1. Discrete-Time Markov Chains
In this lecture, we will begin with the Basics of Markov Chains, providing a foundational understanding of Markov processes, their properties, and significance. Moving forward, the Basics of Discrete-Time Markov Chains will be explored in depth, starting with the transition matrix and graph, followed by detailed discussions on the transition matrix and graph separately. We will then delve into the period of a state, irreducible Markov chains, cyclic classes, matrix structure, return set structure, and further results on periodic Markov chains. This section will also cover the invariant measure, the strong Markov property, and the number of visits to a state.
Next, we will examine Recurrence, Transience, and Invariant Measure, where the concepts of recurrence and transience will be introduced. We will discuss the invariant measure for a recurrent chain, the invariant distribution for a positive recurrence chain, and the ergodic theorems for Markov chains, providing a comprehensive understanding of these critical aspects.
In the section on the Asymptotic Behavior of Markov Chains, we will explore the convergence in variation, the coupling method, and the convergence to the stationary distribution. These topics will help in understanding the long-term behavior of Markov chains and their practical implications.
The Reversed Markov Chain section will cover the concept of reversibility and the Kolmogorov criterion for reversibility. This will provide insights into the conditions under which a Markov chain can be reversed and the implications of such reversibility.
Finally, we will discuss Foster’s Theorem for Positive Recurrence, which is a crucial result for understanding the conditions under which a Markov chain exhibits positive recurrence. This theorem will be presented with its proof and applications, rounding off the lecture with a deep dive into one of the fundamental theorems in the study of Markov chains.
Course Outline:
1 Basics of Markov Chains
2 Basics of Discrete-Time Markov Chains
2.1 Transition Matrix and Graph
2.2 Transition Matrix
2.3 Transition Graph
2.4 Period of a State and an Irreducible Markov Chain
2.5 Cyclic Classes and Matrix Structure
2.6 Return Set Structure
2.7 Further Results on Periodic Markov Chains
2.8 Invariant Measure
2.9 Strong Markov Property
2.10 Number of Visits to a State
3 Recurrence, Transience and Invariant Measure
3.1 Recurrence and Transience
3.2 Invariant Measure for a Recurrent Chain
3.3 Invariant Distribution for a Positive Recurrence Chain
3.4 Ergodic Theorems for Markov Chains
4 Asymptotic Behavior of Markov Chains
4.1 Convergence in Variation
4.2 Coupling Method
4.3 Convergence to the Stationary Distribution
5 Reversed Markov Chain
5.1 Reversibility
5.2 Kolmogorov Criterion for Reversibility
6 Foster’s Theorem for Positive Recurrence
To further enhance your understanding of Discrete-Time Markov Chains, this lecture will delve into essential topics such as the transition matrix and transition graph, which are fundamental in representing the state transitions of Markov processes. We will also explore the period of a state and the concept of an irreducible Markov chain, providing insights into their structural properties. Additionally, the lecture will cover cyclic classes and the return set structure, which are crucial for understanding the periodic nature of these chains. Advanced topics such as the invariant measure and the strong Markov property will be discussed to highlight their significance in long-term behavior analysis. Furthermore, we will examine recurrence and transience, along with the invariant distribution and ergodic theorems, to provide a comprehensive view of the stability and convergence properties of Markov chains. The lecture will also address the asymptotic behavior of these chains, focusing on convergence in variation and the coupling method. Finally, we will discuss the reversed Markov chain, including the conditions for reversibility and the Kolmogorov criterion, and conclude with an in-depth look at Foster’s Theorem for positive recurrence. This structured approach ensures a thorough understanding of both the theoretical and practical aspects of discrete-time Markov chains.
2. Continous-Time Markov Chains
In this lecture on Continuous-Time Markov Chains (CTMCs), we will explore the fundamental concepts, practical realizations, properties, and advanced topics related to these stochastic processes. The lecture is structured into four main sections, each providing a detailed examination of key aspects of CTMCs.
- Basics of Continuous-Time Markov Chains: We begin with the basics of CTMCs, starting with the family of transition matrices that define the probabilities of moving between states over time. Next, we introduce the transition semigroup, a generalized framework that extends the concept of the family of transition matrices.We then delve into the infinitesimal generator, a critical operator that captures the instantaneous rate of change of the process. Kolmogorov’s equations, both forward and backward, are presented. The section concludes with a discussion on the invariant measure, which characterizes the long-term behavior of the chain, and the strong Markov property, which extends the Markov property to stopping times.
- Realization of a Generator: This section details the construction of a CTMC from a specified generator. We explore the concepts of filtration and stopping time, which are essential for understanding the temporal structure of stochastic processes. The strong Markov property is revisited in the context of Poisson point processes, highlighting its applicability. We then discuss how to realize a generator using Poisson point processes, providing a concrete method for constructing CTMCs. The section also covers regular jump Markov chains, emphasizing their regenerative structure, which allows for the decomposition of the process into independent and identically distributed cycles.
- Regular Jump Markov Chain Properties: Here, we examine the properties of regular jump Markov chains. We start with irreducibility, recurrence, and transience, which classify the long-term behavior of the chain. The concept of an invariant measure for a recurrent chain is discussed, followed by the relationship between invariant distribution and positive recurrence. An example of a birth and death process is provided to illustrate these concepts in a concrete setting. The section also examines the asymptotic behavior of the chain, including ergodic theorems that describe the convergence to equilibrium. Finally, we discuss the flow rate between subsets, which quantifies the movement of probability mass within the state space.
- Continuous-Time Reversal: The final section addresses the concept of continuous-time reversal. We begin by defining the reversed chain and extend the discussion to negative times, providing a comprehensive view of the process’s behavior in reverse. The notion of reversibility is introduced, along with the conditions under which a CTMC is reversible. We then examine the behavior of the reversed process and present a reversal test for generators, which helps determine if a given generator corresponds to a reversible chain. The section concludes with a discussion on the restriction of a Markov chain to a subset of states and the product of generators, which allows for the combination of multiple CTMCs into a single process.
Course Outline:
1 Basics of Continuous-Time Markov Chains
1.1 Family of Transition Matrices
1.2 Transition Semigroup
1.3 Infinitesimal Generator
1.4 Kolmogorov’s Equations
1.5 Invariant Measure
1.6 Strong Markov Property
2 Realization of a Generator
2.1 Filtration and Stopping Time
2.2 Strong Markov Property for Poisson Point Processes
2.3 Realization of a Generator Using Poisson Point Processes
2.4 Regular Jump Markov Chain
2.4 Regenerative Structure
3 Regular Jump Markov Chain Properties
3.1 Irreducibility, Recurrence and Transience
3.2 Invariant Measure for a Recurrent Chain
3.3 Invariant Distribution and Positive Recurrence
3.4 Example: Birth and Death Process
3.5 Asymptotic Behavior
3.6 Ergodic Theorems
3.7 Flow Rate Between Subsets
4 Continuous-Time Reversal
4.1 Reversed Chain
4.2 Extension to Negative Times
4.3 Reversibility
4.4 Behaviour of Reversed Process
4.5 Reversal Test for Generators
4.6 Restriction of a Markov Chain
4.7 Product of Generators
For those interested in a deeper understanding of Continuous-Time Markov Chains (CTMC), this lecture provides comprehensive insights into key concepts such as transition matrices, transition semigroup, and the infinitesimal generator. We explore Kolmogorov’s equations and the invariant measure, essential for characterizing the long-term behavior of CTMCs. The strong Markov property, filtration, and stopping time are discussed in detail, along with practical applications involving Poisson point processes and regular jump Markov chains. Key properties like irreducibility, recurrence, and transience are examined, with illustrative examples such as the birth and death process. We also delve into the asymptotic behavior and ergodic theorems, providing a thorough understanding of flow rates and continuous-time reversal. Whether you are studying reversibility, the behavior of reversed chains, or the product of generators, this lecture equips you with the knowledge to master CTMCs.
3. Fundamental Queues
This lecture focuses on the mathematical study of waiting lines or queues. The course is structured to equip students with the foundational knowledge and analytical skills necessary to understand and optimize various queueing systems encountered in telecommunications, computer networks, service industries, and healthcare. The lecture is organized into three main sections, each delving into specific aspects of queueing theory.
The first section introduces the essential terminology and notation used in queueing theory. Students will learn about Kendall’s notation, a standardized method for describing the characteristics of different queueing systems. This section lays the groundwork for understanding more complex models by establishing a common language and framework for analyzing queueing systems.
The second section focuses on processor sharing queues, a class of models where multiple users or tasks share the service capacity of a single server. It begins with an examination of the single-class processor sharing queue, where all users belong to a single class and share the server’s capacity equally. The discussion then extends to multi-class processor sharing queues, which accommodate multiple user classes with distinct arrival and service characteristics. Further, the section explores scenarios with class-dependent service rates, generalized service rate functions, and discriminatory service disciplines, providing a thorough understanding of how different factors influence queue performance.
The third section addresses infinite server queues and capacity constraints. It starts with the M/M/infty queue, a fundamental model for systems with an unlimited number of servers, allowing each user to receive immediate service. The section then examines the Erlang queue (M/M/C/C), where a finite number of servers and no waiting room introduce blocking probabilities. Additionally, the analysis extends to multi-class infinite server queues and multi-class Erlang queues, offering a comprehensive view of systems with limited resources and multiple user classes. This section highlights the ergodic behavior, invariant distributions, and blocking probabilities, providing valuable insights into the performance and reliability of such systems under resource limitations.
Course Outline:
1 Queue Terminology and Notation
2 Processor Sharing Queues
2.1 Single Class Processor Sharing Queue
2.2 Multi-Class Processor Sharing Queue
2.3 Class-Dependent Service Rates
2.4 Generalized Processor Sharing
2.5 Discriminatory Processor Sharing
3 Infinite Server Queues and Capacity Constraints
3.1 Infinite Server Queue
3.2 Erlang Queue
3.3 Multi-Class Infinite Server Queue
3.4 Multi-Class Erlang Queue
4. Whittle Queueing Networks
In this comprehensive lecture on Whittle Queueing Networks, we will explore the theoretical foundations and practical applications of this advanced analytical framework. The course is structured into five main sections, each delving into critical aspects of Whittle networks, from basic definitions to performance analysis and restrictions.
- Basic Definitions and Notations: We begin with an introduction to the essential definitions and notations that form the backbone of Whittle queueing networks. This section ensures that all participants have a solid understanding of the fundamental concepts, such as state spaces, transition rates, and queueing disciplines, which are pivotal for grasping more complex topics later in the lecture.
- Markovian Modeling of Whittle Network: The second section focuses on the Markovian modeling of Whittle networks. We will cover the formulation of traffic equations, which describe the flow of entities through the network. This is followed by an in-depth discussion on the invariant distribution for the Whittle generator, providing insights into the long-term behavior of the network. We will also examine the conditions for irreducibility and ergodicity of the Whittle process, ensuring that the network reaches a steady state. Additionally, the concept of time reversal and its relation to Burke’s theorem will be explored, highlighting the symmetry properties of these networks.
- Examples of Whittle Networks: To contextualize the theoretical concepts, the third section presents practical examples of Whittle networks. We will demonstrate that the balance property of service rates is an intrinsic characteristic. The study of Jackson queueing networks will illustrate the application of Whittle networks in a well-known queueing model. Furthermore, we will examine queues in series, showcasing the versatility of Whittle networks in handling complex, multi-stage processes.
- Performance in Whittle Networks: Performance analysis is a crucial aspect of understanding queueing networks. In this section, we will investigate various performance metrics within Whittle networks. The Gibbs distribution will be discussed as a tool for characterizing the equilibrium state of the network. We will interpret traffic intensity to understand its impact on network performance. The analysis of sojourn time will provide insights into the time entities spend within the network, while user throughput will be examined to assess the efficiency of the network in processing entities.
- Restriction of a Whittle Process: The final section addresses the restriction of a Whittle process. We will explore the concept of a restricted Whittle generator, which modifies the network dynamics under certain constraints. Blocking probabilities in restricted Whittle networks will be calculated to understand the likelihood of congestion. Lastly, we will analyze the number of users in restricted Whittle processes, providing a comprehensive view of how restrictions impact overall network performance.
By the end of this lecture, readers will have a thorough understanding of Whittle queueing networks, equipped with both theoretical knowledge and practical insights to apply these concepts effectively in real-world scenarios.
Course Outline:
1 Basic Definitions and Notations
2 Markovian Modeling of Whittle Network
2.1 Traffic Equations
2.2 Invariant Distribution for Whittle Generator
2.3 Irreducibility and Ergodicity of Whittle Process
2.4 Time Reversal and Burke’s Theorem
3 Examples of Whittle Networks
3.1 Service Rates Balance
3.2 Multiclass Queues as Whittle Networks
3.3 Jackson Queueing Networks
3.4 Queues in Series
4 Performance in Whittle Networks
4.1 Gibbs Distribution
4.2 Interpretation of Traffic Intensity
4.3 Sojourn Time
4.4 User Throughput
5 Restriction of a Whittle Process
5.1 Restricted Whittle Generator
5.2 Blocking Probabilities in Restricted Whittle Networks
5.3 User Number in Restricted Whittle Processes
5. Quasi-reversible Queues and Networks
This lecture provides an in-depth exploration of quasi-reversible queues and networks, focusing on their unique properties and the simplifications they offer in analyzing complex queueing systems. The course is structured into two main sections:
- Quasi-reversible Queues: This section begins with a formal definition of quasi-reversible queues, accompanied by basic examples to illustrate the concept. We then delve into the key properties of quasi-reversible queues, highlighting their significance in simplifying the analysis of queueing systems. Further examples are provided to deepen the understanding of these properties in various contexts, demonstrating their practical applications. The section concludes with a discussion on partial balance equations, which are essential for understanding the flow of entities through the system and maintaining the balance between arrival and departure rates.
- Quasi-reversible Queueing Networks: In this section, we extend the concept of quasi-reversibility to networks of interconnected queues. We start by defining these networks and examining their basic properties, setting the foundation for more advanced topics. A significant part of this section is dedicated to demonstrating that quasi-reversible queueing networks admit an invariant distribution with a product-form structure, and discussing the conditions under which these networks are ergodic. One of the key results presented is the Arrival Theorem, which extends the PASTA property to quasi-reversible settings and provides insights into the state distribution observed by users during transitions. This section offers a thorough exploration of the theoretical underpinnings and practical implications of quasi-reversible queueing networks.
Course Outline:
1 Quasi-reversible Queues
1.1 Definition and Basic Examples
1.2 Properties of Quasi-reversible Queues
1.3 Further Examples
1.4 Partial Balance Equations
2 Quasi-reversible Queueing Networks
2.1 Definition and Basic Properties
2.2 Invariant Distribution and Ergodicity
2.3 The Arrival Theorem
Our comprehensive lecture on quasi-reversible queues and queueing networks is designed for those interested in mastering the intricacies of queueing theory and its applications in various fields such as operations research and applied probability. By exploring key concepts like invariant distribution, ergodicity, and the Arrival Theorem, participants will gain a deep understanding of how Markov processes and Poisson processes underpin the behavior of complex queueing systems. The lecture also covers essential topics such as the PASTA property and partial balance equations, providing valuable insights into network analysis and the practical implications of these theoretical frameworks. Whether you are a student, researcher, or professional, this lecture will equip you with the knowledge and tools needed to analyze and design efficient queueing systems.
6. Insensitivity in Queues and Networks to Service Distribution
This course is designed to provide a comprehensive understanding of the insensitivity property in queueing systems and networks, which ensures that the equilibrium distribution of the system remains unaffected by the specific form of the service distribution, depending only on its mean. The lecture is structured into three main sections, each delving into different aspects of this intriguing property.
The first section, « Equilibrium Distribution and Insensitivity, » lays the groundwork by introducing the fundamental concepts of equilibrium distribution and insensitivity in queueing systems. This section covers the essential principles and theoretical results that form the basis for understanding how and why certain queueing models exhibit insensitivity to service distributions.
In the second section, « Symmetric Queues and Their Insensitivity, » we delve into the specific case of symmetric queues. We begin with an introduction to phase-type distributions, which are instrumental in modeling service times in these systems. The section then explores the Markov renewal process for phase-type distributions and the concept of time reversal in such processes. We introduce the symmetric multiclass M/PH/· queue and discuss its Markovian modeling, leading to the derivation of the invariant distribution of symmetric queues. Finally, we analyze the distributional properties of symmetric queues, demonstrating their insensitivity to the specific form of the service distribution and highlighting the role of symmetry and phase-type distributions in achieving this property.
The third section, « Whittle Networks Insensitivity, » extends the discussion to Whittle networks, a class of queueing networks known for their robust insensitivity properties. We start with preliminary concepts and properties essential for understanding Whittle networks. The section then presents the insensitivity theorem for Whittle networks, showing that the equilibrium distribution of these networks remains unchanged under different service distributions, provided certain conditions are met. We also discuss the converse insensitivity theorem, which establishes the conditions under which the insensitivity property holds. Finally, we extend performance results from traditional Whittle networks to a more general class, demonstrating the applicability of these results to networks with more complex service distributions. This section underscores the robustness and versatility of Whittle networks in diverse queueing scenarios.
Course Outline:
1 Equilibrium Distribution and Insensitivity
2 Symmetric Queues and Their Insensitivity
2.1 Phase-type Distribution
2.2 Markov Renewal Process for Phase-type Distributions
2.3 Time Reversal of Markov Renewal Process
2.4 Symmetric Multiclass M/PH/· Queue
2.5 Markovian Modeling of Symmetric Queues
2.6 Invariant Distribution of Symmetric Queues
2.7 Distributional Properties of Symmetric Queues
3 Whittle Networks Insensitivity
3.1 Preliminary Concepts and Properties
3.2 Insensitivity Theorem for Whittle networks
3.3 Converse Insensitivity Theorem for Whittle networks
3.4 Extension of Performance Results in Whittle Networks
This lecture provides a thorough exploration of the insensitivity property in various queueing systems and networks, ensuring that participants gain a deep understanding of how the equilibrium distribution remains unaffected by the specific form of the service distribution. We cover essential topics such as phase-type distributions and the Markov renewal process, including the concept of time reversal in these processes. The lecture also delves into symmetric queues and the modeling of M/PH/· queues using Markovian modeling techniques. Participants will learn about the invariant distribution and the distributional properties of symmetric queues, as well as the robust insensitivity properties of Whittle networks. Key theorems such as the insensitivity theorem and the converse insensitivity theorem are discussed in detail, along with their implications for performance results in various queueing models. This lecture is ideal for those looking to understand the impact of service times on queueing scenarios and the broader applications of these principles in network analysis.
Exercises with Solutions on Queueing Theory
1. Exercises on Discrete-Time Markov Chains
Exercises with Solutions: Explore our comprehensive set of exercises designed to deepen your understanding of Discrete-Time Markov Chains. These exercises are perfect for students and researchers looking to apply theoretical concepts to practical scenarios. Our exercises cover a wide range of topics, including:
- Irreducibility Conditions: Learn how to determine if a Markov chain is irreducible by analyzing transition matrices and state probabilities.
- Expectation of Visits: Understand how to calculate the expected number of visits to a particular state in a homogeneous Markov chain.
- Return and Hitting Times: Master the first step method to compute return probabilities and hitting times for various states.
- Aggregation of States: Discover the conditions under which aggregated states form a homogeneous Markov chain and how to compute the transition matrix.
- Recurrence and Transience Criteria: Identify whether a Markov chain is recurrent or transient using specific criteria and system solutions.
- Random Walks: Analyze random walks on integer sets and determine their recurrence and transience properties.
- Communication Channels: Study discrete-time communication channels shared between multiple classes of users, and learn how to model and analyze them using Markov chains.
Each exercise is accompanied by detailed solutions to guide you through the problem-solving process, ensuring a thorough understanding of the material. Whether you’re a student preparing for exams or a researcher delving into advanced topics, these exercises provide valuable insights and practical experience with discrete-time Markov chains.
2. Exercises on Continuous-Time Markov Chains
Exercises with Solutions: These exercises serve as applications of the lecture titled Continuous-Time Markov Chains. They cover a range of topics, including the properties of exponential random variables, single-class and multiclass queueing systems, and the analysis of M/M/1 and M/M/∞ queues. Each exercise is designed to reinforce key concepts such as ergodicity, irreducibility, reversibility, and the existence of invariant distributions in continuous-time Markov chains. By working through these problems, students will gain a deeper understanding of the theoretical underpinnings and practical implications of continuous-time Markov processes.