Brill Level Crossing Methods in Stochastic Models

1. Auflage 2008
ISBN: 978-0-387-09421-2
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 123, 480 Seiten

Reihe: International Series in Operations Research & Management Science

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.

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. 

Autoren/Hrsg.

Brill, Percy H.

Inhaltsverzeichnis

1;PREFACE;8
2;ACKNOWLEDGEMENTS;11
3;CONTENTS;12
4;ORIGIN OF LEVEL CROSSING METHOD;23
4.1;1.1 Introduction;23
4.2;1.2 Lindley Recursion for GI/G/1 Wait;25
4.3;1.3 Integral Equation for M/ G/1 Waiting Time Derived Using Lindley Recursion;26
4.4;1.4 Observations and Questions;29
4.5;1.5 Further Properties of Integral Equation for PDF of Waiting Time in M/ G/ 1;30
4.6;1.6 Basic Level Crossing Theorem for M/G/ 1;35
4.7;1.7 Integral Equation for M/ G/1 Waiting Time Using Level Crossing Method;38
5;SAMPLE PATH AND SYSTEM POINT;40
5.1;2.1 Introduction;40
5.2;2.2 State Space and Sample Paths in Continu-ous Time Stochastic Models;40
5.3;2.3 System Point Motion and Jumps;44
5.4;2.4 State Space a Subset of R;49
5.5;2.5 Transition Types Geometrically;66
6;M/ G/ 1 QUEUES AND VARIANTS;70
6.1;3.1 Introduction;70
6.2;3.2 Transient Distribution of Wait;70
6.3;3.3 Waiting Time Properties;87
6.4;3.4 M/ M/ 1 Queue;108
6.5;3.5 M/ G/1 with Service Depending on Wait;119
6.6;3.6 M/ G/1 with Multiple Poisson Inputs;124
6.7;3.7 M/G/1: Wait-number Dependent Service;129
6.8;3.8 M/ D/ 1 Queue;134
6.9;3.9 M/ Discrete/ 1 Queue;141
6.10;3.10 M/{ iD}/ 1 Queue;147
6.11;3.11 M/G/ 1 with Reneging;152
6.12;3.12 M/G/ 1 with Priorities;165
6.13;3.13 M/G/ 1 with Server Vacations;174
6.14;3.14 M/G/ 1 with Bounded System Time;177
6.15;3.15 PDF of Wait and Busy-period Structure;180
6.16;3.16 Discussion;182
7;M/ M/ C QUEUES;184
7.1;4.1 Introduction;184
7.2;4.2 Theorem B for Transient Analysis;185
7.3;4.3 Generalized M/M/ c Model;187
7.4;4.4 Virtual Wait and Server Workload;187
7.5;4.5 System Configuration;189
7.6;4.6 System Point Process;194
7.7;4.7 Example of Steady-state Equations;226
7.8;4.8 Standard M/ M/ c: Steady-state Analysis;232
7.9;4.9 M/M/c/c and Standard M/M/c Queues;241
7.10;4.10 M/M/ c: Zero-waits Get Special Service;246
7.11;4.11 M/M/ 2: Zero-waits Get Special Service;253
7.12;4.12 M/ Mi/ c with Reneging;266
7.13;4.13 Discussion;276
8;G/ M/ c QUEUES;277
8.1;5.1 Single-server G/M/ 1 Queue;277
8.2;5.2 Multiple-Server G/M/ c Queue;302
8.3;5.3 G/ M/ 2: PDF of Virtual and of Actual Wait;314
9;DAMS AND INVENTORIES;321
9.1;6.1 Introduction;321
9.2;6.2 M/ G/ r( · ) Dam;322
9.3;6.3 M/ M/ r( · ) Dam;336
9.4;6.4 M/ M/ r( · ) : Efflux Proportional to Content;338
9.5;6.5 Generalization of M/G/r( · ) Dam;341
9.6;6.6 r( · ) / G/ M Dam;349
9.7;6.7 r( · ) / G/ M Dam: Constant Influx Rate;354
9.8;6.8 (s, S) Inventory Model: Decay;357
9.9;6.9 (s, S) Inventory Model: No Decay;364
10;MULTI- DIMENSIONAL MODELS;370
10.1;7.1 Models with State Space a Subset of R2;370
10.2;7.2 Two Products Sharing Limited Storage;374
10.3;7.3 Two Products Sharing Storage: Model 1;375
10.4;7.4 Two Products Sharing Storage: Model 2;384
11;EMBEDDED LEVEL CROSSING METHOD;391
11.1;8.1 Dams and Queues;391
11.2;8.2 GI/ G/ r( · ) Dam;393
11.3;8.3 GI/ G/ 1 Queue;398
11.4;8.4 M/ G/ 1 with Reneging;403
12;LEVEL CROSSING ESTIMATION;406
12.1;9.1 Introduction;406
12.2;9.2 Theoretical Basis for LC Estimation;407
12.3;9.3 Computer Program for LCE;413
12.4;9.4 LCE for M/ G/ 1 Queue;414
12.5;9.5 LCE Example: M/ M/1 with Reneging;421
12.6;9.6 Discussion;425
13;ADDITIONAL APPLICATIONS;427
13.1;10.1 Introduction;427
13.2;10.2 Renewal Processes;427
13.3;10.3 A Technique for Transient Distributions;436
13.4;10.4 Discrete-Parameter Processes;438
13.5;10.5 Semi-Markov Process;440
13.6;10.6 Non-homogeneous Pure Birth Processes;442
13.7;10.7 Revisit of Transient M/ G/1 Queue;444
13.8;10.8 Pharmacokinetic Model;445
13.9;10.9 Counter Models;448
13.10;10.10 A Damwith Alternating Influx and Efflux;455
13.11;10.11 Estimation of Laplace Transforms;462
13.12;10.12 Simple Harmonic Motion;463
13.13;10.13 Renewal Problem with Barrier;467
14;REFERENCES;477
15;PARTIAL BIBLIOGRAPHY;487
16;INDEX;493