee machine dispenses a cup of coffee in
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exactly 10 seconds and then basically different
performance criteria like what is the avg
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number of people waiting in line what is the
avg number in the system how long does avg
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person wait in line before receiving service
so obviously all these questions can be answered
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very easily let me write down 14-27 lambda
is 4/minute and mu is 6 per minute so why
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6 because it says coffee machine dispenses
coffee every 10 seconds so in 1 minute you
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will have 6 cups of coffee so we have to keep
the units same so if lambda is in minutes
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mu should also be in minutes now let's open
QM for Windows so I'll close this and we will
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not save and then file and new so if you look
at the 2nd one that's constant service times
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so that's the one we want in this problem
we have no cost so we can say ok and then
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lambda is 4 and service is 6 so we can click
solve now when we are using QM for Windows
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one thing you will note is that you have a
column called value and then there is a column
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called minutes then there is a column called
seconds so this will work very good if your
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original units are in hrs so it will convert
hrs into minutes and seconds but in the problem
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27 actually our units are already in minutes
so do not go by these two columns because
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this will be misleading so this 10 here doesn't
mean 10 minutes or 600 doesn't mean 600 seconds
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so you have to be careful this will work only
when original units are in hrs so in this
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case it is minutes so you should ignore those
two so from this you can see that again avg
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number in the queue is 0.67 avg server utilization
is 67% means that amount of time it is busy
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and remaining time it is basically idle or
not in use and avg number in the system means
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queue plus the one in service so there could
be some people in the queue and then add the
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one who is in the service so that's why always
this is the bigger number and same thing for
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time so avg time in queue so this is only
queue and in the system means queue plus service
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so those people who are getting serviced that
is included there so this number will be higher
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that is model-3 now the last model is model-4
finite source so finite source is that limited
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calling population so number of items or people
are coming from a population which is limited
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for this let's look at 14-29 so this is a
manufacturing example so 1 machine services
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5 drilling machines for steel plate manufacturer
machines breakdown on an avg of once every
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6 working days and breakdowns tends to follow
Poisson distribution so which is M mechanic
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can handle an avg of 1 repair job per day
so that is the service rate repairs follow
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exponential distribution so that is again
M in the model so how many machines are waiting
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for service on an avg how many are in the
system on an avg so standard questions of
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waiting lines basically to assess its performance
so let's look at lambda and mu and note that
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there are only 5 machines so this is finite
population and they can service 1 per day
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so with this we can go to QM for Windows let
me close this do not save and open new so
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now I'm going to use the 4th one which is
limited population model and again no cost
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in this example avg rate per customer so 1/6
is 0.17 mu is 1 and population size is 5 so
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with this we can click on solve so using this
we can easily answer the questions like how
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many machines are waiting for service on an
avg so this is avg number in the queue so
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0.54 part-b is how many are in the system
so in the system avg number in the system
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is 1.19 so more than one machine in the system
at any point of time what is the avg waiting
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time in queue so avg waiting time in queue
you can see is 0.83 and this is days so 0.83
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days and again this 50.1 and this other number
doesn't matter because this in days so obviously
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from days this calculation in minutes and
seconds is not correct because these calculations
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are based on original units in hrs so we should
ignore minutes and seconds column part-e is
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what is the avg wait in the system so you
can see that avg wait in the system is 1.83
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days the situations which are given in these
4 models may not always be true like you could
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have a situation which is very different from
what is given in the models and it can make
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things very very complex when you have a very
complex situation which is difficult to model
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using waiting lines so one way to deal with
is with that situation is to go for what is
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called as simulation so simulate the conditions
and then see the performance so simulation
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is the topic of our next chapter
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