ISE 195
Introduction to Industrial &
Systems Engineering
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Queuing Theory
(Topics in ISE 484)
Ref: Applied Management Science, 2nd Ed., Lawrence & Pasternack
Slides based on textbook slides
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Queuing is Familiar
3
Queuing Occurs in Service Systems
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Introduction
• Queuing is the study of waiting lines, or
queues.
• The objective of queuing analysis is to
design systems that enable organizations
to perform optimally according to some
criterion.
• Possible Criteria
– Maximum Profits or Minimize Costs
– Desired Service Level.
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Application Areas for Queuing
• Transportation Systems
– Cars waiting in traffic
– Planes waiting on runways
• Factories
– Jobs waiting for machines
– Machines waiting for repairs
• Service Systems
– Customers at retails stores or restaurants
– Paperwork in a business process
– Patients at a hospital emergency room or waiting for
an operating room
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Essential Tradeoff in Queuing
• What is the value of improved service to a
customer?
– Higher levels of service either require more
resources (people, machines, technology) at
higher cost
– Saving money on resources often leads to
longer waiting times, and lower levels of
service
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Introduction
• Analyzing queuing systems requires a
clear understanding of the appropriate
service measurement.
• Possible service measurements
– Average time a customer spends in line.
– Average length of the waiting line.
– The probability that an arriving customer must
wait for service.
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Elements of the Queuing Process
• A queuing system consists of three basic
components:
– Arrivals: Customers arrive according to some
arrival pattern.
– Waiting in a queue: Arriving customers may
have to wait in one or more queues for service.
– Service: Customers receive service and leave
the system.
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Fundamental Queuing Insights
• Over long periods of time, customers cannot
arrive at a rate that is higher than the overall rate
of service, or queues get large!
– Queuing theory helps us estimate the waiting time
– How many resources are enough to guarantee
reasonable queues?
• During short busy periods, rate of arrivals can
outpace rate of service, but service will suffer.
– How will we manage busy periods?
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Ways to Manage Queues
• Segment the customers
– Split off customers that can be handled
quickly and uniformly
• Inform customers of what to expect
• Divert the customer’s attention while
waiting
• Encourage customers to come during slow
times
• Train servers to be friendly
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The Arrival Process
• There are two possible types of arrival
processes
– Deterministic arrival process.
– Random arrival process.
• The random process is more common in
businesses.
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The Arrival Process
• Under three conditions the arrivals can be
modeled as a Poisson process
– Orderliness : one customer, at most, will arrive
during any time interval.
– Stationarity : for a given time frame, the probability
of arrivals within a certain time interval is the same
for all time intervals of equal length.
– Independence : the arrival of one customer has no
influence on the arrival of another.
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The Poisson Arrival Process
ke- lt
(lt)
P(X = k) =
k!
Where
l = mean arrival rate per time unit.
t = the length of the interval.
e = 2.7182818 (the base of the natural logarithm).
k! = k (k -1) (k -2) (k -3) … (3) (2) (1).
Note: Arrivals according to Poisson Process have exponential interarrivals which turns out to provide quite nice theoretical results. 14
HANK’s HARDWARE
• Arrival Process
– Customers arrive at Hank’s Hardware according
to a Poisson distribution.
– Between 8:00 and 9:00 A.M. an average of 6
customers arrive at the store.
– Hank would like to know what is the probability
that k customers will arrive between 8:00 and
8:30 in the morning (k = 0, 1, 2,…)?
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HANK’s HARDWARE
Illustration of the Poisson distribution.
• Input to the Poisson
distribution
P(X = k)
l = 6 customers per hour.
t = 0.5 hour.
lt = (6)(0.5) = 3.
X = Number of Arrivals in
lt = 3 time units
10k23
(lt)
e
P(X = 01k23 ) =
k2!
1!
0!
3!!
lt
0 1 2 3 4 5 6 7 8
k
= 0.224042
0.149361
0.049787
0.224042
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Waiting Line Characteristics
• Factors that influence the modeling of
queues
–Line configuration
– Priority
–Jockeying
– Tandem Queues
–Balking
– Homogeneity
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Line Configuration
• A single service queue.
• Multiple service queue with single
waiting line.
• Multiple service queue with multiple
waiting lines.
• Tandem queue (multistage service
system).
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Waiting Line Configurations
Single Server
Multiple Server
Single Line
Multiple Server
Multiple Lines
Tandem Queue
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Jockeying and Balking
• Jockeying occurs when customers
switch lines once they perceive that
another line is moving faster.
• Balking occurs if customers avoid
joining the line when they perceive the
line to be too long.
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Priority Rules
• These rules select the next customer for
service.
• There are several commonly used rules:
– First come first served (FCFS).
– Last come first served (LCFS).
– Estimated service time.
– Random selection of customers for service.
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Tandem Queues
• These are multi-server systems.
• A customer needs to visit several service
stations (usually in a distinct order) to
complete the service process.
• Examples
– Patients in an emergency room.
– Passengers prepare for the next flight.
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Homogeneity
• A homogeneous customer population is
one in which customers require
essentially the same type of service.
• A non-homogeneous customer
population is one in which customers
can be categorized according to:
– Different arrival patterns
– Different service treatments (such as
hospital)
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The Service Process
• In most business situations, service time
varies widely among customers.
• When service time varies, it is treated as a
random variable.
• The exponential probability distribution is
used sometimes to model customer
service time.
– Simple model; useful for analysis purposes.
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HANK’s HARDWARE
• Service time
– Hank’s estimates the average service time to
be 1/m = 4 minutes per customer.
– Service time follows an exponential
distribution.
– Define:
• X = Amount of time it takes to serve the next customer
– What is the probability that it will take less
than 3 minutes to serve the next customer?
• What is Pr( X < 3 minutes )?
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HANK’s HARDWARE
• Using Excel for the Exponential
Probabilities
– The mean number of customers served per
minute is ¼ = ¼(60) = 15 customers per
hour.
– P(X < .05 hours) = 1 – e-(15)(.05) = ?
– From Excel we have:
3 minutes = .05 hours
• EXPONDIST(.05,15,TRUE) = .5276
– In ISE 484 you will learn the details of these
calculations; for now we are simply
illustrating how the results would be used 26
HANK’s HARDWARE
• What is your recommendation to HANK’s
management based on this?
– Management wants to have 95% of
customers served in less than 3 minutes
– You recommend a detailed study of the
service process to implement some lean
techniques to improve the level of customer
service
– Notice, this is just the service time, and
doesn’t include any of the queuing time
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The Exponential Distribution
• Characteristics
• The “Lack of Memory” property.
– No additional information about the time left for the
completion of a service, is gained by recording the time
elapsed since the service started.
– For Hank’s, the probability of completing a service within
the next 3 minutes is (0.52763) independent of how long
the customer has been served already.
• The Exponential and the Poisson distributions are
related to one another.
– If customer arrivals follow a Poisson distribution with
mean rate l, their interarrival times are exponentially
distributed with mean time 1/l.
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Performance Measures in Queuing Systems
• Performance can be measured by
focusing on:
– Customers in queue.
– Customers in the system.
• Performance is measured for a system in
steady state.
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Steady State Performance Measures
P0 = Probability that there are no customers in the system.
Pn = Probability that there are “n” customers in the system.
L = Average number of customers in the system.
Lq = Average number of customers in the queue.
W = Average time a customer spends in the system.
Wq = Average time a customer spends in the queue.
Pw = Probability that an arriving customer must wait
for service.
r = Utilization rate for each server
(the percentage of time that each server is busy).
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MARY’s SHOES
• Customers arrive at Mary’s Shoes every 12
minutes on the average, according to a
Poisson process.
• Service time is exponentially distributed with
an average of 8 minutes per customer.
• Management is interested in determining the
performance measures for this service
system.
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MARY’s SHOES
-
• Solution
– Input
l = 1/12 customers per minute = 60/12 = 5 per hour.
m = 1/ 8 customers per minute = 60/ 8 = 7.5 per hour.
– Performance Calculations
m –l = 7.5 – 5 = 2.5 per hr.
P0 = 1 - (l/m) = 1 - (5/7.5) = 0.3333
Pn = [1 - (l/m)](l/m)n = (0.3333)(0.6667)n
L = l/(m - l) = 2
Lq = l2/[m(m - l)] = 1.3333
W = 1/(m - l) = 0.4 hours = 24 minutes
Wq = l/[m(m - l)] = 0.26667 hours = 16 minutes
P(X<10min) = 1 – e-2.5(10/60)
= .565
Pw = l/m = 0.6667
r = l/m = 0.6667
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Excel Queueing Spreadsheet
Queueing Model Calculator
The formulas used here are for the M/M/m system. The arrival
process is assumed to be Poisson (exponential inter-arrival times).
The service process is assumed to have exponential processing
times.
Cells that are this color can be changed. Results for the
Queueing model are displayed in the other cells.
System 1
System 2
1
1
1.00
1.00
1.00000000
1.00000000
0.50
0.50
0.5000
0.5000
1.00000000
1.00000000
2.00000000
2.00000000
0.50000000
0.50000000
1.00000000
1.00000000
Input Parameters
Number of Servers
Average Processing Rate
(jobs per unit time)
Average Processing Time
(time per job)
Average Arrival Rate (jobs
per time unit)
Measures of Performance
Utilization
Average Time in Queue
(time units)
Average Time in System
(time units)
Average Number in
Queue
Average Number in
System
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MARY’s SHOES
• Recommendations to management
– 16 minutes of time in queue seems excessive
– Question 1: Is the model correct?
• Compare model predictions to reality
– How does the 16 minute waiting time compare to the
goals for customer service for Mary’s shoes?
• Make recommendations to improve waiting time
– More resources
– Better processes
– Better information systems
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Tandem Queuing Systems
• In a Tandem Queuing System a customer must
visit several different servers before service is
completed.
Meats
Beverage
• Examples
– All-You-Can-Eat restaurant
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Tandem Queuing Systems
• In a Tandem Queuing System a customer must
visit several different servers before service is
completed.
Meats
Beverage
• Examples
– All-You-Can-Eat restaurant
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Tandem Queuing Systems
• In a Tandem Queuing System a customer must visit
several different servers before service is completed.
Meats
Beverage
• Examples
– All-You-Can-Eat restaurant
– A drive-in restaurant, where first you place your order, then
pay and receive it in the next window.
– A multiple stage assembly line.
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Applications of Queuing
•
•
•
•
•
•
Determine number of servers
Examine line configurations
Evaluate efficiency of process
Determine storage requirements
Various cost benefit analyses
Bound performance of complex systems
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Other Applications
• In ISE 483 (Integrated Systems for
Manufacturing)
– Use Queuing models to decide number of machines,
workers in a system
– Extend queuing models to include impact of downtime
(breakdowns, maintenance)
• In ISE 471 (Simulation)
– Use queuing models to get a quick “guess” at what
the result of a simulation should be close to
– Helps in debugging and validating models
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Questions?
Ref: Applied Management Science, 2nd Ed., Lawrence & Pasternack
Slides based on textbook slides
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