Spreadsheet Modeling
& Decision Analysis
A Practical Introduction to
Management Science
5th edition
Cliff T. Ragsdale
Chapter 13
Queuing Theory
Introduction to Queuing Theory
 It is estimated that Americans spend a total of 37
billion hours a year waiting in lines.
 Places we wait in line...
- stores
- hotels
- post offices
- banks
- traffic lights
- restaurants
- airports
- theme parks
- on the phone
 Waiting lines do not always contain people...
- returned videos
- subassemblies in a manufacturing plant
- electronic message on the Internet
 Queuing theory deals with the analysis and
management of waiting lines.
The Purpose of Queuing Models
 Queuing models are used to:
– describe the behavior of queuing systems
– determine the level of service to provide
– evaluate alternate configurations for providing
service
Queuing Costs
$
Total Cost
Cost of providing service
Cost of customer dissatisfaction
Service Level
Common Queuing System Configurations
Customer
Arrives
Customer
Arrives
...
Waiting Line
...
Waiting Line
...
Waiting Line
Customer
Arrives
Server
Customer
Leaves
Server 1
Customer
Leaves
Server 2
Customer
Leaves
Server 3
Customer
Leaves
Server 1
Customer
Leaves
Server 2
Customer
Leaves
...
Waiting Line
...
Waiting Line
Server 3
Customer
Leaves
Characteristics of Queuing Systems:
The Arrival Process
 Arrival rate - the manner in which customers
arrive at the system for service.
 Arrivals are often described by a Poisson random
variable:
x e  
P( x ) 
, for x = 0, 1, 2, ...
x!
where  is the arrival rate (e.g., calls arrive at a
rate of =5 per hour)
 See file Fig13-3.xls
Characteristics of Queuing Systems:
The Service Process
 Service time - the amount of time a customer
spends receiving service (not including time in
the queue).
 Service times are often described by an
Exponential random variable:
t2
P(t1  T  t2 )   e  x dx  e  ut1  e  t2 , for t1  t2
t1
where  is the service rate (e.g., calls can be
serviced at a rate of =7 per hour)
 The average service time is 1/.
 See file Fig13-4.xls
Comments
 If arrivals follow a Poisson distribution with mean ,
interarrival times follow an Exponential distribution
with mean 1/.
– Example
Assume calls arrive according to a Poisson
distribution with mean =5 per hour.
Interarrivals follow an exponential distribution
with mean 1/5 = 0.2 per hour.
On average, calls arrive every 0.2 hours or
every 12 minutes.
 The exponential distribution exhibits the Markovian
(memoryless) property.
Kendall Notation
 Queuing systems are described by 3 parameters:
1/2/3
– Parameter 1
M = Markovian interarrival times
D = Deterministic interarrival times
– Parameter 2
M = Markovian service times
G = General service times
D = Deterministic service times
– Parameter 3
A number Indicating the number of servers.
 Examples,
M/M/3
D/G/4
M/G/2
Operating Characteristics
Typical operating characteristics of interest include:
U - Utilization factor, % of time that all servers are busy.
P0 - Prob. that there are no zero units in the system.
Lq - Avg number of units in line waiting for service.
L - Avg number of units in the system (in line & being
served).
Wq - Avg time a unit spends in line waiting for service.
W - Avg time a unit spends in the system (in line & being
served).
Pw - Prob. that an arriving unit has to wait for service.
Pn - Prob. of n units in the system.
Key Operating Characteristics
of the M/M/1 Model
1
W

L  W
1
Wq  W 

Lq  Wq
The Q.xls Queuing Template
 Formulas for the operating characteristics of a
number of queuing models have been derived
analytically.
 An Excel template called Q.xls implements the
formulas for several common types of models.
 Q.xls was created by Professor David Ashley
of the Univ. of Missouri at Kansas City.
The M/M/s Model
 Assumptions:
–
–
–
–
–
There are s servers.
Arrivals follow a Poisson distribution and occur
at an average rate of  per time period.
Each server provides service at an average
rate of  per time period, and actual service
times follow an exponential distribution.
Arrivals wait in a single FIFO queue and are
serviced by the first available server.
< s.
An M/M/s Example: Bitway Computers
 The customer support hotline for Bitway Computers is currently
staffed by a single technician.
 Calls arrive randomly at a rate of 5 per hour and follow a
Poisson distribution.
 The technician services calls at an average rate of 7 per hour,
but the actual time required to handle a call follows an
exponential distribution.
 Bitway’s president, Rod Taylor, has received numerous
complaints from customers about the length of time they must
wait “on hold” for service when calling the hotline.
 Rod wants to determine the average length of time customers
currently wait before the technician answers their calls.
 If the average waiting time is more than 5 minutes, he wants to
determine how many technicians would be required to reduce
the average waiting time to 2 minutes or less.
Implementing the Model
See file Q.xls
Summary of Results: Bitway Computers
Arrival rate
Service rate
Number of servers
Utilization
P(0), probability that the system is empty
Lq, expected queue length
L, expected number in system
Wq, expected time in queue
W, expected total time in system
Probability that a customer waits
5
7
1
5
7
2
71.43%
0.2857
1.7857
2.5000
0.3571
0.5000
0.7143
35.71%
0.4737
0.1044
0.8187
0.0209
0.1637
0.1880
The M/M/s Model With
Finite Queue Length
 In some problems, the amount of waiting area is limited.
 Example,
– Suppose Bitway’s telephone system can keep a maximum of 5
calls on hold at any point in time.
– If a new call is made to the hotline when five calls are already in
the queue, the new call receives a busy signal.
– One way to reduce the number of calls encountering busy signals
is to increase the number of calls that can be put on hold.
– If a call is answered only to be put on hold for a long time, the
caller might find this more annoying than receiving a busy signal.
– Rod wants to investigate what effect adding a second technician to
answer hotline calls has on:
 the number of calls receiving busy signals
 the average time callers must wait before receiving service.
Implementing the Model
See file Q.xls
Summary of Results:
Bitway Computers With Finite Queue
Arrival rate
Service rate
Number of servers
Maximum queue length
Utilization
P(0), probability that the system is empty
Lq, expected queue length
L, expected number in system
Wq, expected time in queue
W, expected total time in system
Probability that a customer waits
Probability that a customer balks
5
7
1
5
5
7
2
5
68.43%
0.3157
1.0820
1.7664
0.2259
0.3687
0.6843
0.0419
35.69%
0.4739
0.1019
0.8157
0.0204
0.1633
0.1877
0.0007
The M/M/s Model With Finite Population
 Assumptions:
– There are s servers.
– There are N potential customers in the arrival
population.
– The arrival pattern of each customer follows a
Poisson distribution with a mean arrival rate of 
per time period.
– Each server provides service at an average rate
of  per time period, and actual service times
follow an exponential distribution.
– Arrivals wait in a single FIFO queue and are
serviced by the first available server.
M/M/s With Finite Population Example:
The Miller Manufacturing Company
 Miller Manufacturing owns 10 identical machines that
produce colored nylon thread for the textile industry.
 Machine breakdowns follow a Poisson distribution with an
average of 0.01 breakdowns per operating hour per
machine.
 The company loses $100 each hour a machine is down.
 The company employs one technician to fix these
machines.
 Service times to repair the machines are exponentially
distributed with an avg of 8 hours per repair. (So service is
performed at a rate of 1/8 machines per hour.)
 Management wants to analyze the impact of adding another
service technician on the average time to fix a machine.
 Service technicians are paid $20 per hour.
Implementing the Model
See file Q.xls
Summary of Results:
Miller Manufacturing
Arrival rate
Service rate
Number of servers
Population size
0.01
0.125
1
10
0.01
0.125
2
10
0.01
0.125
3
10
Utilization
P(0), probability that the system is empty
Lq, expected queue length
L, expected number in system
Wq, expected time in queue
W, expected total time in system
Probability that a customer waits
67.80%
0.3220
0.8463
1.5244
9.9856
17.986
0.6780
36.76%
0.4517
0.0761
0.8112
0.8282
8.8282
0.1869
24.67%
0.4623
0.0074
0.7476
0.0799
8.0799
0.0347
Hourly cost of service technicians
Hourly cost of inoperable machines
Total hourly costs
$20.00
$152.44
$172.44
$40.00
$81.12
$121.12
$60.00
$74.76
$134.76
The M/G/1 Model
 Not all service times can be modeled accurately
using the Exponential distribution.
– Examples:
Changing oil in a car
Getting an eye exam
Getting a hair cut
 M/G/1 Model Assumptions:
– Arrivals follow a Poisson distribution with mean .
– Service times follow any distribution with mean 
and standard deviation s.
– There is a single server.
An M/G/1 Example: Zippy Lube
 Zippy-Lube is a drive-through automotive oil change business
that operates 10 hours a day, 6 days a week.
 The profit margin on an oil change at Zippy-Lube is $15.
 Cars arrive at the Zippy-Lube oil change center following a
Poisson distribution at an average rate of 3.5 cars per hour.
 The average service time per car is 15 minutes (or 0.25 hours)
with a standard deviation of 2 minutes (or 0.0333 hours).
 A new automated oil dispensing device costs $5,000.
 The manufacturer's representative claims this device will reduce
the average service time by 3 minutes per car. (Currently,
employees manually open and pour individual cans of oil.)
 The owner wants to analyze the impact the new automated
device would have on his business and determine the pay back
period for this device.
Implementing the Model
See file Q.xls
Summary of Results: Zippy Lube
Arrival rate
Average service TIME
Standard dev. of service time
3.5
0.25
0.0333
3.5
0.2
0.0333
4.371
0.2
0.333
Utilization
P(0), probability that the system is empty
Lq, expected queue length
L, expected number in system
Wq, expected time in queue
W, expected total time in system
87.5%
0.1250
3.1168
3.9918
0.8905
1.1405
70.0% 87.41%
0.3000 0.1259
0.8393 3.1198
1.5393 3.9939
0.2398 0.7138
0.4398 0.9138
Payback Period Calculation
Increase in:
Arrivals per hour
Profit per hour
Profit per day
Profit per week
0.871
$13.06
$130.61
$783.63
Cost of Machine
$5,000
Payback Period
6.381 weeks
The M/D/1 Model
 Service times may not be random in some queuing
systems.
– Examples
In manufacturing, the time to machine an item
might be exactly 10 seconds per piece.
An automatic car wash might spend exactly the
same amount of time on each car it services.
 The M/D/1 model can be used in these types of
situations where the service times are deterministic
(not random).
 The results for an M/D/1 model can be obtained
using the M/G/1 model by setting the standard
deviation of the service time to 0 ( s= 0).
Simulating Queues
 The queuing formulas used in Q.xls describe
the steady-state operations of the various
queuing systems.
 Simulation is often used to analyze more
complex queuing systems.
 See file Fig13-21.xls
End of Chapter 13