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Probability, Markov Chains, Queues, and Simulation: The Mathematical Basis of Performance Modeling - Hardcover
Synopsis
Probability, Markov Chains, Queues, and Simulation provides a modern and authoritative treatment of the mathematical processes that underlie performance modeling. The detailed explanations of mathematical derivations and numerous illustrative examples make this textbook readily accessible to graduate and advanced undergraduate students taking courses in which stochastic processes play a fundamental role. The textbook is relevant to a wide variety of fields, including computer science, engineering, operations research, statistics, and mathematics.
The textbook looks at the fundamentals of probability theory, from the basic concepts of set-based probability, through probability distributions, to bounds, limit theorems, and the laws of large numbers. Discrete and continuous-time Markov chains are analyzed from a theoretical and computational point of view. Topics include the Chapman-Kolmogorov equations; irreducibility; the potential, fundamental, and reachability matrices; random walk problems; reversibility; renewal processes; and the numerical computation of stationary and transient distributions. The M/M/1 queue and its extensions to more general birth-death processes are analyzed in detail, as are queues with phase-type arrival and service processes. The M/G/1 and G/M/1 queues are solved using embedded Markov chains; the busy period, residual service time, and priority scheduling are treated. Open and closed queueing networks are analyzed. The final part of the book addresses the mathematical basis of simulation.
Each chapter of the textbook concludes with an extensive set of exercises. An instructor's solution manual, in which all exercises are completely worked out, is also available (to professors only).
- Numerous examples illuminate the mathematical theories
- Carefully detailed explanations of mathematical derivations guarantee a valuable pedagogical approach
- Each chapter concludes with an extensive set of exercises
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About the Author
William J. Stewart is professor of computer science at North Carolina State University. He is the author of An Introduction to the Numerical Solution of Markov Chains (Princeton).
From the Back Cover
"This is an excellent book on the topics of probability, Markov chains, and queuing theory. Extremely well-written, the contents range from elementary topics to quite advanced material and include plenty of well-chosen examples."--Adarsh Sethi, University of Delaware
"Clear and pleasant to read, this book distinguishes itself from comparable textbooks by its inclusion of the computational aspects of the material."--Richard R. Muntz, University of California, Los Angeles
From the Inside Flap
"This is an excellent book on the topics of probability, Markov chains, and queuing theory. Extremely well-written, the contents range from elementary topics to quite advanced material and include plenty of well-chosen examples."--Adarsh Sethi, University of Delaware
"Clear and pleasant to read, this book distinguishes itself from comparable textbooks by its inclusion of the computational aspects of the material."--Richard R. Muntz, University of California, Los Angeles
Excerpt. © Reprinted by permission. All rights reserved.
PROBABILITY, MARKOV CHAINS, QUEUES, AND SIMULATION
The Mathematical Basis of Performance ModelingBy William J. StewartPRINCETON UNIVERSITY PRESS
Copyright © 2009 Princeton University PressAll right reserved.
ISBN: 978-0-691-14062-9
Contents
Preface and Acknowledgments................................................................xvI PROBABILITY..............................................................................11 Probability..............................................................................32 Combinatorics—The Art of Counting..................................................253 Random Variables and Distribution Functions..............................................404 Joint and Conditional Distributions......................................................645 Expectations and More....................................................................876 Discrete Distribution Functions..........................................................1157 Continuous Distribution Functions........................................................1348 Bounds and Limit Theorems................................................................180II MARKOV CHAINS...........................................................................1919 Discrete- and Continuous-Time Markov Chains..............................................19310 Numerical Solution of Markov Chains.....................................................285III QUEUEING MODELS........................................................................38311 Elementary Queueing Theory..............................................................38512 Queues with Phase-Type Laws: Neuts' Matrix-Geometric Method.............................44413 The z-Transform Approach to Solving Markovian Queues....................................47514 The M/G/1 and G/M/1 Queues..............................................................50915 Queueing Networks.......................................................................559IV SIMULATION..............................................................................61116 Some Probabilistic and Deterministic Applications of Random Numbers.....................61317 Uniformly Distributed "Random" Numbers..................................................62518 Nonuniformly Distributed "Random" Numbers...............................................64719 Implementing Discrete-Event Simulations.................................................68020 Simulation Measurements and Accuracy....................................................697Appendix A: The Greek Alphabet.............................................................719Appendix B: Elements of Linear Algebra.....................................................721Bibliography...............................................................................745Index......................................................................................749
Chapter One
Probability
1.1 Trials, Sample Spaces, and Events
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 and is denoted by Ω. The outcomes that constitute a sample space are also referred to as sample points or elements. We shall use ω to denote an element of the sample space.
Example 1.1 The sample space for coin tossing has two sample points, a head (H) and a tail (T). This gives Ω = {H, T}, as shown in Figure 1.1.
Example 1.2 For throwing a die, the sample space is Ω = {1, 2, 3, 4, 5, 6}, Figure 1.2, which represents the number of spots on the six faces of the die.
Example 1.3 For choosing a card, the sample space is a set consisting of 52 elements, one for each of the 52 cards in the deck, from the ace of spades through the king of hearts.
Example 1.4 If an experiment consists of three tosses of a coin, then the sample space is given by
{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}.
Notice that the element HHT is considered to be different from the elements HTH and THH, even though all three tosses give two heads and one tail. The position in which the tail occurs is important.
A sample space may be finite, denumerable (i.e., infinite but countable), or infinite. 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.
Example 1.5 The sample space derived from an experiment that consists of observing the number of email messages received at a government office in one day may be taken to be denumerable. The sample space is denumerable since we may tag each arriving email message with a unique integer n that denotes the number of emails received prior to its arrival. Thus, Ω = {n|n [member of] N}, where N is the set of nonnegative integers.
Example 1.6 The sample space that arises from an experiment consisting of measuring the time one waits at a bus stop is infinite. Each outcome is a nonnegative real number x and the sample space is given by Ω = {x|x ≥ 0}.
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. For example, it is possible, though perhaps not very likely (i.e., not very probable) that after a very large number of throws of the die, the number 6 has yet to appear.
Notice with emphasis that the sample space is a set, the set consisting of all the elements of the sample space, i.e., all the possible outcomes, associated with a given experiment. Since the sample space is a set, all permissible set operations can be performed on it. For example, the notion of subset is well defined and, for the coin tossing example, four subsets can be defined: the null subset φ the subsets {H}, {T}; and the subset that contains all the elements, Ω = {H, T}. The set of subsets of Ω is
φ, {H}, {T}, {H, T}.
Events
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.
Example 1.7 Consider the event of tossing three coins and getting exactly two heads. There are three outcomes that allow for this event, namely, {HHT, HTH, THH}. The single tail appears on the third, second, or first toss, respectively.
Example 1.8 Consider the event of throwing a die and getting a prime number. Three outcomes allow for this event to occur, namely, {2, 3, 5}. This event comes to pass so long as the throw gives neither one, four, nor six spots.
In these last two examples, we have composed an event as a subset of the sample space, the subset {HHT, HTH, THH} in the first case and the subset {2, 3, 5} in the second. 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. In this case, the elementary events are the singleton subsets of the sample space, the subsets {H}, {5}, and {10 of diamonds}, for example. 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.
Example 1.9 Let A be the event that a throw of the die gives a number greater than 3. This event consists of the subset {4, 5, 6} and we write A = {4, 5, 6}. Event A occurs if the outcome of the trial, the number of spots obtained when the die is thrown, is any one of the numbers 4, 5, and 6. This is illustrated in Figure 1.3.
Example 1.10 Let B be the event that the chosen card is a 9. Event B is the subset containing four elements of the sample space: the 9 of spades, the 9 of clubs, the 9 of diamonds, and the 9 of hearts. Event B occurs if the card chosen is one of these four.
Example 1.11 The waiting time in minutes at a bus stop can be any nonnegative real number. The sample space is Ω = {t [member of] R | t ≥ 0}, and A = {2 ≤ t ≤ 10} is the event that the waiting time is between 2 and 10 minutes. Event A occurs if the wait is 2.1 minutes or 3.5 minutes or 9.99 minutes, etc.
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.
Two events A and B defined on the same sample space are said to be equivalent or identical if A occurs if and only if B occurs. Events A and B may be specified differently, but the elements in their defining subsets are identical. In set terminology, two sets A and B are equal (written A = B) if and only if A [subset] B and B [subset] A.
Example 1.12 Consider an experiment that consists of simultaneously throwing two dice. The sample space consists of all pairs of the form (i, j) for i = 1, 2, ..., 6 and j = 1, 2, ..., 6. Let A be the event that the sum of the number of spots obtained on the two dice is even, i.e., i + j is an even number, and let B be the event that both dice show an even number of spots or both dice show an odd number of spots, i.e., i and j are even or i and j are odd. Although event A has been stated differently from B, a moment's reflection should convince the reader that the sample points in both defining subsets must be exactly the same, and hence A = B.
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.
1. If A is an event, then the complement of A, denoted Ac, is also an event. Ac is the subset of all sample points of Ω that are not in A. Event Ac occurs only if A does not occur.
2. The union of two events A and B, denoted A [union] B, is the event consisting of all the sample points in A and in B. It occurs if either A or B occurs.
3. The intersection of two events A and B, denoted A [intersection] B, is also an event. It consists of the sample points that are in both A and B and occurs if both A and B occur.
4. The difference of two events A and B, denoted by A - B, is the event that A occurs and B does not occur. It consists of the sample points that are in A but not in B. This means that
A - B = A [intersection] Bc.
It follows that Ω - B = Ω n Bc = Bc.
5. Finally, notice that if B is a subset of A, i.e., B ? A, then the event B implies the event A. In other words, if B occurs, it must follow that A has also occurred.
Example 1.13 Let event A be "throw A number greater than 3" and let event B be "throw an odd number." Event A occurs if A 4, 5, or 6 is thrown, and event B occurs if A 1, 3, or 5 is thrown. Thus both events occur if A 5 is thrown (this is the event that is the intersection of events A and B) and neither event occurs if A 2 is thrown (this is the event that is the complement of the union of A and B). These are represented graphically in Figure 1.4. We have
Ac = {1, 2, 3}; A [union] B = {1, 3, 4, 5, 6}; A [intersection] B = {5}; A - B = {4, 6}.
Example 1.14 Or again, consider the card-choosing scenario. The sample space for the deck of cards contains 52 elements, each of which constitutes an elementary event. Now consider two events. Let event A be the subset containing the 13 elements corresponding to the diamond cards in the deck. Event A occurs if any one of these 13 cards is chosen. Let event B be the subset that contains the elements representing the four queens. This event occurs if one of the four queens is chosen. The event A [union] B contains 16 elements, the 13 corresponding to the 13 diamonds plus the queens of spades, clubs, and hearts. The event A [union] B occurs if any one of these 16 cards is chosen: i.e., if one of the 13 diamond cards is chosen or if one of the four queens is chosen (logical OR). On the other hand, the event An B has A single element, the element corresponding to the queen of diamonds. The event A [intersection] B occurs only if a diamond card is chosen and that card is a queen (logical AND). Finally, the event A - B occurs if any diamond card, other than the queen of diamonds, occurs.
Thus, as these examples show, the union of two events is also an event. It is the event that consists of all of the sample points in the two events. Likewise, the intersection of two events is the event that consists of the sample points that are simultaneously in both events. It follows that the union of an event and its complement is the universal event Ω, while the intersection of an event and its complement is the null event φ.
The definitions of union and intersection may be extended to more than two events. For n events A1, A2, ..., An, they are denoted, respectively, by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
In the first case, the event [union]ni = 1 Ai occurs if any one of the events Ai occurs, while the second event, [intersection]ni = 1 Ai occurs only if all the events Ai occur. The entire logical algebra is available for use with events, to give an "algebra of events." Commutative, associative, and distributive laws, the laws of DeMorgan and so on, may be used to manipulate events. Some of the most important of these are as follows (where A, B, and ITLITL are subsets of the universal set Ω):
[TABLE OMITTED]
Venn diagrams can be used to illustrate these results and can be helpful in establishing proofs. For example, an illustration of DeMorgan's laws is presented in Figure 1.5.
Mutually Exclusive and Collectively Exhaustive Events
When two events A and B contain no element of the sample space in common (i.e., A [intersection] B is the null set), the events are said to be mutually exclusive or incompatible. The occurrence of one of them precludes the occurrence of the other. In the case of multiple events, A1, A2, ..., An are mutually exclusive if and only if Ai [intersection] Aj = φ for all i [not equal to] j.
Example 1.15 Consider the four events, A1, A2, A3, A4 corresponding to the four suits in a deck of cards, i.e., A1 contains the 13 elements corresponding to the 13 diamonds, A2 contains the 13 elements corresponding to the 13 hearts, etc. Then none of the sets A1 through A4 has any element in common. The four sets are mutually exclusive.
Example 1.16 Similarly, in the die-throwing experiment, if we choose B1 to be the event "throw a number greater than 5," B2 to be the event "throw an odd number," and A3 to be the event "throw a 2," then the events B1 through B3 are mutually exclusive.
In all cases, an event A and its complement Ac are mutually exclusive. In general, a list of events is said to be mutually exclusive if no element in their sample space is in more than one event. This is illustrated in Figure 1.6.
When all the elements in a sample space can be found in at least one event in a list of events, then the list of events is said to be collectively exhaustive. In this case, no element of the sample space is omitted and a single element may be in more than one event. This is illustrated in Figure 1.7.
(Continues...)
Excerpted from PROBABILITY, MARKOV CHAINS, QUEUES, AND SIMULATIONby William J. Stewart Copyright © 2009 by Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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