| Titre : | Advanced Discrete-Time Queueing Models. |
| Auteurs : | BOUDIA Hadjira, Auteur ; YAHIAOUI Lahcene, Directeur de thèse |
| Type de document : | texte manuscrit |
| Editeur : | Université de Saïda – Dr. Moulay Tahar – Faculté des Mathématiques, de l’Informatique et des Télécommunications, 2025/2026 |
| Format : | 61 ص |
| Accompagnement : | CD |
| Langues: | Anglais |
| Index. décimale : | BUC-M 008561 |
| Catégories : |
Master Mathématiques Specialization: Stochastic Statistical Analysis of Processes and Applications |
| Résumé : |
This thesis is devoted to the study of advanced discrete-time queueing models, which
are widely used to model and analyze modern service systems operating in time-slotted environments such as communication networks and computer systems. The work begins with a presentation of the fundamental theoretical concepts necessary for the analysis of queueing systems. Next, we investigate classical discrete-time queueing models based on Markov chains, including the Geo/Geo/1 and Geo/Geo/s models. For these models, key performance measures such as the stationary distribution, server utilization, and mean queue length are derived and analyzed. The main contribution of this thesis lies in the study of advanced queueing models with retrial phenomena, namely the Geo/Geo/s with retrial and Geo/G/1 with retrial models. These models are more realistic because they account for customers who, upon finding the server busy, leave and retry after a random delay. Finally, a numerical approach has been developed to illustrate the theoretical results and highlight their practical relevance for decision-making and system performance evaluation. This work provides a comprehensive understanding of discrete-time queueing systems, from basic concepts to advanced models, and emphasizes their importance in modeling real-world stochastic systems. |
| Note de contenu : |
Contents
Acknowledgments1 Dedications2 Abstract3 Introduction6 1 Theoretical Foundations of Discrete-Time Queueing Systems 1.1 Probability Theory Review . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Bernoulli Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Geometric Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 memoryless . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Counting Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Processes with independent increments . . . . . . . . . . . . . . . . 1.2.3 Processes with stationary increments . . . . . . . . . . . . . . . . . 1.2.4 Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 Discrete-Time Markov Chains . . . . . . . . . . . . . . . . . . . . . 1.2.6 Transition matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.7 Chapman-Kolmogorov equation . . . . . . . . . . . . . . . . . . . . 1.2.8 Birth and Death Processes . . . . . . . . . . . . . . . . . . . . . . . 1.2.9 Birth and Death Processes In Discrete Time . . . . . . . . . . . . 1.2.10 Bernoulli Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.11 Renewal Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.12 Discrete-Time Renewal Process . . . . . . . . . . . . . . . . . . . . 1.3 Introduction to Queueing Systems . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Arrival flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Queue Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 System Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Service disciplines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Kendall Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Little’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 The Performance Measurement Of a Queueing System . . . . . . . . . . .8 2 Classical Markovian Discrete-Time Queueing System 2.1 Discrete-Time Markovian Queues . . . . . . . . . . . . . . . . . . . . . . . 2.2 Some Models Of Discrete-Time Queues . . . . . . . . . . . . . . . . . . . 2.2.1 The Geo/Geo/1 Model . . . . . . . . . . . . . . . . . . . . . . . . .24 2.2.2 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Associated Markov Chain . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Performance Measurement . . . . . . . . . . . . . . . . . . . . . . . The Geo/Geo/s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Steady-State Distribution . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Performance Measurement . . . . . . . . . . . . . . . . . . . . . . .26 3 Advanced Discrete-Time Queueing Models 3.1 Queueing Systems with Callbacks (retrial) . . . . . . . . . . . . . . . . . . 3.1.1 Description of retrial with a queueing system . . . . . . . . . . . . 3.1.2 Description of a Queueing System with Balking . . . . . . . . . . . 3.1.3 Description of a Queueing System with Retrials, Callbacks, and a Waiting Room . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Terminology and Notation . . . . . . . . . . . . . . . . . . . . . . . 3.2 Some Examples of Queueing Systems with Callbacks . . . . . . . . . . . . 3.2.1 Packet Communication Networks . . . . . . . . . . . . . . . . . . . 3.2.2 Local Area Networks: CSMA . . . . . . . . . . . . . . . . . . . . . 3.3 Geo/Geo/s Queueing System with Callbacks . . . . . . . . . . . . . . . . 3.3.1 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 The stationary distribution of the system state . . . . . . . . . . . . 3.3.3 Busy period analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Waiting time analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Geo/G/1 Queueing System with Callbacks . . . . . . . . . . . . . . . . . . 3.4.1 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Markov Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 System Characteristics . . . . . . . . . . . . . . . . . . . . . . . . .35 4 Model Simulations 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Geo/Geo/1 Queueing Model . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 System characteristics as a function of parameter variation p . . . . 4.2.2 System characteristics as a function of parameter variation λ . . . . 4.3 The Geo/Geo/s Queueing Model . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 The Geo/Geo/s Queueing Model as a function of p . . . . . . . . . 4.4 The Geo/G/1 Queueing Model with Callbacks . . . . . . . . . . . . . . . . 4.4.1 The Geo/Geo/1 Queueing Model with Callbacks . . . . . . . . . .53 Conclusion58 Bibliography |
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