Level crossing methods in stochastic models / Percy H. Brill.
Brill, Percy H., author.
2017
QA274.2 .B75 2017
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Title Level crossing methods in stochastic models / Percy H. Brill.
Author Brill, Percy H., author.
Edition Second edition.
ISBN 9783319503325 (electronic book)
3319503324 (electronic book)
9783319503301
3319503308
DOI https://doi.org/10.1007/978-3-319-50332-5
Published Cham, Switzerland : Springer, [2017]
Copyright ©2017
Language English
Description 1 online resource.
Item Number 10.1007/978-3-319-50332-5 doi
Call Number QA274.2 .B75 2017
Dewey Decimal Classification 519.2
Summary This is a complete update of the first edition of Level Crossing Methods in Stochastic Models, which was published in 2008. Level crossing methods are a set of sample-path based mathematical tools used in applied probability to establish reliable probability distributions. Since the basis for solving any applied probability problem requires a reliable probability distribution, Level Crossing Methods in Stochastic Models, Second Edition is a useful tool for all researchers working on stochastic application problems, including inventory control, queueing theory, reliability theory, actuarial ruin theory, renewal theory, pharmacokinetics, and related Markov processes. The second edition includes a new section with a novel derivation of the Beneš series for M/G/1 queues. It provides new results on the service time for three M/G/I queueing models with bounded workload. It analyzes new applications of queues where zero-wait customers get exceptional service, including several examples on M/G/1 queues, and a new section on G/M/1 queues. Additionally, there are two other important new sections: on the level-crossing derivation of the finite time-t probability distributions of excess, age, and total life, in renewal theory; and on a level-crossing analysis of a risk model in Insurance. The original Chapter 10 has been split into two chapters: the new chapter 10 is on renewal theory, and the first section of the new Chapter 11 is on a risk model. More explicit use is made of the renewal reward theorem throughout, and many technical and editorial changes have been made to facilitate readability. Percy H. Brill, Ph. D., is a Professor emeritus at the University of Windsor, Canada. Dr. Brill is the creator of the level crossing method for analyzing stochastic models. He has published extensively in stochastic processes, queueing theory and related models, especially using level crossing methods.
Bibliography, etc. Note Includes bibliographical references and index.
Access Note Access limited to authorized users.
Digital File Characteristics text file PDF
Source of Description Online resource, title from PDF title page (viewed May 30, 2017).
Series International series in operations research & management science ; v. 250.
Available in Other Form Level Crossing Methods in Stochastic Models.
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Preface; From Preface of First Edition; Acknowledgements; From Acknowledgements of First Edition; Contents; 1 Origin of Level Crossing Method; 1.1 Prologue ; 1.2 Lindley Recursion for GI/G/1 Wait; 1.3 Integral Equation for M/G/1 PDF of Wait #x83;; 1.4 Observations and Questions; 1.5 Further Properties of Integral Equation #x83;; 1.5.1 Connection with Virtual Wait Process; 1.5.2 Looking Upward from Level Zero ; 1.5.3 Integral Equation in Light of the Sample Path; 1.6 Basic Level Crossing Theorem for M/G/1; 1.6.1 Downcrossing and Upcrossing Rates; 1.7 Integral Equation for M/G/1 #x83.

2 Sample Path and System Point2.1 Introduction; 2.2 State Space and Sample Paths; 2.2.1 Sample Paths; 2.2.2 Sample-Path Properties and Jumps; 2.3 System Point Motion and Jumps; 2.3.1 State Space in the Wide Sense; 2.4 State Space a Subset of mathbbR; 2.4.1 Picture of a Subset of S Over Time; 2.4.2 Levels in S; 2.4.3 Sample Path Transitions; 2.4.4 System Point (SP) Transitions; 2.4.5 Continuous and Jump Crossings; 2.4.6 Number of Transitions in a Finite Time Interval; 2.4.7 Principle of Set Balance; 2.4.8 Rate Balance for Down- and Upcrossings; 2.4.9 Continuous and Discrete States.

2.4.10 Hits and Egresses of Levels2.4.11 Principle of Rate Balance for Hits and Egresses; 2.4.12 Hits and Egresses for Discrete States (Atoms); 2.5 Transition Types Geometrically; 3 M/G/1 Queues and Variants; 3.1 Introduction; 3.2 Transient Distribution of Wait; 3.2.1 Derivative E(mathcalDt(x))/t, x0; 3.2.2 Derivative E(mathcalUt(x))/t, x0; 3.2.3 Level Crossings and Transient CDF of Wait; 3.2.4 Relating the Transient CDF and Level Crossings; 3.2.5 Downcrossings and Transient PDF of Wait; 3.2.6 Alternative Proof of limtrightarrowinftyE(mathcalD t(x))/t=f(x).

3.2.7 Upcrossings and Transient PDF of Wait3.2.8 Integro-differential Equation for PDF of Wait; 3.2.9 PDF When Arrivals and Service Are Time Dependent; 3.2.10 Steady-State PDF of Wait from Transient PDF; 3.3 Steady-State Distribution of Wait; 3.3.1 Alternative LC Equations for PDF of Wait; 3.3.2 Relating System and Waiting Times Using LC; 3.4 Waiting Time Properties in Steady State; 3.4.1 Probability of Zero Wait; 3.4.2 Pollaczek-Khinchine (P-K) Formula; 3.4.3 Expected Number in Queue and in System; 3.4.4 Laplace-Stieltjes Transform (LST) of a PDF.

3.4.5 Series for PDF of Wq by Inverting widetildef(s)3.4.6 Another Look at System Time; 3.4.7 Connecting PDFs of System and Waiting Times; 3.4.8 Number in System Probability Distribution; 3.4.9 Renewal Reward Theorem: Statement; 3.4.10 Expected Busy Period in M/G/1; 3.4.11 Equation for f(x) via Renewal Reward Theorem; 3.4.12 Busy Period Structure in Standard M/G/1; 3.4.13 Probability Distribution of the Busy Period; 3.4.14 Expected Number Served in Busy Period; 3.4.15 Inter-Downcrossing Time of a State-Space Level; 3.4.16 Sojourn Below a Level of { W(t)} t0.

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