0:01 [Music]
0:05 hello my name is Blanca Ferguson Yaya
0:07 and welcome to our introductory video on
0:11 the queueing theory so let's start the
0:13 basic question is what is a queue a queue
0:15 queue
0:19 whence of people vehicle or any other
0:21 things awaiting their turn to be
0:24 attended to or to proceed I'm pretty
0:27 sure at least once in your day you are
0:30 always in a queue or you are in a queue
0:32 so what are very examples for these for
0:36 for a queue one good example for this
0:38 one would be a queue or a line awaiting
0:43 for your turn to the toilet or lining up
0:46 to pay in the supermarket and or lining
0:50 up to check your balance accounts on
0:53 automatic teller machines but then
0:56 queues are not only present for people
0:59 it can also be for example for airplanes
1:02 which are waiting for their turn in
1:05 order for them to fly in the airports
1:08 then we have jobs or products which are
1:11 waiting and assembly line and then the
1:13 most annoying of all would be traffic
1:18 within line and traffic so what is
1:20 queueing Theory queueing theory
1:22 determines the measures of performance
1:26 of waiting lines which can be used to
1:29 design services it is not an
1:33 optimization technique thus because
1:35 waiting cannot be eliminated completely
1:38 without incurring inordinate expenses
1:41 the goal of giving theory is to reduce
1:45 its adverse impact tolerable levels so
1:47 queueing theory would not give you
1:50 solution but would give you the
1:54 measurements or the analysis on your
1:57 queue or on your different lines and the
2:01 goal is for us or for the user to reduce
2:05 its adverse impact tolerable levels so
2:06 what are the characteristics of a queue
2:09 there are six characteristics and we
2:10 will go through each one of them as we
2:11 go through the
2:14 different examples one would be input or
2:15 the arrival or inter-arrival
2:18 distribution then we have output or the
2:22 departure service distribution we have
2:24 service channels we have service
2:27 discipline the maximum number of
2:29 customers allowed in the system and
2:32 lastly the calling source again we will
2:34 go through each one of them as we go
2:37 with an example so let's try for example
2:41 you are paying in a grocery store you
2:45 have a server and you have the different
2:50 customers the grocery or the place where
2:53 you bought your food or your grocery
2:56 would be your source of the line order
3:01 the source of the customer so the
3:03 service time would be the amount of time
3:06 that the cashier or the person in the
3:08 point-of-sale system would take care of
3:10 your belongings or to service you
3:14 basically and it is usually under
3:18 negative exponential distribution then
3:20 the inter arrival time between the
3:23 different customers for example 2
3:25 minutes 4 minutes for the next one and
3:27 then five minutes for the next one is
3:29 called the inter arrival time and
3:32 usually it follows the Poisson
3:36 distribution now let's try putting in
3:41 two servers and one line the two servers
3:45 or the two point-of-sale system would be
3:48 your service facility then the one line
3:52 is called the Q or the waiting line the
3:54 people in the queueing or the waiting
3:56 line are called the arriving customers
3:58 and the one who just finished being
4:00 serviced by the POS or by the
4:02 point-of-sale system or the servers or
4:06 the departing customers the Q plus the
4:09 wait in line the Q and waiting line plus
4:11 the service facility is called the
4:17 system or the entire system Q now let's
4:21 try zooming in only to one line the
4:23 queue size is the number of people that
4:25 one queue can
4:29 it can either be finite or infinite then
4:32 these customers would have a different
4:33 discipline or the queue would have
4:36 different discipline on how these people
4:38 arrived it would either be first come
4:42 first served last come last served or
4:45 last come first served service in or in
4:50 random order and or priority there are
4:53 three common reactions are aptitude for
4:55 each of the customer or for each of the
4:57 member of the queue it can either be
5:00 jockeying bulking or reneging so let's
5:04 try to see each one of them jockeying is
5:06 when the customer enters one line and
5:09 then switches to a different one in an
5:11 effort to reduce the waiting line so
5:14 let's check the guy in green shirt for
5:17 example he comes in and then he decides
5:20 to go to queue number one then he saw
5:22 the queue number three is smaller or has
5:25 a fewer number of people so he goes
5:28 there in an effort to reduce the waiting
5:30 time that is called jogging
5:34 the next one is called balking where the
5:36 customer decides not to enter the
5:38 waiting line let's look at again the guy
5:42 in green shirt he saw that all the lines
5:44 are actually pretty long and he doesn't
5:47 want to wait in the line thus he balked
5:50 and then he decides not to enter the
5:53 waiting line and then lastly reneging
5:56 reneging is when the customer enters the
5:59 line but decides to leave before being
6:03 served so let's look at the mom and the
6:07 little boy they saw they saw that line
6:10 number two would be the fastest one
6:12 among the other three point-of-sale
6:16 services and then they decided to leave
6:19 even before they were being served so
6:22 the attitude or the cap the reaction is
6:25 called renaming so how do we put these
6:28 characteristics of a queue in a
6:31 mathematical form here we call it the
6:34 candle notation the candle notation is
6:35 composed of six different
6:37 characteristics which we have discussed
6:40 earlier let's try to see each one again
6:44 one by one letter aim would be the input
6:46 or the arrival or the inter-arrival
6:49 distribution it could either be M or
6:52 plus an arrival and or negative
6:55 exponential service distribution which
6:57 is the most common common among all of
7:01 them M stands for markov actually d is
7:04 deterministic inter arrival or service
7:08 time distribution meaning we know
7:12 deterministically and then ER e sub k
7:14 which is called Erlang gain or the gamma
7:16 inter arrival or service time
7:19 distribution GI would be general
7:21 independent distribution and G would be
7:23 general distribution so it could either
7:27 be on one of the five the letter B is
7:29 the output or the departure or the
7:32 service distribution again it can also
7:35 be either one of the five usually we use
7:38 M or the Markovian or the Poisson
7:40 arrival or here since its service the
7:42 negative exponential service
7:44 distribution letter C is the service
7:50 channels or the number of servers and
7:52 then that idea would be the service
7:55 disappear so it's either FCFS a
7:58 first-come first-serve i'll CFS last
8:00 come first served as IR all or the
8:03 service in random order and lastly a
8:05 general service discipline or the G D
8:08 and then we have laughter e which is the
8:10 maximum number of customers allowed in
8:13 the system so it can either be finite or
8:16 infinite and then lastly the calling
8:19 source which is letter F and again it
8:21 could either be finite or infinite in
8:23 our example earlier it would be the
8:26 grocery store which can be infinite at
8:29 some point so what are the different
8:30 symbols that are usually used in
8:34 queueing theory first we have the small
8:36 letter lambda which is the mean arrival
8:41 rate then the small letter mu which is
8:44 the mean service rate for busy server
8:47 and then the small letter Rho
8:51 which is equal to the division between
8:55 lambda and mu which is our utilization
8:58 factor or the traffic intensity purposes
9:01 server and then we also have letter n
9:03 which is the number of units in the
9:07 system again the system includes vq+ the
9:11 service facility and then the sub n of T
9:14 is the probability of exactly and
9:18 customers in the system at time T then P
9:21 sub n is the probability of exactly and
9:24 customers in the system and C would be
9:29 the number of parallel servers then
9:31 there are four more other symbols that
9:36 we use WS W sub s would be the expected
9:38 waiting time per customer in the system
9:42 and WQ is the expected waiting time per
9:46 customer in the queue so WS and WQ are
9:48 waiting time so it could either be in
9:51 seconds or in minutes or in hours then
9:54 we have L sub s which is expected number
9:57 of customers in the system and al sub Q
10:00 which is expected number of customers in
10:03 the queue so here R sub s and R sub Q
10:05 would be the number of customers so it
10:10 would be integer values so the easiest
10:14 one is to do an example so here we have
10:17 one example a gas station has one pump
10:21 which can serve six customers per hour
10:24 cars arrive at the station at the rate
10:27 of 10 per hour which is exponentially
10:32 distributed so according to our candle
10:35 notation it will look like this our
10:37 input or our arrival distribution will
10:39 be Markovian or as it says it would be
10:42 possible distributed our output is
10:44 exponentially distributed so it's also
10:47 Markovian so M let us see since we only
10:50 have one pump then our service channel
10:54 will be 1 then our service discipline is
10:57 CFS meaning first-come first-serve our
11:00 calling source is the entire population
11:05 so that would be infinite and then the
11:07 maximum number of customers allowed in
11:11 the system would also be infinite so
11:14 let's try looking into or solving for
11:15 the mean arrival rate and the mean
11:21 service rate per busy server so seven
11:25 the problem there would be ten cars per
11:28 60 minute so cars arrive at the station
11:30 at the rate of ten per hour so that
11:33 would be ten cars per 60 minutes or ten
11:35 cars for one hour and that would mean
11:39 one car or every car arrives at an
11:43 average of every six minutes so our mean
11:46 arrival rate is one one car for six
11:49 minutes or six minutes per car then we
11:51 have the mean service rate per busy
11:55 server so as stated here a gas station
11:57 has one pump which then serves six
12:00 customers per hour so that would mean it
12:02 would be six cars for sixty minutes or
12:06 six cars per hour and that means one car
12:10 would be served for ten minutes so our
12:12 lambda or our arrival rate is six
12:16 minutes and our service rate is ten
12:18 minutes so now we can solve for our
12:21 utilization factor which is equal to
12:25 sixty percent then we have to solve for
12:30 each one of our parameter so WS WQ LS
12:33 and L Q so for each type of problem
12:36 these computations would actually differ
12:38 for each of the candle notation so since
12:41 our candle notation is an M one we will
12:44 be following mm1 calculation or mm1
12:47 formula I will show you like later the
12:49 different formulas for different candle
12:51 notations of different problems so you
12:53 have to choose among the different set
12:57 of the formulas which Falls your candle
12:59 notation and that would be the
13:01 appropriate equations that you will use
13:03 in order for you to solve the W as
13:07 WQ l SL q and all the other necessary
13:10 parameters so here let's try to solve
13:15 with the mm one candle notation so WS
13:19 would be one divided by mu minus lambda
13:22 which s we have our values over here at
13:25 the bottom right so that's 1 over 10
13:28 which is our mu minus 6 what that would
13:32 mean 1 over 4 or 0.25 hours that means
13:38 that 15 minutes and then so the expected
13:40 waiting time per customer in the system
13:44 is 15 minutes then we have expected
13:46 waiting time per customer in the queue
13:50 which follows the formula lambda divided
13:54 by mu times mu minus lambda which then
13:58 provides us with 0.15 hours then here is
14:00 our competition for our expected number
14:02 of customers in the system and in the
14:06 queue so LS would be the lambda divided
14:10 by mu minus lambda which means the
14:12 expected number of customers in the
14:14 system is 2 cars and the expected number
14:17 of customers in the queue is 1 car again
14:20 these formulas would only be used for
14:26 mmm one type of candle notation so here
14:28 are the different queueing formulas for
14:31 the different models or four different
14:33 candle notation these are the formula
14:38 for m1 or Markovian Markovian toinsanan
14:41 distribution poisson distribution and
14:43 negative exponential distribution for
14:47 both me effort both arrival and service
14:51 die and then only one server this one is
14:55 for md one meaning you have constant
15:00 service and then this would be the multi
15:02 channel or the multiple number of
15:07 servers so these are the formula for SWS
15:13 alkyl and WQ and then lastly the formula
15:16 for limited population meaning you
15:18 or population or your calling source is
15:23 finite or not infinite basically so we
15:27 have these formula over here so if you
15:28 want to try out one problem this is a
15:32 multi-channel server type of problem and
15:35 you can screenshot the question and then
15:37 send in the answer for my students this
15:40 is your homework problem
15:44 so that was Kimmy theory that was only
15:46 an introduction to queueing theory thank
15:48 you very much for listening and have a
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