Commuting matrices in the sojourn time analysis of MAP/MAP/1 queues

Queues with Markovian arrival and service processes, i.e., MAP/MAP/1 queues, have been useful in the analysis of computer and communication systems and different representations for their sojourn time distribution have been derived. More specifically, the class of MAP/MAP/1 queues lies at the intersection of the class of QBD queues and the class of semi-Markovian queues. While QBD queues have an order $N^2$ matrix exponential representation for their sojourn time distribution, where $N$ is the size of the background continuous time Markov chain, the sojourn time distribution of the latter class allows for a more compact representation of order $N$. In this paper we unify these two results and show that the key step exists in establishing the commutativity of some fundamental matrices involved in the analysis of the MAP/MAP/1 queue. We prove, using two different approaches, that the required matrices do commute and identify several other sets of commuting matrices. Finally, we generalize some of the results to queueing systems with batch arrivals and services

Analysis of a two-class single-server discrete-time FCFS queue : the effect of interclass correlation

In this paper, we study a discrete-time queueing system with one server and two classes of customers. Customers enter the system according to a general independent arrival process. The classes of consecutive customers, however, are correlated in a Markovian way. The system uses a global FCFS service discipline, i.e., all arriving customers are accommodated in one single FCFS queue, regardless of their classes. The service-time distribution of the customers is general but class-dependent, and therefore, the exact order in which the customers of both classes succeed each other in the arrival stream is important, which is reflected by the complexity of the system content and waiting time analysis presented in this paper. In particular, a detailed waiting time analysis of this kind of multi-class system has not yet been published, and is considered to be one of the main novelties by the authors. In addition to that, a major aim of the paper is to estimate the impact of interclass correlation in the arrival stream on the total number of customers in the system, and the customer delay. The results reveal that the system can exhibit two different classes of stochastic equilibrium: a strong equilibrium where both customer classes give rise to stable behavior individually, and a compensated equilibrium where one customer type creates overload

15 september 2010: de internationale dag van de democratie

Since 2008 the International Community has been observing annually the International Day of Democracy. This article examines what exactly the international community celebrates on that day. In other words it is analyzed how the concept of democracy is defined within the UN framework

Performance modelling and analysis of e-commerce systems using class based priority scheduling. An investigation into the development of new class based priority scheduling mechanisms for e-commerce system combining different techniques.

Recently, technological developments have affected most lifestyles, especially with the growth in Internet usage. Internet applications highlight the E-commerce capabilities and applications which are now available everywhere; they receive a great number of users on a 24-7 basis because online services are easy to use, faster and cheaper to acquire. Thus E-commerce web sites have become crucial for companies to increase their revenues. This importance has identified certain effective requirements needed from the performance of these applications. In particular, if the web server is overloaded, poor performance can result, due to either a huge rate of requests being generated which are beyond the server¿s capacity, or due to saturation of the communication links capacity which connects the web server to the network. Recent researches consider the overload issue and explore different mechanisms for managing the performance of E-commerce applications under overload condition. This thesis proposes a formal approach in order to investigate the effects of the extreme load and the number of dropped requests on the performance of E- III commerce web servers. The proposed approach is based on the class-based priority scheme that classifies E-commerce requests into different classes. Because no single technique can solve all aspects of overload problems, this research combines several techniques including: admission control mechanism, session-based admission control, service differentiation, request scheduling and queuing model-based approach. Request classification is based on the premise that some requests (e.g. buy) are generally considered more important than others (e.g. browse or search). Moreover, this research considers the extended models from Priority Scheduling Mechanism (PSM). These models add a new parameter, such as a review model or modify the basic PSM to low priority fair model, after the discovery of ineffectiveness with low priority customers or to add new features such as portal models. The proposed model is formally specified using the ¿ -calculus in early stage of models design and a multi-actor simulation was developed to reflect the target models as accurately as possible and is implemented as a Java-based prototype system. A formal specification that captures the essential PSM features while keeping the performance model sufficiently simple is presented. Furthermore, the simplicity of the UML bridges the gap between ¿-calculus and Java programming language. IV There are many metrics for measuring the performance of E-commerce web servers. This research focuses on the performance of E-commerce web servers that refer to the throughput, utilisation, average response time, dropped requests and arrival rate. A number of experiments are conducted in order to test the performance management of the proposed approaches

Eliminierung negativer Effekte autokorrelierter Prozesse an Zusammenführungen

Im Kern der vorliegenden Arbeit wird eine neue Vorfahrtstrategie zur Steuerung von Materialflüssen an Zusammenführungen vorgestellt. Das Hauptanwendungsgebiet stellen innerbetriebliche Transportsysteme dar, wobei die Erkenntnisse auf beliebige Transport- bzw. Bediensysteme übertragbar sind. Die Arbeit grenzt sich mit der Annahme autokorrelierter Ankunftsprozesse von bisheriger Forschung und Entwicklung ab. Bis dato werden stets unkorrelierte Ströme angenommen bzw. findet keine spezielle Beachtung autokorrelierter Ströme bei der Vorfahrtsteuerung statt. Untersuchungen zeigen aber, dass zum einen mit hoher Konfidenz mit autokorrelierten Materialflüssen zu rechnen ist und in diesem Fall zum anderen von einem erheblichen Einfluss auf die Systemleistung ausgegangen werden muss. Zusammengefasst konnten im Rahmen der vorliegenden Arbeit 68 Realdatensätze verschiedener Unternehmen untersucht werden, mit dem Ergebnis, dass ca. 95% der Materialflüsse Autokorrelation aufweisen. Ferner wird hergeleitet, dass Autokorrelation intrinsisch in Materialflusssystemen entsteht. Die Folgen autokorrelierter Prozesse bestehen dabei in längeren Durchlaufzeiten, einem volatileren Systemverhalten und höheren Wahrscheinlichkeiten von Systemblockaden. Um die genannten Effekte an Zusammenführungen zu eliminieren, stellt die Arbeit eine neue Vorfahrtstrategie HAFI – Highest Autocorrelated First vor. Diese priorisiert die Ankunftsprozesse anhand deren Autokorrelation. Konkret wird die Vorfahrt zunächst so lange nach dem Prinzip First Come First Served gewährt, bis richtungsweise eine spezifische Warteschlangenlänge überschritten wird. Der jeweilige Wert ergibt sich aus der Höhe der Autokorrelation der Ankunftsprozesse. Vorfahrt bekommt der Strom mit der höchsten Überschreitung seines Grenzwertes. Die Arbeit stellt ferner eine Heuristik DyDeT zur automatischen Bestimmung und dynamischen Anpassung der Grenzwerte vor. Mit einer Simulationsstudie wird gezeigt, dass HAFI mit Anwendung von DyDeT die Vorzüge der etablierten Vorfahrtstrategien First Come First Served und Longest Queue First vereint. Dabei wird auch deutlich, dass die zwei letztgenannten Strategien den besonderen Herausforderungen autokorrelierter Ankunftsprozesse nicht gerecht werden. Bei einer Anwendung von HAFI zur Vorfahrtsteuerung können Durchlaufzeiten und Warteschlangenlängen auf dem Niveau von First Come First Served erreicht werden, wobei dieses ca. 10% unter dem von Longest Queue First liegt. Gleichzeitig ermöglicht HAFI, im Gegensatz zu First Come First Served, eine ähnlich gute Lastbalancierung wie Longest Queue First. Die Ergebnisse stellen sich robust gegenüber Änderungen der Auslastung sowie der Höhe der Autokorrelation dar. Gleichzeitig sind die Erkenntnisse unabhängig der Analyse einer isolierten Zusammenführung und der Anordnung mehrerer Zusammenführungen in einem Netzwerk.:1 Einleitung 1 1.1 Motivation 1 1.2 Zielsetzung, wissenschaftlicher Beitrag 4 1.3 Konzeption 5 2 Grundlagen 7 2.1 Automatisierung, Steuern, Regeln 7 2.2 System, Modell 10 2.3 Stochastik, Statistik 14 2.3.1 Wahrscheinlichkeitsverteilungen 14 2.3.2 Zufallszahlengeneratoren 21 2.3.3 Autokorrelation als Ähnlichkeits- bzw. Abhängigkeitsmaß 24 2.4 Simulation 29 2.5 Warteschlangentheorie und -modelle 32 2.6 Materialflusssystem 35 2.7 Materialflusssteuerung 37 2.7.1 Steuerungssysteme 37 2.7.2 Steuerungsstrategien 40 2.8 Materialflusssystem charakterisierende Kennzahlen 46 3 Stand der Forschung und Technik 51 3.1 Erzeugung autokorrelierter Zufallszahlen 51 3.1.1 Autoregressive Prozesse nach der Box-Jenkins-Methode 52 3.1.2 Distorsions-Methoden 54 3.1.3 Copulae 56 3.1.4 Markovian Arrival Processes 58 3.1.5 Autoregressive Prozesse mit beliebiger Randverteilung 61 3.1.6 Weitere Verfahren 64 3.1.7 Bewertung der Verfahren und Werkzeuge zur Generierung 65 3.2 Wirken von Autokorrelation in Bediensystemen 68 3.3 Fallstudien über Autokorrelation in logistischen Systemen 75 3.4 Ursachen von Autokorrelation in logistischen Systemen 89 3.5 Steuerung von Ankunftsprozessen an Zusammenführungen 96 3.6 Steuerung autokorrelierter Ankunftsprozesse 100 4 Steuerung autokorrelierter Ankunftsprozesse an Zusammenführungen 105 4.1 Modellannahmen, Methodenauswahl, Vorbetrachtungen 106 4.2 First Come First Served und Longest Queue First 114 4.3 Highest Autocorrelated First 117 4.3.1 Grundprinzip 117 4.3.2 Bestimmung der Grenzwerte 127 4.3.3 Dynamische Bestimmung der Grenzwerte mittels „DyDeT“ 133 4.4 Highest Autocorrelated First in Netzwerken 150 4.5 Abschließende Bewertung und Diskussion 161 5 Zusammenfassung und Ausblick 167 Primärliteratur 172 Normen und Standards 194 Abbildungsverzeichnis 197 Tabellenverzeichnis 199 Pseudocodeverzeichnis 201 Abkürzungsverzeichnis 203 Symbolverzeichnis 205 Erklärung an Eides statt 209The work at hand presents a novel strategy to control arrival processes at merges. The main fields of application are intralogistics transport systems. Nevertheless, the findings can be adapted to any queuing system. In contrast to further research and development the thesis assumes autocorrelated arrival processes. Up until now, arrivals are usually assumed to be uncorrelated and there are no special treatments for autocorrelated arrivals in the context of merge controlling. However, surveys show with high reliability the existence of autocorrelated arrivals, resulting in some major impacts on the systems\' performance. In detail, 68 real-world datasets of different companies have been tested in the scope of this work, and in 95% of the cases arrival processes significantly show autocorrelations. Furthermore, the research shows that autocorrelation comes from the system itself. As a direct consequence it was observed that there were longer cycle times, more volatile system behavior, and a higher likelihood of deadlocks. In order to eliminate these effects at merges, this thesis introduces a new priority rule called HAFI-Highest Autocorrelated First. It assesses the arrivals\' priority in accordance to their autocorrelation. More concretely, priority initially is given in accordance to the First Come First Served scheme as long as specific direction-wise queue lengths are not exceeded. The particular thresholds are determined by the arrival processes\' autocorrelation, wherein the process with the highest volume gets priority. Furthermore, the thesis introduces a heuristic to automatically and dynamically determine the specific thresholds of HAFI-so called DyDeT. With a simulation study it can be shown that HAFI in connection with DyDeT, combines the advantages of the well-established priority rules First Come First Served and Longest Queue First. It also becomes obvious that the latter ones are not able to deal with the challenges of autocorrelated arrival processes. By applying HAFI cycling times and mean queue lengths on the level of First Come First Served can be achieved. These are about 10% lower than for Longest Queue First. Concomitantly and in contrast to First Come First Served, HAFI also shows well balanced queues like Longest Queue First. The results are robust against different levels of throughput and autocorrelation, respectively. Furthermore, the findings are independent from analyzing a single instance of a merge or several merges in a network.:1 Einleitung 1 1.1 Motivation 1 1.2 Zielsetzung, wissenschaftlicher Beitrag 4 1.3 Konzeption 5 2 Grundlagen 7 2.1 Automatisierung, Steuern, Regeln 7 2.2 System, Modell 10 2.3 Stochastik, Statistik 14 2.3.1 Wahrscheinlichkeitsverteilungen 14 2.3.2 Zufallszahlengeneratoren 21 2.3.3 Autokorrelation als Ähnlichkeits- bzw. Abhängigkeitsmaß 24 2.4 Simulation 29 2.5 Warteschlangentheorie und -modelle 32 2.6 Materialflusssystem 35 2.7 Materialflusssteuerung 37 2.7.1 Steuerungssysteme 37 2.7.2 Steuerungsstrategien 40 2.8 Materialflusssystem charakterisierende Kennzahlen 46 3 Stand der Forschung und Technik 51 3.1 Erzeugung autokorrelierter Zufallszahlen 51 3.1.1 Autoregressive Prozesse nach der Box-Jenkins-Methode 52 3.1.2 Distorsions-Methoden 54 3.1.3 Copulae 56 3.1.4 Markovian Arrival Processes 58 3.1.5 Autoregressive Prozesse mit beliebiger Randverteilung 61 3.1.6 Weitere Verfahren 64 3.1.7 Bewertung der Verfahren und Werkzeuge zur Generierung 65 3.2 Wirken von Autokorrelation in Bediensystemen 68 3.3 Fallstudien über Autokorrelation in logistischen Systemen 75 3.4 Ursachen von Autokorrelation in logistischen Systemen 89 3.5 Steuerung von Ankunftsprozessen an Zusammenführungen 96 3.6 Steuerung autokorrelierter Ankunftsprozesse 100 4 Steuerung autokorrelierter Ankunftsprozesse an Zusammenführungen 105 4.1 Modellannahmen, Methodenauswahl, Vorbetrachtungen 106 4.2 First Come First Served und Longest Queue First 114 4.3 Highest Autocorrelated First 117 4.3.1 Grundprinzip 117 4.3.2 Bestimmung der Grenzwerte 127 4.3.3 Dynamische Bestimmung der Grenzwerte mittels „DyDeT“ 133 4.4 Highest Autocorrelated First in Netzwerken 150 4.5 Abschließende Bewertung und Diskussion 161 5 Zusammenfassung und Ausblick 167 Primärliteratur 172 Normen und Standards 194 Abbildungsverzeichnis 197 Tabellenverzeichnis 199 Pseudocodeverzeichnis 201 Abkürzungsverzeichnis 203 Symbolverzeichnis 205 Erklärung an Eides statt 20

Analysis of Scheduling Policies under Correlated Job Sizes (CMU-CS-10-107)

Correlations in traffic patterns are an important facet of the workloads faced by real systems, and one that has far-reaching consequences on the performance and optimization of the systems involved. While there has been considerable amount of work on understanding the effect of correlations between successive interarrival times, there is very little analytical work in understanding the effect of correlations between successive service requirements (job sizes). All the prior work on analyzing the effects of correlated job sizes is limited to First-Come-First-Served scheduling. This leaves open fundamental questions such as: How do various scheduling policies interact with correlated job sizes? Can scheduling be used to mitigate the harmful effects of correlations? In this paper we take the first step towards answering these questions. We assume a simple and intuitive model for job size correlations and present the first asymptotic analysis of various common size-independent scheduling policies when the job size sequence exhibits high correlation. Our analysis reveals that the characteristics of various scheduling policies, as well as their performance relative to each other, are markedly different under the assumption of i.i.d. job sizes versus correlated job sizes. Further, among the class of size-independent scheduling policies, there is no single scheduling policy that is optimal for all degrees of correlations and thus any optimal policy must learn the correlations. We support the asymptotic analysis with numerical algorithms for exact performance analysis under an arbitrary degree of correlation, with simulations, and finally verify the lessons from our correlation model on real world traces.</p