Table of Contents
Chapter 11 (Queueing Models)
11.1
1. Elements of a Queueing Model (Section 11.1) 11.2–11.12
2. Some Examples of Queueing Systems (Section 11.2) 11.13–11.15
3. Measures of Performance for Queueing Systems (Section 11.3) 11.16–11.19
4. A Case Study: The Dupit Corp. Problem (Section 11.4) 11.20–11.22
5. Some Single-Server Queueing Models (Section 11.5) 11.23–11.32
6. Some Multiple-Server Queueing Models (Section 11.6) 11.33–11.40
7. Priority Queueing Models (Section 11.7) 11.41–11.48
8. Some Insights about Designing Queueing Systems (Section 11.8) 11.49–11.51
9. Economic Analysis of the Number of Servers to Provide (Section 11.9) 11.52–11.55
Chapter 11 (Queueing Models)
11.2
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Learning Objective
1. Describe the elements of a queueing model
2. Identify the characteristics of the probability distributions that are commonly used in queueing models
3. Give many examples of various types of queueing systems that are commonly encountered.
4. Identify the key measures of the performance for queueing system and relationships between these measures
5. Describe the main types of basic queueing models
6. Determine which queueing model is most appropriate from a description of the queueing system being considered
7. Apply a queueing model to determine the key measures of performance for a queueing system
8. Describe how differences in the importance of customers can be incorporate into priority queueing models
9. Describe some key insights that queueing models provide about how queueing systems hould be designed.
10. Apply economic analysis to determine how many servers should be provided in a queueing system
© The McGraw-Hill Companies, Inc., 2008 11.3
A Basic Queueing System
© The McGraw-Hill Companies, Inc., 2008 11.4
Customers
Queue
Served Customers
Queueing System
Service facility
SSSS
CCCC
C C C C C C C
Served Customers
Herr Cutter’s Barber Shop
Herr Cutter is a German barber who runs a one-man barber shop.
Herr Cutter opens his shop at 8:00 A.M.
The table shows his queueing system in action over a typical morning.
© The McGraw-Hill Companies, Inc., 2008 11.5
Customer
Time of
Arrival
Haicut
Begins
Duration
of Haircut
Haircut
Ends
1 8:03 8:03 17 minutes 8:20
2 8:15 8:20 21 minutes 8:41
3 8:25 8:41 19 minutes 9:00
4 8:30 9:00 15 minutes 9:15
5 9:05 9:15 20 minutes 9:35
6 9:43 — — —
Arrivals
The time between consecutive arrivals to a queueing system are called the
interarrival times.
The expected number of arrivals per unit time is referred to as the mean arrival
rate.
The symbol used for the mean arrival rate is
λ = Mean arrival rate for customers coming to the queueing system
where l is the Greek letter lambda.
The mean of the probability distribution of interarrival times is
1 / λ = Expected interarrival time
Most queueing models assume that the form of the probability distribution of
interarrival times is an exponential distribution.
© The McGraw-Hill Companies, Inc., 2008 11.6
Evolution of the Number of Customers
© The McGraw-Hill Companies, Inc., 2008
11.7
20 40 60 80
1
2
3
4
Number of Customers in the System
0
Time (in minutes)
100
The Exponential Distribution for Interarrival Times
© The McGraw-Hill Companies, Inc., 2008
11.8
Mean Time0
Properties of the Exponential Distribution
There is a high likelihood of small interarrival times, but a small chance of a
very large interarrival time. This is characteristic of interarrival times in
practice.
For most queueing systems, the servers have no control over when customers
will arrive. Customers generally arrive randomly.
Having random arrivals means that interarrival times are completely
unpredictable, in the sense that the chance of an arrival in the next minute is
always just the same.
The only probability distribution with this property of random arrivals is the
exponential distribution.
The fact that the probability of an arrival in the next minute is completely
uninfluenced by when the last arrival occurred is called the lack-of-memory
property. © The McGraw-Hill Companies, Inc., 2008 11.9
The Queue
The number of customers in the queue (or queue size) is the number of customers waiting for service to begin.
The number of customers in the system is the number in the queue plus the number currently being served.
The queue capacity is the maximum number of customers that can be held in the queue.
An infinite queue is one in which, for all practical purposes, an unlimited number of customers can be held there.
When the capacity is small enough that it needs to be taken into account, then the queue is called a finite queue.
The queue discipline refers to the order in which members of the queue are selected to begin service.
The most common is first-come, first-served (FCFS).
Other possibilities include random selection, some priority procedure, or even last-come, first-served.
© The McGraw-Hill Companies, Inc., 2008 11.10
Service
When a customer enters service, the elapsed time from the beginning to the end
of the service is referred to as the service time.
Basic queueing models assume that the service time has a particular probability
distribution.
The symbol used for the mean of the service time distribution is
1 / µ = Expected service time
where m is the Greek letter mu.
The interpretation of m itself is the mean service rate.
µ = Expected service completions per unit time for a single busy server
© The McGraw-Hill Companies, Inc., 2008 11.11
Some Service-Time Distributions
Exponential Distribution
The most popular choice.
Much easier to analyze than any other.
Although it provides a good fit for interarrival times, this is much less true for
service times.
Provides a better fit when the service provided is random than if it involves a
fixed set of tasks.
Standard deviation: σ = Mean
Constant Service Times
A better fit for systems that involve a fixed set of tasks.
Standard deviation: σ = 0.
© The McGraw-Hill Companies, Inc., 2008 11.12
Labels for Queueing Models
To identify which probability distribution is being assumed for service times (and
for interarrival times), a queueing model conventionally is labeled as follows:
Distribution of service times
— / — / — Number of Servers
Distribution of interarrival times
The symbols used for the possible distributions are
M = Exponential distribution (Markovian)
D = Degenerate distribution (constant times)
GI = General independent interarrival-time distribution (any distribution)
G = General service-time distribution (any arbitrary distribution) © The McGraw-Hill Companies, Inc., 2008 11.13
Summary of Usual Model Assumptions
1. Interarrival times are independent and identically distributed according to a specified
probability distribution.
2. All arriving customers enter the queueing system and remain there until service has been
completed.
3. The queueing system has a single infinite queue, so that the queue will hold an unlimited number
of customers (for all practical purposes).
4. The queue discipline is first-come, first-served.
5. The queueing system has a specified number of servers, where each server is capable of serving
any of the customers.
6. Each customer is served individually by any one of the servers.
7. Service times are independent and identically distributed according to a specified probability
distribution.
© The McGraw-Hill Companies, Inc., 2008 11.14
Examples of Commercial Service Systems
That Are Queueing Systems
Type of System Customers Server(s)
Barber shop People Barber
Bank teller services People Teller
ATM machine service People ATM machine
Checkout at a store People Checkout clerk
Plumbing services Clogged pipes Plumber
Ticket window at a movie theater People Cashier
Check-in counter at an airport People Airline agent
Brokerage service People Stock broker
Gas station Cars Pump
Call center for ordering goods People Telephone agent
Call center for technical assistance People Technical representative
Travel agency People Travel agent
Automobile repair shop Car owners Mechanic
Vending services People Vending machine
Dental services People Dentist
Roofing Services Roofs Roofer © The McGraw-Hill Companies, Inc., 2008 11.15
Examples of Internal Service Systems
That Are Queueing Systems
Type of System Customers Server(s)
Secretarial services Employees Secretary
Copying services Employees Copy machine
Computer programming services Employees Programmer
Mainframe computer Employees Computer
First-aid center Employees Nurse
Faxing services Employees Fax machine
Materials-handling system Loads Materials-handling unit
Maintenance system Machines Repair crew
Inspection station Items Inspector
Production system Jobs Machine
Semiautomatic machines Machines Operator
Tool crib Machine operators Clerk
© The McGraw-Hill Companies, Inc., 2008 11.16
Examples of Transportation Service Systems
That Are Queueing Systems
Type of System Customers Server(s)
Highway tollbooth Cars Cashier
Truck loading dock Trucks Loading crew
Port unloading area Ships Unloading crew
Airplanes waiting to take
off
Airplanes Runway
Airplanes waiting to land Airplanes Runway
Airline service People Airplane
Taxicab service People Taxicab
Elevator service People Elevator
Fire department Fires Fire truck
Parking lot Cars Parking space
Ambulance service People Ambulance
© The McGraw-Hill Companies, Inc., 2008 11.17
Choosing a Measure of Performance
Managers who oversee queueing systems are mainly concerned with two
measures of performance:
How many customers typically are waiting in the queueing system?
How long do these customers typically have to wait?
When customers are internal to the organization, the first measure tends to be
more important.
Having such customers wait causes lost productivity.
Commercial service systems tend to place greater importance on the second
measure.
Outside customers are typically more concerned with how long they have to wait
than with how many customers are there.
© The McGraw-Hill Companies, Inc., 2008 11.18
Defining the Measures of Performance
L = Expected number of customers in the system, including those being served
(the symbol L comes from Line Length).
Lq = Expected number of customers in the queue, which excludes customers being
served.
W = Expected waiting time in the system (including service time) for an individual
customer (the symbol W comes from Waiting time).
Wq = Expected waiting time in the queue (excludes service time) for an individual
customer.
These definitions assume that the queueing system is in a steady-state condition.
© The McGraw-Hill Companies, Inc., 2008 11.19
Relationship between L, W, Lq, and Wq
Since 1/µ is the expected service time
W = Wq + 1/µ
Little’s formula states that
L = λW
and
Lq = λWq
Combining the above relationships leads to
L = Lq + λ/µ
© The McGraw-Hill Companies, Inc., 2008 11.20
Using Probabilities as Measures of Performance
In addition to knowing what happens on the average, we may also be interested in worst-case
scenarios.
What will be the maximum number of customers in the system? (Exceeded no more than, say, 5%
of the time.)
What will be the maximum waiting time of customers in the system? (Exceeded no more than, say,
5% of the time.)
Statistics that are helpful to answer these types of questions are available for some queueing
systems:
Pn = Steady-state probability of having exactly n customers in the system.
P(W ≤ t) = Probability the time spent in the system will be no more than t.
P(Wq ≤ t) = Probability the wait time will be no more than t.
Examples of common goals:
No more than three customers 95% of the time: P0 + P1 + P2 + P3 ≥ 0.95
No more than 5% of customers wait more than 2 hours: P(W ≤ 2 hours) ≥ 0.95
© The McGraw-Hill Companies, Inc., 2008 11.21
The Dupit Corp. Problem
© The McGraw-Hill Companies, Inc., 2008 11.22
The Dupit Corp. Problem
The Dupit Corporation is a longtime leader in the office photocopier
marketplace.
Dupit’s service division is responsible for providing support to the customers by
promptly repairing the machines when needed. This is done by the company’s
service technical representatives, or tech reps.
Current policy: Each tech rep’s territory is assigned enough machines so that
the tech rep will be active repairing machines (or traveling to the site) 75% of
the time.
A repair call averages 2 hours, so this corresponds to 3 repair calls per day.
Machines average 50 workdays between repairs, so assign 150 machines per rep.
Proposed New Service Standard: The average waiting time before a tech rep
begins the trip to the customer site should not exceed two hours.
© The McGraw-Hill Companies, Inc., 2008 11.23
Alternative Approaches to the Problem
Approach Suggested by John Phixitt: Modify the current policy by decreasing
the percentage of time that tech reps are expected to be repairing machines.
Approach Suggested by the Vice President for Engineering: Provide new
equipment to tech reps that would reduce the time required for repairs.
Approach Suggested by the Chief Financial Officer: Replace the current one-
person tech rep territories by larger territories served by multiple tech reps.
Approach Suggested by the Vice President for Marketing: Give owners of the
new printer-copier priority for receiving repairs over the company’s other
customers. © The McGraw-Hill Companies, Inc., 2008 11.24
The Queueing System for Each Tech Rep
The customers: The machines needing repair.
Customer arrivals: The calls to the tech rep requesting repairs.
The queue: The machines waiting for repair to begin at their sites.
The server: The tech rep.
Service time: The total time the tech rep is tied up with a machine, either
traveling to the machine site or repairing the machine. (Thus, a machine is
viewed as leaving the queue and entering service when the tech rep begins the
trip to the machine site.)
© The McGraw-Hill Companies, Inc., 2008 11.25
Notation for Single-Server Queueing Models
λ = Mean arrival rate for customers
= Expected number of arrivals per unit time
1/ λ = expected interarrival time
µ = Mean service rate (for a continuously busy server)
= Expected number of service completions per unit time
1/ µ = expected service time
ρ = the utilization factor
= the average fraction of time that a server is busy serving customers
= λ / µ
© The McGraw-Hill Companies, Inc., 2008 11.26
The M/M/1 Model
Assumptions
1. Interarrival times have an exponential distribution with a mean of 1/l.
2. Service times have an exponential distribution with a mean of 1/m.
3. The queueing system has one server.
• The expected number of customers in the system is
L = ρ / (1 – ρ) = λ / (µ – λ)
• The expected waiting time in the system is
W = (1 / λ)L = 1 / (µ – λ)
• The expected waiting time in the queue is
Wq = W – 1/ µ = λ / [µ(µ – λ)]
• The expected number of customers in the queue is
Lq = λ Wq = λ 2 / [µ(µ – λ)] = ρ 2 / (1 – ρ)
© The McGraw-Hill Companies, Inc., 2008 11.27
The M/M/1 Model
The probability of having exactly n customers in the system is
Pn = (1 – ρ) ρn
Thus,
P0 = 1 – ρ
P1 = (1 – ρ) ρ
P2 = (1 – ρ) ρ2
:
:
The probability that the waiting time in the system exceeds t is
P(W > t) = e–m(1–ρ)t for t ≥ 0
The probability that the waiting time in the queue exceeds t is
P(Wq > t) = ρe–m(1–ρ)t for t ≥ 0 © The McGraw-Hill Companies, Inc., 2008 11.28
M/M/1 Queueing Model for the Dupit’s Current Policy
© The McGraw-Hill Companies, Inc., 2008 11.29
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B C D E G H
Data Results
3 (mean arrival rate) L = 3
4 (mean service rate) Lq = 2.25
s = 1 (# servers)
W = 1
Pr(W > t) = 0.368 Wq = 0.75
when t = 1
0.75
Prob(Wq > t) = 0.276
when t = 1 n Pn
0 0.25
1 0.1875
2 0.1406
3 0.1055
4 0.0791
5 0.0593
6 0.0445
7 0.0334
8 0.0250
9 0.0188
10 0.0141
John Phixitt’s Approach (Reduce Machines/Rep)
The proposed new service standard is that the average waiting
time before service begins be two hours (i.e., Wq ≤ 1/4 day).
John Phixitt’s suggested approach is to lower the tech rep’s
utilization factor sufficiently to meet the new service
requirement.
Lower ρ = λ / µ, until Wq ≤ 1/4 day,
where
λ = (Number of machines assigned to tech rep) / 50.
© The McGraw-Hill Companies, Inc., 2008 11.30
M/M/1 Model for John Phixitt’s Suggested Approach
(Reduce Machines/Rep)
© The McGraw-Hill Companies, Inc., 2008 11.31
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B C D E G H
Data Results
2 (mean arrival rate) L = 1
4 (mean service rate) Lq = 0.5
s = 1 (# servers)
W = 0.5
Pr(W > t) = 0.135 Wq = 0.25
when t = 1
0.5
Prob(Wq > t) = 0.068
when t = 1 n Pn
0 0.5
1 0.25
2 0.1250
3 0.0625
4 0.0313
5 0.0156
6 0.0078
7 0.0039
8 0.0020
9 0.0010
10 0.0005
M/M/1 Model for John Phixitt’s Suggested Approach
(Reduce Machines/Rep)
Desrease the number of machines assigned to each
tech rep from 150 to 100 require to hire nearly
5000 more tech reps. The addition payroll cost
about $270 Mil annually. The addition cost of
hiring, training, etc equivalent to about $30 mil
annually.
The addition cost of Phixitt’s approach is around
$300 mil annually. © The McGraw-Hill Companies, Inc., 2008 11.32
The M/G/1 Model
Assumptions
1. Interarrival times have an exponential distribution with a mean of 1/λ.
2. Service times can have any probability distribution. You only need the mean (1/µ) and
standard deviation (σ).
3. The queueing system has one server.
• The probability of zero customers in the system is
P0 = 1 – ρ
• The expected number of customers in the queue is
Lq = [λ2 σ 2 + ρ2] / [2(1 – ρ)]
• The expected number of customers in the system is
L = Lq + ρ
• The expected waiting time in the queue is
Wq = Lq / λ
• The expected waiting time in the system is
W = Wq + 1/µ © The McGraw-Hill Companies, Inc., 2008 11.33
The Values of s and Lq for the M/G/1 Model
with Various Service-Time Distributions
Distribution Mean s Model Lq
Exponential 1/ 1/ M/M/1 2 / (1 – )
Degenerate (constant) 1/ 0 M/D/1 (1/2) [2 / (1 – )]
Erlang, with shape parameter k 1/ (1/k) (1/) M/Ek/1 (k+1)/(2k) [2 / (1 – )]
© The McGraw-Hill Companies, Inc., 2008 11.34
VP for Engineering Approach (New Equipment)
The proposed new service standard is that the average waiting
time before service begins be two hours (i.e., Wq ≤ 1/4 day).
The Vice President for Engineering has suggested providing tech
reps with new state-of-the-art equipment that would reduce the
time required for the longer repairs.
After gathering more information, they estimate the new
equipment would have the following effect on the service-time
distribution:
Decrease the mean from 1/4 day to 1/5 day.
Decrease the standard deviation from 1/4 day to 1/10 day.
© The McGraw-Hill Companies, Inc., 2008 11.35
M/G/1 Model for the VP of Engineering Approach
(New Equipment)
© The McGraw-Hill Companies, Inc., 2008 11.36
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B C D E F G
Data Results
3 (mean arrival rate) L = 1.163
0.2 (expected service time) Lq = 0.563
s 0.1 (standard deviation)
s = 1 (# servers) W = 0.388
Wq = 0.188
0.6
P0 = 0.4
M/G/1 Model for the VP of Engineering Approach
(New Equipment)
This approach meet the requirement for Wq =< 0.25 (Wq = 0.188) this is the big reduction from Wq = 0.75 at the beginning.
Thus this approach also expensive with a one-time cost of approximately $500 million (about $50,000 for new equipment per tech rep)
© The McGraw-Hill Companies, Inc., 2008 11.37
Table of Contents
Chapter 11 (Queueing Models)
11.38
1. Elements of a Queueing Model (Section 11.1) 11.2–11.12
2. Some Examples of Queueing Systems (Section 11.2) 11.13–11.15
3. Measures of Performance for Queueing Systems (Section 11.3) 11.16–11.19
4. A Case Study: The Dupit Corp. Problem (Section 11.4) 11.20–11.22
5. Some Single-Server Queueing Models (Section 11.5) 11.23–11.32
6. Some Multiple-Server Queueing Models (Section 11.6) 11.33–11.40
7. Priority Queueing Models (Section 11.7) 11.41–11.48
8. Some Insights about Designing Queueing Systems (Section 11.8) 11.49–11.51
9. Economic Analysis of the Number of Servers to Provide (Section 11.9) 11.52–11.55
Case 11-2 Reducing In-process Inventory
11.39
Jerry Carstairs. Plant’s production manager
Jim Wells. Vice Presedent
Some wing section are in queue waiting for inspection. They need to find the solution to prevent this problem.
Case 11-2 Reducing In-process Inventory
Each of 10 identical presses is being used to form wing section out of
large sheets of specially processed metal.
The sheets arrive randomly at a mean rate of seven per hour.
Time required by a press to form a wing section out of a sheet has an
exponential distribution with a mean of one hour.
A single inspector has a full-time job of inspecting these wing section to
make sure they meet specifications. Each inspection takes her 71/2
minutes so she can inspect 8 wings per hour
11.40
Case 11-2 Reducing In-process Inventory
Cost of in-process inventory is estimate to be $8 per hour for each metal sheet at the
presses of each wing section at the inpection station.
Jerry has made two alternative proposals to reduce the average level if in-process
inventory:
Proposal 1: is to use slightly less power for the presses (which increase their average time to form a wing
section to 1,2 hour) so that inspector can keep up with their output better. This also reduce the cost of each
machine from $7.00 to $6.50 per hour.
Proposal 2: substitute a certain younger inspector for this task, he is somewhat faster with the mean of 7,2
minutes and standard deviation of 5 minutes. This inpector call for a total compensation of $19 per hour in
comparison with $17 per hour pay for current inspector.
11.41
Case 11-2 Reducing In-process Inventory
a) To provide a basis of comparison, begin by evaluating the status quo. Determine the expected amount of in-process
inventory, the presses, and the inspector
b) What would be the effect of proposal 1? Why? Make a specific comparison to the results from part a. Expalin this
outcome to Jerry Carstairs.
c) Determine the effect of proposal 2. Make specific comparison to the results from part a. Expalin this outcome to Jerry
Carstairs.
d) Make your recommendations for reducing the average level of in-process inventory at the inspection station and at the
group of machines. Bespecific in your recommendations and support them with quantitative analysis lit that done in
part a. Make specific comparisons to the results from part a, and cite the improvements that your recommendations
would yield.
11.42
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