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Queueing systems with periodic service Citation for published version (APA): Eenige, van, M. J. A. (1996). Queueing systems with periodic service. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR465324 DOI: 10.6100/IR465324 Document status and date: Published: 01/01/1996 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. Link to publication 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. 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: 10. Oct. 2021 Queueing Systen1s with Periodic Service Queueing Systems with Periodic Service PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de Rector Magnificus, prof.dr. M. Rem, voor een commissie aangewezen door het College van Dekanen in het openbaar te verdedigen op dinsdag 17 september 1996 om 16.00 uur door Michel Johannes Alfonsus van Eenige geboren te Voorburg Dit proefschrift is goedgekeurd door de promotoren: prof.dr. J. Wessels en prof.dr. F.W. Steutel Copromotor: dr.ir. J. van der Wal Eenige, Michel Johannes Alfonsus van Queueing Systems with Periodic Service I Michel Johannes Alfonsus van Eenige. - Eindhoven: Eindhoven University of Technology Thesis Technische Universiteit Eindhoven. - With index, ref. - With summary in Dutch ISBN 90-386-0348-7 © 1996 by M.J.A. van Eenige, Eindhoven, The Netherlands Contents 1 Introduction 1 1.1 Motivation and objective 1.2 Overview of related I iterature . 4 1.2.1 Analytical techniques . 5 1.2.2 Approximation techniques 11 1.2.3 Conclusions 13 1.3 Outline of the monograph ..... 13 2 A Queueing System with Periodic Service 17 2.1 Introduction . 17 2.2 The model and the imbedded queue-length process 18 2.3 Two analytical techniques . .... 20 2.3.1 The method of particular solutions 22 2.3.2 The generating-function technique 25 2.3.3 Conclusions . .. 31 2.4 A numerical approach exploiting the tail behaviour 31 2.5 A moment-iteration technique . 34 2.6 Conclusions . 38 3 A Numerical Technique for a Class of One-Dimensional Markov Chains 41 3.1 Introduction . 41 3.2 The Markov chain and the equilibrium equations 43 3.3 Solving the equilibrium equations analytically 45 3.3.1 The method of particular solutions .. 46 3.3.2 The generating-function technique .. 51 3.3.3 Numerical difficulties of the analytical techniques . 54 3.4 Solving the equilibrium equations numerically . 54 3.5 The G I/ E,/ 1 queueing system ........ 57 3.5.1 The model and the equilibrium equations 58 3.5.2 Solving the equilibrium equations . 59 3.5.3 Numerical examples 60 3.6 Conclusions . ..... 63 4 A Numerical Technique for Queueing Systems with Periodic Service 65 4.1 Introduction . 65 4.2 The model . 67 4.2.1 The periodic service policy 67 ii Contents 4.2.2 A class of arrival processes and service-time distributions . 68 4.2.3 Additional notation and conventions . 70 4.3 The queue-length process . 71 4.3.1 Validation of the applicability of the GT technique 72 4.3.2 The fixed-cycle traffic-light queue . 79 4.4 The sojourn-time distribution . 79 4.5 A matrix-analytic approach . 84 4.5.1 The queue-length process . 84 4.5.2 The sojourn-time distribution 86 4.6 Numerical examples 89 4.7 Conclusions ........... 93 5 A Moment-Iteration Technique for Queueing Systems with Periodic Service 95 5.1 Introduction. 95 5.2 A moment-iteration method for the G I I G I 1 queueing system . 96 5.3 The model . 98 5.4 The queue-length process . 99 5.4.1 The queue-length process at slot boundaries . 100 5.4.2 The MI technique . 101 5.5 Fitting discrete distributions on the first two moments . 103 5.6 The sojourn-time distribution 107 5.7 Numerical examples 108 5.8 Conclusions . 113 6 Make to Stock and Overtime in Queueing Systems with Periodic Service 115 6.1 Introduction. 115 6.2 Queueing systems with periodic service and make to stock 116 6.2.1 The model . 116 6.2.2 The GT technique . 117 6.2.3 The MI technique . 119 6.2.4 Numerical examples 119 6.3 Queueing systems with periodic service and overtime . 120 6.3.1 The model . 122 6.3.2 The GT technique . 122 6.3.3 The MI technique . 125 6.3.4 Numerical examples 126 6.4 Conclusions . 127 7 Regular and Incidental Customers in Queueing Systems with Periodic Service 129 7.1 Introduction . 129 7.2 The model . 130 7.3 Regu1arcustomers. 131 7.4 Incidental customers 131 7.5 A numerical example B2 7.6 Conclusions . 134 Contents iii 8 Conclusions and Suggestions for Future Research 13S Bibliography 139 Samenvatting 14S Curriculum Vitae 149 IV Contents 1 Introduction 1.1 Motivation and objective In this monograph, we study single-server multi-queue systems with periodic service. For these sys­ tems, the time axis consists of intervals of equal length, called cycles. In a cycle, the server attends the different queues to serve customers. The order in which he visits the queues is the same for all cycles. Moreover, the time instants within a cycle at which the server starts switching from one queue to another are fixed and the same for all cycles. Further, switching from one queue to another may take some time, and no customer can be served during these time periods. As an illustration of such a queueing system, consider the case of two queues and deterministic switch-over times. In Figure 1.1, we give an example of a periodic service policy for this system. For this example, the server attends queue 1 for three time units to serve customers in this queue. After this period, he requires two time units to switch to queue 2. Then, he attends queue 2 for four time units to serve customers, after which he requires one time unit to switch back to queue 1. Finally, the service policy starts over again. So, in this example, the time interval (0, lO) can be considered as a cycle. Figure 1.1: A representation of a periodic service policyfor an example(<{ a queueing system with two queues and deterministic switch-over times. Various real-life situations are modelled in a natural way by queueing systems with periodic ser­ vice. We give three examples of such situations. The first example is a fixed-cycle traffic light at an 2 1. Introduction intersection. The second example is a periodic access scheme to allocate the capacity of communica­ th:-n and computer systems. The third example is the manufacturing of products governed by a periodic production rule. For the first example, consider a fixed-cycle traffic light to control an intersection. More precisely, for each direction, cars that approach the intersection alternately face red and green time periods of fixed duration. Clearly, this model can be regarded as a queueing system with periodic service with the traffic light as the server, the different directions as the queues, and the cars as the customers. These models were first studied in the 1950's to evaluate the throughput and delay of cars. Nowadays, fixed­ cycle traffic lights particularly arise in heavy traffic and as a result of phased traffic lights. Since the mid 1970's, queueing systems with periodic service are also used in communication and computer systems. The capacity of these systems for transmitting data or executing tasks has to be shared by different types of data or tasks. In real-time computer systems, some tasks have strict time­ critical requirements. To meet these requirements, these tasks have priority, and their executions are scheduled periodically. Hence, the execution of ordinary tasks is interrupted periodically. For these systems, the quantities of interest are the probability of a buffer overflow and the probability that or­ dinary tasks meet their deadlines. In the last decade, queueing systems with periodic service have been applied to model situations at production centres as well. This application is our main motivation for studying these systems. Con­ sider a machine at a production centre, which manufactures a variety of products.
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