Level Crossing Methods in Stochastic Models
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Beschreibung
Since its inception in 1974, the level crossing approach for analyzing a large class of stochastic models has become increasingly popular among researchers. This volume traces the evolution of level crossing theory for obtaining probability distributions of state variables and demonstrates solution methods in a variety of stochastic models including: queues, inventories, dams, renewal models, counter models, pharmacokinetics, and the natural sciences. Results for both steady-state and transient distributions are given, and numerous examples help the reader apply the method to solve problems faster, more easily, and more intuitively.
The book includes introductory material for readers new to the area, as well as advanced material for experienced users of the method, highlighting its usefulness for analyzing a broad class of models and illustrating its flexibility and adaptivity. The concepts, techniques, examples, applications and theoretical results in this book may suggest potentially new theory and new applications. The result is an essential resource for researchers, students, and professionals in operations research, management science, engineering, applied probability, statistics, actuarial science, mathematics, and the natural sciences.
This volume traces the evolution of level crossing theory for obtaining probability distributions of state variables and demonstrates solution methods in a variety of stochastic models. Results for both steady-state and transient distributions are given.
From 1972 to 1974, I was working on a PhD thesis entitled Multiple Server Queues with Service Time Depending on Waiting Time.The method of analysis was the embedded Markov chain technique, described in the papers [82] and [77]. My analysis involved lengthy, tedious deri- tions of systems of integral equations for the probability density function (pdf) of the waiting time. After pondering for many months whether there might be a faster, easier way to derive the integral equations, I ?nally discovered the basic theorems for such a method in August, 1974. The theorems establish a connection between sample-path level-crossing rates of the virtual wait process and the pdf of the waiting time. This connection was not found anywhere else in the literature at the time. I immediately developed a comprehensive new methodology for deriving the integral equations based on these theorems, and called it system point theory. (Subsequently it was called system point method,or system point level crossing method: SPLC or simply LC.) I rewrote the entire PhD thesis from November 1974 to March 1975, using LC to reach solutions. The new thesis was called System Point Theory in Exponential Queues. On June 12, 1975 I presented an invited talk on the new methodology at the Fifth Conference on Stochastic Processes and their Applications at the University of Maryland. Many queueing theorists were present.
Outlines the original developmental ideas that led to the discovery of the level crossing method in the 1970s and traces their evolution to the present
Demonstrates the flexibility and adaptivity of the level crossing method by applying it to a variety of stochastic models, including: queues, inventories, dams, replacement models, renewal processes, counter models, multi-dimensional models, and models in the natural sciences
Presents both introductory and advanced material that can be applied to practical problem solving and theory building
Autor*in
Percy H. Brill
Themen in »Level Crossing Methods in Stochastic Models«
M/G/1 queue Operations Research Probability distribution Stochastic model Stochastic models dams embedded process estimation inventories level crossing linear optimization queues sample path statistics system point
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From the reviews:
This excellent book by the originator of level crossing methods for stochastic models is a highly welcome addition to the literature on queues and inventories. The level crossing method is very powerful and sometimes results in extremely quick and easy derivations when compared with other methods. This book is in a class by itself for its focus on level crossing and for its breadth of coverage, and abundance of new ideas. Some applications of the method even deserve to be inserted into standard texts on stochastic models. Researchers will find a wealth of ideas in this book just waiting for consideration. I suspect that the book will result in a number of dissertations which develop and apply some of the methods presented.
Dr. Myron Hlynka, Department of Mathematics & Statistics, University of Windsor
"As the title indicates, this monograph explores the LC method thoroughly for obtaining probability distributions of state variables in a variety of stochastic models. … The book is intended for research-and-applications-oriented workers in operations research, management science, engineering, probability and statistics, actuarial science, mathematics and the natural sciences. Intended for a large audience … . The book is a welcome addition to any library frequented by researchers interested in stochastic modelling." (Sreenivasan Ravi, Mathematical Reviews, Issue 2009 m)
“Level Crossing Methods in Stochastic Models is a book that describes how such methods can be used to analyse a variety of stochastic models including M/G/1 queues (and variants), M/M/c queues, G/M/c queues, multi-dimensional models, inventory models, and many others. … Overall, the book is well written … . I recommend the book for anyone doing research in queuing or stochastic models. The book is also helpful as a reference in that it summarizes results for a wide variety of systems.” (J. Shortle, Journal of the Operational Research Society, Vol. 61 (5), 2010)
“The book is aimed at the serious researcher in stochastic models, and is loaded with potential research topics. … If someone is working on a new stochastic model, my advice is to examine this book for a related model and try to adopt the methods of the book to the new model. … I highly recommend the book. … There is a large international community of researchers in queueing and stochastic models. It’s time to queue up to purchase this book.” (Myron Hlynka, Technometrics, Vol. 51 (3), August, 2009)
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