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Queuing Theory Tutorial - Queues/Lines, Characteristics, Kendall Notation, M/M/1 Queues
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[Music]
hello my name is Blanca Ferguson Yaya
and welcome to our introductory video on
the queueing theory so let's start the
basic question is what is a queue a queue
queue
whence of people vehicle or any other
things awaiting their turn to be
attended to or to proceed I'm pretty
sure at least once in your day you are
always in a queue or you are in a queue
so what are very examples for these for
for a queue one good example for this
one would be a queue or a line awaiting
for your turn to the toilet or lining up
to pay in the supermarket and or lining
up to check your balance accounts on
automatic teller machines but then
queues are not only present for people
it can also be for example for airplanes
which are waiting for their turn in
order for them to fly in the airports
then we have jobs or products which are
waiting and assembly line and then the
most annoying of all would be traffic
within line and traffic so what is
queueing Theory queueing theory
determines the measures of performance
of waiting lines which can be used to
design services it is not an
optimization technique thus because
waiting cannot be eliminated completely
without incurring inordinate expenses
the goal of giving theory is to reduce
its adverse impact tolerable levels so
queueing theory would not give you
solution but would give you the
measurements or the analysis on your
queue or on your different lines and the
goal is for us or for the user to reduce
its adverse impact tolerable levels so
what are the characteristics of a queue
there are six characteristics and we
will go through each one of them as we
go through the
different examples one would be input or
the arrival or inter-arrival
distribution then we have output or the
departure service distribution we have
service channels we have service
discipline the maximum number of
customers allowed in the system and
lastly the calling source again we will
go through each one of them as we go
with an example so let's try for example
you are paying in a grocery store you
have a server and you have the different
customers the grocery or the place where
you bought your food or your grocery
would be your source of the line order
the source of the customer so the
service time would be the amount of time
that the cashier or the person in the
point-of-sale system would take care of
your belongings or to service you
basically and it is usually under
negative exponential distribution then
the inter arrival time between the
different customers for example 2
minutes 4 minutes for the next one and
then five minutes for the next one is
called the inter arrival time and
usually it follows the Poisson
distribution now let's try putting in
two servers and one line the two servers
or the two point-of-sale system would be
your service facility then the one line
is called the Q or the waiting line the
people in the queueing or the waiting
line are called the arriving customers
and the one who just finished being
serviced by the POS or by the
point-of-sale system or the servers or
the departing customers the Q plus the
wait in line the Q and waiting line plus
the service facility is called the
system or the entire system Q now let's
try zooming in only to one line the
queue size is the number of people that
one queue can
it can either be finite or infinite then
these customers would have a different
discipline or the queue would have
different discipline on how these people
arrived it would either be first come
first served last come last served or
last come first served service in or in
random order and or priority there are
three common reactions are aptitude for
each of the customer or for each of the
member of the queue it can either be
jockeying bulking or reneging so let's
try to see each one of them jockeying is
when the customer enters one line and
then switches to a different one in an
effort to reduce the waiting line so
let's check the guy in green shirt for
example he comes in and then he decides
to go to queue number one then he saw
the queue number three is smaller or has
a fewer number of people so he goes
there in an effort to reduce the waiting
time that is called jogging
the next one is called balking where the
customer decides not to enter the
waiting line let's look at again the guy
in green shirt he saw that all the lines
are actually pretty long and he doesn't
want to wait in the line thus he balked
and then he decides not to enter the
waiting line and then lastly reneging
reneging is when the customer enters the
line but decides to leave before being
served so let's look at the mom and the
little boy they saw they saw that line
number two would be the fastest one
among the other three point-of-sale
services and then they decided to leave
even before they were being served so
the attitude or the cap the reaction is
called renaming so how do we put these
characteristics of a queue in a
mathematical form here we call it the
candle notation the candle notation is
composed of six different
characteristics which we have discussed
earlier let's try to see each one again
one by one letter aim would be the input
or the arrival or the inter-arrival
distribution it could either be M or
plus an arrival and or negative
exponential service distribution which
is the most common common among all of
them M stands for markov actually d is
deterministic inter arrival or service
time distribution meaning we know
deterministically and then ER e sub k
which is called Erlang gain or the gamma
inter arrival or service time
distribution GI would be general
independent distribution and G would be
general distribution so it could either
be on one of the five the letter B is
the output or the departure or the
service distribution again it can also
be either one of the five usually we use
M or the Markovian or the Poisson
arrival or here since its service the
negative exponential service
distribution letter C is the service
channels or the number of servers and
then that idea would be the service
disappear so it's either FCFS a
first-come first-serve i'll CFS last
come first served as IR all or the
service in random order and lastly a
general service discipline or the G D
and then we have laughter e which is the
maximum number of customers allowed in
the system so it can either be finite or
infinite and then lastly the calling
source which is letter F and again it
could either be finite or infinite in
our example earlier it would be the
grocery store which can be infinite at
some point so what are the different
symbols that are usually used in
queueing theory first we have the small
letter lambda which is the mean arrival
rate then the small letter mu which is
the mean service rate for busy server
and then the small letter Rho
which is equal to the division between
lambda and mu which is our utilization
factor or the traffic intensity purposes
server and then we also have letter n
which is the number of units in the
system again the system includes vq+ the
service facility and then the sub n of T
is the probability of exactly and
customers in the system at time T then P
sub n is the probability of exactly and
customers in the system and C would be
the number of parallel servers then
there are four more other symbols that
we use WS W sub s would be the expected
waiting time per customer in the system
and WQ is the expected waiting time per
customer in the queue so WS and WQ are
waiting time so it could either be in
seconds or in minutes or in hours then
we have L sub s which is expected number
of customers in the system and al sub Q
which is expected number of customers in
the queue so here R sub s and R sub Q
would be the number of customers so it
would be integer values so the easiest
one is to do an example so here we have
one example a gas station has one pump
which can serve six customers per hour
cars arrive at the station at the rate
of 10 per hour which is exponentially
distributed so according to our candle
notation it will look like this our
input or our arrival distribution will
be Markovian or as it says it would be
possible distributed our output is
exponentially distributed so it's also
Markovian so M let us see since we only
have one pump then our service channel
will be 1 then our service discipline is
CFS meaning first-come first-serve our
calling source is the entire population
so that would be infinite and then the
maximum number of customers allowed in
the system would also be infinite so
let's try looking into or solving for
the mean arrival rate and the mean
service rate per busy server so seven
the problem there would be ten cars per
60 minute so cars arrive at the station
at the rate of ten per hour so that
would be ten cars per 60 minutes or ten
cars for one hour and that would mean
one car or every car arrives at an
average of every six minutes so our mean
arrival rate is one one car for six
minutes or six minutes per car then we
have the mean service rate per busy
server so as stated here a gas station
has one pump which then serves six
customers per hour so that would mean it
would be six cars for sixty minutes or
six cars per hour and that means one car
would be served for ten minutes so our
lambda or our arrival rate is six
minutes and our service rate is ten
minutes so now we can solve for our
utilization factor which is equal to
sixty percent then we have to solve for
each one of our parameter so WS WQ LS
and L Q so for each type of problem
these computations would actually differ
for each of the candle notation so since
our candle notation is an M one we will
be following mm1 calculation or mm1
formula I will show you like later the
different formulas for different candle
notations of different problems so you
have to choose among the different set
of the formulas which Falls your candle
notation and that would be the
appropriate equations that you will use
in order for you to solve the W as
WQ l SL q and all the other necessary
parameters so here let's try to solve
with the mm one candle notation so WS
would be one divided by mu minus lambda
which s we have our values over here at
the bottom right so that's 1 over 10
which is our mu minus 6 what that would
mean 1 over 4 or 0.25 hours that means
that 15 minutes and then so the expected
waiting time per customer in the system
is 15 minutes then we have expected
waiting time per customer in the queue
which follows the formula lambda divided
by mu times mu minus lambda which then
provides us with 0.15 hours then here is
our competition for our expected number
of customers in the system and in the
queue so LS would be the lambda divided
by mu minus lambda which means the
expected number of customers in the
system is 2 cars and the expected number
of customers in the queue is 1 car again
these formulas would only be used for
mmm one type of candle notation so here
are the different queueing formulas for
the different models or four different
candle notation these are the formula
for m1 or Markovian Markovian toinsanan
distribution poisson distribution and
negative exponential distribution for
both me effort both arrival and service
die and then only one server this one is
for md one meaning you have constant
service and then this would be the multi
channel or the multiple number of
servers so these are the formula for SWS
alkyl and WQ and then lastly the formula
for limited population meaning you
or population or your calling source is
finite or not infinite basically so we
have these formula over here so if you
want to try out one problem this is a
multi-channel server type of problem and
you can screenshot the question and then
send in the answer for my students this
is your homework problem
so that was Kimmy theory that was only
an introduction to queueing theory thank
you very much for listening and have a
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