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VU Research Portal VU Research Portal Queueing Networks with Shared Resources van der Weij, W. 2009 document version Publisher's PDF, also known as Version of record Link to publication in VU Research Portal citation for published version (APA) van der Weij, W. (2009). Queueing Networks with Shared Resources. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal ? Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. E-mail address: [email protected] Download date: 02. Oct. 2021 Queueing Networks with Shared Resources Wemke van der Weij, 1982 Queueing Networks with Shared Resources ISBN 978-90-8659-298-2 THOMAS STIELTJES INSTITUTE FOR MATHEMATICS The research in this thesis has been carried out in the context of E-Quality Center of Excellence, and has been partly supported by the Basic Research in Informatics for Creating the Knowledge Society (BRICKS) project on Quality of Service in Communication Networks. c W. van der Weij, Amsterdam 2009. All rights reserved. No part of this publication may be reproduced in any form or by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval systems) without permission in writing from the author. Cover designed by Jolanda Oudhoff. Printed by Printpartners Ipskamp B.V., The Netherlands. VRIJE UNIVERSITEIT Queueing Networks with Shared Resources ACADEMISCH PROEFSCHRIFT ter verkrijging van de graad Doctor aan de Vrije Universiteit Amsterdam, op gezag van de rector magnificus prof.dr. L.M. Bouter, in het openbaar te verdedigen ten overstaan van de promotiecommissie van de faculteit der Exacte Wetenschappen op donderdag 23 april 2009 om 10.45 uur in de aula van de universiteit, De Boelelaan 1105 door Wemke van der Weij geboren te Langedijk promotor: prof.dr. R.D. van der Mei DANKWOORD Vier jaar onderzoek zijn voorbij en hier heeft u het eindresultaat in handen. Het proefschrift, een boekwerk waarin mijn onderzoeksresultaten uiteen gezet worden. Bij de totstandkoming van dit proefschrift hebben vele mensen een rol gespeeld. En zoals de titel van deze tekst al zegt wil ik een aantal mensen bedanken. Geen gemakkelijke opgave, want veel mensen hebben op de een of andere manier een bijdrage geleverd, dan wel wetenschappelijk dan wel persoonlijk. Zonder ieder bij toenaam te kunnen noemen, wil ik toch een aantal mensen persoonlijk uitlichten. Allereerst wil ik mijn promotor Rob van der Mei bedanken voor zijn steun en en- thousiasme. Zijn overtuiging van de zelfstandigheid van een promovendus heeft mij vele mogelijkheden tot samenwerkingen in binnen- en buitenland geboden, waarvoor ik hem dankbaar ben. Daarnaast dank aan mijn leescommissie: Hans van den Berg, Sandjai Bhulai, Sem Borst, Nico van Dijk en Michel Mandjes. De suggesties van de commissie voor mijn proefschrift hebben een zeer positieve bijdrage gehad op het eindresultaat. Ook dank aan Bert Zwart en Ger Koole voor het zitting nemen in mijn promotiecommissie. Op het Centrum Wiskunde & Informatica (CWI) maakte ik deel uit van de groep PNA2 en op de Vrije Universiteit maakte ik deel uit van de groep OBP. Beide heb ik als een zeer plezierig en gezellig ervaren, waarvoor ik jullie allen wil bedanken. En in het bijzonder: Maaike, Regina en Pascal als mede- promovendi op het CWI, Marco en Vivek als mijn maatjes op de Vrije Universiteit (VU) en de post-docs Matthieu en Misja met hun wijsheden op wetenschappenlijk en sociaal vlak. Daarnaast mijn grote dank aan Sindo en Sandjai. Jullie hebben niet alleen wetenschappelijk maar ook zeker persoonlijk veel voor mij betekend. Naast mijn ‘wetenschappelijke’ collega’s, vind ik het ook belangrijk de medewerk(st)ers van de ondersteunende diensten te bedanken: Rick, Karin, Micha¨el, Jan, Bikkie, Rob, Susanne en Mike. Jullie stonden altijd voor me klaar om elke vraag te beant- ii woorden en te helpen waar mogelijk was. De gastvrijheid op de VU en de mogelijkheden die mij daar werden geboden om college te geven, studenten te begeleiden en zelfs wat uitstapjes in de wiskunde van callcenters beschouw ik als een zeer leerzame en waardevolle aanvulling op mijn onderzoekswerkzaamheden. Daarnaast heeft de VU alsook het CWI het mogelijk gemaakt dat ik een aantal maanden deel heb mogen uitmaken van de onderzoeks- groep van George Shanthikumar en Rhonda Righter aan de University of California in Berkeley. Naast de wetenschappelijke uitdagingen een zeer mooie ervaring om in Berkeley te zijn geweest. Het OiO-schap bracht ook de nodige congresbezoeken met zich mee, waardoor ik naast Berkeley ook Ilkley, Madrid, Edinburgh, Seattle en Washington heb mogen bewonderen. Onderzoeken is natuurlijk een ontzettend mooi vak, maar om de focus niet te verliezen waren voor mij muzikale uitstapjes een onmisbare afwisseling. Saxofoon en tegenwoordig ook hobo spelen hebben me met beide benen op de grond weten te houden. Veel mensen hebben hier hun bijdrage aan geleverd, wat voor mij erg waardevol is geweest. Zonder hier een lijst met namen te willen produceren wil ik jullie allemaal ontzettend bedanken. Ook heb ik veel steun ontvangen van vrienden en vriendinnen uit mijn geboorte- dorp, mijn middelbare schooltijd en ook mijn studententijd. Ondanks dat het soms onmogelijk leek om te vertellen wat ik nu precies op een werkdag deed, ben ik blij dat jullie mij altijd vertrouwen hebben gegeven. Naast vrienden, collega’s en muzikanten, en velen in een dubbelrol, wil ik mijn familie bedanken. Ze hebben me altijd gesteund en aangemoedigd nieuwe uitdagingen aan te gaan, wat zeer zeker ook heeft bijgedragen tot deze mijlpaal in mijn leven: het proefschrift. Ontzettend bedankt! Last but not least, Bart bedankt voor je begrip en liefde tijdens de laatste fase van het tot stand komen van mijn proefschrift. CONTENTS Dankwoord i Contents iii 1 Introduction 1 1.1 Backgroundandmotivation . 2 1.2 Classicalqueueingmodels . 4 1.2.1 Single-serverqueue. 5 1.2.2 Processor-sharingqueue . 5 1.3 Layeredqueueingmodels .. .. .. .. .. .. .. .. .. .. 6 1.4 Performanceaspects ........................... 8 1.4.1 Stability.............................. 8 1.4.2 Productforms .......................... 9 1.4.3 Scheduling ............................ 10 1.5 Outlineofthethesis ........................... 11 2 Stability analysis 13 2.1 Introduction................................ 13 2.2 Model and stability definitions . 15 2.2.1 Networkmodel .......................... 15 2.2.2 Stabilitydefinitions . 17 2.3 Outputratesandgrowthrates . 19 2.3.1 Definition ............................. 19 2.3.2 Propertiesoftheasymptoticrates . 20 2.4 Ratestabilitynecessaryconditions . ... 22 iv CONTENTS 2.4.1 Trafficinequalities . 22 2.4.2 Necessary conditions for rate stability for convergingrates . 24 2.4.3 Necessary conditions for rate stability . .. 26 2.5 Two-queuetandemmodel . 28 2.6 Parallelqueues .............................. 30 2.6.1 Extendedtrafficequations. 30 2.6.2 Outputratesconvergence . 31 2.6.3 Characterization of the per-queue rate stability . ..... 33 2.7 Conclusion and topics for further research . .... 34 3 Product-form solutions 35 3.1 Introduction................................ 35 3.2 The models and general product-form characterization . ....... 36 3.2.1 Parallelmodel .......................... 37 3.2.2 Tandemmodel .......................... 37 3.2.3 A unifying product-form characterization . .. 39 3.3 Parallelmodel:Examples . 44 3.3.1 ProportionalPSmodel. 45 3.3.2 Unproportional PS model with full capacity to one queue .. 48 3.3.3 α-UnproportionalPSmodel . 50 3.3.4 Statespacerestriction . 52 3.3.5 Two-queueLPSmodel. 52 3.3.6 Truncatedtwo-queueLPSmodel . 54 3.4 Tandemmodel:Examples . 54 3.4.1 ProportionalPSmodel. 55 3.4.2 Unproportional PS model with full capacity to one queue .. 58 3.4.3 α-UnproportionalPSmodel . 60 3.4.4 Statespacerestriction . 62 3.4.5 Two-queueLPSmodel. 62 3.4.6 Truncatedtwo-queueLPSmodel . 63 3.5 Discussion................................. 64 3.6 Conclusion and topics for further research . .... 66 4 Monotonicity properties 67 4.1 Introduction................................ 67 4.2 Modelandpreliminaries . 69 4.3 Themonotonicityresult . 70 4.4 TheLimitedFBdiscipline. 73 4.5 The tail of the sojourn-time distribution . .... 75 4.6 Conclusion and topics for further research . .... 76 CONTENTS v 5 Dynamic scheduling for the LPS queue 79 5.1 Introduction................................ 79 5.2 Modelandpreliminaries . 80 5.3 Optimalschedulingpolicy . 82 5.4 Numericalexperiments. 88 5.5 Conclusion and topics for further research . .... 90 5.6 Appendix: ProofofLemma5.3.3 . 91 6 OptimalserverassignmentinWebservers 99 6.1 Introduction................................ 99 6.2 Modeldescription............................. 102 6.3 Dynamicthreadmanagement . 106 6.4 Numericalexperiments. 111 6.4.1 Comparisonofpolicies . 111 6.4.2 TheApacheWebserver . 112 6.5 Discussion................................. 114 6.6 Conclusion and topics for further research .
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