Pricing
Stat Labs: Mathematical Statistics Through Applications
This book is written at a lower mathematical level than many other introductory stochastic processes books such as those of Taylor and Karlin (1993) and Resnick (1992) but for the mixed group of students that I was teaching, the level would probably have been appropriate.
Abstract
This is a nice book on applied stochastic processes for senior undergraduates or rst-year graduate students. The book aims at getting students to try out the material on a computer as early as possible, and I think that is a great plus! Chapters 1–4 contain a probability introduction that covers univariate and joint distributions, random variables, conditional probability, expectation, conditional expectation, and the like. This is fairly standard with many books on the subject. The chapters provide a nice review and should be understandable to students from other disciplines such as computer science, psychology, anthropology, and so forth who may be in such a course. The core material begins in Chapter 5 by de ning discrete-time Markov chains. The introductory examples are simpli ed models for machine reliability, weather, inventory systems, manufacturing, manpower planning, stock markets, and packet switching in communications. They are carried throughout the chapter and used for introducing transition matrices, state transition diagrams, the Chapman–Kolmogorov equations, occupancy times, and limiting behavior. There is a short section on cost modeling and rst passage times. Although the examples are not based on real data, they are realistic enough to maintain the reader’s interest. Having students work with numerical examples as soon as possible helps to reinforce the concepts. Chapter 6 deals with continuous-time Markov chains. The development is analogous to that of discrete-time Markov chains. Topics introduced are Poisson processes, birth and death processes, rst passage times, mean sojourn time in a state, transient and limiting distributions, and cost models. The examples involve reliability, telephone switching, traf c-control mechanisms in digital telecommunications networks, and inventory management. They are realistically presented and accessible to the student. Chapter 7 brie y covers renewal theory, renewal reward processes, and semi-Markov processes. These topics are dif cult for students in an introductory course, so I think a brief introduction such as this is valuable and appropriate. Chapter 8 is on queuing models. It contains an exhaustive discussion presenting singleand multiple-server Markovian queues with nite and in nite capacity. Both the M/G/1 and G/M/1 queues are examined. The last section considers Jackson networks with a simple example. In a onesemester course, if time were short, I would recommend skipping Chapter 7 but covering 8. Chapters 9 and 10 address design and control. Although it is nice to have these chapters in the book, in a traditional one-semester course I doubt that one could cover them. Nonetheless, they allow the interested student a glimpse into these important areas. Traditionally, stochastic processes courses are not concerned with data analysis. They spend a lot of time on the modeling and analysis of systems with xed parameters. Yet, anyone attempting to use these models for real problems needs to know how to handle data! Chapters 9 and 10 do not focus on estimation, but at least they begin with the rst step of recognizing that the parameters are unknown and developing an understanding of static and dynamic control. The topics covered include optimal number of servers in a queue, optimal replacement in reliability, optimal server allocation, inventory control, and control for Markov decision processes. By happy coincidence, I was about a month into teaching introductory stochastic processes when this book arrived for review, so I tried out some of the examples from Chapters 5–8 on my class. The course was a mixed-level class of graduate students from biology, computer science, psychology, anthropology, and statistics. The examples seemed accessible to out-of-department students and statistics master-level students. This book is written at a lower mathematical level than many other introductory stochastic processes books such as those of Taylor and Karlin (1993) and Resnick (1992) but for the mixed group of students that I was teaching, the level would probably have been appropriate. For courses such as ours, this book should certainly be considered as a possible textbook.