Table of Contents

Title page

Copyright page

Dedication

Preface

CHAPTER ONE: Probability and Statistics

1.1. Basic Definitions and Concepts of Probability

1.2. Discrete Random Variables and Probability Distribution Functions

1.3. Moments of a Discrete Random Variable

1.4. Continuous Random Variables

1.5. Moments of a Continuous Random Variable

1.6. Continuous Probability Distribution Functions

1.7. Random Vector

1.8. Continuous Random Vector

1.9. Functions of a Random Variable

1.10. Basic Elements of Statistics

1.11. Inferential Statistics

1.12. Hypothesis Testing

1.13. Reliability

Exercises

CHAPTER TWO: Transforms

2.1. Fourier Transform

2.2. Laplace Transform

2.3. Z-Transform

2.4. Probability Generating Function

Exercises

CHAPTER THREE: Differential Equations

3.1. Basic Concepts and Definitions

3.2. Existence and Uniqueness

3.3. Separable Equations

3.4. Linear Differential Equations

3.5. Exact Differential Equations

3.6. Solution of the First ODE by Substitution Method

3.7. Applications of the First-Order ODEs

3.8. Second-Order Homogeneous ODE

3.9. The Second-Order Nonhomogeneous Linear ODE with Constant Coefficients

3.10. Miscellaneous Methods for Solving ODE

3.11. Applications of the Second-Order ODE

3.12. Introduction to PDE: Basic Concepts

Exercises

CHAPTER FOUR: Difference Equations

4.1. Basic Terms

4.2. Linear Homogeneous Difference Equations with Constant Coefficients

4.3. Linear Nonhomogeneous Difference Equations with Constant Coefficients

4.4. System of Linear Difference Equations

4.5. Differential–Difference Equations

4.6. Nonlinear Difference Equations

Exercises

CHAPTER FIVE: Queueing Theory

5.1. Introduction

5.2. Markov Chain and Markov Process

5.3. Birth and Death (B-D) Process

5.4. Introduction to Queueing Theory

5.5.Single-Server Markovian Queue, M/M/1

5.6. Finite Buffer Single-Server Markovian Queue: M/M/1/N

5.7. M/M/1 Queue with Feedback

5.8. Single-Server Markovian Queue with State-Dependent Balking

5.9. Multiserver Parallel Queue

5.10. Many-Server Parallel Queues with Feedback

5.11. Many-Server Queues with Balking and Reneging

5.12. Single-Server Markovian Queueing System with Splitting and DELAYED Feedback

Exercises

Appendix

The Poisson Probability Distribution

The Chi-Square Distribution

The Chi-Square Distribution (continued)

The Standard Normal Probability Distribution

The Standard Normal Probability Distribution (continued)

The (Student's) t Probability Distribution

References and Further Readings

Answers/Solutions to Selected Exercises

Index

Copyright © 2013 by John Wiley & Sons, Inc. All rights reserved

Published by John Wiley & Sons, Inc., Hoboken, New Jersey

Published simultaneously in Canada

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Library of Congress Cataloging-in-Publication Data:

Haghighi, Aliakbar Montazer.

Difference and differential equations with applications in queueing theory / Aliakbar Montazer Haghighi Department of Mathematics, Prairie View A&M University, Prairie View, Texas, Dimitar P. Mishev Department of Mathematics, Prairie View A&M University, Prairie View, Texas.

pages cm

Includes bibliographical references and index.

ISBN 978-1-118-39324-6 (hardback) 1. Queuing theory. 2. Difference equations. 3. Differential equations. I. Mishev, D. P. (Dimiter P.) II. Title.

QA274.8.H338 2013

519.8'2–dc23

2013001302

To my better half, Shahin,

who has given me three beautiful grandchildren:

Maya Haghighi, Kayvan Haghighi, and Leila Madison Grant.

—A. M. Haghighi

To my mother, wife, and three children.

—D. P. Mishev

Preface

Topics in difference and differential equations with applications in queueing theory typically span five subject areas: (1) probability and statistics, (2) transforms, (3) differential equations, (4) difference equations, and (5) queueing theory. These are addressed in at least four separate textbooks and taught in four different courses in as many semesters. Due to this arrangement, students needing to take these courses have to wait to take some important and fundamental required courses until much later than should be necessary. Additionally, based on our long experience in teaching at the university level, we find that perhaps not all topics in one subject are necessary for a degree. Hence, perhaps we as faculty and administrators should rethink our traditional way of developing and offering courses. This is another reason for the content of this book, as of the previous one from the authors, to offer several related topics in one textbook. This gives the instructor the freedom to choose topics according to his or her desire to emphasize, yet cover enough of a subject for students to continue to the next course, if necessary.

The methodological content of this textbook is not exactly novel, as “mathematics for engineers” textbooks have reflected this method for long past. However, that type of textbook may cover some topics that an engineering student may already know. Now with this textbook the subject will be reinforced. The need for this practice has generally ignored some striking relations that exist between the seemingly separate areas of a subject, for instance, in statistical concepts such as the estimation of parameters of distributions used in queueing theory that are derived from differential–difference equations. These concepts commonly appear in queueing theory, for instance, in measures on effectiveness in queuing models.

All engineering and mathematics majors at colleges and universities take at least one course in ordinary differential equations, and some go further to take courses in partial differential equations. As mentioned earlier, there are many books on “mathematics for engineers” on the market, and one that contains some applications using Laplace and Fourier transforms. Some also have included topics of probability and statistics, as the one by these authors. However, there is a lack of applications of probability and statistics that use differential equations, although we did it in our book. Hence, we felt that there is an urgent need for a textbook that recognizes the corresponding relationships between the various areas and a matching cohesive course. Particularly, theories of queues and reliability are two of those topics, and this book is designed to achieve just that. Its five chapters, while retaining their individual integrity, flow from selected topics in probability and statistics to differential and difference equations to stochastic processes and queueing theory.

Chapter 1 establishes a strong foundation for what follows in Chapter 2 and beyond. Classical Fourier and Laplace transforms as well as Z-transforms and generating functions are included in Chapter 2. Partial differential equations are often used to construct models of the most basic theories underlying physics and engineering, such as the system of partial differential equations known as Maxwell's equations, from which one can derive the entire theory of electricity and magnetism, including light. In particular, elegant mathematics can be used to describe the vibrating circular membrane. However, our goal here is to develop the most basic ideas from the theory of partial differential equations and to apply them to the simplest models arising from physics and the queueing models. Detailed topics of ordinary and partial differential and difference equations are included in Chapter 3 and Chapter 4 that complete the necessary tools for Chapter 5, which discusses stochastic processes and queueing models. However, we have also included the power series method of solutions of differential equations, which can be applied to, for instance, Bessel's equation.

In our previous book, we required two semesters of calculus and a semester of ordinary differential equations for a reader to comprehend the contents of the book. In this book, however, knowledge of at least two semesters of calculus that includes some familiarity with terminology such as the gradient, divergence, and curl, and the integral theorems that relate them to each other, are needed. However, we discuss not only the topics in differential equation, but also the difference equations that have vast applications in the theory of signal processing, stochastic analysis, and queueing theory.

Few instructors teach the combined subject areas together due to the difficulties associated with handling such a rigorous course with such hefty materials. Instructors can easily solve this issue by teaching the class as a multi-instructor course.

We should note that throughout the book, we use boldface letters, Greek or Roman (lowercase or capital) for vectors and matrices. We shall write P(n) or Pn to mean P as a function of a discrete parameter n. Thus, we want to make sure that students are well familiar with functions of discrete variables as well as continuous ones. For instance, a vibrating string can be regarded as a continuous object, yet if we look at a fine enough scale, the string is made up of molecules, suggesting a discrete model with a large number of variables. There are many cases in which a discrete model may actually provide a better description of the phenomenon under study than a continuous one.

We also want to make sure that students realize that solution of some problems requires the ability to carry out lengthy calculations with confidence. Of course, all of these skills are necessary for a thorough understanding of the mathematical terminology that is an essential foundation for the sciences and engineering. We further note that subjects discussed in each chapter could be studied in isolation; however, their cohesiveness comes from a thorough understanding of applications, as discussed in this book.

We hope this book will be an interesting and useful one to both students and faculty in science, technology, engineering, and mathematics.

Aliakbar Montazer Haghighi

Dimitar P. Mishev

Houston, Texas

April 2013

CHAPTER ONE

Probability and Statistics

The organization of this book is such that by the time reader gets to the last chapter, all necessary terminology and methods of solutions of standard mathematical background has been covered. Thus, we start the book with the basics of probability and statistics, although we could have placed the chapter in a later location. This is because some chapters are independent of the others.

In this chapter, the basics of probability and some important properties of the theory of probability, such as discrete and continuous random variables and distributions, as well as conditional probability, are covered.

After the presentation of the basics of probability, we will discuss statistics. Note that there is still a dispute as to whether statistics is a subject on its own or a branch of mathematics. Regardless, statistics deals with gathering, analyzing, and interpreting data. Statistics is an important concept that no science can do without. Statistics is divided in two parts: descriptive statistics and inferential statistics. Descriptive statistics includes some important basic terms that are widely used in our day-to-day lives. The latter is based on probability theory. To discuss this part of the statistics, we include point estimation, interval estimation, and hypothesis testing.

We will discuss one more topic related to both probability and statistics, which is extremely necessary for business and industry, namely reliability of a system. This concept is also needed in applications such as queueing networks, which will be discussed in the last chapter.

In this chapter, we cover as much probability and statistics as we will need in this book, except some parts that are added for the sake of completeness of the subject.

1.1. Basic Definitions and Concepts of Probability

Nowadays, it has been established in the scientific world that since quantities needed are not quite often predictable in advance, randomness should be accounted for in any realistic world phenomenon, and that is why we will consider random experiments in this book.

Determining probability, or chance, is to quantify the variability in the outcome or outcomes of a random experiment whose exact outcome or outcomes cannot be predicted by certainty. Satellite communication systems, such as radar, are built of electronic components such as transistors, integrated circuits, and diodes. However, as any engineer would testify, the components installed usually never function as the designer has anticipated. Thus, not only is the probability of failure to be considered, but the reliability of the system is also quite important, since the failure of the system may have not only economic losses but other damages as well. With probability theory, one may answer the question, “How reliable is the system?”

Definition 1.1.1. Basics

(a) Any result of performing an experiment is called an outcome of that experiment. A set of outcomes is called an event.

(b) If occurrences of outcomes are not certain or completely predictable, the experiment is called a chance or random experiment.

(c) In a random experiment, sets of outcomes that cannot be broken down into smaller sets are called elementary (or simple or fundamental) events.

(d) An elementary event is, usually, just a singleton (a set with a single element, such as {e}). Hence, a combination of elementary events is just an event.

(e) When any element (or outcome) of an event happens, we say that the event occurred.

(f) The union (set of all elements, with no repetition) of all events for a random experiment (or the set of all possible outcomes) is called the sample space.

(g) In “set” terminology, an event is a subset of the sample space. Two events A₁ and A₂ are called mutually exclusive if their intersection is the empty set, that is, they are disjoint subsets of the sample space.

(h) Let A₁, A₂, … , An be mutually exclusive events such that A₁ ∪ A₂ ∪ … ∪ An = Ω. The set of {A₁, A₂, … , An} is then called a partition of the sample space Ω.

(i) For an experiment, a collection or a set of all individuals, objects, or measurements of interest is called a (statistical) population.
For instance, to determine the average grade of the differential equation course for all mathematics major students in four-year colleges and universities in Texas, the totality of students majoring mathematics in the colleges and universities in the Texas constitutes the population for the study.
Usually, studying the population may not be practically or economically feasible because it may be quite time consuming, too costly, and/or impossible to identify all members of it. In such cases, sampling is being used.

(j) A portion, subset, or a part of the population of interest (finite or infinite number of them) is called a sample.
Of course, the sample must be representative of the entire population in order to make any prediction about the population.

(k) An element of the sample is called a sample point. By quantification of the sample we mean changing the sample points to numbers.

(l) The range is the difference between the smallest and the largest sample points.

(m) A sample selected such that each element or unit in the population has the same chance to be selected is called a random sample.

(n) The probability of an event A, denoted by P(A), is a number between 0 and 1 (inclusive) describing likelihood of the event A to occur.

(o) An event with probability 1 is called an almost sure event. An event with probability 0 is called a null or an impossible event.

(p) For a sample space with n (finite) elements, if all elements or outcomes have the same chance to occur, then we assign probability 1/n to each member. In this case, the sample space is called equiprobable.
For instance, to choose a digit at random from 1 to 5, we mean that every digit of {1, 2, 3, 4, 5} has the same chance to be picked, that is, all elementary events in {1}, {2}, {3}, {4}, and {5} are equiprobable. In that case, we may associate probability 1/5 to each digit singleton.

(q) If a random experiment is repeated, then the chance of occurrence of an outcome, intuitively, will be approximated by the ratio of occurrences of the outcome to the total number of repetitions of the experiment. This ratio is called the .

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!