This book provides an introduction to heavy-traffic stochastic-process limits for queues, with special emphasis on nonstandard scaling and nonstandard limit processes. In addition to limit processes related to Brownian motion, the book discusses limit processes related to stable Levy motion and fractional Brownian motion, which appear with heavy-tailed probability distributions and strong dependence.

To set the stage, the first four chapters present an informal introduction to stochastic-process limits. These chapters convey the unifying power of stochastic-process limits without sinking into the mathematical details. Simulation is used to help build intuition. Throughout, the book emphasizes the applied significance as well as the underlying mathematics. Hence, the book should appeal to a wide audience, including researchers and graduate students in operations research, industrial engineering, electrical engineering, computer science, business, economics and statistics, as well as specialists in probability theory.

See the selected chapters available here on the Internet to get an overview of the book. See the Preface and Contents to get a more detailed description. See Chapters 1, 2 and 5 to see the introductory focus. See Chapter 6 to see motivation for considering limiting stochastic processes with discontinuous sample paths. See Chapters 12 and 13 for a study of the function space D endowed with the Skorohod (1956) M1 topology. In these more theoretical chapters, the book extends and complements the seminal paper: A. V. Skorohod, Limit theorems for stochastic processes, Theory of Probability and its Applications 1 (1956) 261-290. There is also an Internet Supplement to the book available online or in hard copy.

Here are recent papers closely related to the book:

• An Overview of Brownian and Non-Brownian FCLTs for Single-Server Queues. Queueing Systems, vol. 36, 2000, pp. 39-70. [published PDF]
• Limits for Cumulative Input Processes to Queues. Probability in the Engineering and Information Sciences, vol. 14, 2000, pp. 123-150. [published PDF]
• The Reflection Map with Discontinuities. Mathematics of Operations Research, vol. 26, No. 3, 2001, pp. 447-484. [PDF]

Here are other recent papers somewhat related to the book:

• A Nonstationary Offered-Load Model for Packet Networks. Telecommunication Systems vol. 13, Nos. 3,4, March-April 2001, pp. 271-296 (with Nicholas G. Duffield and William A. Massey). [PDF] [published PDF]
• Network Design and Control Using On-Off and Multi-Level Source Traffic Models with Heavy-Tailed Distributions. Chapter in Self-Similar Network Traffic and Performance Evaluation, Kihong Park and Walter Willinger, eds., Wiley, New York, 2000, pp. 421-445. (with Nicholas G. Duffield). [PDF]
• The Impact of a Heavy-Tailed Service-Time Distribution Upon the M/GI/s Waiting-Time Distribution. Queueing Systems, vol. 36, November 2000, pp. 71-87. [published PDF]

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