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1
• A Practical Introduction to Management Science
• 4th edition
• Cliff T. Ragsdale

2
Queuing Theory
Chapter 13
3
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.

4
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

5
Queuing Costs

Total Cost
Cost of providing service
Cost of customer dissatisfaction
Service Level
6
Common Queuing System Configurations
7
Characteristics of Queuing SystemsThe Arrival
Process
• Arrival rate - the manner in which customers
arrive at the system for service.
• where l is the arrival rate (e.g., calls arrive
at a rate of l5 per hour)
• See file Fig13-3.xls

8
Characteristics of Queuing SystemsThe Service
Process
• Service time - the amount of time a customer
spends receiving service (not including time in
the queue).
• where m is the service rate (e.g., calls can be
serviced at a rate of m7 per hour)
• The average service time is 1/m.
• See file Fig13-4.xls

9
• If arrivals follow a Poisson distribution with
mean l, interarrival times follow an Exponential
distribution with mean 1/l.
• Example
• Assume calls arrive according to a Poisson
distribution with mean l5 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.

10
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

11
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.

12
Key Operating Characteristics of the M/M/1 Model
13
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.

14
The M/M/s Model
• Assumptions
• There are s servers.
• Arrivals follow a Poisson distribution and occur
at an average rate of l per time period.
• Each server provides service at an average rate
of m per time period, and actual service times
• Arrivals wait in a single FIFO queue and are
serviced by the first available server.
• llt sm.

15
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
• 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.
• Bitways 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.
• Continued

16
Bitway Computers (continued)
• Rod wants to determine the average length of time
customers currently wait before the technician
• 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.

17
Implementing the Model
• See file Q.xls

18
Summary of Results Bitway Computers
• Arrival rate 5 5
• Service rate 7 7
• Number of servers 1 2
• Utilization 71.43 35.71
• P(0), probability that the system is empty 0.2857
0.4737
• Lq, expected queue length 1.7857 0.1044
• L, expected number in system 2.5000 0.8187
• Wq, expected time in queue 0.3571 0.0209
• W, expected total time in system 0.5000 0.1637
• Probability that a customer waits 0.7143 0.1880

19
The M/M/s Model With Finite Queue Length
• In some problems, the amount of waiting area is
limited.
• Example,
• Suppose Bitways 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
• 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.

20
Implementing the Model
• See file Q.xls

21
Summary of ResultsBitway Computers With Finite
Queue
• Arrival rate 5 5
• Service rate 7 7
• Number of servers 1 2
• Maximum queue length 5 5
• Utilization 68.43 35.69
• P(0), probability that the system is
empty 0.3157 0.4739
• Lq, expected queue length 1.0820 0.1019
• L, expected number in system 1.7664 0.8157
• Wq, expected time in queue 0.2259 0.0204
• W, expected total time in system 0.3687 0.1633
• Probability that a customer waits 0.6843 0.1877
• Probability that a customer balks 0.0419 0.0007

22
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
l per time period.
• Each server provides service at an average rate
of m per time period, and actual service times
• Arrivals wait in a single FIFO queue and are
serviced by the first available server.

23
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.

24
Implementing the Model
• See file Q.xls

25
Summary of Results Miller Manufacturing
• Arrival rate 0.01 0.01 0.01
• Service rate 0.125 0.125 0.125
• Number of servers 1 2 3
• Population size 10 10 10
• Utilization 67.80 36.76 24.67
• P(0), probability that the system is
empty 0.3220 0.4517 0.4623
• Lq, expected queue length 0.8463 0.0761 0.0074
• L, expected number in system 1.5244 0.8112
0.7476
• Wq, expected time in queue 9.9856 0.8282 0.0799
• W, expected total time in system 17.986 8.8282
8.0799
• Probability that a customer waits 0.6780 0.1869
0.0347
• Hourly cost of service technicians 20.00 40.00
60.00
• Hourly cost of inoperable machines 152.44
81.12 74.76
• Total hourly costs 172.44 121.12 134.76

26
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
l.
• Service times follow any distribution with mean m
and standard deviation s.
• There is a single server.

27
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).
• Continued

28
Zippy Lube (continued)
• 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.

29
Implementing the Model
• See file Q.xls

30
Summary of Results Zippy Lube
• Arrival rate 3.5 3.5 4.371
• Average service TIME 0.25 0.2 0.2
• Standard dev. of service time 0.0333 0.0333 0.333
• Utilization 87.5 70.0 87.41
• P(0), probability that the system is
empty 0.1250 0.3000 0.1259
• Lq, expected queue length 3.1168 0.8393 3.1198
• L, expected number in system 3.9918 1.5393 3.9939
• Wq, expected time in queue 0.8905 0.2398 0.7138
• W, expected total time in system 1.1405 0.4398 0.9
138

31
Payback Period Calculation
• Increase in
• Arrivals per hour 0.871
• Profit per hour 13.06
• Profit per day 130.61
• Profit per week 783.63
• Cost of Machine 5,000
• Payback Period 6.381 weeks

32
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).

33
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

34
End of Chapter 13