Queuing_2 Flashcards
what are the 4 elements of qs
s
mu
lambda
FIFO
servers= not enough servers so q
average service rate= too slow so q
average arrival rate= too many ppl so q
First come first serve= too many ppl so q
why do we do queing theory
to design operate and improve perfomance of queuing system
what are some performance measures to evaluate efficincy/ effecticness of queing
- avg number of customers waiting
- avg number of customers in system
- avg waiting time in q
- avg time in system
small business hiring example
-really busy business, customers waiting in line and some leaving business
cost for one more server is 75, cost per lost customer is 10
with two employees what happens?
-no one ever waits!!! no queue (look at the means in output)
how to assess if 2 employees is better than one employee
revenue * change in number of balks compared to the cost of 2nd employee
ex: revenue *( balks with 1 employee- bals with 2 employees)
see if ur final value is greater than cost!! if so then go for it
analyzing those tables, just use logic working through
what does balking w 3 + in system mean
if there are 3 or more customers in the system already 9Including one being serviced and two waiting) the new comer balks
how to model reneging in those tables?
- note when jcustomer joins the line up
- add the max allowable time from the beginning, and note when they leave
- look at customers who arrive after them! will they also renege? or by the time their allowable time is up with the newest customer be oka to wait?
how to do reneging and balking problems in tables
draw it out!!!!! draw out wait times and shit
what is steady state
at beginning of each day, the empty and idle time
gradual business activity builds up and will bring to normal operations level -> this is steady state!!!
different levels of steady state operations during the day (Restaurant busy at lunch adn dinner) (rush hour traffic during diff times)
what is the MM1 a good model for
when customers arrive without appointments
whenn lambda < mu, system is…
IMPORTANT NOTE
stable! custommers fo not arrive to the system faster than they can be served
no balking, no reneging
NOTE: UNITS MUST BE THE SAME BABY GIRL
what does it mean when the system has reached steady state?
it has warmed up since the start up
WHEN 1/LAMBDA > 1/MU SYSTEM IS
STABLE!!! average time between arrivals > average TIME BETWEEN SERVICE
WHAT 9 performance measures can we calculate from MM1
- server utilizations rate
- prob of 0 customers in system
- prob a customer must wait in queue
- avg # of customers in q
- avg # of customers in system
- avg time customer spends in q
- avg time cusotmer spends in system
8/ prob of n customer in system - prob of more than k customers in system
Formula for utilization rate
Meaning
U= lamda/mu
how often is a server busy (—% of the time)
Formula for Probability of 0 customers in system
meaning
Po=1-lamda/mu
how often is the server not busy (—% of the time) cuz no customers in system
Probability a customer must wait in line
meaning
Pw=1-Po
Customers must wait when the server is busy in a MM1 queue WITH A SINGLE SERVER
FORMULA for avg number of customers in the queue
Lq= lambda ^2/ mu(mu-lambda)
or
L- lambda/mu
where L is (# of customers in the system)
Dont round this number btw
Formula for avg number of customers in the system (in queue and being served)
L= Lq + lambda/mu
or
L=lambda/(mu-lambda)
formula for avg time a customer spends in the queue
what are the units?
Wq= Lq/lambda
or
Wq= W-1/mu
or
Wq= lambda/ mu(mu-lambda)
The units are the same as original!! so you may have to convert from hours to mins in the answer
formula for the average time a customer spends in the system (in queue and being served)
W= L/lambda
or
W= 1/mu-lambda
pr
W= Wq + 1/mu
watch the units!!
Formula the prob of exactly n customers in systetm
Pn= (1-lambda/mu)*(lamda/mu)^n
this tells you how often you will have x amount of customers in system (and has implications for how big ur waiting area should be)
Formula for the prob of more tahn k customers in the system
Pn>k= (lambda/mu)^k+1
Theis will tell you how often you will have more than x amount of customer sin the system, this will influence how big ur waiting area should be
how cna we calculate values for MM1 queue
formulas by hand
how can we calculate values for MM/# any number of servers
we can use the Q.xls softawre!!! will calculate 7/9 measures but not the last 2 cuz they deal with n
how cna you have a long average time waiting in the queue
because of the variable arriavals and service time!!!! it is not perfect
how could queuing theory help decisionmaking in business
1) how busy should a server be
2) should we set up multiple single server qs or one multiple server q
3) should we add workers
4) should we double the service speed or number of servers?
trade off for how busy a server should be
benefit of high server utilization vs cost of long customer wait
tradeoff in adding workers
costs of speefing up service and the benefifts of shorter waits
insights from doctor example- how busy should a server be?
- we have a set consultation time (mu, service time)
-we can compare different arrival rates (lambdas)
-the higher the lambda, the higher the utilization (good) but higher average wait time and queue lengths (bad)
is high utilizaiton a good thing
-what should we do
avoids wasting resources but too high is system congestion
-look at the chart of different service levels you can have and see if 95% service level is an appropriate amount of q length (Consider the context)
in hostpials 70% service is okay because q is decent
how can doctors reduce the q length that is in their control?
reduce the variation in consultation time, will reduce the waiting time
In insurance application processing and high-volume manufacturing, high utilization is usually good as it is items and not people in the queue
Compare 3xMM1 and MM3
mm1 is like grocery store (3 single server queues w 3 lineups)
mm3 is like bank ( three server queue w one line up)
lambda=15
mu=6
how would you put 3 mm1 in Q.xls
lambda=5 (15/3=5, cuz 5 people arrive in one line at a time)
mu=6
of servers=3
lambda=15
mu=6
how would you put mm3 inQ.xls
lambda 15
mu 6
number of servers 3
which server configurstation performs better 3xmm1 or mm3
explain
MM3
-cuz in 3xMM1 people might not split themselves evenly, one server idle while others busy
-cuz in 3xMM1 someone in queeue might take super long and others wait (long processing time)
MM3 is better cuz if there is one person taking long ass time, go to the next available agent
why do grocery stores keep 3xmm1 even if mm3 better
-cuz space issues (banks can have one single long line but hard for stores)
-they want u to wait to do those impulse purchase!
-it takes prep time to put stuff on conveyoer belt
customers intuitively get that mm3 is better, and they will enforce mm3 often
important case we use
queing at eCycle services
eCycle services case
-owner of store has 30,000 waiting fee from the city from last month cuz of cityy trailers being idle at our facility
-city charges 60.hour for the aount oof time their trucks have to wait or unload at eCycle services
eCycle services q1: is the invoice for 30k correct?
-use the mean/average time in system (days) to calculate how long the trucks are in the system
-use the avg number of trailers in system to calcuulate how many there are
then do daily # of traielrs * hours * cost per hour * 5 days in a week * 4.33 weeks in a month to get cost
yes!!! 30,000 invoice is reasonable
what are the current costs of the unloading system? eCycle services
calculate cost of crew per day
workers * hourly pay * hours per day= daily cost
daily cost * 5 days a week= weekly cost
weekly cost *4.33 weeks a month= monthly cost
compare how much of total cost is unloading crew and how much is city trailers!!!! most of the costs come from city trialers!!
what is the optimal crew size? ecycle
think about the utilization formula in Q.xls (how much are we currently utilizing?)
think about that green line and red line chart
-since most of the cost come from city trailers waiting, lets provide a higher service level as this would lower our cost
-provide a higher service level by looking at utilization rate formula (either make lambda bigger or mu smaller)
if you ekeep adding more people to your crew will you keep going faster and faster 2x 3x?
no! you will only increase by one unit cuz diminshing returns!!
how to do optimal crew size analysis?
-compare different sizes of crew
-get the eman unload rate for each
-calcualte daily cost
-calcaulte L (avg # of trailers iin system)
-Calculate daily cost of traielrs
-sum daily cost
pick the size of crew w lowest daily cost
with an increased crew size will you have higher utilization rate?
no!! low cost of staff are not wokring
BUT
better to have lower cos staff idle then having higher cost trailers idle
how to do sensitivity analysis
-increase/decrease wage rate (what hourly wage rate should be at the optimal crew size)
-trailer waiting rate- what hourly trailer waiting rate should we stay at w optimal crew
-arrival rate: what is the optimal crew size if # of trailer arrivals increase
conculsion of ecylce
The optimal solution of a four-person unloading crew is not very sensitive to the hourly rates for both employees and trailers.
The four-person crew is optimal unless wage rates rise above $30 an hour or the city charges over $96 an hour for their waiting trailers.
Currently $24/hour for crew and $60/hour for trailers
The optimal solution IS VERY sensitive to the trailer arrival rate. Any average arrival increase beyond a ½ trailer (from the current 3) will cause the optimal solution to change.
WHAT are osme other process improvement alts for eclcycle?
- unload faster
(have pallets are dorp off locatins)
(have conveyer belts// rollers extended into trucks) - rent second loading dock
(2nd server so mm2) - get own trucks or trailers
(unloading time costs more than 10k per month)
(leasing buying may be cheaper)
-set delivery times
(Appointment could eleimentae variability)
-flexible crew size
(have dissamebly team memebrs help unload)
should we double the speed or double the # of servers?
meaning
is lambda goes form 4 to 8 (trucks arrive 8 per day) should u double service speed (serve 10 per hour) or double the number of servers (from 1 to 2)?
if you do nothing, q goes to infinity cuz lambda> mu
analysis of doubling service rate
spend less time in system overall
analysis doubling # of servers
spend less time in quueue waiting for service
what should a manager choose (doubling service rate or doubling # of servers)
WHATEVRE CUSTOMER WANTS!!!! (IF BOTH ALTS ARE THE SAME)
if customer doesnt like waiting in queue/wait times: add a server
if cusotmer doesnt like service time more: speed up the service
analyzing lineups: compare queuing theory and simulation in
ACCURACY
PERF MEASURES
FLEXIBILITY
CONVENIENCE
RESOURCES
Queing theory:
-excelling if assumptions are OK, and has good approximations if assumptions are good
-limited choice (using averages only)
-limited (useless for many situiations)
-very convenient
-very fast
simulation
-exclelent for accuracy
-extensive performance measures
-useful for many situtaions (flexibile)
-programming excel or using @risk or arena
-timec onsuming computer intensive
