The document describes applying a multi-server queue model (M/M/C) to analyze waiting lines at bank ATMs. Data was collected on customer arrival and service times at 5 ATMs over 5 days. The M/M/C model was used to calculate performance metrics like average wait times and server utilization. It was found that the busy time was 2.6 hours while idle time was 7.4 hours, showing efficient service. The utilization rate of 26% indicated no need for additional servers. The study demonstrated how queue models can help banks evaluate ATM performance and waiting lines.
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
A Case Study of Employing A Single Server Nonpreemptive Priority Queuing Mode...IJERA Editor
This paper discusses a case study of employing a single server nonpreemptivepriorityqueuing model [1]at ATM
machine which originally operates on M/M/1 model. In this study we have taken two priority classes of people
in following order:-
.Priority class 1- woman
.Priority class 2- man
Sometimea long queue is formed at ATMmachine (single server)but the bank management don’t have enough
money to invest on installing new ATM machine.In this situation we want to apply single server nonpreemptive
priority queuing model.The security guard at the ATM will divide the customers in two category and arrange the
customers in the above said priority order Thuspriority class 1 people willreceive theatm service ahead of
priority class 2 people.This will reduce the waiting time of priority class 1 people. Of course by doing this the
waiting time of priority class 2will increase.
A Queue Fairness Model of Automated Teller Machine ATM System A Case Study of...ijtsrd
Until recently, consensus of opinions, thoughts and theories on the studies of ATM queuing system, and the analysis of its influence on customers’ behaviour were predicated on the average time spent by a tagged customer on the queue. Very few were poised to consider the unfairness or injustice experienced by customers in the systems in terms of whether the actual services rendered to the customers commensurate with their delay probabilities. To this end we employ the Resource Allocation Queue Fairness RAQF metrics to study and appraise “fairness†perceptions as indicator for service delivery quality and hence customers’ satisfaction in financial institution in Kaduna metropolis. Tentatively, we evaluate both the performance measures and the unfairness characteristics of the system, with respect to Poisson arrival process and exponentially distributed service time in M M 4 single queue system. Summarily, the results of the analyses show that, both the unfairness and the discrimination coefficients of the system in general increases with the probability of having more k m customers in the system and vice versa. The high negative discrimination index as well as its corresponding high unfairness coefficient, shows that though the ATM system may be necessary to improving service delivery quality, but queuing all customers with varying job size service requirement differences in a single queue structure served under FCFS job seniority service policy is unfair and hence may provoke customers’ dissatisfaction. To reverse the situation, the study recommended that i since all customers arriving at the system at any epoch have varying job size, then scheduling of services based on service requirement differences, and dedicating separate ATM machines to customers’ with similar job size will reduce the unfairness index and hence enhance customers’ satisfaction. ii To reduce the ATM overutilization factor during peak periods, as well as the high delay probability associated with the system, the deployment of few ATM machines with optimal processing speed would be more cost effective than more ATM machines with relatively slow processing speed and iii Since the efficient utilization of the ATM machine is a function of good internet connectivity, steady electricity supply and good computer literacy level of customers, therefore, the provision of good internet connectivity, alternative steady electricity supply solar , as well as provision of basic ATM operation guides to customers would reduce customer’s sojourn time at the service point, reduce the system unfairness index and hence guarantee customer’s satisfaction. Israel Udoh | Eno-bong Ukoh | Harmony Igoru | Linus Okafor "A Queue Fairness Model of Automated Teller Machine (ATM) System: (A Case Study of Financial Institutions in Kaduna Nigeria)" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-6 | Issue-1 , December 2021, URL: https://www
Currently the operators in the telecommunications market present offers of subscription to the consumers,and given that competition is strong in this area, most of these advertising offers are prepared to attract and / or keep customers. For this reason, customers face problems in choosing operators that meet their needs in terms of price, quality of service (QoS), etc..., while taking into account the margin between what is advertising and what is real. Therefore, we are led to solve a problem of decision support. Mathematical modeling of this problem led to the solution of an inverse problem. Specifi-cally, the inverse problem is to find the real Quality of Service (QoS) function knowing the theoretical QoS. To solve this problem we have reformulated in an optimization problem of minimizing the difference between the real quality of service (QoS) and theoretical (QoS). This model will help customers who seek to know the degree of sincerity of Their operators, as well as it is an opportunity for operators who want to maintain their resources so that they gain the trust of customers. The resulting optimization problem is solved using evolutionary algorithms. The numerical results showed the reliability and credibility of our inverse model and the performance and effectiveness of our approach.
Solving Of Waiting Lines Models in the Bank Using Queuing Theory Model the Pr...IOSR Journals
Waiting lines and service systems are important parts of the business world. In this article we describe several common queuing situations and present mathematical models for analyzing waiting lines following certain assumptions. Those assumptions are that (1) arrivals come from an infinite or very large population, (2) arrivals are Poisson distributed, (3) arrivals are treated on a FIFO basis and do not balk or renege, (4) service times follow the negative exponential distribution or are constant, and (5) the average service rate is faster than the average arrival rate. The model illustrated in this Bank for customers on a level with service is the multiple-channel queuing model with Poisson Arrival and Exponential Service Times (M/M/S). After a series of operating characteristics are computed, total expected costs are studied, total costs is the sum of the cost of providing service plus the cost of waiting time. Finally we find the total minimum expected cost.
Queuing is the common activity of customers or people to avail the desired service, which could be processed or distributed one at a time. Bank ATMs would avoid losing their customers due to a long wait on the line. The bank initially provides one ATM in every branch. But, one ATM would not serve a purpose when customers withdraw to use ATM and try to use other bank ATM. Thus the service time needs to be improved to maintain the customers. This paper shows that the queuing theory used to solve this problem. We obtain the data from a bank ATM in a city. We then derive the arrival rate, service rate, utilization rate, waiting time in the queue and the average number of customers in the queue based on the data using Little’s theorem and M/M/I queuing model. The arrival rate at a bank ATM on Sunday during banking time is 1 customer per minute (cpm) while the service rate is 1.50 cpm. The average number of customer in the ATM is 2 and the utilization period is 0.70. We conclude the paper by discussing the benefits of performing queuing analysis to a busy ATM.
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
A Case Study of Employing A Single Server Nonpreemptive Priority Queuing Mode...IJERA Editor
This paper discusses a case study of employing a single server nonpreemptivepriorityqueuing model [1]at ATM
machine which originally operates on M/M/1 model. In this study we have taken two priority classes of people
in following order:-
.Priority class 1- woman
.Priority class 2- man
Sometimea long queue is formed at ATMmachine (single server)but the bank management don’t have enough
money to invest on installing new ATM machine.In this situation we want to apply single server nonpreemptive
priority queuing model.The security guard at the ATM will divide the customers in two category and arrange the
customers in the above said priority order Thuspriority class 1 people willreceive theatm service ahead of
priority class 2 people.This will reduce the waiting time of priority class 1 people. Of course by doing this the
waiting time of priority class 2will increase.
A Queue Fairness Model of Automated Teller Machine ATM System A Case Study of...ijtsrd
Until recently, consensus of opinions, thoughts and theories on the studies of ATM queuing system, and the analysis of its influence on customers’ behaviour were predicated on the average time spent by a tagged customer on the queue. Very few were poised to consider the unfairness or injustice experienced by customers in the systems in terms of whether the actual services rendered to the customers commensurate with their delay probabilities. To this end we employ the Resource Allocation Queue Fairness RAQF metrics to study and appraise “fairness†perceptions as indicator for service delivery quality and hence customers’ satisfaction in financial institution in Kaduna metropolis. Tentatively, we evaluate both the performance measures and the unfairness characteristics of the system, with respect to Poisson arrival process and exponentially distributed service time in M M 4 single queue system. Summarily, the results of the analyses show that, both the unfairness and the discrimination coefficients of the system in general increases with the probability of having more k m customers in the system and vice versa. The high negative discrimination index as well as its corresponding high unfairness coefficient, shows that though the ATM system may be necessary to improving service delivery quality, but queuing all customers with varying job size service requirement differences in a single queue structure served under FCFS job seniority service policy is unfair and hence may provoke customers’ dissatisfaction. To reverse the situation, the study recommended that i since all customers arriving at the system at any epoch have varying job size, then scheduling of services based on service requirement differences, and dedicating separate ATM machines to customers’ with similar job size will reduce the unfairness index and hence enhance customers’ satisfaction. ii To reduce the ATM overutilization factor during peak periods, as well as the high delay probability associated with the system, the deployment of few ATM machines with optimal processing speed would be more cost effective than more ATM machines with relatively slow processing speed and iii Since the efficient utilization of the ATM machine is a function of good internet connectivity, steady electricity supply and good computer literacy level of customers, therefore, the provision of good internet connectivity, alternative steady electricity supply solar , as well as provision of basic ATM operation guides to customers would reduce customer’s sojourn time at the service point, reduce the system unfairness index and hence guarantee customer’s satisfaction. Israel Udoh | Eno-bong Ukoh | Harmony Igoru | Linus Okafor "A Queue Fairness Model of Automated Teller Machine (ATM) System: (A Case Study of Financial Institutions in Kaduna Nigeria)" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-6 | Issue-1 , December 2021, URL: https://www
Currently the operators in the telecommunications market present offers of subscription to the consumers,and given that competition is strong in this area, most of these advertising offers are prepared to attract and / or keep customers. For this reason, customers face problems in choosing operators that meet their needs in terms of price, quality of service (QoS), etc..., while taking into account the margin between what is advertising and what is real. Therefore, we are led to solve a problem of decision support. Mathematical modeling of this problem led to the solution of an inverse problem. Specifi-cally, the inverse problem is to find the real Quality of Service (QoS) function knowing the theoretical QoS. To solve this problem we have reformulated in an optimization problem of minimizing the difference between the real quality of service (QoS) and theoretical (QoS). This model will help customers who seek to know the degree of sincerity of Their operators, as well as it is an opportunity for operators who want to maintain their resources so that they gain the trust of customers. The resulting optimization problem is solved using evolutionary algorithms. The numerical results showed the reliability and credibility of our inverse model and the performance and effectiveness of our approach.
Solving Of Waiting Lines Models in the Bank Using Queuing Theory Model the Pr...IOSR Journals
Waiting lines and service systems are important parts of the business world. In this article we describe several common queuing situations and present mathematical models for analyzing waiting lines following certain assumptions. Those assumptions are that (1) arrivals come from an infinite or very large population, (2) arrivals are Poisson distributed, (3) arrivals are treated on a FIFO basis and do not balk or renege, (4) service times follow the negative exponential distribution or are constant, and (5) the average service rate is faster than the average arrival rate. The model illustrated in this Bank for customers on a level with service is the multiple-channel queuing model with Poisson Arrival and Exponential Service Times (M/M/S). After a series of operating characteristics are computed, total expected costs are studied, total costs is the sum of the cost of providing service plus the cost of waiting time. Finally we find the total minimum expected cost.
Queuing is the common activity of customers or people to avail the desired service, which could be processed or distributed one at a time. Bank ATMs would avoid losing their customers due to a long wait on the line. The bank initially provides one ATM in every branch. But, one ATM would not serve a purpose when customers withdraw to use ATM and try to use other bank ATM. Thus the service time needs to be improved to maintain the customers. This paper shows that the queuing theory used to solve this problem. We obtain the data from a bank ATM in a city. We then derive the arrival rate, service rate, utilization rate, waiting time in the queue and the average number of customers in the queue based on the data using Little’s theorem and M/M/I queuing model. The arrival rate at a bank ATM on Sunday during banking time is 1 customer per minute (cpm) while the service rate is 1.50 cpm. The average number of customer in the ATM is 2 and the utilization period is 0.70. We conclude the paper by discussing the benefits of performing queuing analysis to a busy ATM.
Performance measures for internet server by using m m m queueing modeleSAT Journals
Abstract This paper deals with the performance measurement of single queue multiple server model. This gives the performance measure for internet server for highly dynamic traffic conditions. Our previous work is related to performance measurement of single queue single server model. This is achieved by analyzing the performance measures and capacity planning of internet server using different queuing models by comparing the parameters like queuing length, response time, waiting time for different links. Keywords- Internet server, waiting time, response time, queue length, queuing models
Performance measures for internet server by using m mm queueing modeleSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
Waiting Line Model is one of the decision line model.Waiting Line Model is one of the decision line model.Waiting Line Model is one of the decision line model.Waiting Line Model is one of the decision line model.
A New Data Stream Mining Algorithm for Interestingness-rich Association RulesVenu Madhav
Frequent itemset mining and association rule generation is
a challenging task in data stream. Even though, various algorithms
have been proposed to solve the issue, it has been found
out that only frequency does not decides the significance
interestingness of the mined itemset and hence the association
rules. This accelerates the algorithms to mine the association
rules based on utility i.e. proficiency of the mined rules. However,
fewer algorithms exist in the literature to deal with the utility
as most of them deals with reducing the complexity in frequent
itemset/association rules mining algorithm. Also, those few
algorithms consider only the overall utility of the association
rules and not the consistency of the rules throughout a defined
number of periods. To solve this issue, in this paper, an enhanced
association rule mining algorithm is proposed. The algorithm
introduces new weightage validation in the conventional
association rule mining algorithms to validate the utility and
its consistency in the mined association rules. The utility is
validated by the integrated calculation of the cost/price efficiency
of the itemsets and its frequency. The consistency validation
is performed at every defined number of windows using the
probability distribution function, assuming that the weights are
normally distributed. Hence, validated and the obtained rules
are frequent and utility efficient and their interestingness are
distributed throughout the entire time period. The algorithm is
implemented and the resultant rules are compared against the
rules that can be obtained from conventional mining algorithms
Talks about what is Queuing and its application, practical life usage, with a complex problem statement with its solution. Pre-emptive and non-preemptive queue models and its algorithm.
A Review on Performance of Toll Plaza by using Queuing Theoryijtsrd
One of the key problems in the study of any stream of traffic system is the analysis of delay. Delay is a more delicate concept. It may be well defined as the difference between the actual travel time on a given section and some ideal travel time of that section. This raises the question as what is the ideal travel time. In practice, the ideal travel time chosen will depend on the situation. There are two particular travel times that seem best suited as benchmarks for assessment with the actual performance of the system. These are the travel times under free flow conditions and travel time at capacity. Most recent research has found that for highway systems, there is reasonably little difference between these two speeds. The analysis of delay normally focuses on the delay when demand exceeds its capacity. Such delay is known as queuing delay, and may be studied by means of queuing theory. This theory involves the analysis which is known as a queuing system, which is composed of a server, a stream of customers who demand service, and a queue or line of customers waiting to be served. A. N. V. Ravindra | Mr. S. Siva Gowri Prasad "A Review on Performance of Toll Plaza by using Queuing Theory" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-6 , October 2019, URL: https://www.ijtsrd.com/papers/ijtsrd28068.pdf Paper URL: https://www.ijtsrd.com/engineering/transport-engineering/28068/a-review-on-performance-of-toll-plaza-by-using-queuing-theory/a-n-v-ravindra
Data Mining on Customer Churn ClassificationKaushik Rajan
Implemented multiple classifiers to classify if a customer will leave or stay with the company based on multiple independent variables.
Tools used:
> RStudio for Exploratory data analysis, Data Pre-processing and building the models
> Tableau and RStudio for Visualization
> LATEX for documentation
Machine learning models used:
> Random Forest
> C5.0
> Decision tree
> Neural Network
> K-Nearest Neighbour
> Naive Bayes
> Support Vector Machine
Methodology: CRISP-DM
Customer churn classification using machine learning techniquesSindhujanDhayalan
Advanced data mining project on classifying customer churn by
using machine learning algorithms such as random forest,
C5.0, Decision tree, KNN, ANN, and SVM. CRISP-DM approach was followed for developing the project. Accuracy rate, Error rate, Precision, Recall, F1 and ROC curve was generated using R programming and the efficient model was found comparing these values.
A Survey on Patient Queue Management SystemIJAEMSJORNAL
In this paper, we study the various types of Queue and Queue Management System. Queue system can successfully reduce waiting time of the patients in the hospitals. We aim to implement a model that initializes alert notification via SMS to patients of a hospital. It will minimize the queue of patients in the waiting area of hospital and also patient can book an appointment from anywhere at a given time. Patients can book an appointment via android application. and accordingly patients and also it will provide navigation towards that nearest hospital. Hence, for developing such a model a survey has been done on various types of queues.
Performance measures for internet server by using m m m queueing modeleSAT Journals
Abstract This paper deals with the performance measurement of single queue multiple server model. This gives the performance measure for internet server for highly dynamic traffic conditions. Our previous work is related to performance measurement of single queue single server model. This is achieved by analyzing the performance measures and capacity planning of internet server using different queuing models by comparing the parameters like queuing length, response time, waiting time for different links. Keywords- Internet server, waiting time, response time, queue length, queuing models
Performance measures for internet server by using m mm queueing modeleSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
Waiting Line Model is one of the decision line model.Waiting Line Model is one of the decision line model.Waiting Line Model is one of the decision line model.Waiting Line Model is one of the decision line model.
A New Data Stream Mining Algorithm for Interestingness-rich Association RulesVenu Madhav
Frequent itemset mining and association rule generation is
a challenging task in data stream. Even though, various algorithms
have been proposed to solve the issue, it has been found
out that only frequency does not decides the significance
interestingness of the mined itemset and hence the association
rules. This accelerates the algorithms to mine the association
rules based on utility i.e. proficiency of the mined rules. However,
fewer algorithms exist in the literature to deal with the utility
as most of them deals with reducing the complexity in frequent
itemset/association rules mining algorithm. Also, those few
algorithms consider only the overall utility of the association
rules and not the consistency of the rules throughout a defined
number of periods. To solve this issue, in this paper, an enhanced
association rule mining algorithm is proposed. The algorithm
introduces new weightage validation in the conventional
association rule mining algorithms to validate the utility and
its consistency in the mined association rules. The utility is
validated by the integrated calculation of the cost/price efficiency
of the itemsets and its frequency. The consistency validation
is performed at every defined number of windows using the
probability distribution function, assuming that the weights are
normally distributed. Hence, validated and the obtained rules
are frequent and utility efficient and their interestingness are
distributed throughout the entire time period. The algorithm is
implemented and the resultant rules are compared against the
rules that can be obtained from conventional mining algorithms
Talks about what is Queuing and its application, practical life usage, with a complex problem statement with its solution. Pre-emptive and non-preemptive queue models and its algorithm.
A Review on Performance of Toll Plaza by using Queuing Theoryijtsrd
One of the key problems in the study of any stream of traffic system is the analysis of delay. Delay is a more delicate concept. It may be well defined as the difference between the actual travel time on a given section and some ideal travel time of that section. This raises the question as what is the ideal travel time. In practice, the ideal travel time chosen will depend on the situation. There are two particular travel times that seem best suited as benchmarks for assessment with the actual performance of the system. These are the travel times under free flow conditions and travel time at capacity. Most recent research has found that for highway systems, there is reasonably little difference between these two speeds. The analysis of delay normally focuses on the delay when demand exceeds its capacity. Such delay is known as queuing delay, and may be studied by means of queuing theory. This theory involves the analysis which is known as a queuing system, which is composed of a server, a stream of customers who demand service, and a queue or line of customers waiting to be served. A. N. V. Ravindra | Mr. S. Siva Gowri Prasad "A Review on Performance of Toll Plaza by using Queuing Theory" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-6 , October 2019, URL: https://www.ijtsrd.com/papers/ijtsrd28068.pdf Paper URL: https://www.ijtsrd.com/engineering/transport-engineering/28068/a-review-on-performance-of-toll-plaza-by-using-queuing-theory/a-n-v-ravindra
Data Mining on Customer Churn ClassificationKaushik Rajan
Implemented multiple classifiers to classify if a customer will leave or stay with the company based on multiple independent variables.
Tools used:
> RStudio for Exploratory data analysis, Data Pre-processing and building the models
> Tableau and RStudio for Visualization
> LATEX for documentation
Machine learning models used:
> Random Forest
> C5.0
> Decision tree
> Neural Network
> K-Nearest Neighbour
> Naive Bayes
> Support Vector Machine
Methodology: CRISP-DM
Customer churn classification using machine learning techniquesSindhujanDhayalan
Advanced data mining project on classifying customer churn by
using machine learning algorithms such as random forest,
C5.0, Decision tree, KNN, ANN, and SVM. CRISP-DM approach was followed for developing the project. Accuracy rate, Error rate, Precision, Recall, F1 and ROC curve was generated using R programming and the efficient model was found comparing these values.
A Survey on Patient Queue Management SystemIJAEMSJORNAL
In this paper, we study the various types of Queue and Queue Management System. Queue system can successfully reduce waiting time of the patients in the hospitals. We aim to implement a model that initializes alert notification via SMS to patients of a hospital. It will minimize the queue of patients in the waiting area of hospital and also patient can book an appointment from anywhere at a given time. Patients can book an appointment via android application. and accordingly patients and also it will provide navigation towards that nearest hospital. Hence, for developing such a model a survey has been done on various types of queues.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
Chapter 3 - Islamic Banking Products and Services.pptx
07_AJMS_392_22_Revised.pdf
1. www.ajms.com 75
ISSN 2581-3463
RESEARCH ARTICLE
Application of Multi-server Queue Model (m/m/c) for Waiting Lines Management
in Banking System
S. A. Akande, U. N. Mohammed, A. M. Ibrahim, O. A. Adedayo
Department of Mathematics, Federal University Technology, Minna, Nigeria
Received: 20-01-2021; Revised: 10-02-2022; Accepted: 05-03-2022
ABSTRACT
Queues or waiting lines arise when the demand for service exceeds the capacity of a service facility.
One of the major challenges bank customers encounter in banks is the waiting lines in automated teller
machines (ATMs). This study formulated a Multi-Server Queue Model (M/M/C) for Queue Management
in Banking ATM. The performance level of a typical bank ATM has been effectively investigated using
the M/M/S queuing model. It was observed that the busy time of the machine is 2.6 h while the idle time
is 7.4 h in the 10 h of banking time which is attributed to the availability of many servers in the system.
The utilization factor is 0.26 or 26.0% shows that the service delivery of the machine is very efficient and
there is no urgent need for an additional server. The researcher thereby recommended that banks should
consider the use of the queue model to test the performance of waiting lines in the ATMs.
Key words: Multi-server, Queue model, Waiting line, Banking system
Address for correspondence:
S. A. Akande,
siqlam@yahoo.com
INTRODUCTION
The act of joining a line or waiting is referred to
as a queue. Customers (arrivals) wanting service
must wait because the number of servers available
exceeds the number of servers available, or the
facility does not perform smoothly or takes longer
than the time allotted to serve a client.[1]
It is a
common occurrence in gas stations, supermarkets,
and banks, among other places.
Electronic banking is one of the many
technological achievements in the banking
industry. In the banking business, an automated
teller machine (ATM) is one of various electronic
banking channels. In the banking business, ATMs
are one of the most crucial service facilities.[2]
ATMs first introduced in Nigeria in 1989 and
have since gained widespread acceptance and
usage. Nigeria exchange group is a company
based in Nigeria.[3]
More than half of respondents
in a study stated that preference for the situation
becomes compounded during festive periods and
month finishes, when demand for cash is at its
peak.
According[4]
to they described, queuing theory is
a research that aims to assist business owners in
analyzing the percentage of time customers wait
for services to be delivered to them and improving
the percentage of services delivered in this way.
Erlang, a Danish mathematician, conducted the
first investigation on queuing theory, which led
to the development of the world-renowned Erlang
telephone model. He looked at the phone network
system and sought to figure out what influence
variable service demands had on call volume and
the use of automatic dialing equipment.[5]
Queuing theory, often known as congestion
theory, is an area of operational research that
investigates the relationship between demand for
a service system and the delays experienced by its
users.[6]
Other researchers[7,8]
worked on improving
performance inside the banking hall.
The application of queuing theory in banking
operations is still in its early stages of research, with
notable studies being conducted in various areas.[1]
Salami et al.[9]
investigated strategies for Chinese
commercial banks to increase their efficiency in
terms of customer queuing in bank halls. The focus
was on using queuing theory to investigate the
queue problem, which was based on supposed data
with an arrival rate of 32 and a service rate of 20.
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Kumaran et al.[10]
did a study on three ATM
facilities from three different banks that were
evaluated at Veliore Institute of Technology using
simulation and queuing theory. Simulation was
used to collect data, which was done throughout
a daily free period and on weekends. They used
transformed data to calculate client wait times
in the queue and in the system, after which they
compared consumer satisfaction with the services
offered by variousATM facilities. In the study of[4]
a method for calculating queue length and waiting
time was established. The appropriate queue
models are utilized to estimate queue length,
which is then used to determine gateway server
memory size. In Nigeria’s financial system, ATMs
and ATM queues are relatively new.
The present study sought to investigate the
application of multi-server queue model (M/M/C)
for waiting lines management of Union Bank
(Union Bank) Bida branch ATMs.
MATERIALS AND METHODS
Research Method
The research method used in this study was
quantitative and no qualitative data were collected.
Because the goal of this study is to look into the
performance of a Union Bank ATM located on
Bida Road in Bida Niger State. The development
of a queuing model for the analysis of the queuing
system at Union Bank ATM service point, as well
as the establishment of a strategy to tackle the
problem of customer arrival rate, was among the
methodologies employed in this research effort.
Method of Data Collection
The input data were manually collected with the
help of an electronic instrument (e-stopwatch)
and a study assistant using direct observation. The
informationforthisstudywasgatheredattheUnion
Bank Bank PIc ATM service station on Bida Road
Bida, Niger State, for a period of 5 working days,
from 7:30 a.m. to 7:00 p.m., respectively. Direct
observation of consumers was used to collect data,
and their arrival time, “the time service begins,”
and “the time service finishes” were all collected
and entered in real-time onto a form created
specifically for this experiment. To calculate,
we used data that were well-fitting. According
to queue theory, the following assumptions were
made for the queuing system. They are:
1. Arrivals follow a Poisson probability
distribution and are drawn from an unlimited
population
2. No balking or reneging occurs because there is
only one waiting line and each arrival waits to
be serviced regardless of the length of the line
3. Queue discipline is first come first serve
(FCFS)
4. With an average of customers per unit of time,
service time follows an exponential distribution
5. The average rate of service is higher than the
average rate of arrival
6. The system can accommodate an infinite
number of customers
7. Service providers do not go faster because
the line is longer; service providers do not go
faster because the line is longer.
Method of Data Analysis
Based on the data collected from the bank ATM
service point, we conduct an analysis.
i. The time arrival of each customer
ii. The inter-arrival time between customers
iii. The time service commenced and ends for
each customer using the M/M/S model and a
single queue.
To appropriately assess the data recorded and
generate the performance measurements, an
analytical approach/method or queuing theory
(formula based) was used.
Formulation of Model from Kendall’s
Notation
Let us introduce a notation created by Kendall
to define a queuing system before we begin our
research of simple queuing systems. Let’s call it a
system if A/B/m/K/n/D
• A: Distribution function of the inter arrival
times
• B: Distribution function of the service times
• m: number of servers
• K: Capacity of the system, the maximum
number of customers in the system including
the one being serviced
• n: Population size, number of sources of
customers,
• D: Service discipline.
M, which stands for Markovian or memoryless, is
used to denote exponentially distributed random
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variables. They are also omitted if the population
size and capacity are unlimited, and the service
discipline is FIFO.
With the value of s=5, the Kendall notation
becomes M/M/5, which exactly fits the queuing
model utilized in this research to analyze the
performance metrics of the queue system at Union
Bank Bida. M/M/r/K/n also refers to a system in
which consumers enter from a finite-source with n
elements and stay for an exponentially dispersed
period, service times are exponentially distributed,
service is delivered according to the arrival of the
request by r servers, and the system capacity is K.
Queuing Model with Single Queue and
Multiple ATMs [(M/M/S): (∞/FCFS)] M/M/S
Model
The model shown here can capture all of the
characteristics involved in a multi-server queuing
system for an infinite calling population with a
first-come and first-served multiple server queuing
system (/∞FCFS). In this example, multiple and
identical servers are connected in a parallel line
to deliver the same service to clients. When a
customer arrives, he is placed in a single server
queue and remains there until he is served. Here,
μ is the service rate of one server, but we have s
number of servers; therefore, sμ will be the service
rate. If there are n consumers in the queue at any
given time, one of the following two scenarios
may occur:
i. If ns (number of customers in the system is
less than the number of servers), then there
will be no queue. However, (s−n) numbers of
servers are not busy. The combined service
rate will be: μn
=nμ; ns
ii. If n≥s (number of customers in the system is
more than or equal to the number of servers)
then all servers will be busy and the maximum
number of customers in the queue will be
(n−s). The combined service rate will be:
𝜇n
=sμ; n=s. Thus, we have λn
=λ for all n≥0.
{ ; ,
n
n
n s n s n s
=
= ≥ . Figure 1 shows the
diagram queuing system model.
The Basic Indexes/Parameters of the Queuing
system
• n = Number of customers in the system at
time t
• λ = Mean arrival rate (number of arrival per
unit of time)
• 𝜇 = Mean Service rate per busy server (number
of customers served per unit of time)
• 𝜌 = Expected fraction of time for which server
is busy
• s = number of service channel channels
(service facilities or servers).
Some Queuing Notations used in this Multi-
Server Queuing Model
i.
n = Number of total customers in the system
(in queue plus in service)
ii. λ = Mean arrival rate [1/(average number of
customers arriving in each queue in a system
in 1 h)]
iii. 𝜇 = Mean service rate [1/(average number of
customers being served at a server per hour)]
iv. s = Number of parallel servers (service units in
queue system)
v. s𝜇 = Servicing rate when s 1 in a system
vi. 𝜌 = System intensity or load, utilization factor
(ρ=λ/sμ=) (the expected factor of time the
server is busy that is service capability being
utilized on the average arriving customers)
vii. Ls
= Length of system
viii. Lq
= Length of queue
ix. Ws
= Waiting time in the system
x. Wq
= Waiting time in the queue
xi.
Pn
= Probability of having n customers in the
queuing system
xii.
P0
(t) = Probability that there are no customers
in queuing system at time t
xiii.
Ps
=Theprobabilitythatonarrival,acustomer
must wait for services.
Operating Characteristics of a Multiple
Server Queue Model
a. The probability Pn
of n customers in the
queuing system is given by
Pn
P n s
n
P n s
n
n s
n
=
−
≤
s!s
;
!
;
0
0
(1)
b. The probability that the ATM is idle (P0
) that
is, the probability of no customer in the ATM
1
1
0
1 1
! s!
n s
s
o
n
P
n s
−
−
=
= +
−
∑
4. Akande, et al.: Application of Multi-Server Queue Model (m/m/c) for Waiting Lines Management In Banking System
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c. Expected number of customers waiting in the
queue (i.e., queue length)
L
s s
P
q
s
=
−
( )
−
( )
1
1 5 0
!
λ
µ
λµ
µ λ
d. Expected number of customers in a system
L L
s q
= +
λ
µ
e. Expected waiting time of a customer in the
queue
W
s s
P
L
q
s
q
=
−
( )
−
( )
=
1
1 5 0
!
λ
µ
λµ
µ λ λ
f. Expected waiting time that a customer spends
in a system
W W
s q
= +
1
g. Utilization factor, that is, the fraction of time
servers is busy
ρ
λ
µ
=
s
h. The probability that on arrival a customer
must wait for service
P
s
s
s
P
s
s
=
−
1
0
!
λ
µ
µ
µ λ
RESULTS AND DISCUSSION
This section discusses that the analysis of data
obtained on daily observation taking atATM stand
of Union Bank, Bida. Table 1 shows the summary
of data collected on inter-arrival time, service time
for ATM 1, ATM 2, ATM 3, ATM 4, and ATM 5 as
well as total number of customers.
From the Table 1, we can obtain the following
1. Average inter-arrival time for each day =
total inter arrival for eachday
total no of customer for eachday
−
.
Hence, average inter-arrival time for Monday is =
1089
742
=1468
.
Average inter-arrival time for Tuesday is =
1069
692
=1545
.
Average inter-arrival time for Wednesday is =
954
733
0
=13 2
.
Average inter-arrival time for Thursday is =
1072
854
=1255
.
Average inter-arrival time for Friday is =
933
7688
=1215
.
2. The average service time for each server is =
total servicetime for eachday
total no of customer for eachday
.
a. The average service time for ATM 1 is =
total servicetime for eachday
total no of customer for eachday
.
therefore, Average service time for Monday is =
499
742
0
= .673
AverageservicetimeforTuesdayis=
526
692
0 0
= .76
Average service time for Wednesday is =
405
733
0
= .553
Table 1: Summary of data collected on total inter-arrival
time and total service time of the ATMs for period of 5
working day (7:30 am–7:00 pm)
ATM Day Total
1 2 3 4 5
1 499 526 405 364 352 2146
2 435 422 402 431 407 2097
3 375 406 418 397 404 2000
4 454 412 436 429 418 2149
5 465 439 403 427 474 2208
Inter-Arrival Time (min) 1089 1069 954 1072 933 5117
Total No. of customers 742 692 733 854 768 3789
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Average service time for Thursday is =
364
854
0
= .426
Average service time for Friday is =
352
768
0
= .458
b. The average service time for ATM 2 is =
total servicetime for eachday
total no of customer for eachday
.
therefore, average service time for Monday is =
435
742
0
= .586
Average service time forTuesday is =
422
692
0 0
= .61
Average service time for Wednesday is =
402
733
0
= .548
Average service time for Thursday is =
431
854
0 0
= .5 5
Average service time for Friday is =
407
768
0 0
= .53
c. The average service time for ATM 3 is =
total servicetime for eachday
total no of customer for eachday
.
therefore, average service time for Monday is =
375
742
0 0
= .5 5
AverageservicetimeforTuesdayis=
406
692
0
= .587
Average service time for Wednesday is =
41
733
0 0
= .57
Average service time for Thursday is =
397
854
0
= .465
Average service time for Friday is =
404
768
0
= .526
d. The average service time for ATM 4 is =
total servicetime for eachday
total no of customer for eachday
.
Therefore, average service time for Monday is =
454
742
0
= .612
AverageservicetimeforTuesdayis=
412
692
0
= .595
Average service time for Wednesday is =
436
733
0
= .595
Average service time for Thursday is =
429
854
0 0
= .5 2
Average service time for Friday is =
418
768
0
= .544
e. The average service time for ATM 5 is =
total servicetime for eachday
total no of customer for eachday
.
therefore, average service time for Monday is =
465
742
0
= .627
AverageservicetimeforTuesdayis=
439
692
0
= .634
Average service time for Wednesday is =
403
733
0 0
= .55
Average service time for Thursday is =
427
854
0 00
= .5
Average service time for Friday is =
474
768
0
= .617
3. Obtaining the total average service time =
sumof averageservicetime for eachday
nunber of day
� � � � � � �
� �
a. Obtaining the total average service time for
ATM1
=
+ + + +
=
0 673 0 600 0 553 0 426 0 458
5
0
. . . . .
.574
b. Obtaining the total average service time for
ATM2
6. Akande, et al.: Application of Multi-Server Queue Model (m/m/c) for Waiting Lines Management In Banking System
AJMS/Jan-Mar-2022/Vol 6/Issue 1 80
=
+ + + +
=
0 56 0 610 0 548 0 505 0 530
5
0
. . . . .
.556
c. Obtaining the total average service time for
ATM3
=
+ + + +
=
0 505 0 587 0 570 0 465 0 526
5
0
. . . . .
.531
d. Obtaining the total average service time for
ATM4
=
+ + + +
=
0 612 0 595 0 595 0 502 0 544
5
0 0
. . . . .
.57
e. Obtaining the total average service time for
ATM5
=
+ + + +
=
0 627 0 634 0 550 0 500 0 617
5
0
. . . . .
.586
4. Obtaining the total average service time for
system
= =
+ +
sumof average
servicetime for
each ATM
nunber of ATM
0 574 0 556
. .
0
0 531 0 570
0 56
5
0
. .
.
.
+ +
= 563
5. Obtaining the average inter arrival time for
system
= =
+ +
+
sumof
number of days
inter
arrival time
1 468 1 545 1
302 1 255
. .
. . +
+
=
1 215
5
.
.
1375
From the expression, it can be deduce that average
service time is 0.56 while average inter arrival
time is 1.375
Then, the service rate and arrival rate are calculated as;
Service rate (μ)=
1 1
0 563
8
averageservice time
1776 17
= = ≅
.
. .
customers per minute = 106.8 cust/h
average inter arrival time (λ)
= = = ≅
1 1
1 375
0 0
averageinter arrival time
737 74
.
. .
customers per minute = 44.4 cust/h.
Since the major queue parameters are already
known, we can proceed to obtain the performance
measure of the system such as;
1. Utilization factor
ρ
λ
µ
= = =
s
0 74
5 1 78
0
.
.
. . %.
×
262 or 26 2
2. Probability of number of customer in the
system;
P
n s
o
n s
n
s
=
+
−
=
−
−
∑
1 1
0
1
1
! s!
λ
µ
λ
µ
λµ
µ λ
Po =
+
×
× −
1
0
0 74
1 78
1
5
0 74
1 78
5 1 78
5 1 78 0 74
0 5
!
.
. !
.
.
.
. .
=
−
∑
n 0
4
1
Po =
+
+
1
0
0 74
1 78
1
1
0 74
1 78
1
2
0 74
1 78
0 1
!
.
. !
.
. !
.
.
2
2
3 4
1
3
0 74
1 78
1
4
0 74
1 78
1
5
0 74
1 78
+
+
+
!
.
. !
.
.
!
.
.
5
5
0
4
1
5 1 78
5 1 78 0 74
×
×
× −
=
−
∑
.
. .
n
Po
n
=
+ + + +
+
=
∑
1 0 41573 0 086416 0 011975
0 001245 0 00011287
0
4
. . .
. .
−1
Po =
+ + + +
+
−
1 0 41573 0 086416 0 011975
0 001245 0 00011287
1
. . .
. .
Po = [ ]−
1 515478911
1
.
Po =
1
1 515478911
.
P0
= 0.659857
3. The excepted number of customers waiting in
the queue
L
s s
P
q
s
=
−
( )
−
( )
1
1 5 0
!
λ
µ
λµ
µ λ
7. Akande, et al.: Application of Multi-Server Queue Model (m/m/c) for Waiting Lines Management In Banking System
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Lq =
−
( )
×
× −
( )
×
1
5 1
0 74
1 78
0 74 1 78
5 1 78 0 74
0
5
5
!
.
.
. .
. .
.
.659857
Lq =
( )
( )
−
( )
×
1
24
1 3172
8 9 0 74
0 41573 0 659857
5
5
. .
.
. .
L E
q = [ ]×
× −
0 000517 3 64083 05 0 659857
. . .
L E
q = −
1 243072163361630 08
.
4. Expected number of customers in a system
L L
s q
= +
λ
µ
L L
s q
= +
λ
µ
L E
s = +
−
1 243072163361630 08
0 74
1 78
.
.
.
=0.43529413 customers
5. Expected waiting time of a customer in the
queue
W
s s
P
L
q
s
q
=
−
( )
−
( )
=
1
1 5 0
!
λ
µ
λµ
µ λ λ
W
E
q =
−
1 243072163361630 08
0 74
.
.
Wq
= 1.679827247785980E-08 min
6. Expected waiting time that a customer spends
in a system
W W
s q
= +
1
W E
s = +
−
1 679827247785980 08
8
1
1 7
.
.
Ws = 0 562524627
. min
ρ
λ
µ
=
s
7. The probability Pn
of n customers in the
queuing system is given by
Pn
P n s
n
P n s
n
n s
n
=
−
≤
s!s
;
!
;
0
0
where ρ
λ
µ
=
s
(2)
Probability when n 5, n = 0, 1, 2,…5.
At n = 0
P
n
P
n
n
= = ( ) =
!
( )
!
.
. .
0
0
0 262
0 659857 0 659857
0
At n = 1
P P
1
1
0
1
1 1
0 262
0 659857 0 172882534
= = ( ) =
!
( )
!
.
. .
At n = 2
P P
2
2
0
2
2 2
0 262
0 659857 0 045295224
= = ( ) =
!
( )
!
.
. .
At n = 3
P P
3
3
0
3
3 3
0 262
0 659857 0 011867349
= = ( ) =
!
( )
!
.
. .
At n = 4
P P
n
4 0
4
4 4
0 262
0 659857 0 003109245
= = ( ) =
!
( )
!
.
. .
At n = 5
P P
5
5
0
5
5 5
0 262
0 659857 0 000814622
= = ( ) =
!
( )
!
.
. .
The probability when n5 i.e. n= 6, 7,….10
At n = 6
P
s s
P P
n
n s
6 0
6
6 5 0
6
5 5
5 5
0 262
0 659857 0 000814622
= = =
( ) =
− −
! !
( )
!
.
. .
At n = 12
P
s s
P P
n
n s
2 0
12
12 5 0
12
7
12 12 12 5
0 262
0 659857 0 00
= = =
( ) =
− −
! !
( )
!
.
. . 0
0814622
8. Akande, et al.: Application of Multi-Server Queue Model (m/m/c) for Waiting Lines Management In Banking System
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8. The probability that on arrival a customer
must wait for service:
P
s
s
s
P
s
s
=
−
1
0
!
λ
µ
µ
µ λ
P P
s =
×
× −
1
5
0 74
1 78
5 1 78
5 1 78 0 74
5
0
!
.
.
.
. .
Ps =
×
× −
×
1
120
0 74
1 78
5 1 78
5 1 78 0 74
0 659857
5
.
.
.
. .
.
Ps
= 9.2488E-07
9. Busy time of the system:
To calculate the busy time of the machine, we
multiply the banking hours of the ATM machines
used by utilization factor, that is,
B = Banking hours of ATM ×
λ
µ
s
B =100.262 = 2.62 h
10. Idle time of the system:
To calculate the idle time of the machine, we
subtract busy time from banking hours of theATM
I = Banking hours of ATM – Busy time
I = 10–2.62 = 7.38 h.
Figure 2 disclosed the probability of n customer
in the system.
Table 2showsthesummaryofvaluesofparameters
and queue formula.
Discussion of Result
Considering the analytical solution, the capacity
of the system under study is 1617 customers and
the arrival rate is 0.74 while the service rate is
1.78. This shows that the service rate of the system
is greater than the arrival rate, this implies that
customers would not have to queue up so much
waiting to be served. Probability that the servers
are idle is 0.6599 which implies a probability of
65.99% idle server and 34.01% busy server and
the utilization factor or traffic intensity is 0.262.
The expected number in the waiting line is 0.0904.
The expected number in the system is 0.43529413.
The expected waiting time in the queue is
1.679827247785980E-08 min and the expected
waiting time in the system is 0.55524627 min.
CONCLUSION
The performance level of the Union Bank ATM
has been effectively investigated using the M/M/S
queuing model. It was observed that the busy time
of the machine is 2.62 hours while the idle time is
7.38 h in the 10 h of banking time which attributed
to available of much server in the system. The
utilization factor is 0.262 or 26.2% shows that the
service delivery of the machine is very efficient
and there is no urgent need for an additional server.
RECOMMENDATION
Based on the conclusion of the study, it is
recommended that banks should consider the use
Table 2: Summary of values of parameters and queue
formula
Probability Value
Arrival rate λ 0.737
Service rate µ 1.776
Utilization factor ρ 0.262
Expected number of customers in system Ls
0.43529413
Average Length of queue Lq
1.2430721336130E‑08
Expected waiting time the in system Ws
0.55524627
Expected waiting time the in queue Wq
1.679827247785980E‑08
Probability of zero customers in the system P0
0.43529413
Probability that customers must wait for
service on arrival Ps
9.2488E‑07
Figure 2: Probability of n customer in the system
Figure 1: Design queuing system model for the study
9. Akande, et al.: Application of Multi-Server Queue Model (m/m/c) for Waiting Lines Management In Banking System
AJMS/Jan-Mar-2022/Vol 6/Issue 1 83
of queue model to test and control waiting lines in
the ATMs.
REFERENCES
1. Weiss E, Tucker C. Queue management: Elimination,
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