Supplement C
Waiting Line Models
Operations Management
by
R. Dan Reid & Nada R. Sanders
4th Edition © Wiley 2010
Learning Objectives
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Describe the elements of a waiting line
problem.
Use waiting line models to estimate system
performance.
Use waiting line models to make managerial
decisions.
Elements of Waiting Lines
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“Queue” is another name for a waiting line.
A waiting line system consists of two
components:
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The customer population (people or objects to be
processed)
The process or service system
Whenever demand exceeds available
capacity, a waiting line or queue forms
There is a tradeoff between cost and service
level.
Customer Population
Characteristics

Finite versus Infinite populations:
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Balking
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When an arriving customer chooses not to enter a queue because
it’s already too long.
Reneging
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Is the number of potential new customers materially affected by
the number of customers already in queue?
When a customer already in queue gives up and exits without
being serviced.
Jockeying

When a customer switches between alternate queues in an effort
to reduce waiting time.
Service System

The service system is defined by:
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The
The
The
The
The
number of waiting lines
number of servers
arrangement of servers
arrival and service patterns
waiting line priority rules
Number of Lines

Waiting lines systems can have single
or multiple queues.

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Single queues avoid jockeying behavior
and perceived fairness is usually high.
Multiple queues are often used when
arriving customers have differing
characteristics (e.g. paying with cash, less
than 10 items, etc.) and can be readily
segmented.
Servers

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Single servers or multiple, parallel servers
providing multiple channels
Arrangement of servers (phases)

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Multiple phase systems require customers to visit
more than one server
Example of a multi-phase, multi-server system:
1
Arrivals
C
C
C
2
4
C
C
5
3
6
Phase 1
Phase 2
Depart
Example Queuing Systems
Arrival & Service Patterns

Arrival rate:
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
The average number of customers arriving per
time period
Modeled using the Poisson distribution
Arrival rate usually denoted by lambda ()
Example: =50 customers/hour; 1/=0.02 hours
between customer arrivals (1.2 minutes between
customers)
Arrival & Service Patterns con’t
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Service rate:
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The average number of customers that can be served
during the period of time
Service times are usually modeled using the exponential
distribution
Service rate usually denoted by mu (µ)
Example: µ=70 customers/hour; 1/µ=0.014 hours per
customer (0.857 minutes per customer).
Even if the service rate is larger than the arrival
rate, waiting lines form!

Reason is the variation in specific customer arrival and
service times.
Waiting Line Priority Rules
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First come, first served
Best customers first (reward loyalty)
Highest profit customers first
Quickest service requirements first
Largest service requirements first
Earliest reservation first
Emergencies first
Etc.
Waiting Line Performance
Measures
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Lq = The average number of customers
waiting in queue
L = The average number of customers in the
system
Wq = The average waiting time in queue
W = The average time in the system
p = The system utilization rate (% of time
servers are busy)
Single-Server Waiting Line
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Assumptions
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Customers are patient (no balking, reneging, or
jockeying)
Arrivals follow a Poisson distribution with a mean arrival
rate of . This means that the time between successive
customer arrivals follows an exponential distribution with
an average of 1/ 
The service rate is described by a Poisson distribution
with a mean service rate of µ. This means that the
service time for one customer follows an exponential
distribution with an average of 1/µ
The waiting line priority rule is first-come, first-served
Infinite population
Formulas: Single-Server Case
  lambda  mean arrival rate
  mu  mean service rate

p   average system utilizatio n

Note :    for system stability. If this is not the case,
an infinitly long line will eventually form.
Formulas: Single-Server
Case con’t
L

 
 average number of customers in system
Lq  pL  average number of customers in line
1
W
 average time in system including service 
 
Wq  pW  average time spent waiting
Pn  1  p  p n  probabilit y of n customers in the system
at a given point in time
State Univ Computer Lab
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A help desk in the computer lab serves
students on a first-come, first served basis.
On average, 15 students need help every
hour. The help desk can serve an average of
20 students per hour.
Based on this description, we know:
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µ = 20 students/hour (average service time is 3
minutes)
 = 15 students/hour (average time between
student arrivals is 4 minutes)
Average Utilization
 15
p 
 0.75 or 75%
 20
Average Number of Students
in the System, and in Line

15
L

 3 students
   20  15
Lq  pL  0.753  2.25 students
Average Time in the System & in
Line
1
1
W

 0.2 hours
   20  15
or 12 minutes
Wq  pW  0.750.2  0.15 hours
or 9 minutes
Probability of n
Students in the Line
P0  1  p  p  1  0.751  0.25
0
P1  1  p  p  1  0.750.75  0.188
1
P2  1  p  p  1  0.750.75  0.141
2
2
P3  1  p  p  1  0.750.75  0.105
3
3
P4  1  p  p  1  0.750.75  0.079
4
4
Single Server: Spreadsheet
Approach
Key Formulas
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
B
C
B9: =1/B5
Queuing Analysis: Single Server
B10: =1/B6
Inputs
Time unit
Arrival Rate (lambda)
Service Rate (mu)
hour
15
20
B13: =B5/B6
customers/hour
customers/hour
B14: =1-B13
B15: =B5/(B6-B5)
Intermediate Calculations
Average time between arrivals
Average service time
Performance Measures
Rho (average server utilization)
P0 (probability the system is empty)
L (average numberin the system)
Lq (average number waiting in the queue)
W (average time in the system)
Wq (average time in the queue)
0.066667
0.05
0.75
0.25
3
2.25
0.2
0.15
B16: =B13*B15
hour
hour
B17: =1/(B6-B5)
B18: =B13*B17
B22: =(1-B$13)*(B13^B21)
customers
customers
hour
hour
Probability of a specific number of customers in the system
Number
2
Probability
0.140625

Use Data Table (tracking
B22) to easily compute the
probability of n customers in
the system.
Single Server: Probability of n
Students in the System
Probability of Number in System
0.3000
0.2000
0.1500
0.1000
0.0500
Number in System
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
0.0000
0
Probability
0.2500
Multiple Server Case
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Assumptions
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Same as Single-Server, except here we
have multiple, parallel servers
Single Line
When server finishes with customer, first
person in line goes to the idle server
All servers are identical
Multiple Server Formulas
  lambda  mean arrival rate
  mu  mean service rate for one server
s  number of parallel, identical servers

p
 average system utilizatio n
s
Note : s   for system stability. If this is not the case,
an infinitly long line will eventually form.
Multiple Server Formulas con’t
1
  /    /    1 

  probabilit y of zero
P0  

s!  1  p 
 n  0 n!
customers in the system at a given point in time
s 1
n
s
  /  n
P0 for n  s

Pn   n! n
 probabilit y of n customers
  /   P for n  s
 s!s n  s 0
in the system at a given point in time
Multiple Server Formulas con’t
P0  /   p
Lq 
 average number of customers in line
2
s!1  p 
Wq  Lq   average time spent waiting in line
s
W  Wq 
1

 average time in system including service 
L  W  average number of customers in system
Example: Multiple Server
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Computer Lab Help Desk
Now 45 students/hour need help.
3 servers, each with service rate of 18
students/hour
Based on this, we know:
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µ = 18 students/hour
s = 3 servers
 = 45 students/hour
Flexible Spreadsheet Approach
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Formulas are somewhat complex to set up initially, but you
only need to do it once!
For other multiple-server problems, can just change the input
values.
This approach also makes sensitivity analysis possible.
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
B
C
hour
45
18
3
customers/hour
customers/hour
servers
Queuing Analysis: Multiple Servers
Inputs
Time unit
Arrival Rate (lambda)
Service Rate per Server (mu)
Number of Servers (s)
Intermediate Calculations
Average time between arrivals
Average service time per server
Combined service rate (s*mu)
0.022222
hour
0.055556
hour
54
customers/hour
Performance Measures
Rho (average server utilization)
P0 (probability the system is empty)
L (average numberin the system)
Lq (average number waiting in the queue)
W (average time in the system)
Wq (average time in the queue)
0.833333
0.044944
6.011236
3.511236
0.133583
0.078027
customers
customers
hour
hour
Probability of a specific number of customers in the system
Number
5
Probability
0.081279
E
F
G
H
3 Working Calculations, mainly for P0 Calculation
4
5 lambda/mu
2.5
6 s!
6
7
8
n
(/)^n
n!
Sum
9
0
1
1
1
10
1
2.5
1
3.5
11
2
6.25
2
6.625
12
3
15.625
6
9.229166667
13
4
39.0625
24
10.85677083
14
5
97.65625
120
11.67057292
15
6
244.14063
720
12.00965712
16
7
610.35156
5040
12.13075862
17
8
1525.8789
40320
12.16860284
18
9
3814.6973
362880 12.17911512
19
10
9536.7432 3628800 12.18174319
20
11
23841.858 39916800 12.18234048
21
12
59604.645 479001600 12.18246492
22
13
149011.61 6.227E+09 12.18248885
23
14
372529.03 8.718E+10 12.18249312
24
15
931322.57 1.308E+12 12.18249383
25
16
2328306.4 2.092E+13 12.18249394
26
17
5820766.1 3.557E+14 12.18249396
27
18
14551915 6.402E+15 12.18249396
108
99
2.489E+39 9.33E+155 12.18249396
109
100
6.223E+39 9.33E+157 12.18249396
Key Formulas for Spreadsheet
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F10: =F$5^E10 (copied down)
G10: =E10*G9 (copied down)
H10: =H9+(F10/G10) (copied down)
F5: =B5/B6
F6: =INDEX(G9:G109,B7+1)
B10: =1/B5
B11: =1/B6
B12: =B7*B6
B15: =B5/B12
B16: = (INDEX(H9:H109,B7)+ (((F5^B7)/F6)*((1)/(1-B15))))^(-1)
B17: =B5*B19
B18: =(B16*(F5^B7)*B15)/(INDEX(G9:G109,B7+1)*(1-B15)^2)
B19: =B20+(1/B6)
B20: =B18/B5
B24: =IF(B23<=B7, ((F5^B23)*B16)/INDEX(G9:G109,B23+1),
((F5^B23)*B16)/ (INDEX(G9:G109,B7+1)*(B7^(B23-B7))))
Probability of n students in the
system
Probability of Number in System
0.1600
0.1400
0.1000
0.0800
0.0600
0.0400
0.0200
Number in System
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
0.0000
0
Probability
0.1200
Changing System Performance

Customer Arrival Rates


Number and type of service facilities


Try to smooth demand through non-peak discounts or price
promotions
Increase or decrease number of servers, or dedicate specific
servers for certain tasks (e.g., express line for under 10
items)
Change Number of Phases

Can use multi-phase system instead of single phase. This
spreads the workload among more servers and may result in
better flow (e.g., fast food restaurants having an order
phase, pay phase, and pick-up phase during busy hours)
Changing System Performance

Server efficiency



Change priority rules


Add resources to each phase (e.g., bagger helping
a checker at the grocery store)
Use technology (e.g. price scanners) to improve
efficiency
Example: implement a reservation protocol
Change the number of lines


Reduce multiple lines to single queue to avoid
jockeying
Dedicate specific servers to specific transactions
Waiting Lines Models within
OM: How it all fits together


Although it is unlikely that you calculate performance
measures for the lines you wait in on a day-to-day
basis, you should now be aware of the potential for
mathematical analysis of these systems. More
importantly, management has a tool by which it can
evaluate system performance and make decisions as
to how to improve the performance while weighing
performance against the costs to achieve that
performance.
Waiting line models are important to a company
because they directly affect customer service
perception and the costs of providing a service.
Supplement C Highlights
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The elements of a waiting line system include the customer population
source, the patience of the customer, the service system, arrival and
service distributions, waiting line priority rules, and system performance
measures. Understanding these elements is critical when analyzing
waiting line systems.
Waiting line models allow us to estimate system performance by
predicting average system utilization, average number of customers in
the service system, average number of customers waiting in line, average
time a customer waits in line, and the probability of n customers in the
service system.
The benefit of calculating operational characteristics is to provide
management with information as to whether system changes are
needed. Management can change the operational performance of the
waiting line system by altering any or all of the following: the customer
arrival rates, the number of service facilities, the number of phases,
server efficiency, the priority rule, and the number of lines in the system.
Based on proposed changes, management can then evaluate the
expected performance of the system.
Homework Hints

Problems C.3 and C.4: these are based on
the single-server model, the “additional
server” is one who works within the single
server system. C.3 asks for the utilization
rate and average number of customers
waiting in the system and in line. C.4 asks
for the average time in the system and in
line and the probability of more than 3 and
4 customers in the system