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Analysis of Structured Multi-Dimensional Markov Processes Analysis of structured multi-dimensional Markov processes Citation for published version (APA): Selen, J. (2017). Analysis of structured multi-dimensional Markov processes. Technische Universiteit Eindhoven. Document status and date: Published: 27/09/2017 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. 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Oct. 2021 Analysis of structured multi-dimensional Markov processes This work was financially supported by The Netherlands Organisation for Scientific Research (NWO) through the Free Competition grant QueQuaR. c Jori Selen, 2017 Analysis of structured multi-dimensional Markov processes A catalogue record is available from the Eindhoven University of Technology Library ISBN: 978-90-386-4322-9 Painting on cover: You and me (in the centre of the universe), Frank Selen Printed by EPS Amsterdam Analysis of structured multi-dimensional Markov processes PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de rector magnificus prof.dr.ir. F.P.T. Baaijens, voor een commissie aangewezen door het College voor Promoties, in het openbaar te verdedigen op woensdag 27 september 2017 om 16:00 uur door Jori Selen geboren te Tegelen Dit proefschrift is goedgekeurd door de promotoren en de samenstelling van de promotiecommissie is als volgt: voorzitter: prof.dr.ir. B. Koren 1e promotor: prof.dr. J.S.H. van Leeuwaarden 2e promotor: prof.dr.ir. I.J.B.F. Adan leden: dr. B.H. Fralix (Clemson University) dr. F.M. Spieksma (Universiteit Leiden) prof.dr.ir. O.J. Boxma prof.dr.ir. G.J.J.A.N. van Houtum prof.dr. R.J. Boucherie (Universiteit Twente) Het onderzoek dat in dit proefschrift wordt beschreven is uitgevoerd in overeenstemming met de TU/e Gedragscode Wetenschapsbeoefening. Acknowledgements As a mechanical engineering student I craved for more mathematical rigour. Ivo Adan and Johan van Leeuwaarden allowed me to satisfy this hunger by offering me a Ph.D. position in an area that is close to my heart. The main result of that position is this thesis. Along the path towards the completion of the thesis I have experienced support and encouragement from many. I would like to take this opportunity to thank them. I am incredibly grateful for the supervision that I have received. Johan, your guidance, ambition, and love for analysis have not only shaped me mathematically, but also as a person. I thoroughly enjoyed our close collaboration on the book and the off-topic discussions that we had during those meetings. Ivo, the enthusiasm, intuition and rigour with which you tackle problems is phenomenal. I have been in the fortunate position to witness these qualities first-hand and I hope that I succeeded in making some of them my own. I would not have been able to do all the work in this thesis by myself. In addition to my supervisors, I wish to thank Stella Kapodistria for the joint effort that resulted in Chapter6 and the intensive period of six weeks filled with daily meetings that led to Chapter7. I am also indebted to Guido Janssen for his help with a particularly nasty proof in Chapter6. Brian Fralix, thank you for teaching me about so many new techniques, for hosting me at Clemson, and for the Friday afternoon Skype calls that ultimately led to Chapter5. Vidyadhar Kulkarni, Andrei Sleptchenko, Geert-Jan van Houtum, your contributions to Chapters2 and4 are also highly appreciated. To Brian Fralix, Floske Spieksma, Onno Boxma, Geert-Jan van Houtum and Richard Boucherie I would like to extend my thanks for being part of my defense committee, reading my dissertation and providing valuable feedback. For four years the Stochastics group has been my second home. I have thoroughly enjoyed its open and pleasant atmosphere and the bright and v vi Acknowledgements motivated people inside of it. I am also thankful for all the help that I received from the support staff. A special mention goes to my current and former office mates Alessandro, Debankur, Gianmarco, Kay and Marta for creating the best work environment that I could have ever hoped for. Outside work, I have had the pleasure to spend time with some great friends. Daan, Don, Etienne, Guus, Jos, Kay H, Kay P, Koen H, Koen J, and Rob, thank you for the great time at the Vestdijk, the boardgames, weekends away, festivals, and all the other things we did throughout the years. Mom, dad, Janne, Dennis, Imo, Karlijn, Lina, thank you for all the interest that you showed in my work, for helping me in whatever way possible and for always being there. I am also very appreciative of the support that I received from Sjors and Margriet. The final paragraph is dedicated to you, Simone. Thank you for your unconditional support, the comfort at home, and all the fun we have together. I am truly happy that I got to share this experience with you and all the ones that are to come. Contents Notation index 1 1 Introduction 3 1.1 Machine with setup times ................................... 5 1.2 Single-server priority system ................................ 9 1.3 Join-the-shortest-queue routing ............................ 13 1.4 Outline of this thesis ........................................ 18 2 The snowball effect of customer slowdown 19 2.1 Introduction.................................................. 20 2.2 Key features of threshold slowdown systems .............. 25 2.3 Model description............................................ 33 2.4 Obtaining the equilibrium distribution .................... 34 2.5 A QED-type regime ......................................... 41 2.6 Conclusion ................................................... 44 2.A Proof of Lemma 2.2 ......................................... 44 2.B Proof of Lemma 2.3 ......................................... 46 3 Product-form solutions for a class of Markov processes 49 3.1 Introduction.................................................. 50 3.2 Modeling assumptions....................................... 54 3.3 Analysis of the balance equations .......................... 56 3.4 Symmetric processes......................................... 63 3.5 Aggregated state concept ................................... 69 3.6 First passage times .......................................... 71 3.7 Conclusion ................................................... 76 4 Single-server priority systems 79 4.1 Introduction.................................................. 80 4.2 Matrix-analytic method ..................................... 82 vii viii Contents 4.3 Application in spare parts logistics......................... 93 4.4 Conclusion ................................................... 97 5 Multi-server priority systems 99 5.1 Introduction.................................................. 100 5.2 Model description and outline of approach ................ 104 5.3 A slight modification of the CAP method ................. 109 5.4 Deriving an explicit expression for vi;j ..................... 112 5.5 Laplace transforms for states in the horizontal boundary 117 5.6 The single-server case ....................................... 131 5.7 Conclusion ................................................... 133 5.A Combinatorial identities..................................... 134 5.B Random-product representation ............................ 138 5.C Results for M=M=1 queues ................................. 139 6 Shortest-expected-delay routing 145 6.1 Introduction.................................................. 146 6.2 Balance equations ........................................... 149 6.3 Evolution of the compensation approach and our contri- bution ........................................................ 153 6.4 Numerical results ............................................ 158 6.5 Applying the compensation approach ...................... 162 6.6 Conclusion ................................................... 190 6.A Proof of step (c) of Lemma 6.8(i) .........................
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