107,99 €
Mehr erfahren.
- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Presents an introduction to differential equations, probability, and stochastic processes with real-world applications of queues with delay and delayed network queues
Featuring recent advances in queueing theory and modeling, Delayed and Network Queues provides the most up-to-date theories in queueing model applications. Balancing both theoretical and practical applications of queueing theory, the book introduces queueing network models as tools to assist in the answering of questions on cost and performance that arise throughout the life of a computer system and signal processing. Written by well-known researchers in the field, the book presents key information for understanding the essential aspects of queues with delay and networks of queues with unreliable nodes and vacationing servers.
- Beginning with simple analytical fundamentals, the book contains a selection of realistic and advanced queueing models that address current deficiencies. In addition, the book presents the treatment of queues with delay and networks of queues, including possible breakdowns and disruptions that may cause delay. Delayed and Network Queues also features:
- Numerous examples and exercises with applications in various fields of study such as mathematical sciences, biomathematics, engineering, physics, business, health industry, and economics
- A wide array of practical applications of network queues and queueing systems, all of which are related to the appropriate stochastic processes
- Up-to-date topical coverage such as single- and multiserver queues with and without delays, along with the necessary fundamental coverage of probability and difference equations
- Discussions on queueing models such as single- and multiserver Markovian queues with balking, reneging, delay, feedback, splitting, and blocking, as well as their role in the treatment of networks of queues with and without delay and network reliability
Delayed and Network Queues is an excellent textbook for upper-undergraduate and graduate-level courses in applied mathematics, queueing theory, queueing systems, probability, and stochastic processes. The book is also an ideal reference for academics and practitioners in mathematical sciences, biomathematics, operations research, management, engineering, physics, business, economics, health industry, and industrial engineering.
Aliakbar Montazer Haghighi, PhD, is Professor and Head of the Department of Mathematics at Prairie View A&M University, USA, as well as founding Editor-in-Chief of Applications and Applied Mathematics: An International Journal (AAM). His research interests include probability, statistics, stochastic processes, and queueing theory. Among his research publications and books, Dr. Haghighi is the coauthor of Difference and Differential Equations with Applications in Queueing Theory (Wiley, 2013).
Dimitar P. Mishev, PhD, is Professor in the Department of Mathematics at Prairie View A&M University, USA. His research interests include differential and difference equations and queueing theory. The author of numerous research papers and three books, Dr. Mishev is the coauthor of Difference and Differential Equations with Applications in Queueing Theory (Wiley, 2013).
Sie lesen das E-Book in den Legimi-Apps auf:
Seitenzahl: 582
Veröffentlichungsjahr: 2016
Ähnliche
Table of Contents
Cover
Title Page
Copyright
Dedication
Preface
Chapter 1: Preliminaries
1.1 Basics of Probability
1.2 Discrete Random Variables and Distributions
1.3 Discrete Moments
1.4 Continuous Random Variables, Density, and Cumulative Distribution Functions
1.5 Continuous Random Vector
1.6 Functions of Random Variables
1.7 Continuous Moments
1.8 Difference Equations
1.9 Methods of Solving Linear Difference Equations with Constant Coefficients
Exercises
Chapter 2: Stochastic Processes
2.1 Introduction and Basic Definitions
2.2 Markov Chain
2.3 Markov Process
2.4 Random Walk
2.5 Up-and-Down Biased Coin Design as a Random Walk
Exercises
Chapter 3: Birth and Death Processes
3.1 Overviews of the Birth and Death Processes
3.2 Finite B–D Process
3.3 Pure Birth Process (Poisson Process)
3.4 Pure Death Process (Poisson Death Process)
Exercises
Chapter 4: Standard Queues
4.1 Introduction of Queues (General Birth and Death Process)
4.2 Remarks on Non-Markovian Queues
4.3 Stationary
M/M/
1 Queueing Process
4.4 A Parallel
M
/
M
/
C/K
with Baking and Reneging
4.5 Stationary
M/M/
1/
K
Queueing Process
4.6 Busy Period of an
M
/
M
/1/
K
Queue
4.7 Stationary
M/M/
1 and
M
/
M
/1/
K
Queueing Processes with Feedback
4.8 Queues with Bulk Arrivals and Batch Service
4.9 A Priority Queue with Balking and Reneging
4.10 Discrete Time
M/M/
1 Queueing Process, Combinatorics Method (Lattice Paths)
4.11 Stationary
M/M/
C Queueing Process
Exercises
Chapter 5: Queues with Delay
5.1 Introduction
5.2 A Queuing System with Delayed Service
5.3 An
M
/
G
/1 Queue with Server Breakdown and with Multiple Working Vacation
5.4 A Bulk Queuing System Under
N
-Policy with Bilevel Service Delay Discipline and Start-Up Time
5.5 Interrelationship between
N
-policy
M/G/
1/
K
and
F
-policy
G/M/
1/
K
Queues with Start-up Time
5.6 A Transient
M
/
M
/1 Queue Under (
M
,
N
)-Policy, Lattice Path Method
5.7 Stationary
M/M/
1 Queuing Process with Delayed Feedback
5.8 Single-Server Queue with Unreliable Server and Breakdowns with an Optional Second Service
5.9 A Bulk Arrival Retrial Queue with Unreliable Server
5.10 Multiserver Queue with Retrial Feedback Queuing System with Two Orbits
5.11 Steady-State Stability Condition of a Retrial Queuing System with Two Orbits, Reneging, and Feedback
5.12 Batch Arrival Queue with General Service in two Fluctuating Modes and Reneging During Vacation and Breakdowns
Exercises
Chapter 6: Networks of Queues with Delay
6.1 Introduction to Networks of Queues
6.2 Historical Notes on Networks of Queues
6.3 Jackson's Network of Queues
6.4 Robustness of Networks of Queues
6.5 A
MAP
Single-Server Queueing System with Delayed Feedback as a Network of Queues
6.6 Unreliable Networks of Queueing System Models
6.7 Assessment of Reliability of a Network of Queues
6.8 Effect of Network Service Breakdown
Exercises
References
Index
End User License Agreement
Pages
xi
xii
xiii
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
Guide
Cover
Table of Contents
Preface
Begin Reading
List of Illustrations
Chapter 2: Stochastic Processes
Figure 2.1 One-dimensional random walk.
Figure 2.2 Treatment rule, biased coin design I (BCD I).
Chapter 3: Birth and Death Processes
Figure 3.1 Transition diagram of a birth and death processes. Arrows represent the possible transitions between states and the labels on the arrows represent the state transition rates between states.
Figure 3.2 Transition diagram of the dual process of the birth–death process of Figure 3.1. Arrows represent the possible transitions between states and the labels on the arrows represent the state transition rates between states.
Figure 3.3 Transition diagram of a birth and death process with staying put allowed. Arrows represent the possible transitions between states and the labels on the arrows represent the state transition rates between states.
Figure 3.4 Transition diagram of the dual birth–death process depicted in Figure 3.3. It is an absorbing B–D at states −1 and
K
. Arrows represent the possible transitions between states and the labels on the arrows represent the state transition rates between states.
Chapter 4: Standard Queues
Figure 4.1 Single-server Markovian queueing process, exponential interarrival times, and exponential service times; = average arrival rate, = average service rate, = average service time.
Figure 4.2 Lattice path for the ballot problem with eight and five votes for candidates A and B, respectively
Figure 4.3 Lattice path representation of random walk.
Chapter 5: Queues with Delay
Figure 5.1 Probability of busy period approaching 1 as time becomes large;
λ
= 1,
μ
= 2.
Figure 5.2 Probability of busy period approaching 1 as time becomes large;
λ
= 1,
μ
= 10.
Figure 5.3 Difference between busy period of model discussed and standard
M
/
M
/1;
λ
= 1,
μ
= 2,
τ
= 0.09.
Figure 5.4 Difference between busy period of model discussed and standard
M
/
M
/1;
λ
= 1,
μ
= 10,
τ
= 0.08.
Figure 5.5 Cumulative probabilities approaching 1, regardless of values of
τ
;
λ
= 1,
μ
= 2.
Figure 5.6 Cumulative probabilities approaching 1, regardless of values of
τ
;
λ
= 2,
μ
= 6.
Figure 5.7 Cumulative probabilities approaching 1, regardless of values of
τ
;
λ
= 1,
μ
= 10.
Figure 5.8 Cumulative probabilities approaching 1, regardless of values of
τ
;
λ
= 0.02,
μ
= 2.
Figure 5.9 Single processor with splitting and delayed feedback
Figure 5.10 The mean orbit size, versus
δ
.
Figure 5.11 The idle probability decreases for the increasing the service loss probability; versus
θ
.
Figure 5.12 The surface displays an upward trend; versus
λ
and
δ
.
Figure 5.13 Increase of the number of vacations and decrease of the probability of idle, cause increase of the probability of idle during retrial time and increase of probability of that server on vacation; versus
p
and
b
.
Figure 5.14 Increase of repair rate and the probability of server idle will cause decrease of the mean orbit size and the probability of server under repair; versus
α
and
γ
.
Figure 5.15 Decrease of mean orbit size when the first type probability increases; versus
p
1
and .
Figure 5.16 A queuing model with two redial and two reconnect orbits.
Figure 5.17 A queuing model with two orbits, abandonment, and feedback.
Chapter 6: Networks of Queues with Delay
Figure 6.1 Cyclic queueing system.
Figure 6.2 Busy period process state transition for station 1.
Figure 6.3 Basic operational model of a hospital system.
Figure 6.4 Jennings and Véricourt's model.
Figure 6.5 The IW model as a semi-open queueing network.
Figure 6.6 IW model as a closed Jackson network.
Figure 6.7 Single processor with infinite server-buffer, splitting and delay-general batch feedback.
Figure 6.8 Density and distribution functions of a busy period with small values of
h
.
Figure 6.9 Graph of the joint probability distribution of the number of tasks in the system.
Figure 6.10 Graph of the cumulative joint probability distribution of the number of tasks in the system.
Figure 6.11 Three-dimensional graph of the cumulative joint probability distribution of the number of tasks in the system.
Figure 6.12 Three-dimensional graph of differences between two consecutive probabilities of the queue length with
τ
= 700 and 600.
Figure 6.13 A CoginfoCom scenario.
Figure 6.14 A retrial queue with components.
Figure 6.15 Mean orbit size versus service's failure rate.
Figure 6.19 Wasted time of Intelligent entities versus service's failure rate.
Figure 6.16 Probability that the server is in limited state versus service's failure rate.
Figure 6.17 Probability that the server fails versus service's failure rate.
Figure 6.18 Mean response time of Intelligent entities versus service's failure rate.
Figure 6.20 Queueing network of Exercise 6.7.
List of Tables
Chapter 1: Preliminaries
Table 1.1 pmf
Table 1.2
p
X
,
Y
=
P
(
X
=
x
and
Y
=
y
)
Table 1.3 Laplace Transforms of Some Elementary Functions
Chapter 2: Stochastic Processes
Table 2.1 First Passage Time Probabilities
Table 2.2 Response Function Values Under Logistic CDF
Table 2.3 Target Quantile Values,
μ
, Under Logistic CDF
Table 2.4 Target Quantile Values,
μ
, Under Weibull CDF,
α
= 5.5, and
β
= 1.5
Table 2.5 Target Quantile Values,
μ
, Under Weibull CDF,
α
= 7.0, and
β
= 3
Chapter 3: Birth and Death Processes
Table 3.1 Transient Probability Distribution of the Number of Tasks in the System with
K
= 39 and
c
= 3
Table 3.2 Transient Probability Distribution of the Number of Tasks in the System with
K
= 7 and
c
= 3
Table 3.3 Transient Probability Distribution of the Number of Tasks in the System by Randomization Method with
K
= 7,
c
= 3, and
m
being Preassigned According to
Table 3.4 Execution Time to Obtain the Steady-State Solution for the Present Method for Various Values of
N
with
c
= 3
Chapter 4: Standard Queues
Table 4.1 Distribution of a Busy Period with
N
= 7 and
c
= 3
Table 4.2 Summary of Values for Case
M
1
= 1 and
M
2
= 7
Table 4.3 Calculation of
n
(
x
,
y
, 1) for 0 ≤
y
≤
x
≤ 10
Table 4.4 Calculation of
B
(
x
,
y
) for
x
,
y
≤ 10
Table 4.5 Probability that Throughout the Counting Number of Votes Registered for Candidate A,
x
, is Always Greater than the Number of Votes Registered for Candidate B,
y
, for
x
,
y
≤ 10
Chapter 5: Queues with Delay
Table 5.1 Arrival Rate,
λ
, versus Mean System Size,
L
b
, in Regular Service Period
Table 5.2 Arrival Rate,
λ
, versus Mean Waiting Time,
W
b
, in Regular Service Period
Table 5.3 Arrival Rate,
λ
, versus Mean System Size,
L
ν
in Regular Service Period
Table 5.4 Arrival Rate,
λ
, versus Mean Waiting Time,
W
ν
in Regular Service Period
Table 5.5 Interrelationship between the
N
-Policy and
F
-Policy Problems
Table 5.6
q
s
= 0,
q
o
= 0.6,
p
f
= 0.4,
a
=
q
o
μ
,
b
=
p
f
μ
,
d
= 0.6, and
r
=
b
/
μ
= 0
Table 5.7 Influence of the Reliability Factors
α
1
and
α
2
Table 5.8 Effect of Negative Arrival Probability,
δ
, on
P
0
L
q
and
F
Loss
Table 5.9 Effect of Service Loss Probability,
θ
, on
P
0
,
L
q
, and
P
Table 5.10 Effect of Failure Probability, , on
P
0
,
L
q
, and
P
Table 5.11 Effect of Number of Vacations,
J
, on
P
0
,
P,
and
Ω.
Table 5.12 Effect of Repair rate on FTS,
ξ
1
, on
P
0
,
L
q
, and
R
1
Chapter 6: Networks of Queues with Delay
Table 6.1 Expected Value and Variance of Length of a Busy Period for Three Sets of Data Points
Table 6.2 Number of Busy Periods for Example 6.5.2
Table 6.3 Expected Value and Variance of Length of a Busy Period for Three Sets of Data Points
Table 6.4 Distribution of the Number of Tasks in the System (Both Stations), the First 18 Rows and 12 Columns of the Probability Distribution Matrix
Table 6.5 Differences between Two Consecutive pdfs of the Queue with
τ
= 700 and
τ
= 600
Table 6.6 Maximum Differences between Two Consecutive Probabilities in the Distribution of Queue Size
Table 6.7 Data Used for Numerical Example
Delayed and Network Queues
Aliakbar Montazer Haghighi
Department of Mathematics Prairie View A&M University Member of the Texas A&M University System Prairie View, Texas, USA
Dimitar P. Mishev
Department of Mathematics Prairie View A&M University Member of the Texas A&M University System Prairie View, Texas, USA
Copyright © 2016 by John Wiley & Sons, Inc. All rights reserved
Published by John Wiley & Sons, Inc., Hoboken, New Jersey
Published simultaneously in Canada
No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permissions.
Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.
For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002.
Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com.
Library of Congress Cataloging-in-Publication Data:
Names: Haghighi, Aliakbar Montazer, author. | Mishev, D. P. (Dimiter P.), author.
Title: Delayed and network queues / Aliakbar Montazer Haghighi, Prairie View A&M University, member of Texas A&M University System, Prairie View, Texas, USA, Dimitar P. Mishev, Prairie View A&M University, member of Texas A&M University System, Prairie View, Texas.
Description: Hoboken, New Jersey : John Wiley & Sons, 2016. | Includes bibliographical references and index.
Identifiers: LCCN 2016015749 (print) | LCCN 2016020734 (ebook) | ISBN 9781119022138 (cloth) | ISBN 9781119022145 (pdf) | ISBN 9781119022152 (epub)
Subjects: LCSH: Routing (Computer network management)-Mathematics. | Computer networks--Mathematical models. | Telecommunication--Traffic. | Queuing networks (Data transmission)
Classification: LCC TK5105.5487 .H34 2016 (print) | LCC TK5105.5487 (ebook) | DDC 004.601/51982-dc23
LC record available at https://lccn.loc.gov/2016015749
I have always appreciated my lovely wife, Shahin Hamidi, for putting up with me during our about 45 years of marriage. A significant part of what was supposed to be our “quality time” together, I have squandered in research and publication. Fully aware of her involuntary sacrifice, I have always devoted my scholarly work and achievement to her, along with others.
This volume (my 4th in English Language) is also in her honor which she has consented to gracefully share with my late parents, Roghayyeh Jafari and Mohammad Ali Montazer-Haghighi. I am confident, yet continue to pray, that both remain in the peaceful and merciful hands of the Almighty Allah in whom they lived their lives.
“To Allah we belong and to HIM we will return”
Aliakbar Montazer HaghighiHouston, Texas, USADecember 31, 2015
I would like to dedicate the book to my wife, Tatiana, for her patience and encouragement, to my children Elizabeth, Antoan and Ilia, and to my brother Iordan and his family.
Dimitar P. MishevHouston, Texas, USADecember 31, 2015
Preface
Queueing theory is considered a part of Operations Research, in which the concept of stochastic processes is widely applied. With advancements in manufacturing and production systems, public transportation, logistics, and health care, in this era of technological development made possible by wireless communication and the Internet, performance evaluation of such systems is best accomplished by modeling the system as a network of queues. A network of queues is applied to model a real-world system when a set of resources is shared by some components of the system. Thus, queueing network modeling is a methodology for the analysis of systems such as computer systems. Many examples of networks of queues that appear in our day-to-day lives sometimes go unnoticed. The assembly of an automobile in a car manufacturing plant is a simple example.
In the 1960s, the computer industry was not very aware of the need of queueing network analysis; instead, the time sharing idea was used in the data processing industry. The networks only became fashionable by the end of that decade. New challenges developed in the 1970s and continued when the wireless communication system was studied in the 1980s and has still continued to the present time. By now, the algorithms for evaluating queueing network models are well developed and the mathematical sophistication to analyze systems such as computer systems is not so essential.
However, for long, the networks were based on the assumption of reliable nodes. Breaking down the nodes soon proved the assumption unrealistic in both the Internet and wireless and the industry such as production lines and the automobile industry. This fact is, indeed, a source of delay in the performance analysis of network of queues. This delay is different from the standard waiting time in a service line. Because of the importance of delay that not only affects the time delay but also being an economic factor in a production line, for instance, we have allocated part of Chapter 6 to discuss the unreliability of networks.
This book is an attempt to present some queueing models and networks of queues with delay, in a style that is between a monograph and a textbook. We also hope that it will satisfy the need of a handbook for researchers and practitioners.
The principal audience of the book are senior undergraduates, masters and doctorate students, as well as professionals and researchers in the mathematical sciences, stochastic processes, operations research, and engineering. The materials included in the book should also be of interest to those involved in computer network performance analysis. These materials actually instruct the application of queueing network models as tools to assist in the answering of questions on cost and performance that arise throughout the life of a computer system. Our intention is to provide enough information that readers can fully understand the essential aspects of queues with delay and networks of queues with unreliable nodes that may cause delay. We have omitted numerical examples for some models as the development of the fundamentals of models are sufficiently detailed to understand them.
We have included results of some recent research studies to ensure the contents of the book are current enough for all uses.
Particular advantages of this text are as follows:
1.
Treatment of queues with delay and networks of queues with possible breakdown and disruption that may cause delay.
2.
Inclusion of recently published materials on the subject so that the book is very Topical.
3.
Complementation by modern technology (including current software packages).
We have divided the book into six chapters. In Chapter 1, basics of probability and some preliminary materials are discussed as mind refreshers and background materials for later use. Chapter 2 contains topics in stochastic processes that are the basis for queueing theory. Birth and death processes are discussed, not in detail, in Chapter 3. Some standard queues such as single-server and multiserver queues and queues with feedback with some variations and generalizations are presented in Chapter 4. Chapter 5 is devoted to queues with delay. Different types of models that may cause delay in performance and service are discussed in this chapter. In the final chapter, Chapter 6, networks of queues with delay and models with reliable and unreliable nodes, particularly Jackson's types, are discussed. Additionally, included in this chapter are discussions of assessment of reliability of a network of queues and effect of network service breakdown. The splitting (relatively new and still developing) and blocking features are discussed in both Chapters 5 and 6.
In addition to the calculus and differential equations (to the level taught in most engineering undergraduate schools), some knowledge of difference equations is necessary to better comprehend the contents of this book. Related questions are included in the “Exercises” section at the end of each chapter.
It is our wish that readers of this text have an exciting experience going through this lavish collection of queueing models. We would seize one final opportunity to express our gratitude to John Wiley & Sons, Inc., New Jersey, for offering and awarding us the contract to write this book, as well as those who helped us in this work.
Aliakbar Montazer HaghighiDimitar P. Mishev
Houston, Texas, USAJanuary 25, 2016
Chapter 1Preliminaries
1.1 Basics of Probability
1.1.1 Introduction
In this chapter, we introduce some basics of probability that will be needed in the later chapters. We also take the liberty in stating some theorems without presenting proofs and emphasize that the contents of this chapter, by no means, represent all topics of probability that deserve a detailed discussion.
A chance or random experiment is an experiment whose outcomes or results of its performance are uncertain. A set of outcomes is called an event. The set of all possible outcomes is referred to as the sample space, denoted by . Thus, an event is a subset of the sample space and an element of the sample space is a sample point.
Two events and are referred to as mutually exclusive if their intersection is empty. A set is called a partition of if the events are mutually exclusive, such that .
Probability of an event E, denoted by , indicates a number between 0 and 1 (inclusive), describing the likelihood of occurrence of the event E. There are two particular events: an event with probability 1 (referred to as almost sure event) and another with probability 0 (referred to as null or impossible event).
For a finite sample space with n elements, if all outcomes have the same chance to occur, each member is assigned a probability of 1/n and the sample space is called equiprobable. In case of an infinite sample space, elements are with uniform measure.
If a chance experiment is repeated, the chance of occurrence of an outcome is the ratio of the number of occurrences of the outcome to the total number of repetitions. Hence, for a sample space with n equiprobable points, the probability of an event with k points is k/n, referred to as relative frequency, and it is an approximation of the probability of the event. In other words, for a sample space with equiprobable sample points, if E is an event with k points, the probability of the event E is given by
The number of elements of the event E is referred to as the “size” of E. Thus, probability of the event E may be defined as
The triplet is called the probability space associated with the random experiment, where
a.
is the sample space, that is, the set of all outcomes of the random experiment.
b.
is the set function containing all possible events drawn from
, which has the structure of the
field. This means that
satisfies the following conditions:
i.
the empty set is in
,
ii.
if
, then the complement of
E
is also in
, and
iii.
if
, then
.
c.
P
is the probability (measure) of an event. In fact,
P
is a function that associates a number
for each element
E
of
with the following properties (called
axioms of probability
):
Axiom 1
, for each event
E
is
.
Axiom 2
.
Axiom 3
For any sequence
of
mutually exclusive
events (disjoint sets, that is,
if
) in
,
1.1.2 Conditional Probability
For the probability space , let B be an event (that is, ) and P(B) > 0. Then, given the probability of an event A, the conditional probability of B, denoted by , defined on , is given by
If , then is not defined.
It should be noted that conditional probability exhibits properties similar to ordinary probability, but restricted to a smaller space.
One of the concepts often needed is the “independence” of events. We offer the definition for two events that can be easily expanded. Hence, we have the following:
Two events A and B are independent if and only if
In other words, occurrence of one does not affect the chance of occurrence of the other. Relation (1.4) can be expanded for an arbitrary number of events. Hence, the arbitrary family of events , where is the set of natural numbers, is independent if
for every finite subset of indices .
As a consequence of Equation (1.4), it can be easily proved that if two events A and B are independent and then
and conversely, if and Equation (1.6) holds to be true, then A and B are independent.
As another application of independence, the following is called the multiplicative law. Although we offer the definition for only two events, it can be expanded for any finite number of events. Thus, for any two events A and B with conditional probability or and as long as or is nonnegative, we have
Another property of conditional probability that can be easily verified called the law of total probability or total probability theorem is that if is a partition of the sample space , that is, n mutually exclusive events whose sum is unity, then for any given arbitrary event E, we have
Using Equation (1.8) and the conditional probability, a very important theorem called the Bayes' theorem can be proved. It can be stated as follows: If E is an event and a partition of the sample space , then
Relation (1.9) is referred to as Bayes' formula. The conditional probability is called the posterior probability. The original probability of is called the prior probability of .
1.2 Discrete Random Variables and Distributions
Although sample points are numerical in many practical problems, there are nonnumerical in many others. For practical purposes, a numerical sample space is more desirable. The tool to quantify a sample space to a numerical one is the random variable.
Before defining a random variable, we define a countable set. A set is called countable if it has the same number of elements (cardinality) as some subset of the set of natural numbers . In case of a finite subset of , the set is sometimes referred to as finitely countable or at most countable. For the case of the same cardinality as , it is sometimes referred to as countably infinite. A set that is not countable is called uncountable or denumerable. Throughout the book, we may use any of the terms as may be appropriate. Thus for us, the set of natural numbers , the set of natural numbers including zero , the set of integers , and the set of rational numbers (reactions) are all countable or infinitely countable sets. However, is at most countable or finitely countable. We note that all infinitely countable sets are of the same size, infinite. However, the set of real numbers and an interval are not countable. The latter was first proved by George Cantor using the diagonal argument method.
Thus, a random variable is defined as a function (mapping) that assigns a numerical value (or a set of values) to a sample point. Hence, If X is a random variable, it assigns a value to each outcome in . If the function X is on an at most countable sample space into the set of real numbers, , the random variable X is called a discrete random variable. Thus, a random variable is discrete if it takes at most countably many values. In other words, if there is a finitely countable set of real numbers, say A, such that .
For the sake of convenience, we may, allow a discrete random variable to assume positive infinity, as one of its values such as waiting time for an event to occur for the first time. This is because the event may never occur.
We leave it as an exercise for the reader to prove the following properties of discrete random variables. If X and Y are two discrete random variables, then , , and are also random variables, for the last one, provided Y is nonzero.
Probability distribution of a discrete random variable indicates the assignment of probabilities over the entire values of a random variable.
It is important to note that probabilities are nonnegative real numbers and total assignment of probabilities must be 1. Hence, these two simple properties establish two conditions for a function to be a probability distribution function.
Let the discrete random variable X be defined on a sample space with a typical element x. Then, the probability mass function(pmf) of X, denoted by , is defined as . If no specific value is given, we denote it by .The pmf is sometimes described by a table or matrix. For instance, if X is a random variable with the general vale x and specific values , that are assigned probabilities , then the pmf can be written as in Table 1.1.
Table 1.1 pmf
X
It is important to note that according to the second axiom of probability, , where x varies over all possible values of X.
When two random variables X and Y have the same distribution, say and , we say X and Y as equally distributed.
Example 1.2.1 Bernoulli Distribution
The simplest chance experiment (that is, an experiment whose outcomes are determined by chance, that is also called a trial) is an experiment with only two outcomes. Such an experiment is called a Bernoulli Trial. Independent repetitions of such trials are referred to as Bernoulli Trials. The random variable representing a Bernoulli trial is called a Bernoulli random variable. Thus, the sample space for a Bernoulli random variable has two sample points, referred to as success and failure. Denoting the probability of a success by p, , the probability of failure will be . Thus, if X is a Bernoulli random variable with values 1 and 0, corresponding to success and failure, respectively, then distribution of X is given by
with .
Relation (1.10) can also be expressed as
In order to determine whether Equation (1.11) defines a distribution function, in this case, the Bernoulli probability distribution function, we note that
Example 1.2.2 The Indicator Function
For an event A in the probability space, the indicator function of A is defined as the random variable
Hence, for every and . In other words, distribution function of an indicator function is a Bernoulli distribution with parameter . Because of this fact, sometimes, a Bernoulli random variable is called the indicator random variable. The random variable X, in this case, is an indicator of the event . This is because, when , is in the event A, the first part of Equation (1.12). For example, suppose two dice are rolled. Let and be random variables representing the numbers shown on the first and second rolling, respectively. Thus, the sample space for each of these discrete random variables is , and for both will be the cross-product , containing 36 elements, which are ordered pairs. In other words, . Suppose that we are interested in the sum of and be at least 7.Thus, we have . Let the sum of interest be denoted by Z. That is, , or
Thus, we have to compute . Now, for the sum to be at least 7, we have to achieve the following ordered pairs: (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1). That is, the sample space of Z, which is a subset of contains (1,6), (2,5), (3,4), (4,3), (5,2) and (6,1). Probability of achieving each of these pairs is 1/36. Thus, .
Example 1.2.3 Binomial Distribution
Consider Example 1.2.1 Let us assume that a Bernoulli trial is independently repeated n times and also that the random variable X represents the number of successes occurred in the n trials. Obviously, the sample space, in this case, will have 2n sample points. The number of ways k success can occur in n trials is . Requiring k success in n trials leaves n − k failures. With the trials being independent, the probability of a succession of successes and failures will be the product . Thus, the pmf of X, denoted by , and called the binomial random variable is given by
Relation (1.14) is, indeed, a probability distribution function, called the binomial distribution function with parameters n and p. We leave the proof as an exercise to the reader.
Example 1.2.4 Poisson Distribution
The random variable X with probability distribution function (or pmf)
where is a constant, is called a Poisson random variable and Equation (1.15) is called Poisson distribution function (or Poisson pmf) with parameter. We leave it as an exercise to show that Equation (1.15) is, actually, a distribution function.
The discrete joint pmf of the random variables , with real values , denoted by , is defined by
where is nonnegative and
From Equation (1.16), each individual probability, , is called a marginal mass function, that is,
where the summation is over all possible n-tuples with the ith component held fixed at a specific value, say .
We list two properties of a random variable.
Property 1.2.1
We consider two random variables X and Y, with their respective pmfs and and their joint mass function is denoted by . Then, the conditional mass function of X given Y is defined by
provided that .
As observed in defining the conditional probability function (1.18), random variables can be substituted for events. Thus, Equation (1.18) produces a different probability distribution for each value of y.
The pmf of the conditional random variable (read as “X given Y = y”) is given by
for values x of X.
Example 1.2.5 Conditional pmf
Let X and Y be jointly distributed as shown in Table 1.2.
Table 1.2pX, Y = P(X = x and Y = y)
Y
X
2
4
6
Marginal pmf of
Y
0
0.10
0.03
0.04
1
0.01
0.16
0.07
2
0.15
0.06
0.00
3
0.05
0.09
0.24
Marginal pmf of
X
1
FromTable 1.1, . Then, the conditional mass function of X given Y = 1 is
Keeping Equation (1.5) for events in mind, we leave it as an exercise to prove that two random variables X and Y, with their respective pmfs and , and their joint mass function , are independent if and only if
Let X, Y, and Z be discrete random variables. Then, X and Y are said to be conditionally independent, given Z, if
for all x, y, and z such that . As a consequence of Equation (1.22), if X and Y are conditionally independent, given Z, then
for all x, y, and z such that .
Property 1.2.2 Functions of a Random Variable
Let X be a discrete random variable with pmf, denoted by , where x is a real number representing the values of X. Let y be a real-valued function of the real variable x, denoted by . Hence, transforms the random variable X into the random variable Y, whose values are denoted by y. Thus, as X maps a sample point s to a real number, so does . This indicates that cumulative distribution function (cdf) of Y depends on and cdf of X.
We note that the domain of g should contain the range of X. Hence, pmf of Y can be expressed as
where is the indicator function.
1.3 Discrete Moments
For k integers , the arithmetic average, denoted by , is defined as the sum of integers divided by k. In other words, the arithmetic average of the k integers is obtained by multiplying each number by 1/k and add, that is,
Similarly, let X be a random variable with values , and with probabilities , and , respectively. Then, the mean (or weighted average or expected value or expectation or mathematical expectation) of X, denoted by , is defined as
Note that each , is the weight for each value , and , respectively.
Example 1.3.1 The Indicator Function
We leave it as an exercise to prove that
Relation (1.26) may be expanded for a random variable with infinite values. In other words, let X be a random variable with values , and respective probabilities , that is . Then, the mean (or weighted average or expected value or expectation or mathematical expectation) of X is defined as
provided that the infinite series in Equation (1.28) converges absolutely; otherwise, the series does not exist, and hence, the expected value of X does not exist.
The following are some of the properties of Equation (1.28):
1.
The only way Equation (
1.28
) would not exist is when
2.
Let
X
and
Y
be random variables and
a
,
b
, and
c
constants, then
3.
Let X and Y be two independent random variables with marginal pmf and , respectively. Assume and exist. Then,
Relation (1.31) can be expanded for a finite number of independent random variables.
4.
For random variables X and Y,
where x and y are values of X and Y, respectively.
It is important to note that Equation (1.32) is a function of y, which implies that is a random variable in its own right.
Example 1.3.2 Conditional Expected Value
Consider Example 1.2.5 and Table 1.2. In Example 1.2.5, we found the conditional pmf of X given Y = 1. Now we want to find the expected value of this conditional variable. From Table 1.2 and Equation (1.20), we have
5.
For two random variables
X
and
Y
, we have
6.
For a random variable
X
, with
x
representing its values, the
nth moment
, denoted by
, is defined as
where
n
is a nonnegative integer.
a.
If
, then Equation (
1.34
) reduces to the expected value or the first moment of
X
. With this note in mind, the variance of the random variable
X
, denoted by
, can be defined as
where
x
represents values of
X
. The following are some properties of variance (relation (
1.35
)):
b.
Let
X
and
Y
be random variables and
a
,
b
, and
c
constants, then
c.
For
, then Equation (
1.34
) reduces to the second moment and that in turn leads to the variable of
X
, that is
7.
Central Moment.
The expected value of the random variable of
X
is denoted by
, that is,
. The
n
th moment of the random variable
, that is,
, is called the
central moment
of
X
. The random variable
measures the
deviation of X from its expectation
. This deviation can be positive or negative depending on the values of
. Hence, its absolute value gives an
absolute measure of deviation
of the random variable
X
from its mean
. Yet a better measure, the
mean square deviation
, that is,
, called the
second central moment of X
, is, indeed, the variance of
X
. Thus, the variance measures the
average deviation
or
dispersion
of the random variable from its mean. As the variance measures the deviation from the mean by squares, in order to adjust for this squaring process, the square root is used. The positive square root of the variance of a random variable
X
is called the
standard deviation
.
8. Generating Function.
Let
X
be a discrete random variable with nonnegative integer values with
as the pmf of
X
. The
probability generating function
(
pgf
) of
X
, denoted by
G
(
z
),where
z
is a complex number, that was defined in
Chapter 1
, can be redefined as
where the power series in Equation (
1.38
) converges absolutely at least for all
z
, such that
. The idea of applying generating function to probability is intended to encapsulate all the information about the random variable.
a.
When
X
takes
k
as its value,
takes
as its value.
b.
The power series in Equation (
1.38
) follows all the rules of convergence of power series with nonnegative coefficients.
c.
The product of two generating functions and is given by
where
The sequence defined in Equation (1.40) is called the convolution of the two sequences and . In general, the convolution initiates two independent random variables X and Y with pmfs and , respectively. Here it is defined as follows. Let X and Y be two independent discrete random variables with pmfs of and , respectively. Then, the convolution of and is a pmf such that
The pmf defined in Equation (1.41) is distribution function of the random variable .
d.
For a random variable
X
that takes
, as its values, pmf of
X
is recovered by the derivatives of the generating function as follows:
e.
If
is the pmf of a random variable
X
, then
f.
If
z
= 1, then
g.
For a random variable
X
,
h.
The nth factorial moment, denoted by , is defined as
and for a random variable X, it is given by
where
All the moments of X can be obtained from Equation (1.48). For instance, if n = 1, then we obtain Equation (1.43). If n = 2, we obtain , which leads to the variance given in Equation (1.44).
9.
The Moment generating function of a random variable X, denoted by , with the pmf is defined as
where is the generating function of X and t is a nonnegative real number.
It is important to note the following points:
a.
It is possible that
t
is a complex number with nonnegative real part.
b.
The moment generating function generates all the moments of
X
, as did the generating function.
c.
If
t
= 0, then
Recall that for generating function, we had relation (1.42), that is, if , then .
d.
The nth moment of the random variable X can be obtained from the moment generating function as
Recall that for generating function, we had a similar relation (1.49), that is, to obtain the nth factorial moment of a random variable X, we need to evaluate the nth derivative of the generating function at 1. Hence, for higher moments than the second, the moment generation yields direct results.
e.
For computational purposes, it is more convenient to use the following relations:
and
1.4 Continuous Random Variables, Density, and Cumulative Distribution Functions
A sample space is called continuous if the outcomes of a chance experiment can assume real values, that is, is the entire real number line or a part of it. However, in practice, a smaller set of subsets can be taken to contain all events of our interest. The smallest such set, denoted by , is referred to as the Borel set. For instance, the set of all rational numbers in the interval [0, 1] is a Borel set for [0, 1]. The Borel set leads to a new probability space, where P is the probability of evens in the Borel set.
We define the function on the set of real numbers, , such that , for all real x, and . Then, is called a continuous probability density function (pdf) on. Let Ω be a continuous sample space. A random variable X, which takes its values for such , is called a continuous random variable. In other words, a continuous random variable maps to a subset of the real line, that is, X is a real-valued function defined on such that . In terms of pdf, , we have the following properties:
Suppose X is a continuous random variable defined on a sample space with a probability density function . Then, the cdf of X, denoted by , is defined as
As it can be seen, if the density function is continuous, then .
We note that the distribution is nondecreasing, right continuous, and exhibits the following properties:
If there is no danger of confusion, we will suppress the subscript X from and . Equation (1.55) can be rewritten using Equation (1.55) as
It is important to note that
