Level Crossing Methods in Stochastic Models (eBook)

Percy H. Brill (Autor)

eBook Download: PDF

2008 | 2008
XXV, 480 Seiten
Springer US (Verlag)
978-0-387-09421-2 (ISBN)

Infos

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.

 

Inhalt

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

Produkt-Details

• Erscheint lt. Verlag: 3.12.2008
• Reihe/Serie: International Series in Operations Research & Management Science
• Reihe/Serie: International Series in Operations Research & Management Science
• Zusatzinfo: XXV, 480 p.
• Verlagsort: New York
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Informatik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Technik ► Bauwesen
• Themenwelt: Wirtschaft ► Betriebswirtschaft / Management ► Logistik / Produktion
• Themenwelt: Wirtschaft ► Betriebswirtschaft / Management ► Planung / Organisation
• Schlagworte: Dams • embedded process • estimation • inventories • level crossing • linear optimization • M/G/1 queue • Operations Research • Probability Distribution • queues • sample path • Statistics • stochastic model • stochastic models • system point
• ISBN-10: 0-387-09421-0 / 0387094210
• ISBN-13: 978-0-387-09421-2 / 9780387094212