Queues and risk models Citation for published version (APA): Badila, E. S. (2015). Queues and risk models. Technische Universiteit Eindhoven. Document status and date: Published: 01/01/2015 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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If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement: www.tue.nl/taverne Take down policy If you believe that this document breaches copyright please contact us at: [email protected] providing details and we will investigate your claim. Download date: 27. Sep. 2021 Queues and Risk Models This work is part of the research project Queues and Risk Models (QUARM 613.001.017) funded by c Emil S¸erban B˘adil˘a,2015. Queues and Risk Models / by E.S¸. B˘adil˘a.{ Mathematics Subject Classification (2010): Primary 60K25 (Queueing theory), 91B30 (Risk theory, Insurance), Secondary 60G51 (Processes with independent increments; L´evyprocesses), 62P05 (Applications to actuarial sciences and financial mathematics), 90B22 (Queues and service). A catalogue record is available from the Eindhoven University of Technology Library. ISBN: 978-90-386-3877-5 Printed by: Gildeprint Queues and Risk Models 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 maandag 22 juni 2015 om 14.00 uur door Emil S¸erban B˘adil˘a geboren te T^argu-Mure¸s,Roemeni¨e Dit proefschrift is goedgekeurd door de promotoren en de samenstelling van de promotiecommissie is als volgt: voorzitter: prof.dr. E.H.L. Aarts 1e promotor: prof.dr.ir. O.J. Boxma co-promotor: dr. J.A.C. Resing leden: prof.dr.ir. I.J.B.F. Adan prof.dr. H. Albrecher (Universit´ede Lausanne) prof.dr. J.S.H. van Leeuwaarden prof.dr. M.R.H. Mandjes (University of Amsterdam) prof.dr. Z. Palmowski (University of Wroc law) Acknowledgments The contents of this book are the result of the four-year work on the topic presented in the title. Intensive as it was, it wouldn't have been possible without the contribution and support of many people, to whom I am most grateful. Firstly, I am indebted to my supervisors. Onno was always kind, careful and eager to help and inspire me with a fantastic mathematical common sense and intuition; And on top of this with a great sense of humor. I am a lucky fellow to have you as a supervisor. I want to thank Jacques for all the fruitful discussions we had. Throughout all our projects, you always managed to make the right remarks and raise just objections, especially when I was being sloppy. When I had a question, the doors of your offices were always open. I would also like to thank Hansjoerg Albrecher, Zbigniew Palmowski, Michel Mandjes, Ivo Adan and Johan van Leeuwaarden for agreeing to be part of the committee. Special thanks go to Hansjoerg for inviting me to visit his group in Lausanne and to Jevgenij for taking me around the city. I also want to thank Zbyszek for the visit I paid to Wroc law and for the good work we have done together; This visit has also been a great opportunity to have some nice discussions with Thomasz Rolski. EURANDOM felt like a big family. Without trying to be precise, I witnessed here two successive generations of PhD students. I have to mention the pleasant evenings at Botond, Robert and Elena's house - the EURANDOM house as we used to call it. The barbecues, the games and the pub quizzing have always been a good distraction. It was great to have around the "old" bunch of jolly fellows (in random order): Tim, Stella, Paulo, Julia, Peter, Jan-Pieter, Eleni, Enrico, Carlo, Martin, Florence, Sander. And a special mention for Jaron whom I've never seen performing live yet and Alessandro, with whom I had the trip to the other end of the world. Then there is the new generation; I had many nice evenings, pizzas and board games with Britt, Jori, Alessandro, Maria Luisa, Fabio, Thomas, Gianmarco, Murtaza, Szilard, Clara, Rick, Bart, Fiona and Mirko. Besides, there were also the squash and the basketball games with Fabio. Thank you all for the great time. v vi Acknowledgments I also wish to thank R˘azvan, Ada, Andrei, Andreea, Drago¸s,Ionut¸, Delia, Radu; I could not write this without mentioning you here. I wish I would see you more often. Finally, I am grateful to Maria, Daniela and to my parents for their unconditional love and support. S¸erban B˘adil˘a Eindhoven, May 2015 Contents Acknowledgmentsv 1 Introduction1 1.1 Single server queues and Sparre-Andersen risk reserve processes, with correlations.................................4 1.2 Queueing systems and risk reserve processes with multiple components8 1.3 Duality................................... 10 1.4 Outline and contributions......................... 12 2 Duality 15 2.1 The embedded workload as a potential loss for the risk reserve process 16 2.2 Multivariate duality............................ 18 2.3 Siegmund duality for coupled processors models............. 21 3 Single server queues and Sparre-Andersen risk reserve processes with correlations 31 3.1 Model description and analysis of waiting times............ 32 3.2 The steady-state workload......................... 36 3.3 Duality between the insurance and queueing processes......... 38 3.4 Examples and numerical results...................... 41 3.5 Appendix A................................. 50 4 Integral representations for one-dimensional random walks 55 4.1 On Hewitt's inversion formula....................... 57 4.1.1 Preliminaries............................ 57 4.2 The GI/G/1 queue with correlations................... 61 4.2.1 The number of arrivals during an excursion........... 66 4.3 Examples.................................. 68 4.4 The time to ruin when starting at a positive level............ 73 vii viii Contents 5 Queues and risk processes with multivariate Poisson input 77 5.1 Model description.............................. 78 5.2 The analysis of the two-dimensional problem.............. 79 5.3 Relation with other models........................ 83 5.4 The k-dimensional problem........................ 87 5.5 The general two-dimensional workload/reinsurance problem...... 93 5.6 Appendix B................................. 97 6 Two parallel insurance lines with simultaneous arrivals 99 6.1 Model description............................. 100 6.2 A functional equation........................... 102 6.3 Wiener-Hopf analysis of the stochastic recursion............ 104 6.3.1 Preparations............................ 105 6.3.2 A Wiener-Hopf factorization................... 107 6.3.3 The main result.......................... 110 6.4 A probabilistic decomposition....................... 112 6.5 Examples and numerical inversion.................... 114 6.6 Appendix C................................. 124 7 Proportional reinsurance with subexponential claims 127 7.1 Introduction................................. 127 7.2 Main results................................. 128 7.3 Proof of the main result.......................... 129 8 A coupled processor model with simultaneous arrivals 137 8.1 Model description.............................. 138 8.2 Recursive equations for the amount of work in the coupled system.. 139 8.3 The transform of the equilibrium amount of work at arrival epochs.. 141 8.4 The k-dimensional model......................... 144 8.5 Conclusions and final remarks....................... 148 Bibliography 151 Summary Curriculum Vitae Chapter 1 Introduction There are many daily life examples related to traffic and the problem of congestion, which occur whenever a given resource cannot keep up with the rate of arrival of service requests. Examples of offered resources include Internet bandwidth, the width of a highway as part of a traffic network, the speed of the central processing unit (CPU) in a computer, or the working speed of a server at a counter in a supermarket. A natural way to mitigate the unavoidable congestion issues is to consider a buffer in which arriving customers/jobs can queue up waiting for service. The study of phenomena related to congestion relies significantly on the mathematics of Queueing Theory, and this includes related problems such as the design and optimization of queueing systems. There is a dual perspective which makes queueing processes appear as mirror images of so-called risk reserve processes (or better called surplus processes). These latter describe the dynamics of the surplus process of an insurance portfolio, or the evolution of assets on the stock market. The amount of processing time demanded from a CPU is not constant among the various requested jobs. Even more so, one cannot expect fixed deterministic time points at which cars arrive at an intersection.