INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,  
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)  
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XIV, Issue XII, December 2025  
Modeling Customer Service Delivery in Banks Using Queuing  
Theory.  
1Abubakar Muhammad Auwal., 1Chaku Shammah Emmanuel., 1Saleh Ibrahim Musa, 2 Tiletswen  
Timothy Terhemba  
1 Department of Statistics, Nasarawa State University, Keffi. Nasarawa State. Nigeria.  
2 Department of Statistics, Akawe Torkula Polytechnic, Makurdi. Benue State. Nigeria  
DOI : https://doi.org/10.51583/IJLTEMAS.2025.1412000047  
Received: 16 December 2025; Accepted: 23 December 2025; Published: 02 January 2026  
ABSTRACT  
Efficient customer service delivery in banking operations is critical for maintaining customer satisfaction and  
resource utilization. Queuing theory is an approach to analyzing waiting lines, it provides a robust framework  
for modeling and improving service delivery in banks. This research explores the application of queuing theory  
to model customer service delivery processes in banks, focusing on service efficiency, customer wait times, and  
resource allocation. The data for this research was taken for a period of four weeks, covering all work days from  
Mondays to Fridays and hours from 8am to 4pm. The data analysis was done with the aid of EXCEL package  
for descriptive statistics and TORA for optimization system. From the performance measures researched, it  
shows that increasing the teller points to 3 would reduce the waiting time in the queue and system to  
0.01481hours (53.32 seconds) and 0.05829hours (3.50 minutes) respectively as against the present situation  
where each customer has to wait in the queue and system for 0.30095hours (18.06 minutes) and 0,34443hours  
(20.67 minutes) respectively. This will result to each teller being busy for 62.3% of the time while remaining  
idle for 37.7% of the time.  
Keywords: Multi-Server, Service efficiency, Waiting times, Banking Operations, Resource Optimization,  
Queue length and Arrival Rate.  
BACKGROUND TO THE STUDY  
A good number of establishments encounter queues in their effort to offer certain services to their customers.  
These establishments include large organizations such as banks, hospitals, post offices, super markets, fuel  
stations etc, to smaller scenarios such as students queuing up to submit assignments. The different establishments  
vary in scope and complexity but they all consist of a set of activities and procedures that require queuing, in  
which a customer or client must undergo in order to receive the needed services. The resources (or servers) in  
these systems (queuing system) include trained personnel and specialized equipment required for effective  
service delivery. Often, customers get to these servers to receive the needed services only to find that they are  
not attended to as soon as they get there due to one reason or the other. This causes the customers to wait for the  
service delivery for an unknown period of time.  
According to David (1985), waiting is frustrating, demoralizing, agonizing, aggravating, annoying, time  
consuming and incredibly expensive. The truth of this assertion cannot be denied. There should be just a few  
consumers of services in modern society who have not felt, at one time or another, each of the emotions identified  
above. These experiences can also attest to the fact that the waiting line experience in a service facility  
significantly affects customers overall perceptions of the quality of service provided. Once customers are being  
served, their transaction with the service organization may be efficient, courteous and complete; but the bitter  
taste of how long it took to get attention pollutes the overall judgment that they make about the quality of service  
and leaves a very negative impression.  
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Increasing criticisms, cost pressure and increasing demand of quality and efficiency from highly aware and  
educated customers have started putting more pressure on the many organizations urging them to improve on  
the quality of service they offer. The urge to study queues is prompted by two obvious features. Owing to the  
ebb and flow of customers, there would be some occasions when the service facility is not fully employed, i.e  
where there are more servers and fewer customers such that the servers wait idly for a period of time and others  
where it is under continuous pressure, with a long queue of customers waiting to be attended to. Costs are  
involved when the service is under-employed (low productivity), and in the congested period, loss of productive  
time for queuing members. According to Singh (2006), if the organization decides to increase the level of service  
provided, cost of providing service would increase, if it decides to limit the same, costs associated with waiting  
for service would increase. So the manager has to balance the two costs and make a decision about the provision  
of optimum level of service. Hence it is one of the tasks of queuing theory to try to see how these costs can be  
reduced by modifications to the mechanics of the system.  
Statement of the Problem  
This study is important because Benysta microfinance Bank, Makurdi has always experienced failure in terms  
of customer satisfaction due to the population of customers who constantly need different services to be rendered.  
This sometimes leads to chaos in the banking hall. Therefore, modeling customer service delivery in banks using  
queuing theory is relevant.  
Aim and Objectives of the Study  
The aim of this dissertation is to model customer service delivery using queuing theory to the operations of  
Benysta Microfinance Bank, Makurdi. The specific objectives of this dissertation are to:  
1. Determine the average service rate and intensity of the bank per unit time.  
2. Determine the average number of transactions in a given time and compute the average time a customer  
spends for a transaction.  
3. Determine the average waiting time a customer waits in the system and the utilization services of the  
bank at different number of server.  
4. Find the optimal number of server for the transactions in the system.  
Significance of the Study  
Overcrowding and long waiting lines in many organizations are a common occurrence. Any arrangement that  
alters the system and reduces waiting time would go a long way to reduce customer’s anxiety as they wait to get  
services. The findings would be useful to systems which make use of queue in a bid to reduce their waiting  
times. e.g. Banks, IT Centres, Hospitals, telecommunication service providers etc. In addition to this, if this  
arrangement helps in minimizing cost to the establishment, it would have improved on the smooth running of  
the organization and increase system’s revenue. Thus results of this research would provide innovations that  
would assist organizations which make use of queue to improve on their services while minimizing costs.  
Conclusively, the study will guide Benysta Microfinance Bank and related areas in making policies that affect  
the management of queues especially when customers are in season. This will in turn create efficiency and  
effectiveness in services rendered to customers.  
Scope of the Study  
This work focuses on the operations of Benysta Microfinance Bank, Makurdi. Customers come in and queue up  
to make transactions. There is usually more than one server available and customers queue up waiting to go in  
and consult on a first come first served (FCFS) basis. The data is a primary data and was collected over a period  
of four weeks and covers all the working days from Mondays to Fridays during the peak farming season. The  
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scope of this project is limited to Benysta Microfinance Bank and any other related banking institution in terms  
of services; the data is of a primary source for an observational period.  
METHODOLOGY  
Research Design  
The Poisson and Exponential processes are used to model the arrival rates and service rates. This is due to the  
following reasons:  
1. There is a certain amount of regularity in the arrivals of customers in the Bank. Although individual  
arrivals are impossible to predict, there is perhaps some statistical regularity in the sense that when we  
observe customers arrivals during a period of say one month, without knowing the absolute time frame,  
then we have no way to decide whether we observe the time period of January, February or June. In  
probability terms, the process is stationary in time. In other words, the course of time should not change  
the probability properties of the process.  
2. The fact that there is an occurrence at the particular time, says nothing about the probability of an  
occurrence at or around a later or earlier time. In other words, there seems to be some kind of  
independence with respect to various occurrences.  
3. The next occurrence cannot be predicted from past or current information. In other words, the process of  
occurrences seems to have no memory. The fact that something happened in the past has no effect on the  
probabilities for future occurrences.  
4. There is no accumulation of occurrences at anytime. That is, in each finite time interval, there are only  
finitely many occurrences.  
Data Collection  
For the purpose of this research work, the primary source is used. The data was obtained from Benysta  
Microfinance Bank, in Makurdi, Benue State, Nigeria. The Bank opens for services at 8:00am and closes at  
4:00pm from Mondays to Fridays.  
The data that is used for this research work is of primary source, it was obtained for four weeks and the collections  
were done from 8:00am to 4:00pm each day. The arrival time and the number of customers serviced per period  
were duly recorded by observations.  
Models and Technique for Analysis  
Queuing Model Specification  
According to John (2010), Queuing theory modelling is classified by using special or standard notations  
described by D.G. Kendell in 1953 in the form of (a/b/c). Later, A.M. Lee added the symbols d and e to the  
Kendell notation. In the literature of queuing theory, the standard format used to describe queuing models is as  
follows:  
Inputs  
Service  
Facility  
Outputs  
Fig 2.1: Component of a basic queuing system  
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Departure  
Server  
Queue  
Arrival  
Fig 2.2: A multiple server queuing system, M/M/C  
{(a/b/c): (d/e)}  
Where a = arrival distribution  
b = service time distribution  
c = number of servers (service channels)  
d = capacity of the system (queue plus service)  
e = queue (or service discipline)  
In place of notation a and b, the following descriptive notations are used for the arrival and service times  
distribution:  
M = Markovian (or exponential) inter-arrival time or service-time distribution.  
D = Deterministic (or constraint) inter-arrival time or service topic.  
G = General distribution of service time, i.e. no assumption is made about the type of distribution with means  
and variance.  
Gl = General probability distribution normal or uniform for inter-arrival time.  
Ek= Erlang-k distribution for inter-arrival or service time with parameter k (i.e. if k = 1, Erlang is equivalent to  
exponential and if k = ∞, Erlang is equivalent to deterministic)  
Method of Analysis  
The analysis is done with the aid of EXCEL Package for descriptive statistics (averages) and TORA for  
Optimization System.  
푘
휆
휇
휆휇 ( )  
휆
휇
퐿푠 =  
2 푃0 +  
(2.1)  
(
) ( )  
ꢀ − 1 ! ꢀ휇 − 휆  
퐿푠  
푊 =  
푠
(2.2)  
휆
푘
휆
휆휇 ( )  
휇
퐿푞 =  
2 푃0 표푟 퐿푞 = 퐿푠 − 휌  
(2.3)  
(
) ( )  
ꢀ − 1 ! ꢀ휇 − 휆  
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푘
휆
휇 ( )  
휇
퐿푞  
휆
1
휇
푊 =  
푃0 표푟 푊 = 푊 −  
=
(2.4)  
푞
푞
푠
2
(
)( )  
ꢀ − 1 ꢀ휇 − 휆  
ג
휌 =  
(2.5)  
휇푐  
Where,  
Ls is the expected number of customers in the system at any point in time  
푊 is the average time spent on queue  
푞
Lq is the average queue length  
Ws is the expected time spent in the system  
휌 is the utilization of server in the bank  
휆 is the average arrival  
휇 is the average service rate  
푃0 is the Initial Probability i.e the probability that no customer at the system  
휌 is the rho (utilization)  
Economic Analysis  
The two basic types of costs associated with queuing systems are the costs involved in operating each service  
facility like the costs for equipment (including maintenance), materials, labor, etc. These cost increases as the  
number of service facilities put into operation increases and the opportunity costs associated with causing  
customers to wait in the system. . The total of these two basic types of costs goes to a minimum at some specific  
number of facilities. This then is the optimum number of service facilities which should be operated by the  
manager- optimum because it minimizes the total cost of both operating the service facilities and waiting time  
in the system. The total cost model includes the cost of waiting and the cost of service:  
푇퐶 = 퐶푤퐿푠 + 퐶푠ꢀ  
where:  
Cw= the waiting cost per time period for each customer  
Ls=average number of customers in the system  
Cs= the service cost per time period for each channel  
k = the number of channels  
TC= the total economic cost per time period  
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RESULTS AND DISCUSSION  
Summary of inter-arrivals time of customers for four weeks between the intervals of one hour.  
Inter-Arrival  
Time/hour  
Days/week  
1
2
3
4
5
Total Mean  
8:00 – 9:00  
9:00 – 10:00  
10:00 – 11:00  
11:00 – 12:00  
12:00 – 1:00  
1:00 – 2:00  
2:00 – 3:00  
3:00 – 4:00  
Total  
74  
108  
87  
63  
61  
57  
39  
38  
78  
67  
86  
52  
67  
63  
40  
40  
103  
68  
86  
78  
72  
27  
32  
32  
126  
72  
64  
72  
31  
39  
31  
38  
58  
109  
76  
74  
44  
55  
52  
68  
2527  
63.18  
8:00 – 9:00  
9:00 – 10:00  
10:00 – 11:00  
11:00 – 12:00  
12:00 – 1:00  
1:00 – 2:00  
2:00 – 3:00  
3:00 – 4:00  
Total  
82  
89  
75  
87  
53  
73  
46  
19  
107  
88  
57  
97  
84  
0
100  
78  
88  
73  
77  
58  
68  
69  
52  
48  
63  
100  
96  
65  
63  
67  
0
121  
67  
53  
65  
54  
0
51  
13  
28  
2574  
64.35  
8:00 – 9:00  
9:00 – 10:00  
10:00 – 11:00  
11:00 – 12:00  
12:00 – 1:00  
1:00 – 2:00  
13  
2
18  
16  
0
29  
47  
26  
62  
0
13  
50  
2
78  
183  
112  
72  
24  
7
21  
16  
11  
0
17  
11  
4
19  
6
17  
16  
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2:00 – 3:00  
3:00 – 4:00  
Total  
6
1
22  
0
69  
1
5
1
12  
105  
1114  
27.85  
8:00 – 9:00  
9:00 – 10:00  
10:00 – 11:00  
11:00 – 12:00  
12:00 – 1:00  
1:00 – 2:00  
2:00 – 3:00  
3:00 – 4:00  
Total  
16  
38  
22  
19  
1
1
12  
47  
0
0
1
3
21  
53  
20  
24  
24  
22  
74  
26  
35  
14  
0
28  
44  
53  
21  
25  
16  
31  
3
13  
18  
12  
4
0
6
3
6
0
756  
18.9  
Mean arrival rate/hour (흀) = 43 customers  
Summary of inter-service time of customers for four weeks between the intervals of one hour.  
Inter-Arrival  
Time/hour  
Days/week  
1
2
3
4
5
Total Mean  
8:00 – 9:00  
9:00 – 10:00  
10:00 – 11:00  
11:00 – 12:00  
12:00 – 1:00  
1:00 – 2:00  
2:00 – 3:00  
3:00 – 4:00  
Total  
26  
35  
42  
46  
30  
22  
32  
11  
33  
42  
34  
59  
44  
31  
25  
25  
0
36  
0
40  
32  
41  
60  
35  
36  
43  
17  
31  
57  
53  
33  
14  
31  
16  
27  
42  
24  
37  
15  
23  
1280  
32  
8:00 – 9:00  
9:00 – 10:00  
10:00 – 11:00  
36  
29  
41  
28  
19  
33  
3
22  
34  
46  
13  
2
8
38  
47  
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11:00 – 12:00  
12:00 – 1:00  
1:00 – 2:00  
2:00 – 3:00  
3:00 – 4:00  
Total  
48  
53  
48  
33  
31  
38  
51  
0
61  
47  
44  
42  
15  
53  
57  
72  
47  
21  
25  
53  
41  
21  
27  
0
0
1327  
33.18  
15.05  
10.7  
8:00 – 9:00  
9:00 – 10:00  
10:00 – 11:00  
11:00 – 12:00  
12:00 – 1:00  
1:00 – 2:00  
2:00 – 3:00  
3:00 – 4:00  
Total  
29  
0
21  
0
6
13  
13  
17  
19  
6
28  
14  
22  
22  
12  
12  
32  
105  
25  
24  
23  
0
0
22  
26  
0
0
26  
15  
14  
1
0
17  
12  
1
16  
0
14  
5
12  
602  
8:00 – 9:00  
9:00 – 10:00  
10:00 – 11:00  
11:00 – 12:00  
12:00 – 1:00  
1:00 – 2:00  
2:00 – 3:00  
3:00 – 4:00  
Total  
6
15  
18  
16  
0
0
0
15  
19  
2
13  
21  
14  
17  
21  
3
0
13  
22  
26  
12  
16  
0
0
0
14  
0
0
15  
21  
29  
21  
8
0
13  
13  
3
22  
0
0
428  
Mean service rate/hour (흁) = 23 customers  
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Performance Measures of Multiserver Queuing Model at the Benysta Microfinance Bank  
Scenario  
C
2
3
4
5
Lambda  
Mu  
L'da eff  
P0  
Ls  
Lq  
Ws  
Wq  
1
2
3
4
43.00000 23.00000 43.00000 0.03371 14.81061 12.94104  
0.34443  
0.05829  
0.04641  
0.04413  
0.30095  
0.01481  
0.00293  
0.00065  
43.00000 23.00000 43.00000 0.13320  
43.00000 23.00000 43.00000 0.15010  
43.00000 23.00000 43.00000 0.15339  
2.50628  
1.99546  
1.89741  
0.63671  
0.12589  
0.02785  
Summary analysis of the Multiserver Queuing Model of Benysta Microfinance Bank  
Performance Measure  
Arrival rate (λ)  
Service rate (µ)  
System Utilization (p)  
Ls  
2 Tellers  
43  
3 Tellers  
43  
4 Tellers  
43  
5 Tellers  
43  
23  
23  
23  
23  
93.5%  
62.3%  
2.50628  
0.63671  
0.05829  
0.01481  
13.32%  
N550.63  
46.7%  
1.99546  
0.12589  
0.04641  
0.00293  
15.01%  
N599.55  
37.4%  
1.89741  
0.02785  
0.04413  
0.00065  
15.339%  
N689.74  
14.81061  
12.94104  
0.34443  
0.30095  
3.371%  
N1681.06  
Lq  
Ws(Hours)  
Wq(Hours)  
P0  
Total System Cost/hr  
From the queue performance measures, increasing the number of teller points to 3 indicates that the waiting time  
in the queue and system would reduce to 0.01481hours (53.32 seconds) and 0.05829 hours (3.50 minutes)  
respectively as against the present situation where each customer has to wait in the queue and system for 0.30095  
hours (18.06 minutes) and 0.34443 hours (20.67 minutes) respectively. As a result of this, each teller will be  
busy for 62.3% while the remaining 37.7% of the time would be idle. Furthermore, the total economic cost will  
also decrease from N1681.06 with 2 tellers to N550.63 with 3 tellers which is economically optimal.  
From the analysis, it can be observed that the number of tellers necessary to serve customers in the case study  
of Benysta Microfinance Bank, Makurdi is 3 teller points. This has been proven in the tables above, it is the  
appropriate number of tellers that can serve the customers as at when due without waiting for long before  
customers are been served at the actual time necessary for the service.  
FINDINGS  
From the results, the Bank is expected to increase the service facility because the utilization factor is very high  
at 2 servers. The results further showed that:  
1. The mean rate is 43 customers/hour and the mean service rate is 23 customers/hour.  
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2. The average number of customers in the system at any point in time in the scenario 1, 2, 3 and 4 are  
14.81061, 2.50628, 1.99546 and 1.89741. i.e 15 customers, 3 customers, 2 customers and 2 customers  
respectively.  
3. The average queue length for the scenario 1, 2, 3 and 4 are 12.94104, 0.63671, 0.12589 and 0.02785. i.e  
there will be 13 customers in the queue at first scenario, 1 customer at the second scenario while no  
queues at the third and fourth scenario.  
4. The average time spent in the system at scenario 1, 2, 3 and 4 are 0.34443x60 = 20.67 minutes,  
0.05829x60 = 3.50 minutes, 0.04641x60 =2.78 minutes and 0.04413x60 = 2.64 minutes.  
5. The expected time spent in the system by customer for the scenario 1, 2, 3 and 4 are 0.30095x60 = 18.06  
minutes, 0.01481x60 = 53.31 seconds, 0.00293x60 = 10.54seconds and 0.00065x60 = 1.04 seconds  
respectively.  
6. The probabilities that there are no customers in the system are 0.03371, 0.13320, 0.15010 and 0.15339  
respectively for the scenarios.  
7. The utilization at scenario 1, 2, 3 and 4 was calculated to be 0.93478 = 93.48%, 0.62319 = 62.32%,  
0.46739 = 46.74% and 0.15339 = 15.34% respectively; this showed that the Bank was below efficiency  
at the first scenario and also showed that when there is increase in the server there will be increase in the  
work efficiency and satisfying the customer’s needs.  
8. The total system cost for the scenarios are N1681.06, N550.63, N599.55 and N689.74 respectively. That  
indicates that, the optimal server for the Bank is 3.  
SUMMARY AND CONCLUSION  
Summary  
The situation or conditions experienced in this bank during the period of study are similar to what operates in  
other banks in the country. Excessive waste of time in the banking hall would have a negative impact on the  
economy of the country in terms of the opportunity cost, which is the excess time that would have been used  
elsewhere. Also from the M/M/C model, we see that there is a drop in waiting time as more servers are added.  
There is also a drop in queue length, this would probably come with some degree of happiness and satisfaction  
to the customers, but heavy cost would be incurred by the Bank if they have to employ more servers for the  
operation unit of the bank. For this study, given the queue characteristics, the optimal number of servers which  
would minimize the operational cost is five for this particular establishment with time saving and the reduction  
of operational cost as the only factors considered.  
Conclusion  
From the findings, we conclude that the number of customers patronizing the bank was very higher at first and  
second weeks; this is because of the planting season during which some customers need money to make certain  
purchases and others come in to deposit money from sales made. Providing customers with timely access to the  
needed services while minimizing cost to the establishment is one of the major goals of every organization.  
Generally, an excess of demand over supply i.e. more customers than servers causes waiting. Since we obviously  
have more customers than servers, the complete elimination of waiting line is impossible; we only try to  
minimize the waiting time. With the results obtained from this study, we see that the use of three servers can  
help to improve the operations of Benysta Microfinance Bank. Finally, we advice that a similar study should be  
carried out for systems which may have different service times or service rates in order to ascertain if pooling  
could also be beneficial to such systems as this work is limited only to the Operation Unit of Benysta  
Microfinance Bank, Makurdi, Benue State, Nigeria.  
www.ijltemas.in  
Page 524  
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,  
MANAGEMENT & APPLIED SCIENCE (IJLTEMAS)  
ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XIV, Issue XII, December 2025  
REFERENCES  
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