Level Crossing Methods in Stochastic Models

1. Front Matter

   Pages 1-21

   PDF

2. Origin Of Level Crossing Method

   • Percy H Brill

   Pages 1-17

3. Sample Path And System Point

   • Percy H Brill

   Pages 1-30

4. M/G/1 Queues And Variants

   • Percy H Brill

   Pages 1-114

5. M/M/C Queues

   • Percy H Brill

   Pages 1-93

6. G/M/c Queues

   • Percy H Brill

   Pages 1-44

7. Dams and Inventories

   • Percy H Brill

   Pages 1-49

8. Multi-Dimensional Models

   • Percy H Brill

   Pages 1-21

9. Embedded Level Crossing Method

   • Percy H Brill

   Pages 1-15

10. Level Crossing Estimation

    • Percy H Brill

    Pages 1-21

11. Additional Applications

    • Percy H Brill

    Pages 1-50

12. Back Matter

    Pages 1-26

    PDF

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.

• 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

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 theOperational 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)

• Faculty of Business, Dept. Management Science,, University of Windsor, Windsor, Canada

  Percy H. Brill

Percy Brill holds a BSc in mathematics and physics from Carleton University, Canada, an MA in mathematical statistics from Columbia University, USA, and a PhD in industrial engineering with a minor in mathematics from the University of Toronto, Canada. He has held an NSERC (Natural Sciences and Engineering Research Council of Canada) grant continuously for 26 years. He served as a consultant for an NSF (National Science Foundation, USA) grant for 4 years. In addition to his academic career, he worked in industry for 12 years as a research scientist, statistical programmer and data processing consultant. He is presently a Professor Emeritus in the Departments of Management Science and Mathematics & Statistics at the University of Windsor, Canada.

 

He has published over 99 articles in refereed journals, conference Proceedings, books and technical reports. In addition, he has presented over 110 talks and seminars at conferences, universities and professional society meetings. He served as president of the Southeastern-Michigan chapter of INFORMS from 1994 to 1996. He is a former associate editor of the journal INFOR. He received the CanQueue 2000 Award of Distinction in recognition of his work in Applied Probability especially in the area of Level Crossing Techniques, at the Canadian queueing conference held at the University of Western Ontario, Canada in September 2000. His research interests include: level crossing theory and methods, applied probability, stochastic processes, stochastic modeling, queueing theory, renewal theory, applied mathematics, and nonparametric statistical inference.

 

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