| Titre : | Working Vacation Policy in Queueing Systems |
| Auteurs : | Djebli Fouzia, Auteur ; Mokhtar Kadi, Directeur de thèse |
| Type de document : | texte manuscrit |
| Editeur : | Université de Saida - Dr Moulay Tahar. Faculté des Sciences. Département d" l’Informatique, 2025/2026 |
| Format : | 54 ص |
| Accompagnement : | CD |
| Langues: | Anglais |
| Index. décimale : | BUC-M 008488 |
| Catégories : |
Master Mathématiques Spécialité: Analyse stochastique, statistique des processus et applications |
| Résumé : |
General Introduction
Waiting has become a common phenomenon in modern life. It can be observed in many situations such as banks, supermarkets, post offices, hospitals, transportation systems, communication networks, and many other service environments. The study of such phe- nomena has attracted considerable attention from researchers and has led to the devel- opment of queueing theory, which provides mathematical tools for analyzing congestion and service systems. The origins of queueing theory date back to the pioneering work of the Danish engineer Agner Krarup Erlang in the early twentieth century. His studies on telephone traffic in Copenhagen aimed to determine the number of circuits required to provide an acceptable level of telephone service, laying the foundation for modern queueing theory [43]. Since then, queueing theory has evolved considerably and has become an essential tool for modeling and analyzing a wide variety of stochastic service systems [8]. With the rapid development of computer systems, communication networks, and infor- mation technologies, queueing models have found numerous applications in performance evaluation and resource management. Nowadays, applications related to mobile commu- nications, the Internet, multimedia systems, and computer networks represent some of the most active research areas in queueing theory [5]. Despite the significant theoretical advances achieved in this field and the availability of numerous analytical models, several challenges remain. Real-life service systems often involve random customer arrivals, varying service rates, and complex server behaviors that cannot always be accurately represented by classical queueing models. Consequently, researchers have proposed various extensions of traditional queueing systems in order to better reflect practical operating conditions. One of the most important extensions is the Working Vacation Policy, introduced by Servi and Finn [34]. Unlike classical vacation models, where the server completely stops serving customers during vacation periods, the working vacation policy allows the server to continue providing service at a reduced service rate. This mechanism offers a more realistic representation of many practical systems, including manufacturing systems, computer networks, telecommunication systems, and customer service centers. Since its introduction, the working vacation concept has attracted considerable research interest. Wu and Takagi extended this policy to more general queueing systems, in- cluding M/G/1 models [12]. Later, Kim investigated the queue length distribution in M/G/1 working vacation systems [31], while Li and Tian analyzed GI/M/1 and GI/Geo/1 queueing systems under different vacation policies [33]. More recently, Azhagappan stud- ied Markovian queueing systems with working vacations, reneging customers, and server 8 waiting mechanisms, providing a more realistic representation of service environments [3]. The objective of this thesis is to study and analyze queueing systems operating under the Working Vacation Policy, with particular emphasis on the M/M/1 queue with working vacation. The main purpose is to investigate the influence of system parameters on various performance measures and to evaluate the effectiveness of this policy in improving system performance. This thesis is organized into three chapters. • Chapter 1 presents the fundamental concepts required for the study of queueing systems. It introduces stochastic processes, including counting processes, renewal processes, Poisson processes, and birth-and-death processes, which constitute the mathematical foundation of queueing theory. • Chapter 2 is devoted to the theory of queueing systems. It presents the principal concepts and notations used in queueing analysis, including Kendall’s notation and Little’s law. Furthermore, the main performance measures of queueing systems, such as the average number of customers in the system, queue length, and waiting times, are discussed. • Chapter 3 focuses on the analysis of the M/M/1 queueing model with Working Va- cation Policy. The steady-state behavior of the system is investigated, equilibrium equations are derived, and several performance measures are evaluated. Numer- ical results are provided to illustrate the impact of model parameters on system performance and to highlight the effectiveness of the working vacation mechanism. Through this study, we aim to provide a better understanding of the behavior of queueing systems operating under working vacation conditions and to contribute to the development of more efficient service systems in various practical applications. 9 |
| Note de contenu : |
Contents
List of Figures 6 List of Tables 7 General Introduction 8 1 Stochastic Processes 10 1.1 Definition and classification . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2 Continuous-time stochastic processes . . . . . . . . . . . . . . . . . . . . . 11 1.2.1 Elementary definitions and properties . . . . . . . . . . . . . . . . . 11 1.3 Counting process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4 Renewal process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.5 Poisson process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.5.1 Definition and properties . . . . . . . . . . . . . . . . . . . . . . . . 15 1.5.2 Main properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.5.3 Homogeneous Poisson process . . . . . . . . . . . . . . . . . . . . . 15 1.6 Relationship between Counting Process, Renewal Process and Poisson Pro- cess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.7 Poisson and exponential distributions . . . . . . . . . . . . . . . . . . . . . 16 1.7.1 Poisson distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.7.2 Exponential distribution . . . . . . . . . . . . . . . . . . . . . . . . 17 1.7.3 Relationship between the two distributions . . . . . . . . . . . . . . 17 1.7.4 Memoryless property . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.7.5 Erlang process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.8 Birth and death process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.8.1 Birth process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.8.2 Death process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.8.3 Definition (birth and death process) . . . . . . . . . . . . . . . . . . 20 2 Queueing Theory 21 2.1 Description of a Simple Queue . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.1.1 Arrival Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.1.2 Service Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.1.3 Queue Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.2 Kendall Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.3 Little’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.4 Modeling of a Queueing System . . . . . . . . . . . . . . . . . . . . . . . . 27 2.4.1 Markovian Queueing Systems . . . . . . . . . . . . . . . . . . . . . 27 2.4.2 Non-Markovian Queueing Systems . . . . . . . . . . . . . . . . . . 27 2.5 Queueing System Performance . . . . . . . . . . . . . . . . . . . . . . . . . 28 4 2.6 Some Queueing Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.6.1 M/M/1 Queueing Model . . . . . . . . . . . . . . . . . . . . . . . . 28 2.6.2 M/M/C Queueing Model . . . . . . . . . . . . . . . . . . . . . . . . 31 2.6.3 M/M/1/K Queueing Model . . . . . . . . . . . . . . . . . . . . . . 33 2.6.4 M/M/∞ Queueing Model . . . . . . . . . . . . . . . . . . . . . . . 36 3 Working Vacation Policy in Queueing Systems 38 3.1 Working Vacation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.1.0.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.1.0.2 The history . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.1.0.3 Classical Vacation vs Working Vacation . . . . . . . . . . 40 3.1.0.4 Applications of Working Vacation Models . . . . . . . . . 40 3.2 Types of Vacations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2.1 Simple Vacation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2.2 Multiple Vacation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2.3 Synchronized Vacation . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.3 M/M/1 Queue With Working Vacation . . . . . . . . . . . . . . . . . . . . 42 3.3.0.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.3.0.2 State Space . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.3.0.3 Transition Rate Diagram . . . . . . . . . . . . . . . . . . . 43 3.3.1 Steady-State Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.3.2 Measures of Performance . . . . . . . . . . . . . . . . . . . . . . . . 46 3.3.3 Model Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . 47 3.3.3.1 Impact of Arrival Rate on System Metrics . . . . . . . . . 47 General Conclusion 51 Bibliography 52 5 |
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