Masterarbeit, 2010
99 Seiten, Note: A
CONTENT
DEDICATION
ABSTRACT
TABLE OF CONTENT
LIST OF TABLES
LIST OF FIGURES
LIST OF ABBREVIATION
ACKNOWLEDGEMENT
CHAPTER ONE INTRODUCTION
1.0 Background of the Study
1.1 Statement of the Problem
1.2 Objectives of the Study
1.3 Methodology
1.4 Justification
1.5 Limitation
1.6 Organization of the Thesis
CHAPTER TWO LITERATURE REVIEW
2.0 Introduction
2.1 Queuing Researches In The Banking Industry
2.2 Historical Perspective of Queuing Theory
CHAPTER THREE METHODOLOGY OF THE STUDY
3.0 Introduction
3.1 Primary Data Collection
3.2 Description of Queuing System
3.2.1 Queuing Systems in the Case of Study
3.3 Fundamental Queuing Relations
3.4 Customer Arrival and Inter Arrival Distribution
3.4.1 Probability Distribution
3.4.2 Exponential Distribution
3.4.3 Poisson Distribution
3.5 Teller Utilization Factor (p)
3.6 Little's Law
3.7 Queuing Model Description
3.7.1 M/M/1 Model
3.7.2 M/M/s Model
3.8 Data Sheet Processing
CHAPTER FOUR ANALYSIS AND DISCUSSION OF RESULTS
4.0 Introduction
4.1 Primary Data Examination
4.2 Discussion Of Findings
4.3 Queuing Model Analysis
4.3.1 Performance Analysis
CHAPTER FIVE SUMMARY, RECOMMENDATIONS AND CONCLUSION
5.0 Introduction
5.1 Summary of Findings
5.2 Conclusion
5.3 Recommendation to Stakeholders
5.4 Recommendation to Future Researchers
BIBLIOGRAPHY
APPENDICES
Appendix 1.0
Appendix 2.0
Appendix 3.0
Appendix 4.0
This work is dedicated to anyone who has been my teacher, I have always been a learner and I am grateful to them all.
The endless customers queuing for service delivery in the banking industry is a phenomenon that bothers both management and customers alike. The central element of the queuing system is a teller who provides some services to a population of customers. This study determined whether the present capacity level in the banking industry strike a balance between waiting and service time using Barclays Bank, Tafo Branch and Agricultural Development Bank, Kumasi Adum Branch as a case of interest. Primary data on Six Hundred and Thirteen (613) customers arriving at the case of study throughout the selected hours and days were collected, taken into consideration; the arrival, processing and departure times of each customer. The study then showed how the data collected at the respective dates possesses the Markovian properties, that is, Poisson and Exponential Distributions, hence the use of two “M's” in the M/M/s queuing model. It determined the probabilistic analysis that the teller(s) is idle and also determined the probability of certain number of arrivals occurring at a given time. By using the queuing rule First-come, First-serve as practiced by the case study and M/M/s queuing model, the performance measures were calculated uncovering the applicability and extent of usage of queuing models in achieving customer satisfaction at lowest cost by minimizing waiting times, idle times, capacity utilization and queues at the bank. It was observed that any time the number of tellers at BBG was reduced to two during a peak day, the queue elongated resulting in high capacity utilization factor and hence a high waiting time. It is therefore recommended that management makes better decisions relating to number of tellers that would be necessary to serve customers during peak and off- peak hours, days and weeks.
4.1 Data Collection at ADB from 18th January to 9th February, 2010
4.2 Data Collection at BBG from 18th January to 9th February, 2010
4.3 Probability of the number of customers or fewer that would have arrived using confident interval of 95%
4.4 Probability of the highest turning points of the Poisson Distribution Graphs in Appendix 3.0
4.5 Probability that a customer and five customers are in the Queuing system
4.6 Presentation of results for considering one to four tellers at a given point in time
A2.0 Poisson Model for data collected on 20th January, 2010
A3.0 Queuing Analysis on data collected between 18th January and 9th February, 2010
A3.1 Microsoft Office Excel spreadsheet working calculations, mainly for Po calculation.
A3.2 Spreadsheet working calculations for queuing analysis: Multiple Tellers
3.1 Hypothetical structure of a Queuing system in banks
3.2 Multiple Tellers, Single-Stage Queuing System
3.3 Multiple Single Tellers Queuing System
3.4 Inter-arrivals and time at which the jth customer arrives
3.5 Transition diagram for a single teller possessing
3.6 Transition diagram for a multiple teller possessing
4.1 Graphical presentation of data collected at BBG on 9th February,
4.2 Graphical presentation of data collected at ADB on 25th January,
4.3 Poisson Arrival Distribution Graph for data collected on 20th January, 2010
4.4 Exponential Distribution graph of BBG, 18th Jan, 2010
4.5 Probability of a number of customers in the System for 22nd Jan,
A1.0 Graphical presentation of data collected at BBG on 18th Jan., 2010
A1.1 Graphical presentation of data collected at BBG on 20th January,2010
A1.2 Graphical presentation of data collected at BBG on 22nd Jan., 2010
A1.3 Graphical presentation of data collected at BBG on 25th Jan., 2010
A1.4 Graphical presentation of data collected at ADB on 18th Jan., 2010
A1.5 Graphical presentation of data collected at ADB on 20th Jan., 2010
A1.6 Graphical presentation of data collected at ADB on 22nd Jan., 2010
A1.7Graphical presentation of data collected at ADB on 9th Feb., 2010
A2.0 Poisson Arrival Distribution for data collected on 18th Jan., 2010
A2.1 Poisson Arrival Distribution for data collected on 22nd Jan., 2010
A2.2 Poisson Arrival Distribution for data collected on 25th Jan., 2010
A2.3 Poisson Arrival Distribution for data collected on 09th Feb., 2010
A3.0 Probability of a number of customers in the System for 18th Jan,2010
A3.1 Probability of a number of customers in the System for 20th Jan,2010
A3.2 Probability of a number of customers in the System for 22nd Jan,2010
A3.3 Probability of a number of customers in the System for 25th Jan,2010
A3.4 Probability of a number of customers in the System for 09th Feb,2010
Abbildung in dieser Leseprobe nicht enthalten
It is never my might, rather, the will of God Almighty that has brought me this far. I am forever grateful to HIM.
Special thanks go to Dr. S. K. Amponsah whose resourceful knowledge has helped me to come up with this thesis.
I wish to express my heartfelt thanks to Mrs Lydia Bedu-Addo, Ashanti Area Manager of Agricultural Development Bank (ADB) and Edward Sasu Adofo, the Branch Manager, Barclays Bank (BBG) Tafo Branch for given me the permission to use their respective outfit as the case of study.
Finally, to all my friends and my relatives, it is through your prayers and inspirations that have brought me to this level. I wish you in return abundance of love, peace, success and prosperity.
Oppong-Gyebi Emmanuel
Banking, in fact is as primitive as human society, for ever since man came to realise the importance of money as a medium of exchange. Banking is the business of receiving payments on behalf of their customers, accepting deposits, and making short-term loans to private individuals, companies, and other organisations. Commercial bank is a monetary institution owned by either government or private businessmen for the purpose of profit.
Ankaah (1995) testified to the fact that before 1953, Ghana had only two banks; the Bank of British West Africa, now Stanchart Bank, established in 1886 and the Barclays Bank (Dominion, Colonial and Overseas), now Barclays Bank of Ghana (BBG), which was established in 1917. He stated that on the eve of independence, the banking industry had only one indigenous bank, Bank of the Gold Coast, now Ghana Commercial Bank, established in 1953, and that most of the many departments integral to it today, were not in existence. The necessity of a controlling or regulating agency or institution was naturally felt. After independence the Bank of Ghana (BOG) was established in 1957. The Agricultural Development Bank (ADB) was set up by an Act of Parliament (Act 286) in 1965 to promote and modernize the agricultural sector through appropriate but profitable financial intermediation. Its original name then was the Agricultural Credit and Co-operative Bank and the establishing Act gave its main objective as "to provide credit facilities to agriculturists and persons for connected purposes" (www.agricbank.com). Incoom (1998) cites the Banking Act of 1970, section two as source of power that made the Bank of Ghana (BOG) responsible for regulating and supervising the banking system in Ghana. He also notes that, the purpose of financial regulation, whether self to protect the stability of the financial industry and also to preserve confidence in the economy as a whole.
With the immergence of new banks, since the year 2000, Ghana can boast of 25 commercial banks as at 2009, numerous emerging and existing Rural Banks and Credit Unions, the banking industries have largely implemented service delivery technology as a way of augmenting the services traditionally provided by bank personnel. Most of these banks are operating in the country under African Ownership and African management. In the 1980's, the question of banking regulation and supervision had become a global problem. Kapstein (1991), writing on the origin of international banks, noted that the roots of international regulatory cooperation were formed in this period. He claims that the effort to protect both customers and their own institutions in face of systematic stocks like inflation, erratic exchange rates and volatile interest rates, the industrial countries found it necessary to become more active in asset and liability management.
Managers in virtually all industries understand that providing quality customer service is a key strategic component in firm profitability. The competition in Ghanaian banking sector is getting more intense, partly due to regulatory imperatives of universal banking and also due to customers' awareness of their rights. Bank customers have become increasingly demanding, as they require high quality, low priced and immediate service delivery. They want additional improvement of value from their chosen banks (Olaniyi, 2004). Since service delivery in banks is personal, customers are either served immediately or join a queue if the system is busy. A queue occurs when facilities are limited and cannot satisfy demand made against them at a particular period. However, most customers are not comfortable with waiting or queuing (Olaniyi, 2004). The danger of keeping customers in a queue is that their waiting time may amount to or could become a cost to them (i.e. bank customers).
As with most other service providers, banks have moved quickly to invest in technology as a way of controlling costs, attracting new customers and meeting the convenience and technical innovation expectations of their customers. Profit maximization objective may not be easy to achieve in banking, without a good level of customer base, as this customer base enhances the effectiveness and efficiency of the services rendered to the customers. In other words, the faster they get attended to, the more the customer would be encouraged to keep their money with a bank.
As information technology becomes more sophisticated, banks in many parts of the world are adopting a multiple-channel strategy. Changes in banking industry as a result of challenges posed by technologically innovative competitors such as those resulting from deregulation, rapid global networking, and the rise in personal wealth have thus made the implementation of sophisticated delivery systems. The technology of electronic banking, for examples; Automated Teller Machines (ATMs), Internet, Mobile and Telephone Banking, etc, are aimed to render faster and convenient banking services to customers in anywhere and at any time, a strategy necessary in many cases. These electronic banking has already advanced in the developed countries, but only few customers in Ghana subscribe to it. However, it, cannot entirely replace the more traditional channels. Research indicates that a substantial portion of the customer base may always demand the type of personal interaction that can only be provided by individual branch personnel (Lewis et al., 1994). In other relation, it has been observed that, as the country population increases, the customer population also increases. Hence, queue(s) may be witness in the banking halls. At time, the subsidiary channel like, the ATMs do witness queues by the customers, how much less queues at the respective banking halls in Ghana?
“To queue or not to queue?” as pointed out by (Larson, 1987), is a question we face every day. When we encounter a queue, we often make a quick estimate of the expected waiting time and decide whether to join the queue based on the amount of time we are willing to wait. Basically, queuing is a core of almost every transaction undertaken by customers within Ghanaian banking industry. However, several concerns are being raised by customers over the unsatisfactory service condition as a result of queue(s) delay. There may be several factors arising to this fact, but not withstanding those accessions, the ultimate aim for every customer is to spend limited time at the banking halls. As a result of these constraints, most managers attempt to solve queuing problems procuring additional facilities or hiring more workers to reduce the waiting time.
Two of the leading banks, thus, ADB and BBG have branches in all the regions in Ghana. ADB has fifty (50) branches nationwide with their locations reflecting on the level of agricultural activities and the flow of deposits. It also has four (4) Farm Loan Offices and ten (10) Agencies with Kumasi alone having four (4) branches (www.agricbank.com). Branches of BBG have tremendously increased from thirty-two (32) in 2007 to over 140 branches with 118 ATMs, including nine (9) world class Prestige Centres, and a Premier Centre as at September, 2009 (www.barclays.com/africa/ghana). A newly joint commissioned branches at Tanoso, Kumasi increased branches in the metropolis from four (4) in April 2007 to nineteen (19) branches as at March, 2008 (www.ghanaian- chronicles.com). Notwithstanding these branches being possessed by the case study, there are still overwhelming queues at their respective branches, especially, at BBG, Old Tafo and ADB, Kumasi Adum branches. Hence, these options must be considered with proper analysis of the queuing system to prevent increase in staffing cost, idleness of both human and material resources and ultimately, for efficient time management to improve banking services and reduce waiting time of customers in the banking halls.
Queue is a social phenomenon and if managed well, it would be beneficial to the society, especially to both the unit that waits and the one that serves. Queue is a line of customers waiting their turn for service. The word queue comes from the Latin word cauda, meaning tail. Most researchers in the field prefer the spelling 'queuing' over 'queueing', although the latter is somewhat more common in other contexts. Queuing occurs when customers arrive faster than they can be served and the system temporarily buffers them in queues. Queuing system is a system that includes the customer population source, a queuing discipline as well as the service system. The primary tool for studying the phenomena of standing or sitting, waiting, and serving these problems of congestions is known as Queuing Theory. Queuing Theory is a branch of mathematics that studies and models the act of waiting in lines.
The ultimate objective of this analysis of queuing systems is to determine whether the present capacity level in the banking industry, strikes a balance between waiting and service time in the case study so that informed and intelligent decisions can be made in their management. This study is based on a mathematical building process as well as designing and implementing of an appropriate experiment involving that model. This encompasses the study of arrival, behaviour, and service times, service discipline, service capacity and the departure of customers at the case of interest, which is the marketing strategy of many banks as well. Queuing theory embodies the full range of such models covering all perceivable systems which incorporate characteristics of a queue. The problems are categorised into three main parts and can be identified as:
1. Arrival Problems. Usually, there is an assumption that service times are independent and identically distributed, and that they are independent of the inter arrival times. For example, the service times can be deterministic or exponentially distributed. It can also occur that service times are dependent of the queue length. Most queuing models assume that the inter arrival times are independent and have a common distribution. In many practical situations customers arrive according to a Poisson stream (i.e. exponential inter arrival times).
If the occurrence of arrivals and service delivery are strictly according to schedule, a queue can be avoided. But in practice, this does not happen. In most cases the arrivals are the product of external factors. Therefore, the best one can do is to describe the input process in terms of random variables which can represent either the number arriving during a time interval or the time interval between successive arrivals. Customers may arrive one by one, or in batches. If customers arrive in groups, their size can be a random variable as well. An example of batch arrivals is the Customs Office at Aflao where travel documents of bus passengers have to be checked.
Customer population can be considered as finite or infinite. When potential new customers for the queuing system are affected by the number of customers already in the system, the customer population is finite. When the number of customers waiting in line does not significantly affect the rate at which the population generates new customers, the customer population is considered infinite. Banking customer population is infinite since they do not restrict a specific number of customers to be serviced each day and work strictly with time.
2. Behavioural problems. The study of behavioural problems of queuing at banks is intended to understand how customers behave under various conditions. This normally affects the smooth service delivery irrespective of the facilities that a bank may have if much attention is not given to it. Customer's behaviour at the banking hall may destruct attention of the one that serves and can bring serving to a halt. A customer may balk, renege, or jockey and their queuing analysis is based on behavioural problem researches. Balking occurs when the customer decides not to enter the queue. Reneging occurs when the customer enters the queue but leaves before being serviced. Jockeying occurs when a customer changes from one line to another, hoping to reduce the waiting time. A good example of this is picking a line at ADB, Adum Kumasi Branch and changing to another line with the hope of serving quicker. The models used in this supplement assumed that customers are patient; they do not balk, renege, or jockey; and the customers come from an infinite population.
3. Operational problems. The operational system is characterized by the number of queues, the number of tellers, the arrangement of the tellers, the arrival and service patterns, and the service priority rules. Under this heading, all problems that are inherent in the operation of queuing systems are included. How many customers can wait at a time if a queuing system is a significant factor for consideration? There may be a single teller or a group of tellers helping the customers. If the banking hall is large, one can assume that for all practical purposes, it is infinite. In many queues, it is useful to determine various waiting times and queue sizes for particular components of the system in order to make judgments about how the system should be run. Some of such problems are statistical in nature. Others are related to the design, control, and the measurement of effectiveness of the systems. In many situations customers in some classes get priority in service over others. Also, there are other factors of customer behaviour such as balking, reneging, and jockeying that require consideration as well. The slow pace and the number of tellers also affect the time a customer waits in the queuing system.
The primary objective of this study in line with the identified problems is to determine whether the present capacity level in the banking industry, strikes a balance between waiting and service time using BBG, Tafo branch and ADB, Kumasi Adum branch as case study. This would be carried out by measuring customers;
a) The arrival time.
b) The processing time.
c) The departure time.
The study specifically aims to determine:
i) The probabilistic analysis that the teller(s) will be idle.
ii) The amount of waiting time a customer is likely to experience in a system;
iii) How the waiting time will be affected if there are changes in the facilities and
iv) Make policy recommendation base on the findings from the study.
In an observational way, primary data would be collected at the same time and date at the two cases of study in some selected days within January and February, 2010. Microsoft Office Excel 2007 would be used to assess and interpret data with the help of various method of queuing analysis. Secondary data used to execute the literature review of the study was gathered from the internet, professional magazines, research papers, journals, and textbooks.
Queues are so commonplace in society that it is highly worthwhile to study them, even if one waits in the check line for a few seconds. It may take some creative thinking, but if there is any sort of scenario where time passes before a particular event occurs, there is probably some way to develop it into a queuing model.
Some selected banks like Ghana Commercial Bank, Adum Branch had a fear of comparing customers exhausted time at their banking hall with others and were reluctant to give in as a case study. It was a big task in observing and taking customers arrival, processing and departure times, especially at ADB, Kumasi Adum branch which use several single teller with single stage queues arranged in parallel.
Chapter one highlights on the background of study, the objectives, some of the challenges faced during the study and the organisation of the study. Chapter two shows the origin of queuing theory and also highlight on the works of some of its contributors that lead to the study of this thesis topic. In chapter three, the methodology used to achieve the objectives under study would be clearly stated. Component of the basic queuing system, fundamental queuing relations and various assumptions related to model development are explicitly mentioned, and overall model structure would be explained. Analysis and interpretation of data collected would be done in chapter four. Summary of findings, recommendation and conclusion would also appear in chapter five.
This chapter shows the origin of queuing theory and also highlights on the works of some of its contributors that lead to the study of this thesis topic.
Literature review on queuing processes in banking halls tends to be tilted towards experiences and queuing packages outside Ghana, in particular, Europe, Asia, U.S.A and some part of Africa which has been part of their marketing strategy. Researches into banking operations have been tremendous and overwhelming but very little has been done to analyze this problem using queuing theory.
The financial industry has been developing rapidly and receiving more attention in recent years. Financial institutions introduced queuing-based teller staffing models in the late 1960s and early 1970s primarily to control increasing labour expenses (Brewton, 1989). Until the introduction of Automated Teller Machines (ATMs), transactions such as deposits, withdrawals, and cash-checking were handled exclusively by human tellers. Although automated banking and on-line banking have decreased the need for human tellers, many retail banks still rely on them to provide timely and personalized customer service. Agboola and Salawa (2008) identified various Information and Communication (ICT) in use and determined how they could be utilized for optimal performance on business transactions in the banking industry.
Price Waterhouse Cooper's publication (1999) indicates that the primary aim of the introduction of modern electronic delivery channels was to cut costs and congestions by attracting lower-value customers to non-branch channels. However, the result was quite different in the experimented areas from that expected: high- value customers started to use these channels, while the lower-value customers continued to use branches. This is never the same in a developing country like Ghana where every customer want to correspond directly with the personal bankers.
Wenny and Whitney (2004) determined bank teller scheduling using simulation with arrival rates. They investigated scheduling of banks at a branch in Indonesia and the model accounts for real system behaviour including changing arrival rates, customer balking and reneging for randomly selected hours in the day. Travis and Michael (2007) assume that all servers at retail banking have the same service time distribution and that this distribution is exponential. These assumptions are motivated more by operational and analytical convenience than supported by data.
Oladapo (1988) study conducted in Nigeria revealed a positive correlation between arrival rates of customers and bank's service rates. He concluded that the potential utilization of the banks service facility was 3.18% efficient and idle 68.2% of the time. However, Ashley (2000) asserted that even if service system can provide service at a faster rate than customer's arrival rate, queues can still form if the arrival and service processes are random. Emuoyibofarhe, et al., (2005) studied the queuing problem of banks in Nigeria, taking First Bank plc, Marina branch Lagos as a case of interest, and apply queuing theory to solve the multiple server problem (M/M/s/./ <x/<x queuing system) which yielded results upon which the management of the bank could optimality distribute servers (cashier) to minimize waiting times, idle times and queues in the bank.
One week survey conducted by Elegalam, (1978) revealed that 59.2% of the 390 persons making withdrawals from their accounts spent between 30 to 60 minutes while 7% spent between 90 and 120 minutes. Baale (1996) while paraphrasing Alamatu and Ariyo (1983) observed that the mean time spent was 53 minutes but customers prefer to spend a maximum of 20 minutes. Their study revealed worse service delays in urban centres (average of 64.32 minutes) compared to (average of 22.2 minutes) in rural areas. To buttress these observations, Juwah (1986) found out that customers spend between 55.27 to 64.56 minutes making withdrawal from their accounts.
Efforts in this study are directed towards application of queuing models in capacity planning to reduce customer waiting time and total operating costs.
Queues, evolves through centuries, but still leaves some of its techniques and history more imperishable than the creatures of geology. It has been one of the primitive ways of optimising some real life problems up to date. Perhaps the major reason for using a queue at all is to provide fair service to the customers. Furthermore, experimental psychology studies show that fair scheduling in queuing systems is indeed highly important to humans.
It can be traced to the days of the primitive man Noah around 2900BC, a Biblical Hebrew Patriarch who, at God's command built an arc and saved himself, his family, and a pair of every kind of animal from the flood. The above is an early queue, which is described in the Bible (Genesis 6-9). Queues are still in used in solving real life and technological problems in our daily activities.
For queuing theory, it has been found convenient, if possible to work with mathematical theory of probability distribution which exhibits the memorylessness property, as this commonly simplifies the mathematics involved. According to legend, probability theory began as a branch of mathematics with the correspondence between Blaise Pascal and Pierre de Fermat in 1654. Years before Pascal and Fermat ever thought of defining the “True worth of a chance,” isolated problems of probabilistic nature had been tackled by some mathematicians. It would be more appropriate to say that Pascal and Fermat supplied vital links in a chain of reasoning that gave us probability theory as we know it. The difficulty in trying to trace this chain to its origin is that probability theory started essentially as empirical science and developed only later on the mathematical side. Probability had its twin roots in two fairly distinct lines of investigation: the solution of wagering problems connected with games of chance, and the processing of statistical data for such matters as insurance and mortality tables (Burton, 2003).
The memorylessness property is often denoted as a Markovian property and a process with a Markovian property is called a Markov process, named after Andrej Markov (1856-1922), which means that the probability distribution of future states of the process, given the present state and all past states, depends only upon the present state and not on any past states. As a result, queuing models are frequently modelled as Poisson processes through the use of the exponential distribution. The Poisson process, discovered by the French mathematician Simeon-Denis Poisson (1781-1840), is a discrete probability process.
However, the development of queuing theory was not in place until 255 years after the study of probability theory began in 1654. It grew largely out of the need to determine the optimum amount of telephone switching equipment required to serve a given area and population. Its history goes back nearly 102 years when Johannsen's “Waiting Times and Number of Calls” (an article published in 1907 and reprinted in Post Office Electrical Engineers Journal, London, 1910) seems to be the first paper on the subject. But the method used in this paper was not mathematically exact and therefore, from the point of view of exact treatment, the paper that has historic importance is A. K. Erlang's, “The Theory of Probabilities and Telephone Conversations” (Nyt tidsskrift for Matematik, 1909). In his paper, he lays the foundation for the place of Poisson (hence, exponential) distribution in queuing theory (Brockmayer et al., 1960). Erlang's switchboard problem laid the path for modern queuing theory and is also now considered the father of the field. His papers written in the next twenty (20) years contain some of the most important concepts and techniques; the notion of statistical equilibrium and the method of writing down balance of state equations (later called Chapman-Kolmogorov equations) are two such examples. Special mention should be made of his paper “On the Rational Determination of the Number of Circuits” (Brockmeyer et al., 1960), in which an optimization problem in queuing theory was tackled for the first time.
In Erlang's work, as well as the work done by others in the twenties and thirties, the motivation has been the practical problem of congestion. Notable example is the works of (Molina, 1927; Fry, 1928). During the next two decades several theoreticians became interested in these problems and developed general models which could be used in more complex situations. Some of the authors with important contributions are Crommelin, Feller, Jensen, Khintchine, Kolmogorov, Palm, and Pollaczek. A detailed account of the investigations made by these authors may be found in books by (Syski, 1960; Saaty, 1961). Kolmogorov and Feller's study of purely discontinuous processes laid the foundation for the theory of Markov processes as it developed in later years.
Noting the inadequacy of the equilibrium theory in many queue situations, (Pollaczek, 1934) began investigations of the behaviour of the system during a finite time interval. Since then and throughout his career, he did considerable work in the analytical behavioural study of queuing systems, (Pollaczek, 1965). The trend towards the analytical study of the basic stochastic processes of the system continued, and queuing theory proved to be a fertile field for researchers who wanted to do fundamental research on stochastic processes involving mathematical models.
To this day a large majority of queuing theory results used in practice are those derived under the assumption of statistical equilibrium. Nevertheless, to understand the underlying processes fully, a time dependent analysis is essential. But the processes involved are not simple and for such an analysis sophisticated mathematical procedures become necessary.
Thus the growth of queuing theory can be traced on two parallel tracks:
a. Using existing mathematical techniques or developing new ones for the analysis of the underlying processes; and
b. Incorporating various system characteristics to make the model closely represent the real world phenomenon.
Queuing theory as an identifiable body of literature was essentially defined by the foundational research of the 1950's and 1960's. The queue M/M/ 1 (Poisson arrival, exponential service, single server) is one of the earliest systems to be analyzed. The first of such solution was given by (Bailey, 1954) using generating functions for the differential equations governing the underlying process, while Lederman and Reuter (1956) used spectral theory in their solution. Laplace transforms were later used for the same problem, and their use together with generating functions has been one of the standard and popular procedures in the analysis of queuing systems ever since.
A probabilistic approach to the analysis was initiated by Kendall (1951, 1953) when he demonstrated that imbedded Markov chains can be identified in the queue length process in systems M/G/1 and GI/M/s. Lindley (1952) derived integral equations for waiting time distributions defined at imbedded Markov points in the general queue GI/G/1. These investigations led to the use of renewal theory in queuing systems analysis in the 1960's. Identification of the imbedded Markov chains also facilitated the use of combinatorial methods by considering the queue length at Markov points as a random walk.
Mathematical modelling is a process of approximation. A probabilistic model brings it a little bit closer to reality; nevertheless it cannot completely represent the real world phenomenon because of involved uncertainties. Therefore, it is a matter of convenience where one can draw the line between the simplicity of the model and the closeness of the representation. In the 1960's several authors initiated studies on the role of approximations in the analysis of queuing systems. Because of the need for useable results in applications various types of approximations have appeared in the literature (Bhat et al., 2002).
By the end of 1960's most of the basic queuing systems that could be considered as reasonable models of real world phenomena had been analyzed and the papers coming out dealt with only minor variations of the systems without contributing much to methodology. There were even statements made to the effect that queuing theory was at the last stages of its life. But such predictions were made without knowing what advances in the computer technology would mean to queuing theory. At later instance, in the seventies, its application has been extended to computer performance evaluation and manufacturing. The need to analyze traffic processes in the rapidly growing computer and communication industry is the primary reason for the resurgence of queuing theory after the 1960's. Research on queuing networks by (Jackson, 1957; Coffman and Deming, 1973; Kleinrock 1975) laid the foundation for a vigorous growth of the subject. Some of the special queuing models of the 1950's and 1960's have found broader applicability in the context of computer and communication systems and some other real life problems such as banking, manufacturing systems, to mention but a few.
In any theory of stochastic modelling statistical problems naturally arise in the applications of the models. Identification of the appropriate model, estimation of parameters from empirical data and drawing inferences regarding future operations involve statistical procedures. These were recognized even in earlier investigations in the studies by Erlang. The first theoretical treatment of the estimation problem was given by (Clarke, 1957) who derived maximum likelihood estimates of arrival and service rates in an M/M/1 queuing system. Billingsley's (1961) treatment of inference in Markov processes in general and Wolff's (1965) derivation of likelihood ratio tests and maximum likelihood estimates for queues that can be modelled by birth and death processes are other significant advances that have occurred in this area.
The first paper on estimating parameters in a non-Markovian system is by (Goyal and Harris, 1972), who used the transition probabilities of the imbedded Markov chain to set up the likelihood function. Since then significant progress has occurred in adapting statistical procedures to various systems. For instance, (Basawa and Prabhu, 1988) considered the problem of estimation of parameters in the queue GI/G/1; (Rao, et al. 1984) used a sequential probability ratio technique for the control of parameters in M/Ek/1 and Ek/M/1; and Armero (1994) used Bayesian techniques for inference in Markovian queues, to identify only a few. More recent investigations are by (Bhat and Basawa, 2002) who use queue length as well as waiting time data in estimating parameters in queuing systems.
Hillier's (1963) paper on economic models for industrial queue problems is, perhaps, the first paper to introduce standard optimization techniques to queuing problems. While Hillier considered an M/M/1 queue, (Heyman, 1968) derived an optimal policy for turning the server on and off in an M/G/1 queue, depending on the state of the system. Since then, operations researchers trained in mathematical optimization techniques have explored their use in much greater complexity to a large number of queuing systems. These have paved way for us to study the queuing systems in some selected banks by using queuing theory.
Queuing Discipline, that is to say, when an arrival occurs, it is added to the end of the queue and service is not performed on it until all of the arrivals that came before it are served in the order they arrived. Although this is a very common method for queues to be handled, it is far from the only way. Bank queues are typical example that outlines a first-come-first-serve discipline, or an FCFS discipline. Barrer (1957) compared this with a situation where the customers are served at random, and found that the steady state probability of service is slightly less for random selection. Another situation of interest has two classes of customer with different priorities. Other possible disciplines include last-come-first-served or LCFS, and service in random order, or SIRO. While the particular discipline chosen will likely greatly affect waiting times for particular customers (nobody wants to arrive early at an LCFS discipline), the discipline generally doesn't affect important outcomes of the queue itself, since arrivals are constantly receiving service regardless.
A probabilistic approach to the queuing capacity analysis was initiated by (Kendall, 1951, 1953) when he demonstrated that embedded Markov chains can be identified in the queue length process in systems M/G/ 1 and GI/M/s. Lindley (1952) derived integral equations for waiting time distributions defined at embedded Markov points in the general queue GI/G/ 1.
Models with dependencies between inter arrival and service times have been studied by several authors. Models with a linear dependence between the service time and the preceding inter arrival time have been studied in (Cidon, et. al, 1991). Mitchell, et al (1977) analyzes the M/M/1 queue where the service time and the preceding inter arrival time have a bivariate exponential density with a positive correlation. The linear and bivariate exponential cases are both contained in the correlated M/G/1 queue studied by (Borst,et al, 1993). The correlation structure considered in Borst,et al (1993) arises in the following framework: customers arrive according to a Poisson stream at some collection point, where after exponential time periods, they are collected in a batch and transported to a service system.
Single-server queues with Markovian Arrival Processes (MAP), MAP/G/1 queue provides a powerful framework to model dependences between successive inter arrival times at the bank, but typically for the case where the arrival process is Poisson and the service times are independent and identically distributed (i.i.d.) random variables with a general distribution function G, has been investigated by (Newel, 1966). The present study concerns single-server queues where the inter arrival times and the service times depend on a common discrete time Markov Chain; i.e., the so-called semi-Markov queues. As such, the model under consideration is a generalization of the MAP/G/1 queue, by also allowing dependencies between successive service times and between inter arrival times and service times. The more challenging problem of customers' behaviour on the waiting time was first addressed by (Barrer 1957; Gnedenko and Kovalenko, 1968) in the M/M/ 1 setting. There are also works on the similarly challenging problem of customers' behaviour on waiting plus service time, including (Loris-Teghem 1972; Gavish and Schweitzer, 1977; Hokstad; 1979; Van Dijk, 1990).
In order to serve the customers at faster rates, there must be good customer advisors, faster computers and better networks provided the computers are
networked to avoid queuing or jamming networks. The need to analyze service mechanism in the rapidly growing computer and communication industry is the primary reason for the strengthening of queuing theory after the 1960's. Research on queuing networks and books such as (Coffman and Deming, 1973; Kleinrock, 1976) laid the foundation for a vigorous growth of the subject. In tracking this growth, one may cite the following survey type articles from the journal Queuing Systems: (Coffman and Hoffri, 1986), describing important computer devices and the queuing models used in analyzing their performance.
Traffic processes in computers and computer networks have necessitated the development of mathematical techniques to analyse them. The first article on queuing networks is by (Jackson, 1957). Mathematical foundations for the analysis of queuing networks are due to (Whittle, 1968; Kingman, 1969), who treated them in the terminology of population processes. Complex queuing network problems have been investigated extensively since the beginning of the 1970's.
Customers finally leave one by one after being served. This is because each teller can serve one person at a time. The interest of many researchers has not been in customers' departure, resulting in little or no research on departure.
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