table of contents chapter 11 (queueing...

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


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


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.


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.


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.


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


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.


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.


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



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


Examples of Transportation Service Systems



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


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.


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.


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


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 Dupit Corp. Problem


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.


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.)


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 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 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


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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.


M/M/1 Model for John Phixitt’s Suggested Approach

(Reduce Machines/Rep)


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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 – )]


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.


M/G/1 Model for the VP of Engineering Approach

(New Equipment)


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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)


Table of Contents


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.


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


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


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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