Solving performance models based on basic queueing theory formulas

Table of Contents

1 Introduction

2 Foundation

2.1 Queues

2.2 Queueing Networks

2.3 Solving System-Level Performance Models

2.3.1 Performance Measures and Typical Assumptions

2.3.2 Generalized System-Level Model

3 Goals and Approach

3.1 Goals

3.2 Approach

4 Markovian Queue Solver

4.1 Performance Formulas of Markovian Queues

4.1.1 M/M/1 Queue

4.1.2 M/M/m Queue

4.1.3 M/M/1/∞/N Queue

4.1.4 M/M/m/∞/N Queue

4.1.5 M/M/1/K Queue

4.1.6 M/M/m/K Queue

4.1.7 M/M/1/K/N Queue

4.1.8 M/M/m/K/N Queue

4.2 Implementation

5 Mapping Performance Formulas of Markovian Queues on Queueing Networks

5.1 Open Queueing Networks

5.1.1 Queueing Networks with Tandem Topology

5.1.2 Queueing Networks with Routing Probabilities

5.1.3 Summary

5.2 Closed Queueing Networks

5.2.1 Discussion of Applicability of Basic Queueing Formulas

5.2.2 Overview of Algorithms for Product-Form Queueing Networks

6 Evaluation

6.1 Performance Measures of Queues

6.2 Performance Measures of Queueing Networks

6.3 Savings in Computational Costs

7 Conclusion

Objectives and Topics

The primary objective of this thesis is to explore the applicability of basic queueing theory formulas for the analytical performance evaluation of Queueing Network models. The research focuses on identifying the possibilities and limitations of mapping these formulas to different network topologies and evaluating the efficiency of a self-developed analytical solver compared to traditional simulation methods.

• Theoretical derivation of performance formulas for various Markovian queue types.
• Implementation of a Java-based analytical tool for solving performance metrics.
• Theoretical analysis of workload propagation in open and closed Queueing Networks.
• Practical comparison of performance metrics and computational costs between analytical and simulation-based approaches.

Excerpt from the Book

Summary of Chapters

1 Introduction: Provides an overview of the importance of IT system performance evaluation and outlines the thesis's focus on analytical versus simulation-based approaches.

2 Foundation: Introduces fundamental queueing concepts, Queueing Network models, and essential performance measures and assumptions.

3 Goals and Approach: Defines the specific subgoals regarding the automation of performance calculations and the application of formulas to Queueing Networks.

4 Markovian Queue Solver: Details the derivation of performance formulas for eight specialized queue models and explains the implementation of the Markovian Queue Solver.

5 Mapping Performance Formulas of Markovian Queues on Queueing Networks: Discusses the theoretical aspects and constraints of applying basic queueing formulas to open and closed network structures.

6 Evaluation: Compares the results of the analytical solver with simulation results and analyzes the computational time savings.

7 Conclusion: Summarizes the findings regarding the feasibility of analytical modeling and suggests directions for future work.

Keywords

Queueing Theory, Queueing Networks, Performance Evaluation, Analytical Solver, Simulation, Markovian Queues, Workload Propagation, Resource Efficiency, Tandem Topology, Computational Costs, Product-Form, Quality-of-Service, Kendall's Notation, IT System Modeling, Performance Metrics

Frequently Asked Questions

What is the core focus of this thesis?

The work primarily deals with using basic queueing theory formulas to perform analytical performance evaluations of IT systems, specifically focusing on Queueing Network models.

What are the central themes discussed?

The thesis centers on performance modeling, mathematical derivation of queueing formulas, workload propagation, and the practical implementation of an analytical solving tool.

What is the primary research goal?

The main goal is to automate the calculation of performance metrics like utilization and throughput using analytical methods and to test their applicability to complex Queueing Network topologies.

Which scientific method is utilized?

The research employs a mix of theoretical derivation of mathematical formulas and a practical empirical comparison of these analytical results against those generated by a simulation tool.

What is covered in the main body?

The main body covers the theoretical foundations, the derivation of formulas for various Markovian queue types, the implementation of the Markovian Queue Solver, and an evaluation of its accuracy and efficiency.

Which keywords best characterize the work?

Key concepts include Queueing Networks, Analytical Solver, Markovian Queues, Performance Evaluation, and Computational Efficiency.

Why are closed Queueing Networks difficult to solve with basic formulas?

In closed networks, workload is variable and there is no external source; furthermore, departure rates are strongly correlated between nodes, making simple propagation difficult.

What are the main advantages of the analytical solver compared to simulation?

The primary advantage is a significant reduction in computational costs, with the analytical solver being dozens to hundreds of times faster than the simulation tool tested.

In what cases does the analytical solver deviate significantly from the simulation?

Large deviations occur in cases involving multiple-server queues regarding utilization definitions, and in closed models with finite capacity where model assumptions differ between the two approaches.