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## Probability Markov Chains Queues and Simulation

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## Probability, Markov chains, queues and simulation ( edition) | Open Library

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Performance modeling is of fundamental importance to many branches of the mathematical sciences and engineering as well as to the social and economic sciences. Advances in methodology and technology have now provided the wherewithal to build and solve sophisticated models. The purpose of this book is to provide the student and teacher with a modern approach for building and solving probability based models with confidence.

The eight chapters of Part I provide the student with a comprehensive and thorough knowledge of probability theory. Part I is self-contained and complete and should be accessible to anyone with a basic knowledge of calculus. Newcomers to probability theory as well as those whose knowledge of probability is rusty should be equally at ease in their progress through Part I. The first chapter provides the fundamental concepts of set-based probability and the probability axioms.

Chapter 2 introduces combinatorics—the art of counting—which is so important for the correct evaluation of probabilities. Chapter 3 introduces the concepts of random variables and distribution functions including functions of a random variable and conditioned random variables. This chapter prepares the ground work for Chapters 4 and 5: Chapter 4 introduces joint and conditional distributions and Chapter 5 treats expectations and higher moments.

Discrete distribution functions are the subject of Chapter 6 while their continuous counterparts, continuous distribution functions, are the subject of Chapter 7. Particular attention is paid to phase-type distributions due to the important role they play in modeling scenarios and the chapter also includes a section on fitting phase-type distributions to given means and variances. The final chapter in Part I is devoted to bounds and limit theorems, including the laws of large numbers and the central limit theorem.

Part II contains two rather long chapters on the subject of Markov chains, the first on theoretical aspects of Markov chains, and the second on their numerical solution. In Chapter 9, the basic concepts of discrete and continuous-time Markov chains and their underlying equations and properties are discussed. Special attention is paid to irreducible Markov chains and to the potential, fundamental, and reachability matrices in reducible Markov chains.

This chapter also contains sections on random walk problems and their applications, the property of reversibility in Markov chains, and renewal processes. Chapter 10 deals with numerical solutions, from Gaussian elimination and basic iterative-type methods for stationary solutions to ordinary differential equation solvers for transient solutions.

Block methods and iterative aggregation-disaggregation methods for nearly completely decomposable Markov chains are considered. A section is devoted to matrix geometric and matrix analytic methods for structured Markov chains. Algorithms and computational considerations are stressed throughout this chapter. Queueing models are presented in the five chapters that constitute Part III.

Elementary queueing theory is presented in Chapter This is then generalized to birth-death processes, which are queueing systems in which the underlying Markov chain matrix is tridiagonal. Chapter 12 deals with queues in which the arrival process need no longer be Poisson and the service time need not be exponentially distributed. Instead, interarrival times and service times can be represented by phase-type distributions and the underlying Markov chain is now block tridiagonal.

The following chapter, Chapter 13, explores the z-transform approach for solving similar types of queues. The approach used is that of the embedded Markov chain. The Pollaczek-Khintchine mean value and transform equations are derived and a detailed discussion of residual time and busy period follows. A thorough discussion of nonpreemptive and preempt-resume scheduling policies as well as shortest-processing-time-first scheduling is presented.

Closed queueing networks are treated using both the convolution algorithm and the mean value approach. The flow-equivalent server approach is also treated and its potential as an approximate solution procedure for more complex networks is explored. The chapter terminates with a discussion of product form in queueing networks and the BCMP theorem for open, closed, and mixed networks. The final part of the text, Part IV, deals with simulation. Chapter 16 explores how uniformly distributed random numbers can be applied to obtain solutions to probabilistic models and other time-independent problems—the Monte Carlo aspect of simulation.

Chapter 17 describes the modern approaches for generating uniformly distributed random numbers and how to test them to ensure that they are indeed uniformly distributed and independent of each other. The topic of generating random numbers that are not uniformly distributed, but satisfy some other distribution such as Erlang or normal, is dealt with in Chapter A large number of possibilities exist and not all are appropriate for every distribution. The next chapter, Chapter 19, provides guidelines for writing simulation programs and a number of examples are described in detail.

Chapter 20 is the final chapter in the book. It concerns simulation measurement and accuracy and is based on sampling theory. Special attention is paid to the generation of confidence intervals and to variance reduction techniques, an important means of keeping the computational costs of simulation to a manageable level.

The text also includes two appendixes; the first is just a simple list of the letters of the Greek alphabet and their spellings; the second is a succinct, yet complete, overview of the linear algebra used throughout the book. This book saw its origins in two first-year graduate level courses that I teach, and have taught for quite some time now, at North Carolina State University.

This course is required for our networking degrees. It follows then that this book has been designed for students from a variety of academic disciplines in which stochastic processes constitute a fundamental concept, disciplines that include not only computer science and engineering, industrial engineering, and operations research, but also mathematics, statistics, economics, and business, the social sciences—in fact all disciplines in which stochastic performance modeling plays a primary role. A calculus-based probability course is a prerequisite for both these courses so it is expected that students taking these classes are already familiar with probability theory.

However, many of the students who sign up for these courses are returning students, and it is often the case that it has been several years and in some cases a decade or more, since they last studied probability.

A quick review of probability is hardly sufficient to bring them up to the required level. Part I of the book has been designed with them in mind. It provides the prerequisite probability background needed to fully understand and appreciate the material in the remainder of the text. The presentation, with its numerous examples and exercises, is such that it facilitates an independent review so the returning student in a relatively short period of time, preferably prior to the beginning of class, will once again have mastered probability theory.

Part I can then be used as a reference source as and when needed. The entire text has been written at a level that is suitable for upper-level undergraduate students or first-year graduate students and is completely self-contained. A two-semester sequence is appropriate for classes in which students have limited or no exposure to probability theory. In such cases it is recommended that the first semester be devoted to the Chapters 1—8 on probability theory, the first five sections of Chapter 9, which introduce the fundamental concepts of discrete-time Markov chains, and the first three sections of Chapter 11, which concern elementary queueing theory.

With this background clearly understood, the student should have no difficulty in covering the remaining topics of the text in the second semester. The complete content of Parts II—IV might prove to be a little too much for some one-semester classes.

### More about this book

In this case, an instructor might wish to omit the later sections of Chapter 10 on the numerical solution of Markov chains, perhaps covering only the basic direct and iterative methods. In this case the material of Chapter 12 should also be omitted since it depends on a knowledge of the matrix geometric method of Chapter Because of the importance of computing numerical solutions, it would be a mistake to omit Chapter 10 in its entirety. Some of the material in Chapter 18 could also be eliminated: for example, an instructor might include only the first three sections of this chapter.

These students often take simulation as an individual course later on. When teaching the computer science and engineering course, I omit some of the material on the numerical solution of Markov chains so as to leave enough time to cover simulation. Numerous examples with detailed explanations are provided throughout the text. A solution manual is available for teachers who adopt this text for their courses. This manual contains detailed explanations of the solution of all the exercises. Where appropriate, the text contains program modules written in Matlab or in the Java programming language.

These programs are not meant to be robust production code, but are presented so that the student may experiment with the mathematical concepts that are discussed. To free the student from the hassle of copying these code segments from the book, a listing of all of the code used can be freely downloaded from the web page:.

As mentioned just a moment ago, this book arose out of two courses that I teach at North Carolina State University. It is, therefore, ineluctable that the students who took these courses contributed immeasurably to its content and form. I would like to express my gratitude to them for their patience and input. One person who deserves special recognition is my daughter Kathryn, who allowed herself to be badgered by her father into reading over selected probability chapters.

I would like to thank Vickie Kearn and the editorial and production staff at Princeton University Press for their help and guidance in producing this book. It would be irresponsible of me not to mention the influence that my teachers, colleagues, and friends have had on me.

I owe them a considerable debt of gratitude for helping me understand the vital role that mathematics plays, not only in performance modeling, but in all aspects of life. Finally, and most of all, I would like to thank my wife Kathie and our four children, Nicola, Stephanie, Kathryn, and William, for all the love they have shown me over the years. The notions of trial, sample space, and event are fundamental to the study of probability theory. Tossing a coin, rolling a die, and choosing a card from a deck of cards are examples that are frequently used to explain basic concepts of probability.

Each toss of the coin, roll of the die, or choice of a card is called a trial or experiment. We shall use the words trial and experiment interchangeably. Each execution of a trial is called a realization of the probability experiment. At the end of any trial involving the examples given above, we are left with a head or a tail, an integer from one through six, or a particular card, perhaps the queen of hearts.

The result of a trial is called an outcome. The set of all possible outcomes of a probability experiment is called the sample space.

### About the author

The outcomes that constitute a sample space are also referred to as sample points or elements. Example 1. The position in which the tail occurs is important. A sample space may be finite, denumerable i. Its elements depend on the experiment and how the outcome of the experiment is defined. The four illustrative examples given above all have a finite number of elements. The sample space is denumerable since we may tag each arriving email message with a unique integer n is the set of nonnegative integers. Each outcome is a nonnegative real number x.

If a finite number of trials is performed, then, no matter how large this number may be, there is no guarantee that every element of its sample space will be realized, even if the sample space itself is finite. This is a direct result of the essential probabilistic nature of the experiment.

## Probability, Markov Chains, Queues, and Simulation: The Mathematical Basis of Performance Modeling

For example, it is possible, though perhaps not very likely i. The word event by itself conjures up the image of something having happened, and this is no different in probability theory. We toss a coin and get a head, we throw a die and get a five, we choose a card and get the ten of diamonds.

Each experiment has an outcome, and in these examples, the outcome is an element of the sample space. These, the elements of the sample space, are called the elementary events of the experiment. However, we would like to give a broader meaning to the term event.

The single tail appears on the third, second, or first toss, respectively. This event comes to pass so long as the throw gives neither one, four, nor six spots. This is how we define an event in general. Rather than restricting our concept of an event to just another name for the elements of the sample space, we think of events as subsets of the sample space.

More complex events consist of subsets with more than one outcome. Defining an event as a subset of the sample space and not just as a subset that contains a single element provides us with much more flexibility and allows us to define much more general events.

The event is said to occur if and only if, the outcome of the experiment is any one of the elements of the subset that constitute the event. They are assigned names to help identify and manipulate them. This is illustrated in Figure 1. To summarize, the standard definition of an event is a subset of the sample space. It consists of a set of outcomes.

The null or empty subset, which contains none of the sample points, and the subset containing the entire sample space are legitimate events—the first is called the null or impossible event it can never occur ; the second is called the universal or certain event and is sure to happen no matter what the outcome of the experiments gives.

The execution of a trial, or observation of an experiment, must yield one and only one of the outcomes in the sample space. If a subset contains none of these outcomes, the event it represents cannot happen; if a subset contains all of the outcomes, then the event it represents must happen.

In general, for each outcome in the sample space, either the event occurs if that particular outcome is in the defining subset of the event or it does not occur. Viewing events as subsets allows us to apply typical set operations to them, operations such as set union, set intersection, set complementation, and so on. The union. It occurs if either occurs. The intersection and occurs if both occur.

The difference.