Queueing Theory and Scheduling

2.

Mathematical Foundations

2.1.

Probability Theory Review

2.1.1.

Random Variables

2.1.2.

Probability Distributions

2.1.3.

Expectation and Variance

2.1.4.

Moment Generating Functions

2.1.5.

Conditional Probability

2.1.6.

Law of Total Expectation

2.2.

Stochastic Processes

2.2.1.

Definition and Classification

2.2.2.

Random Variables and Processes

2.2.3.

2.2.3.1.

Strict Stationarity

2.2.3.2.

Wide-Sense Stationarity

2.2.4.

2.2.5.

Sample Paths and Realizations

2.3.

The Poisson Process

2.3.1.

Definition and Properties

2.3.2.

Counting Process Formulation

2.3.3.

Properties of Poisson Process

2.3.3.1.

Independent Increments

2.3.3.2.

Stationary Increments

2.3.3.3.

2.3.4.

Inter-arrival Times

2.3.4.1.

Exponential Distribution

2.3.4.2.

Memoryless Property

2.3.5.

Compound Poisson Process

2.3.6.

Non-homogeneous Poisson Process

2.4.

Exponential Distribution

2.4.1.

Probability Density Function

2.4.2.

Cumulative Distribution Function

2.4.3.

Moments and Properties

2.4.4.

Memoryless Property

2.4.5.

Minimum of Exponential Random Variables

2.4.6.

Applications in Queueing

2.5.

2.5.1.

Definition and Basic Concepts

2.5.2.

State Space and Transition Probabilities

2.5.3.

Discrete-Time Markov Chains

2.5.3.1.

Transition Probability Matrix

2.5.3.2.

Chapman-Kolmogorov Equations

2.5.3.3.

Classification of States

2.5.3.3.1.

Transient and Recurrent States

2.5.3.3.2.

Periodic and Aperiodic States

2.5.3.3.3.

Communicating Classes

2.5.3.4.

Limiting Probabilities

2.5.3.5.

Stationary Distribution

2.5.4.

Continuous-Time Markov Chains

2.5.4.1.

Transition Rate Matrix

2.5.4.2.

Generator Matrix

2.5.4.3.

Kolmogorov Equations

2.5.4.3.1.

Forward Equations

2.5.4.3.2.

Backward Equations

2.5.4.4.

Uniformization Technique

2.6.

Birth-Death Processes

2.6.1.

Definition and Structure

2.6.2.

State Transition Diagrams

2.6.3.

Birth and Death Rates

2.6.4.

Balance Equations

2.6.4.1.

2.6.4.2.

2.6.4.3.

Detailed Balance

2.6.5.

Steady-State Analysis

2.6.5.1.

Existence Conditions

2.6.5.2.

Calculation Methods

2.6.5.3.

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1. Introduction to Queueing Systems

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3. Basic Markovian Queueing Models