Delayed and Network Queues

Presents an introduction to differential equations, probability, and stochastic processes with real–world applications of queues with delay and delayed network queues

Featuring recent advances in queueing theory and modeling, Delayed and Network Queues provides the most up–to–date theories in queueing model applications. Balancing both theoretical and practical applications of queueing theory, the book introduces queueing network models as tools to assist in the answering of questions on cost and performance that arise throughout the life of a computer system and signal processing. Written by well–known researchers in the field, the book presents key information for understanding the essential aspects of queues with delay and networks of queues with unreliable nodes and vacationing servers.

Beginning with simple analytical fundamentals, the book contains a selection of realistic and advanced queueing models that address current deficiencies. In addition, the book presents the treatment of queues with delay and networks of queues, including possible breakdowns and disruptions that may cause delay.  Delayed and Network Queues also features:

Numerous examples and exercises with applications in various fields of study such as mathematical sciences, biomathematics, engineering, physics, business, health industry, and economics A wide array of practical applications of network queues and queueing systems, all of which are related to the appropriate stochastic processes Up–to–date topical coverage such as single– and multi–server queues with and without delays along with the necessary fundamental coverage of probability and difference equations Discussions on queueing models such as single– and multi–server Markovian queues with balking, reneging, delay, feedback, splitting, and blocking as well as their role in the treatment of networks of queues with and without delay and network reliability

Delayed and Network Queues is an excellent textbook for upper–undergraduate and graduate–level courses in applied mathematics, queueing theory, queueing systems, probability and statistics, and stochastic processes. The book is also an ideal reference for academics and practitioners in mathematical sciences, biomathematics, operations research, management, engineering, physics, business, economics, health industry, and industrial engineering.

Aliakbar Montazer Haghighi, PhD, is Professor and Head of the Department of Mathematics at Prairie View A&M University, as well as founding Editor–in–Chief of Applications and Applied Mathematics: An International Journal (AAM). His research interests include probability, statistics, stochastic processes, and queueing theory. Among his research publications and books, Dr. Haghighi is the coauthor of Difference and Differential Equations with Applications in Queueing Theory, also published by Wiley.

Dimitar P. Mishev, PhD, is Professor in the Department of Mathematics at Prairie View A&M University. His research interests include differential and difference equations and queueing theory. The author of numerous research papers and three books, Dr. Mishev is the coauthor of Difference and Differential Equations with Applications in Queueing Theory, also published by Wiley.

Dedication i

Preface i

1. Preliminaries 1

1.1. Basics of Probability 1

1.1.1. Introduction 1

1.1.2. Conditional Probability 2

1.2. Discrete Random Variables and Distributions 4

1.3. Discrete Moments 9

1.4. Continuous Random Variables, Density and Cumulative Distribution Functions 15

1.5. Continuous Random Vector 19

1.6. Functions of Random Variables 22

1.7. Continuous Moments 26

1.8. Difference Equations 28

1.8.1. Introduction 28

1.8.2. Basic Definitions and Properties 29

1.9. Methods of Solving Linear Difference Equations with Constant Coefficients 31

1.9.1. Characteristic Equation Method 31

1.9.2. Recursive Method 33

1.9.3. Generating Function Method 34

1.9.4. Laplace Transform Method 37

Exercises 41

2. Stochastic Processes 1

2.1. Introduction and Basic Definitions 1

2.2. Markov Chain 6

2.2.1. Classification of States 5

2.3. Markov Process 21

2.3.1. Markov Process with Discrete Space State 21

2.4. Random Walk 25

2.5. Up–and–Down Biased Coin Design as a Random Walk 33

Exercises 39

3. Birth and Death Processes 1

3.1. Overviews of the Basis of the Chapter, Birth and Death Processes 1

3.2. Finite Birth and Death Process 10

3.3. Pure Birth Process (Poisson Process) 17

3.4. Pure Death Process (Poisson Death Process) 20

Exercises 23

4 Standard Queues 1

4.1. Introduction of Queues (General Birth and Death Process) 1

4.1.1. Mechanism, Characteristics and Types of Queues 3

4.2. Remarks on Non–Markovian Queues 7

4.2.1. Takács s Waiting Time Paradox 8

4.2.2. The Virtual Waiting Time and Takács s Integro–Differential Equation

9

4.2.3. The Unfinished Work 15

4.3. The Stationary M/M/1 Queueing Process 1

4.4. A Parallel M/M/c/K with Baking and Reneging 4

4.5. The Stationary M/M/1/K Queueing Process 6

4.6. Busy Period of a Transient M/M/1/K 7

4.7. The Stationary M/M/1 and M/M/1/K Queueing Processes with Feedback

11

4.7.1. The Stationary Distribution of the Sojourn Time of a Task 11

4.7.2. The Distribution of the Total Time of Service by a Task 14

4.7.3. The Stationary Distribution of the Feedback Queue Size 15

4.7.4. The Stationary Distribution of (the Sojourn Time of the nth task) 16

4.8. Queues with Bulk Arrivals and Batch Service 17

4.9. A Priority Queues with Balking and Reneging 20

4.10. The Discrete–Time M/M/1 Queueing Process, Combinatorics Method (Lattice Paths) 24

4.10.1 The Basic Ballot Problem 25

4.10.2. Ballot Problem [based on Takács (1997)] 27

4.10.3. Transient Solution of the M/M/1 by Lattice Path Method 35

4.11. The Stationary M/M/c Queueing Process 40

4.11.1. A Stationary Multi–server Queue 40

Exercises 43

5 Queues with Delay 1

5.1. Introduction 1

5.2. A Queueing System with Delayed–Service 5

5.3. An M/G/1 Queue with Server Breakdown and with Multiple Working Vacation 11

5.3.1. Mathematical Formulation of the Model 12

5.3.2. The Steady–State Mean Number of Tasks in the System 12

5.3.3. A Special Case 22

5.4. A Bulk Queueing System under N–Policy with Bilevel Service Delay Discipline and Start–up Time 25

5.4.1. Analysis of the Model 26

5.5. The Interrelationship between N–policy M/G/1/K and F–policy G/M/1/K Queues with Startup Time 28

5.5.1. N–policy M/G/1/K Queueing System with Exponential Startup Time 29

5.5.2. F–policy G/E/1/K Queueing System with Exponential Startup Time 37

5.6. A Transient M/M/1 Queue under (M, N)–Policy, Lattice Path Method 42

5.6.1. The Solution in Discrete Time 43

5.6.2. The Solution in Continuos Time 49

5.7. The Stationary M/M/1 Queueing Process with Delayed–Feedback 1

5.7.1. Distribution of the Queue Length 2

5.7.2. Mean Queue Length and Waiting Time 6

5.8. A Single–server Queue with Unreliable Server and Breakdowns with an Optional Second Service 17

5.9. A Bulk Arrival Retrial Queue with Unreliable Server 1

5.9.1. The Model 2

5.9.2. The Model Analysis 4

5.9.3. The Steady–State System Analysis 9

5.9.4. Performance Measures 17

5.9.5. Numerical Illustration 21

5.10. A Multi–Server Queue with Retrial Feedback Queueing System with two Orbits 25

5.11. Steady–state Stability Condition of a Retrial Queueing System with two Orbits, Reneging and Feedback 30

5.11.1. Necessary Stability Condition for the Steady–State System 31

5.12. A Batch Arrival Queue with General Service in Two Fluctuating Modes and Reneging during Vacation and Breakdowns 35

5.12.1. The Model 35

5.12.2. Analysis 38

Exercises 35

6 Networks of Queues with Delay 1

6.1. Introduction of Networks of Queues 1

6.2. Historical Notes on Networks of Queues 4

6.3. Jackson s Network of Queues 5

6.3.1. Jackson s Model 6

6.4. Robustness of Networks of Queues 33

6.5. A MAP Single–server Queueing System with Delayed Feedback as a Network of Queues 1

6.5.1. Description of the Model 3

6.5.2. The Service–Station 6

6.5.2.a. Number of Tasks in the Service–Station 6

6.52.b. Busy Period of the Service–Station 8

6.5.2.c. Number of Busy Periods 12

6.5.2.d. Computation of Takács s Renewal Equation (5.5.35), Mean Number of Busy Periods 16

6.5.3. Stepwise Explicit Joint Distribution of the Number of Tasks in the System: General Case when batch sizes vary between a minimum k and a maximum K

19

6.6. Unreliable Networks of Queueing System Models 1

6.6.1. Unreliable Network Model of Goodman and Massey 1

6.6.2. Unreliable Network of Queues Model of Mylosz and Daduna 5

6.6.3. Unreliable Network of Queues Model of Gautam Choudhury, Jau–Chuan Ke and Lotfi Tadj: A Queueing System with two Network Phases of Services, Unreliable Server, Repair–time Delay under N–policy 13

6.7. Assessment of Reliability of a Network of Queues 29

6.8. Effect of Network Service Breakdown 32

6.8.1. The Model (CoginfoCom system) 33

6.8.2. Analysis 5

6.8.3. Numerical Example 37

Exercises 42

References 1

Index 1