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7/31/2019 Chap008A-Waiting Line Analysis
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8A-1
McGraw-Hill/Irwin Copyright 2009 by The McGraw-Hill Companie
s, Inc. All rights re
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Chapter 8A
Waiting Line Analysis
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Waiting Line Characteristics
Suggestions for ManagingQueues
Examples (Models 1, 2, 3, and 4)
OBJECTIVES
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Components of the Queuing System
Customer
Arrivals
Servers
Waiting Line
Servicing System
Exit
Queue or
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Customer Service Population Sources
Population Source
Finite Infinite
Example: Number of
machines needingrepair when a
company only has
three machines.
Example: Number of
machines needingrepair when a
company only has
three machines.
Example: The
number of peoplewho could wait in
a line for
gasoline.
Example: The
number of peoplewho could wait in
a line for
gasoline.
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Service Pattern
Service
Pattern
Constant Variable
Example: Items
coming down an
automated
assembly line.
Example: Items
coming down anautomated
assembly line.
Example: People
spending time
shopping.
Example: People
spending timeshopping.
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The Queuing System
Queue Discipline
Length
Number of Lines &
Line Structures
Service Time
Distribution
Queuing
System
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Examples of Line Structures
Single Channel
Multichannel
Single
PhaseMultiphase
One-personbarber shop
Car wash
Hospital
admissions
Bank tellers
windows
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Degree of Patience
No Way!
BALK
No Way!
RENEG
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Suggestions for Managing Queues
1. Determine an acceptable waitingtime for your customers
2. Try to divert your customers
attention when waiting
3. Inform your customers of what to
expect
4. Keep employees not serving the
customers out of sight
5. Segment customers
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Suggestions for Managing Queues (Continued)
6. Train your servers to be friendly
7. Encourage customers to come
during the slack periods
8. Take a long-term perspective
toward getting rid of the queues
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Waiting Line Models
Model Layout
Source
Population Service Pattern
1 Single channel Infinite Exponential
2 Single channel Infinite Constant
3 Multichannel Infinite Exponential
4 Single or Multi Finite Exponential
These four models share the following characteristics: Single phase
Poisson arrival
FCFS
Unlimited queue length
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Notation: Infinite Queuing: Models 1-3
lineintingnumber waiAverage
serversingleafor
ratesevicetoratearrivaltotalofRatio==
arrivalsbetweentimeAverage
timeserviceAverage
rateService=
rateArrival=
1
1
=
=
=
Lq
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Infinite Queuing Models 1-3 (Continued)
lineinwaitingofyProbabilit
systeminunitsexactlyofyProbabilit
channelsserviceidenticalofNumber=
systemin theunitsofNumber
served)betotime(includingsystemintimetotalAverage
lineinwaitingtimeAverage=
served)beingthose(including
systeminnumberAverage=s
=
=
=
=
Pw
nPn
S
n
Ws
Wq
L
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Assume a drive-up window at a fast food restaurant.
Customers arrive at the rate of 25 per hour.
The employee can serve one customer every two
minutes.
Assume Poisson arrival and exponential service rates.
Determine:
A) What is the average utilization of the employee?
B) What is the average number of customers in line?
C) What is the average number of customers in the
system?D) What is the average waiting time in line?
E) What is the average waiting time in the system?
F) What is the probability that exactly two cars will be
in the system?
Determine:
A) What is the average utilization of the employee?
B) What is the average number of customers in line?
C) What is the average number of customers in the
system?D) What is the average waiting time in line?
E) What is the average waiting time in the system?
F) What is the probability that exactly two cars will be
in the system?
Example: Model 1
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= 25 cust / hr
=1 customer
2 mins (1hr / 60 mins)= 30 cust / hr
= =25 cust / hr
30 cust / hr= .8333
Example: Model 1
A) What is the average utilization of the
employee?
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Example: Model 1
B) What is the average number of customers in
line?
4.167=25)-30(30
(25)=)-(
=
22
Lq
C) What is the average number of customers in
the system?
5=
25)-(30
25=
-
=
Ls
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Example: Model 1
D) What is the average waiting time in line?
mins10=hrs.1667==
LqWq
E) What is the average waiting time in the system?
mins12=hrs.2==
LsWs
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Example: Model 1
F) What is the probability that exactly two cars willbe in the system (one being served and the other
waiting in line)?
p = (1 -n
n
)( )
p = (1- =2
225
30
25
30)( ) .1157
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Example: Model 2
An automated pizza vending machine
heats anddispenses a slice of pizza in 4 minutes.
Customers arrive at a rate of one every 6
minutes with the arrival rate exhibiting a
Poisson distribution.
Determine:
A) The average number of customers in line.
B) The average total waiting time in the system.
Determine:
A) The average number of customers in line.
B) The average total waiting time in the system.
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Example: Model 2
A) The average number of customers in line.
.6667=10)-(2)(15)(15
(10)=
)-(2=
22
Lq
B) The average total waiting time in the system.
mins4=hrs.06667=
10
6667.=
=
LqWq
mins8=hrs.1333=15/hr
1+hrs.06667=
1+=
WqWs
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Example: Model 3
Recall the Model 1 example:
Drive-up window at a fast food restaurant.Customers arrive at the rate of 25 per
hour.
The employee can serve one customer
every two minutes.Assume Poisson arrival and exponential
service rates.
If an identical window (and an identically trainedserver) were added, what would the effects be on
the average number of cars in the system and the
total time customers wait before being served?
If an identical window (and an identically trainedserver) were added, what would the effects be on
the average number of cars in the system and the
total time customers wait before being served?
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Example: Model 3
Average number of cars in the system
ion)interpolatlinear-using-TN7.11(Exhibit
1760= .Lq
1.009=30
25+.176=+=
LqLs
Total time customers wait before being served
)(=mincustomers/25
customers.176== Wait!No
LqWq mins.007
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Notation: Finite Queuing: Model 4
channelsserviceofNumber
lineinunitsofnumberAverage)(system
queuinginthoselesssourcePopulation=
servedbeingunitsofnumberAverage
lineinwaittohaving
ofeffecttheofmeasureafactor,Efficiency
lineinmust waitarrivalany thatProbabilit=
=
=
=
=
S
Ln-N
J
H
F
D
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Finite Queuing: Model 4 (Continued)
requiredtimeserviceofproportionorfactor,ServicelineintimewaitingAverage
tsrequiremenservicecustomerbetweentimeAverage
servicetheperformtotimeAverage=
systemqueuinginunitsexactlyofyProbabilit
sourcepopulationinunitsofNumber
served)beingonethe(including
systemqueuinginunitsofnumberAverage=
=
=
=
=
=
XW
U
T
nPn
N
n
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Example: Model 4
The copy center of an electronics firm has four copy
machines that are all serviced by a single technician.
Every two hours, on average, the machines requireadjustment. The technician spends an average of 10
minutes per machine when adjustment is required.
Assuming Poisson arrivals and exponential service,how many machines are down (on average)?
The copy center of an electronics firm has four copy
machines that are all serviced by a single technician.
Every two hours, on average, the machines requireadjustment. The technician spends an average of 10
minutes per machine when adjustment is required.
Assuming Poisson arrivals and exponential service,how many machines are down (on average)?
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Example: Model 4
N, the number of machines in the population = 4
M, the number of repair people = 1T, the time required to service a machine = 10 minutes
U, the average time between service = 2 hours
X =T
T + U
10 min
10 min +120 min
= .077=
From Table TN7.11, F = .980 (Interpolation)From Table TN7.11, F = .980 (Interpolation)
L, the number of machines waiting to be
serviced = N(1-F) = 4(1-.980) = .08 machines
L, the number of machines waiting to be
serviced = N(1-F) = 4(1-.980) = .08 machines
H, the number of machines being
serviced = FNX = .980(4)(.077) = .302 machines
H, the number of machines being
serviced = FNX = .980(4)(.077) = .302 machines
Number of machines down = L + H = .382 machinesNumber of machines down = L + H = .382 machines
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Queuing Approximation
This approximation is quick way to analyze a queuing
situation. Now, both interarrival time and service time
distributions are allowed to be general.
In general, average performance measures (waiting
time in queue, number in queue, etc) can be very well
approximated by mean and variance of the distribution
(distribution shape not very important). This is very good news for managers: all you need is
mean and standard deviation, to compute average
waiting time
( )
=
= =22
2
Define:
Standard deviation of Xcoefficient of variation for r.v. X =Mean of X
Variancesquared coefficient of variation (scv) =
mean
x
x x
C
C C
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Queue Approximation
2( 1) 2 2
1 2
S
a s
q
C C
L
+
+=
= +s qL L S
ComputeS
=
2 2,a sC CInputs: S, , ,
(Alternatively: S, , , variances of interarrival and service time distributions)
as before, , andq sq sL L
W W
= =
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Approximation Example
Consider a manufacturing process (for example making
plastic parts) consisting of a single stage with fivemachines. Processing times have a mean of 5.4 days
and standard deviation of 4 days. The firm operates
make-to-order. Management has collected date on
customer orders, and verified that the time between
orders has a mean of 1.2 days and variance of 0.72days. What is the average time that an order waits
before being worked on?
Using our Waiting Line Approximation spreadsheet we
get:Lq = 3.154 Expected number of orders waiting to be
completed.
Wq = 3.78 Expected number of days order waits.
= 0.9 Expected machine utilization.
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Question Bowl
The central problem for virtually all queuing
problems is which of the following?a. Balancing labor costs and equipment costs
b. Balancing costs of providing service with the
costs of waiting
c. Minimizing all service costs in the use ofequipment
d. All of the above
e. None of the above
Answer: b. Balancing costs of providing
service with the costs of waiting
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Question Bowl
Customer Arrival populations in aqueuing system can be
characterized by which of the
following?
a. Poissonb. Finite
c. Patient
d. FCFSe. None of the above
Answer: b. Finite
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Question Bowl
Customer Arrival rates in a queuingsystem can be characterized by which
of the following?
a. Constant
b. Infinitec. Finite
d. All of the above
e. None of the above
Answer: a. Constant
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Question Bowl
An example of a queue discipline in a
queuing system is which of the
following?
a. Single channel, multiphase
b. Single channel, single phasec. Multichannel, single phase
d. Multichannel, multiphase
e. None of the above
Answer: e. None of the above (These are the rules for
determining the order of service to customers, which
include FCFS, reservation first, highest-profit customer
first, etc.)
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Question Bowl
Withdrawing funds from an automatedteller machine is an example in aqueuing system of which of thefollowing line structures?
a. Single channel, multiphaseb. Single channel, single phasec. Multichannel, single phased. Multichannel, multiphase
e. None of the above
Answer: b. Single channel, single phase
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Question Bowl
Refer to Model 1 in the textbook. If
the service rate is 15 per hour,what is the average service timefor this queuing situation?
a. 16.00 minutesb. 0.6667 hoursc. 0.0667 hoursd. 16% of an hour
e. Can not be computed from dataabove
Answer: c. 0.0667 hours (1/15=0.0667)
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Question Bowl
Refer to Model 1 in the textbook. If the
arrival rate is 15 per hour, what is theaverage time between arrivals for this
queuing situation?
a. 16.00 minutes
b. 0.6667 hours
c. 0.0667 hours
d. 16% of an hour
e. Can not be computed from data above
Answer: c. 0.0667 hours (1/15=0.0667)
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Question Bowl
Refer to Model 4 in the textbook. If the
average time to perform a service is 10minutes and the average time betweencustomer service requirements is 2minutes, which of the following is theservice factor for this queuing
situation?a. 0.833b. 0.800c. 0.750d. 0.500
e. None of the above
Answer: a. 0.833 (10/(10+2)=0.833)
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End of Chapter 8A