Level crossing methods in stochastic models. ([2017])
- Record Type:
- Book
- Title:
- Level crossing methods in stochastic models. ([2017])
- Main Title:
- Level crossing methods in stochastic models
- Further Information:
- Note: Percy H. Brill.
- Authors:
- Brill, Percy H
- Contents:
- 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 DerivativePreface; 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. … (more)
- Publisher Details:
- Cham, Switzerland : Springer
- Publication Date:
- 2017
- Copyright Date:
- 2017
- Extent:
- 1 online resource
- Subjects:
- 519.2
Business
Stochastic models
Stochastic processes
Queuing theory
MATHEMATICS -- Applied
MATHEMATICS -- Probability & Statistics -- General
Queuing theory
Stochastic models
Stochastic processes
Business & Economics -- Operations Research
Operational research
Management science
Operations research
Electronic books - Languages:
- English
- ISBNs:
- 9783319503325
3319503324 - Related ISBNs:
- 9783319503301
3319503308 - Notes:
- Note: Includes bibliographical references and index.
Note: Online resource, title from PDF title page (EBSCO, viewed May 30, 2017). - Access Rights:
- Legal Deposit; Only available on premises controlled by the deposit library and to one user at any one time; The Legal Deposit Libraries (Non-Print Works) Regulations (UK).
- Access Usage:
- Restricted: Printing from this resource is governed by The Legal Deposit Libraries (Non-Print Works) Regulations (UK) and UK copyright law currently in force.
- View Content:
- Available online (eLD content is only available in our Reading Rooms) ↗
- Physical Locations:
- British Library HMNTS - ELD.DS.340374
- Ingest File:
- 01_289.xml