Differential and Difference Equations with Applications in Queueing Theory

Aliakbar Montazer Haghighi, PhD, is Regent Professor, Professor, and former Head of the Department of Mathematics at Prairie View A&M University. He’s the Co-founder and Founding Editor-in-Chief of Applications and Applied Mathematics: An International Journal (AAM). Dimitar P. Mishev, PhD, is Professor in the Department of Mathematics at Prairie View, A&M University. His research is focused on differential and difference equations and queueing theory.

• About the Authors xiiiPreface to the Second Edition xv1 Introduction 11.1 Introduction 11.2 Functions of a Real Variable 11.3 Some Properties of Differentiable Functions 31.4 Functions of More Than One Real Variable 31.5 Function of a Complex Variable 71.6 Differentiation of Functions of Complex Variables 121.7 Vectors 152 Transforms 312.1 Introduction 312.2 Fourier Series 322.3 Convergence of Fourier Series 392.4 Fourier Transform 402.5 Laplace Transform 502.6 Integral Transform 682.7 Z-Transform 693 Ordinary Differential Equations 813.1 Introduction and History of Ordinary Differential Educations 813.2 Basics Concepts and Definitions 813.3 Existence and Uniqueness 873.4 Separable Equations 893.5 Linear Ordinary Differential Equations 983.6 Exact Ordinary Differential Equations 1023.7 Solution of the First ODE by Substitution Method 1123.8 Applications of the First-Order ODEs 1173.9 Second-Order Homogeneous Ordinary Differential Equation 1223.10 The Second-Order Nonhomogeneous Linear Ordinary Differential Equation with Constant Coefficients 1383.11 Laplace Transform Method 1503.12 Cauchy–Euler Equation Differential Equation 1573.13 Elimination Method to Solve Differential Equations 1603.14 Solution of Linear ODE Using Power Series 1634 Partial Differential Equations 1734.1 Introduction 1734.2 Basic Terminologies for Partial Differential Equations 1744.3 Some Particular Functions Used in Partial Differential Equations 1764.4 Types of Boundary Conditions for a Partial Differential Equation 1784.5 Solution for a Partial Differential Equation 1814.6 Linear, Semi-linear, and Quasi-linear Partial Differential Equations 1844.7 Solution of Wave Partial Differential Equation, First and Second Orders, with Different Methods 1974.8 A One-Dimensional, Second-Order Heat (or Parabolic) Equations 2115 Differential Difference Equations 2235.1 Introduction 2235.2 Basic Terms 2255.3 Linear Homogeneous Difference Equations with Constant Coefficients 2285.4 Linear Nonhomogeneous Difference Equations with Constant Coefficients 2355.5 System of Linear Difference Equations 2475.6 Differential-Difference Equations 2555.7 Nonlinear Difference Equations 2606 Probability and Statistics 2696.1 Introduction and Basic Definitions and Concepts of Probability 2696.2 Discrete Random Variables and Probability Distribution Functions 2756.3 Moments of a Discrete Random Variable 2836.4 Continuous Random Variables 2876.5 Moments of a Continuous Random Variable 2916.6 Continuous Probability Distribution Functions 2936.7 Random Vector 3076.8 Continuous Random Vector 3126.9 Functions of a Random Variable 3146.10 Basic Elements of Statistics 3176.11 Inferential Statistics 3316.12 Hypothesis Testing 3386.13 Reliability 3417 Queueing Theory 3557.1 Introduction 3557.2 Markov Chain and Markov Process 3577.3 Birth and Death Process 3697.4 Introduction to Queueing Theory 3717.5 Single-Server Markovian Queue, M/M/1 3747.6 Finite Buffer Single-Server Markovian Queue: M/M/1/N 3907.7 M/M/1 Queue with Feedback 3947.8 Single-Server Markovian Queue with State-Dependent Balking 3957.9 Multiserver Parallel Queue 3987.10 Many-Server Parallel Queues with Feedback 4117.11 Many-Server Queues with Balking and Reneging 4147.12 Single-Server Markovian Queueing System with Splitting and Delayed Feedback 420Appendix 443The Poisson Probability Distribution 443The Chi-Square Distribution 447The Standard Normal Probability Distribution 449The (Student) t Probability Distribution 451Bibliography 453Answers/Solutions to Selected Exercises 461Index 469