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DEVELOPMENT OF A GENERALISED SOLUTION FOR THE PERFORMANCE MEASURES OF QUEUING SYSTEMS
This study developed computer software with the capacity to provide a generalized solution to queuing problems involving laborious, time consuming and complex mathematical approach of evaluating the performance measures of queuing systems when it was applied in the analysis of the six (6) fundamental performance measures of thirty (30) different organizations representing the six (6) queuing systems; the result was compared with that of mathematical computations and 97.22% compliance was observed from the 180 computations made with 2.78% discrepancy. It was also observed that the number of servers required to attain steady state for the waiting cost is linearly proportional to the arrival rate (λ) and inversely proportional to the service rate (µ). It also he effect of λ and µ on the cost variable profiles depend on the magnitude of both parameters such that: when λ = μ, Identical cost variables profiles is observed, when λ > µ, Unpredictable random fluctuations and when λ < µ, Defined fluctuations of cost variables profiles is observed. In addition, the cost variables profiles experience discontinuities at number of servers less than 2 but experience continuities at number of servers from 2 upwards. Therefore, the software can analyze finite and infinite queuing systems; single and multi-server queuing systems; all the six queuing systems; all the fundamental performance measures; many derived performance measures and with the added advantage of cost evaluations. It is therefore recommended to both individuals and organisations with businesses involving the analysis of queuing systems.
Keywords: Queuing Systems, Performance Measures, Service Rate, Arrival Rate, Cost Variables, Steady State, MATLAB, Markov Chain, Finite and Infinite Systems.
TABLE OF CONTENTS
Certification ii
Dedication iii
Acknowledgement iv
Abstract v
Table of Contents vi
List of Tables x
List of Figures and Charts xi
List of Symbols xiv
CHAPTER ONE INTRODUCTION
1.1 Background 1
1.2 Problem Statement 4
1.3 Objectives of the Study 5
1.4 Significance of the Study 5
1.5 Scope of the Study 6
CHAPTER TWO LITERATURE REVIEW
2.1 Preamble 7
2.2 Elements of waiting line 13
2.2.1 The customer’s population 13
2.2.2 The service system 15
2.2.2.1 The Number of Waiting Lines 16
2.2.2.2 The Number of Servers 17
2.2.2.3 The Arrangement of the Servers 17
2.2.3 Arrival and Service Patterns 17
2.2.3.1 Exponential distribution 18
2.2.3.2 Poisson distribution 19
2.2.3.3 Arrival patterns 19
2.2.3.4 Size of arrival units 19
2.2.3.5 Degree of patience 19
2.2.4 Waiting Line Priority Rules 20
2.3 Waiting line performance measures 20
2.4 Factors affecting waiting line 21
2.4.1 Length 21
2.4.2 Number of lines 22
2.4.3 Queue discipline 22
2.4.4 Service Time Distribution 23
2.4.5 Line structures 24
2.4.6 Exit 26
2.5 Overview of Markov Chain 27
2.6 Review of Varying Studies on Queuing Concepts 34
2.7 Summary of Literature Review 40
CHAPTER THREE METHODOLOGY
3.1.1 Materials for the Study 42
3.1.2 Description of the Study Area 42
3.1.2.1 The Fundamental performance Measures 42
3.1.2.2 The Derivable Performance Measures: 42
3.1.3 Organisations for the Study 43
3.1.4 Procedure for Data Collection 43
3.1.5 Procedure for Data Analysis 43
3.1.6 The Sample Size 44
3.1.6.1 M/M/1 Queuing System 44
3.1.6.2 M/M/s Queuing System 45
3.1.6.3 M/M/1/K Queuing System 45
3.1.6.4 M/M/S/K Queuing Model 46
3.1.6.5 M/M/S/S Queuing Model 46
3.1.6.6 M/M/Inf Queuing Model 47
3.1.7 Mathematical Theories used for Computations 48
3.1.8 Preferred Scholars of Queue Concept 49
3.1.9 Design 50
3.1.9.2 The Step by Step Installations of the GUI 51
3.1.9.4 The Graphics User Interface 53
3.1.10 Basic Softwares for Solving Queuing Related Problems 53
3.1.11 Theoretical Relationships of the Basic Performance Measures of
Queuing Systems 54
3.2 Data Analysis 56
3.2.1 Performance Measures 56
3.2.2 The Source Code 56
3.2.3 The Graphics User Interface (GUI) Software 57
3.2.4.1 M/M/1 58
3.2.4.2 M/M/S 59
3.2.4.3 M/M/1/K 60
3.2.4.4 M/M/S/K 61
3.2.4.5 M/M/S/S 62
3.2.4.6 M/M/∞ 63
3.2.5 Computations using Case Studies 63
CHAPTER FOUR: RESULTS AND DISCUSSION
4.1 Results 77
4.1.1 Tabulated Comparative Values 77
4.1.1.1 M/M/1 77
4.1.1.2 M/M/1/K 80
4.1.1.3 M/M/S 84
4.1.1.4 M/M/S/S 87
4.1.1.5 M/M/S/K 91
4.1.1.6 M/M/∞ 94
4.1.2 Application of OguomaONQueue 1.0.0 for Cost Analysis 98
4.2 Discussion 99
4.2.1 Observations from Performance Measure Analysis of Queuing
Concepts Using Matlab 99
4.2.2 Observations from Cost Analysis of Queuing Concepts Using Matlab 101
4.2.3 Deductions from Results 105
4.2.4 Contribution to Knowledge 106
CHAPTER FIVE: CONCLUSION AND RECOMMENDATION
5.1 Conclusion 108
5.2 Recommendation for Further Work 108
References 110
Appendices 127
| LIST OF TABLES | |||
| Table 4.1 System Data Case I | 77 | ||
| Table 4.2 Mathematical Approach versus Model Case I | 77 | ||
| Table 4.3 System Data Case II | 80 | ||
| Table 4.4 Mathematical Approach versus Model Case II | 80 | ||
| Table 4.5 Mathematical Computations | 84 | ||
| Table 4.6 Model Computations | 84 | ||
| Table 4.7 System Data Case IV | 87 | ||
| Table 4.8 Mathematical Approach versus Model Case IV | 87 | ||
| Table 4.10 Mathematical Computations | 91 | ||
| Table 4.9 System Data Case IV | 91 | ||
| Table 4.11 Model Computations Case IV | 91 | ||
| Table 4.12 System Data Case V | 94 | ||
| Table 4.13 Mathematical Approach versus Model Case V | 94 | ||
| Table 4.14 COST EFFECT OF SERVERS IN THE SYSTEM | 98 | ||
| Table 4.15 M/M/S/S System 2 of Mathematical Vs Model | 100 |
Table 4.16 Discrepancy in the Ls, Lq, Ws & Wq Computed Values of M/M/S 100
APPENDICES 1-7 Data from Organizations 127
CHAPTER ONE
INTRODUCTION
1.1 BACKGROUND
In general we do not like to wait. But reduction of the waiting time usually requires extra investments. To decide whether or not to invest, it is important to know the effect of the investment on the waiting time hence the need for models and techniques to analyze such situations as submitted by Ivo and Jacque (2002).
Sztrik (2012) citing Erlang (1909, 1918) observed that queuing theory deals with one of the most unpleasant experiences of life, waiting. It is quite common in many fields, for example, in telephone exchange, in a supermarket, at a petrol station, at computer systems. The first problem of Queuing theory was raised by calls and Erlang (1909) was the first who treated congestion problems in the beginning of 20th century. QUEUE in 15th century was originally referred to the “tail of the beast” or “a tail piece.” Not to be confused with a piece of tail. In the 17th century a queue became “a braid of hair.” Later it was used to refer to as “a pigtail”. In the 18th century a billiard stick became a queue, later changed to “que” and then to “cue”. In the early 19th century England, to queue was “to line up.” And that is how it is used today in England. According to Shakya (2001), queuing theory is the branch of operations research concerned with waiting lines (delays or congestion) even in a two or other multi queuing systems. A queuing system consists of a user source, a queue and a Service facility with one or more identical parallel servers. A queuing network is a set of interconnected queuing systems. Fundamental parameters of a queuing system include: Demand rate capacity (service rate), Demand inter-arrival times, Service times, Queue capacity and Discipline (Finite vs. Infinite; FIFO, FCFS, SIRO, LIFO, Priorities) and Myriad details (Feedback effects, “jockeying”). Queuing theory is the mathematical study of waiting lines, or queues where a model is constructed so that queue lengths and waiting times can be predicted hence generally considered as a branch of operation research because the results are often used when making business decisions about the resources needed to provide a service, Sundarapandian (2009). Queuing theory has its origins in research by Agner Krarup Erlang when he created models to describe the Copenhagen telephone exchange as the ideas have since seen applications in telecommunication, traffic engineering, computing and the design of factories, shops, offices and hospitals (Sun et al., 2010; Takine 2001; and Mittal & Tiwari 2010). Tijms (2003) citing Kendall (1953) and Markov (1906) observes that Single Queuing nodes are usually described using Kendall’s notation in the form A/S/C where A describes the time between arrivals to the queue, S the size of jobs and C the number of servers at the node. Many theorems in queue theory can be proved by reducing queues to mathematical systems known as Markov chains, first described by Andrey Markov in his 1906 paper. Asmussen & Boxma (2009) submits that Agner Krarup Erlang, a Danish engineer who worked for the Copenhagen Telephone Exchange, published the first paper on what would now be called Queuing theory in 1909. He also modelled the number of telephone calls arriving at an exchange by a Poisson process and solved the M/D/1 queue in 1917 and M/D/k queuing model in 1920. (Kingman, 2009). In Kendall’s notation:
- M stands for Markov or memoryless and means arrivals occur according to a Poisson process
- D stands for deterministic and means jobs arriving at the queue require a fixed amount of service
- k describes the number of servers at the Queuing node (k = 1, 2,…). If there are more jobs at the node than there are servers then jobs will queue and wait for service.
The M/M/1 queue is a simple model where a single server serves jobs that arrive according to a Poisson process and have exponentially distributed service requirements while in an M/G/1 queue the G stands for general and indicates an arbitrary probability distribution. The M/G/1 model was solved by Felix Pollaczek in 1930, a solution later recast in probabilistic terms by Aleksandr Khinchin and now known as the Pollaczek– Khinchine formula as noted by Kingman(2009). After I Queuing theory became an area of research interest to mathematicians, Whittle (1983). Kingman(2009) citing Little(1961) and Atiyah(1961) showed how work on Queuing theory used in modern packet switching networks was performed in the early 1960s by Leonard Kleinrock. It was in this period that John Little gave a proof of the formula which now bears his name: Little’s law while in 1961 John Kingman gave a formula for the mean waiting time in a G/G/1 queue now known as Kingman’s formula. The matrix geometric method and matrix analytic methods have allowed queues with phase-type distributed interarrival and service time distributions to be considered, (Ramaswami, 1988). Problems such as performance standard for the M/G/k queue remain an open problem (Kingman, 2009). Agner Krarup Erlang, a Danish engineer who worked for the Copenhagen Telephone Exchange, published the first paper on Queuing theory in 1909 while David G. Kendall introduced an A/B/C Queuing notation and in 1953 Kendall notation was introduced, then in 1969, Little’s formula came on board while in 1986 as the first, The Journal of Queuing Systems was published and in1995, the 1st international symposium THO was held (Bosse, 2002; Adan & Resing, 2001; and Willig, 1999). A queuing system can be described as customers arriving for service, waiting for service if it is not immediate, and if having waited for service, leaving the system after being served. In this research, a number of different areas of queuing systems are treated, but attention is paid to methods for the analysis of these systems and also to the applications of queuing models to practical daily life activities.
However, important areas of application for queuing models include Production systems, Transportation and Stocking systems, Communication systems and Information processing systems. Queuing models are particularly useful for the design of these systems in terms of layout, capacities and control.
Any time there is more customer demand for a service than can be provided, a waiting line occurs. Customers can be either humans or inanimate objects. Examples of objects that must wait in lines include a machine waiting for repair, a customer order waiting to be processed, subassemblies in a manufacturing plant (that is, work-in-process inventory), electronic messages on the Internet, and ships or railcars waiting for unloading among others as illustrated in figure 1.0
Fig 1.0 Typical Servicing System
Source: Hillier, F. S., et al.,Queuing Tables and Graphs. New York: Elsevier 63.210
1.2 PROBLEM STATEMENT
The degree of useful productive time generally lost due to long queues is quite enormous to ignore, particularly when considering the consequent reduction in productivity and increases in lost sales and services. Unfortunately the various approaches for analyzing the performance measures of queuing systems presently in existence have certain challenges with precision of values, flexibility and speed in computation. More so, most sensitive organizational, managerial, and even personal decisions are often based on results generated from these analyses. Hence, there is the need for a better approach for the analysis of Performance Measures of Queuing systems.
1.3 OBJECTIVES OF THE STUDY:
The major objective of this study is the development of a generalized solution for the performance measures of queuing systems. The specific objectives of the study include:
- To design programmable software using MATLAB file and a Graphic User Interface (GUI) modular platform for analyzing the performance measures of queuing systems. ii. To analyse the performance measures of the finite and infinite queuing systems mathematically.
iii. To analyse the performance measures of the finite and infinite queuing systems using the designed MATLAB program and the Graphic user interface (GUI) modular platform. iv. To validate the results using the designed Graphic User Interface (GUI) Modular platform designed.
- To apply the software to cost analysis.
1.4 SIGNIFICANCE OF THE STUDY
The study will be useful in areas like Work studies, motion studies, time and such related scientific management applications. The model designed will enhance quick generation of results needed for prompt decisions making both by organizations and individual. It will be a vital reference material for research in operations research and related areas of study. The study will enhance accuracy, uniformity, flexibility and speed when analysing the performance measures of queuing systems. It will standardize the analytical approach of evaluating the performance measures of queuing system globally.
1.5 SCOPE OF THE STUDY This study covers:
- The mathematical approach of analyzing performance measures of queuing systems.
- The programmable approach of analyzing performance measures of Queuing systems
- The six Queuing systems namely;
M/M/1: Single Server, Single Queue System
M/M/S: Multi Server, Single Queue System
M/M/1/K: Multi Server, Multi Queue Single Capacity System
M/M/S/K: Multi Server, Multi Queue Fixed Capacity System
M/M/S/S: Multi Server, Multi Queue, Loss System
M/M/∞: Infinite Server, Queue System
- The Six basic Fundamental Performance Measures viz
The utilization factors or Traffic Intensity
The Mean Number of items in the System, L
The Mean Number of items in the Queue, Lq
The Mean Time Spent in the System, W The Mean Time Spent in the Queue, Wq
The Probability of Idle Item in the System
DEVELOPMENT OF A GENERALISED SOLUTION FOR THE PERFORMANCE MEASURES OF QUEUING SYSTEMS
