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Performance Analysis of Networks on Chips Performance analysis of networks on chips Citation for published version (APA): Beekhuizen, P. (2010). Performance analysis of networks on chips. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR657033 DOI: 10.6100/IR657033 Document status and date: Published: 01/01/2010 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. 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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: 29. Sep. 2021 Performance Analysis of Networks on Chips THOMAS STIELTJES INSTITUTE FOR MATHEMATICS CIP-DATA LIBRARY TECHNISCHE UNIVERSITEIT EINDHOVEN Beekhuizen, Paul Performance analysis of networks on chips / by Paul Beekhuizen – Eindhoven : Technische Universiteit Eindhoven, 2009. Proefschrift. – ISBN 978-90-386-2144-9 NUR 919 Subject headings : Queueing Theory / Networks on Chips 2010 Mathematics Subject Classification : 60K25, 68M20, 90B18 Printed by Eindhoven University of Technology Printservice. Cover design by Paul Verspaget. Performance Analysis of Networks on Chips proefschrift ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de rector magnificus, prof.dr.ir. C.J. van Duijn, voor een commissie aangewezen door het College voor Promoties in het openbaar te verdedigen op donderdag 4 februari 2010 om 16.00 uur door Paul Beekhuizen geboren te Rotterdam Dit proefschrift is goedgekeurd door de promotor: prof.dr.ir. O.J. Boxma Copromotor: dr. J.A.C. Resing Acknowledgements There are many people who, in one way or another, made it possible for me to complete this thesis, and I would like to thank a few of them in particular. First of all, I would like to thank my daily supervisor and co-promotor Jacques Resing. We worked together on most of the research in this thesis, and I enjoyed our cooperation very much. I have especially benefitted a lot from Jacques’ continuous search for the simplest non-trivial model, which proved insightful many times. Next, I would like to thank Dee Denteneer from Philips Research for his guidance and supervision during my period as a PhD and an MSc student. Dee has had a prominent influence on my research over the last five years and I doubt that, without his efforts, I would have pursued a PhD. I also wish to thank Onno Boxma for his supervision. Onno somehow manages to find time for anyone who can benefit from his help and he provided me with valuable feedback on multiple occasions, for which I am very grateful. I am very grateful as well for the funding I received from Philips Research and for the freedom that was given to me. Thanks go out to my fellow PhD students at Philips for making my stay there a pleasant one. During my period as a PhD student, I had an additional affiliation with Eu- random, and I would like to thank everyone at Eurandom for the great time I had there. Eurandom has been a very lively and socially active environment, and working in such an environment seems almost indispensable for the successful completion of a PhD thesis. I will greatly miss the countless diners, poker nights, football and foosball matches, coffee breaks, social excursions, Eurandom lunches, and all other events which I cannot think of at the moment. As there cannot be a PhD defense without a defense committee a word of thanks is due to its president and its members: Ivo Adan, Harm Dorren, Rob van der Mei, Sindo N´u˜nez Queija, and my three supervisors. Ivo is also one of the co-authors of [13] and I wish to thank him for our cooperation on that paper as well. On a more personal note, I owe thanks to (all) my parents, the rest of my family, and my friends for their support and interest. Finally, I thank my wife Ivonne for her unconditional trust in my abilities and for supporting me in whatever I choose to do. v Contents 1 Introduction 1 1.1 Networksonchips ............................ 2 1.1.1 Qualityofservice......................... 3 1.1.2 Flowcontrol ........................... 4 1.1.3 Networktopologies.. .. .. .. .. .. .. .. .. .. 5 1.2 Switches.................................. 6 1.2.1 Bufferingstrategies. 7 1.2.2 Throughput............................ 9 1.2.3 Switchesinnetworksonchips . 11 1.3 Queueingtheory ............................. 11 1.3.1 Generalqueueingsystems . 11 1.3.2 Arrival models in discrete-time queueing systems . .... 13 1.4 Models................................... 16 1.4.1 Single-switchmodels . 16 1.4.2 Concentrating tree networks of polling stations . .... 19 1.5 Keyresultsandorganisationofthethesis . ... 22 2 Uniform switches 25 2.1 Model ................................... 26 2.2 Approximations for K =1........................ 27 2.2.1 TheKHMapproximation . 28 2.2.2 TheKKLapproximation . 29 2.2.3 Geometricapproximation . 29 2.3 Service time approximation for K > 1 ................. 31 2.4 Networkanalysis ............................. 32 2.4.1 Arrivalsattheswitch . 33 2.4.2 ArrivalsattheNI ........................ 33 2.5 Approximationcomparison . 34 2.6 Validation of the geometric distribution . .... 38 2.7 Conclusion ................................ 40 3 Non-uniform switches 41 3.1 Model ................................... 42 vii viii Table of contents 3.2 Saturatedswitch ............................. 42 3.3 Stabilityconditions............................ 43 3.4 Throughput................................ 47 3.5 Waitingtimeandservicerate . 50 3.5.1 Waitingtime ........................... 51 3.5.2 Servicerate............................ 51 3.6 Analysisoftherunningexample . 55 3.6.1 Throughput and stability conditions . 56 3.6.2 Servicerateandmeanwaitingtime . 57 3.7 Correlatedtraffic ............................. 60 3.7.1 Correlatedarrivals . 60 3.7.2 Correlateddestinations . 61 3.8 Numericalanalysis ............................ 62 3.9 Conclusion ................................ 67 4 Reduction of polling tree networks 69 4.1 Introduction................................ 70 4.2 Formalisation ............................... 72 4.2.1 Theoriginalsystem . .. .. .. .. .. .. .. .. .. 72 4.2.2 Thereducedsystem . .. .. .. .. .. .. .. .. .. 75 4.3 Proofofthemainresults. .. .. .. .. .. .. .. .. .. .. 76 4.3.1 Thecoupledsystem . .. .. .. .. .. .. .. .. .. 76 4.3.2 Theoriginalandthecoupledsystem . 77 4.3.3 Thereducedandthecoupledsystem . 80 4.3.4 Waitingtimes........................... 82 4.4 Discussion................................. 83 4.5 Conclusion ................................ 84 5 End-to-end delays in polling tree networks 85 5.1 Model ................................... 86 5.2 Analysisofthetree............................ 88 5.2.1 Overallend-to-enddelay. 88 5.2.2 End-to-enddelaypertype. 89 5.3 AccuracyofApproximation5.2.2 . 91 5.4 Applicationtonetworksonchips . 93 5.4.1 Description ............................ 93 5.4.2 Analysis.............................. 94 5.4.3 Numericalresults. .. .. .. .. .. .. .. .. .. .. 96 5.5 Exactresults ............................... 98 5.5.1 Symmetricstations. 98 5.5.2 Symmetrictrees ......................... 99 5.6 Conclusion ................................ 100 6 Polling systems with Bernoulli service and Markovian routing 103 6.1 Background................................ 104 Table of contents ix 6.2 Modeldescription............................. 105 6.3 Relevantliterature . .. .. .. .. .. .. .. .. .. .. .. 106 6.4 Theapproximation. .. .. .. .. .. .. .. .. .. .. .. 107 6.5 Numericalresults............................. 112 6.6 Large-scalenumericalstudy . 115 6.7 Implementation with Kronecker products . 118 6.8 Conclusion ................................ 121 7 Polling tree networks with flow control 123 7.1 Model ................................... 124 7.2 Markovchainanalysis . 125 7.3 Throughputcomputation . 129 7.4 Bernoulliserviceand Markovianrouting . 130 7.5 Numericalanalysis ............................ 134 7.6 Conclusion ................................ 136 References 139 Summary 151 About the author 153 x Table of contents Chapter 1 Introduction Modules on a chip, such as processors and memories,
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