Stochastic-Process Limits

Stochastic Process Limits are useful and interesting because they generate simple approximations for complicated stochastic processes and also help explain the statistical regularity associated with a macroscopic view of uncertainty.
This book emphasizes the continuous-mapping approach to obtain new stochastic-process limits from previously established stochastic-process limits. The continuous-mapping approach is applied to obtain heavy-traffic-stochastic-process limits for queueing models, including the case in which there are unmatched jumps in the limit process. These heavy-traffic limits generate simple approximations for complicated queueing processes and they reveal the impact of variability upon queueing performance. The book will be of interest to researchers and graduate students working in the areas of probability, stochastic processes, and operations research. In addition this book won the 2003 Lanchester Prize for the best contribution to Operation Research and Management in English, see: http://www.informs.org/Prizes/LanchesterPrize.html

From the reviews:

"The material is self-contained, but it is technical and a solid foundation in probability and queueing theory is beneficial to prospective readers. However, the first five chapters do provide an informal introduction to stochastic-process limits and their applications to queues, and is intended to be accessible to those with less background. This book is a must to researchers and graduate students interested in these areas."

ISI Short Book Reviews, Vol. 22/3, December 2002

"Wonderful work has been done by numerous mathematicians providing the fundamentals of the theory of ‘function fields in one variable’ – at least over finite fields – in equivalence to the established theory of numbers. ... Everybody who wants to get in touch with these results should have a look into this book. For example, Chapter VII ... gives a beautiful application of the Riemann-Hurwitz theorem to the proof of the ABC-conjecture ... . The audience eagerly awaits a second part of this book ... ." (Bernd Richter, Zentralblatt MATH, Vol. 1043 (18), 2004)

"The book covers a reasonable choice of material concerning the arithmetic of function fields that can be treated in a one-year introductory course on the graduate level. ... The text is largely self-contained. ... Both in the large (choice and arrangement of the material) and in the details (accuracy and completeness of proofs, quality of explanations and motivating remarks), the author did a marvellous job. ... Number theory in function fields will be the distinguished textbook in the field for many years." (Ernst-Ulrich Gekeler, Mathematical Reviews, 2003 d)

"The interplay over the years between global function fields and number fields has been intense and extremely fruitful. Number theory in function fields does an excellent job of introducing these ideas. It will be a welcome resource for any number theorist, and it should become a standard text for graduate students in the area. There is a great pedagogical advantage in viewing difficult classical problems first in the function field arena, where they often have very clear, precise, and suggestive solutions." (David Goss, Bulletin of the London Mathematical Society, Vol. 35, 2003)

"Recently, an unusually large number of new textbooks in the area of applied probability/stochastic operations research have appeared. ... The in-depth and integrated treatment of both theoretical and applied issues makes SPL a unique book. It is not only a research monograph, but also aims to serve as an introduction to stochastic process limits and their applications. I believe that both goals have been achieved. ... The list of references is very impressive, which makes the book a very useful reference for researchers." (Bert Zwart, Operations Research Letters, Vol. 33, 2005)

"This is a comprehensive monograph on weak convergence and functional central limit theorem (FCLT) results related to random walk models. Overall, this book is a very welcome addition to the applied probability literature ... . it is a magnificent effort at putting recent issues of the past decade into accessible book form and providing strong motivation on why the study of such limits has important practical implications. It is a ‘tour de force’ which will be a valuable reference for many years to come." (Ravi Mazumdar, Mathematical Reviews, 2003 f)

"This book gives an introduction to functional central limit theorems and their applications to queues. ... With appropriate scaling there may be a nondegenerate stochastic process limit for the queue length process. These heavy-traffic limits can provide useful insight into systems performance. Since mathematical concepts are introduced clearly and results are carefully explained, this book should also be interesting for non-mathematicians working on queueing problems." (F. Hofbauer, Monatshefte für Mathematik, Vol. 139 (4), 2003)

"This book deals with the topic of stochastic-process limits, i.e. limits in which a sequence of stochastic processes converges to another stochastic process, with particular attention to such limits for queues. ... What distinguishes this book from other books on this topic is the author’s focus on stochastic-process limits with non-standard scaling and non-standard limit processes. ... This book is a must to researchers and graduate students interested in these areas." (S. Drekic, Short Book Reviews, Vol. 22 (3), 2002)