### O.D. Adubisi, Dr. E. Otuonye, Dr. J. Ohakwe

**Abstract**

*This study applies queuing theory to assess the utility function of the ATM service delivery. In other to achieve this, the basic characteristics of the case ATM as a queuing system were defined. The primary source of data collection was adopted for this study and is limited to Ecobank plc, Okigwe records on direct observation of the number of customers including each customer’s arrival and service time respectively, during the peak hours of 8:00am to 2:00pm for a period of ten (10) working days. The study revealed that the service pattern did not follow the exponential distribution as described by queue theory which lead to the application of Pollaczek-Klintcline formula. The M/G/1 queuing model best described the ATM queuing system. The study further revealed that the traffic intensity of the system is 0.521 (52.1%), which implied that the system was operating under the steady-state condition. The average waiting time in the system and in the queue are 6.1 and 3.2 minutes which implies that on arrival customers are expected to wait in the system for approximately six (6) minutes at the ATM Service facility.*

**Keywords:**

Bank ATM; M/G/1 model; Queuing theory; Waiting time; Service rate; Arrival rate; Pollaczek-Khintchine formula

**Introduction**

Every business organization faces the problem of waiting line or queue, irrespective of the kind of service they offer to the public. Queuing theory had its beginning in the research study of a Danish engineer named A.K Erlang in which he experimented with fluctuating demand in telephone traffic (Hillier and Lieberman, 2015; Taha, 2007). Queuing theory is the mathematical study of waiting lines. The theory enables mathematical analysis of several related processes, including arriving at the back of the queue, waiting in the queue and being served by the service facility server(s) at the front of the queue (Taha, 2007) while (Murthy, 2007) stated that queuing theory is the present system of tying a belt with time to the hands of a customer.

Given the rapid growth in population and the advancement in technology, queuing theory has become very helpful in so many areas such as bank management. In queuing theory, a queuing model is used to approximate a real queuing situation, so that the queuing behaviour can be analyzed mathematically. The automated teller machine (ATM) is a computerized telecom device designed to provide effective and efficient financial services to bank customers at the shortest possible time but due to variation in arrival and service time, customers still have to wait for a long period of time before they are served which eventually leads to queue formation (Bakari et.al, 2014).

**1.1 Statement of the problem**

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 waiting line. Hence it becomes pertinent that a study of this kind be conducted in order to assist the bank management in making certain decisions in an effort to minimize the time customers spend waiting in line at the ATM service facility.

**Queuing theory**

Queuing theory is the study of waiting lines. It uses queuing models to represent the various types of queuing system (systems that involves queues of some kind) that arise in practice (Hillier and Lieberman, 2015). Therefore, queuing theory is used extensively to analyze any system exhibiting random variability in terms of arrival and service times. According to (Bakari et.al, 2014), it also provides techniques for maximizing a system capacity to meet the service demand to drastically reduce the waiting time.

**2.1.1 Queuing system**

A queue system is a birth-death process with a population consisting of customers either waiting for service or currently in service. A birth occurs when there is an arrival at the service facility while a death occurs when there is a departure from the facility. A queuing system can best be described by customer’s arrival pattern, the service pattern (service mechanism), and the queuing discipline as well as by the customer’s behaviour. The assumptions of the birth-and-death process indicate that probabilities involving how the process will evolve in the future depend only on the current state of the process, and so are independent of events in the past (Hillier and Lieberman, 2015).

**2.1.2 Little’s Queuing formula**

It is impossible to deal with queuing systems without mentioning and introducing the Little’s law. In order to determine the waiting time and the size of a queue for a particular system, the Little’s law is normally used. John D. C. Little provided the first rigorous proof of, this equation sometimes is referred to as Little’s formula. (Moshe, 2013), Consider a system to which continuously and indefinitely arrivals come, stay for a while, and then leave. In particular, nobody gets stuck there for good. Let be the limit average rate of arrivals (which of course coincides with the limit departure rate), let be the limit average time spent in the system per arrival, and let L be the limit average number in the system in a steady state condition. Assume that all these limits exist. Hence, the Little’s equation can be written as (System) (1)

(Queue) (2)

**2.1.3 Queuing Discipline**

Queues are a part of everyday life events. We all wait in lines or join a queue to buy a movie ticket, make a bank deposit, pay for groceries, mail a package, obtain food in a cafeteria, and start a ride in an amusement park. The order in which arrivals to these queues are selected for service is what we termed queuing discipline. The most common queuing discipline is the first come, first served (FCFS) as defined in this study; other disciplines include last come, first served (LCFS) and the service in random order (SIRO). Individuals in the queue may also be selected based on some order of priority. It is important to express that the queuing discipline chosen by the service facility would invariably affect the waiting time of customers in the system (Hillier and Lieberman, 2015).

**2.2 Queue Characteristics**

(Murthy, 2007) stated that queuing systems can be described by four (4) basic elements. These elements are: the arrival pattern, queue discipline, capacity of the service system and the customer’s behaviour. According to (Taha, 2007), a queuing system is characterized by five (5) basic elements which includes: the customer’s arrival pattern, the system service pattern, the numbers of servers in the service facility, capacity of the service system and the queuing discipline.

**2.2.1 Source Population**

It represents the set of customers that are expected to patronize a service facility. It is from the source population that customers are generated to join the queue. The source is said to be finite, if it has a limit on the number of customers it will accommodate while it is infinite if there is no limit on the number of customers. The number of arrivals over a specified time interval follows a Poisson distribution with mean, that is

**2.2.2 The Arrival Pattern**

It represents the pattern in which customers arrive and enter the system from a calling population. It is most times referred to as the input process of the system. The inter-arrival time is used most times to specify the input process (which is the time between consecutive arrivals to the service facility). This can be deterministic or it may be a random variable whose probability distribution is assumed to be exponentially distributed. The arrival process is also known to follow the Poisson process, specifically the Poisson distribution. It is also of interest to know whether customers arrive in a single form or in batches and whether balking, reneging or jockeying is permitted.

**2.2.3 The Service Pattern**

The service pattern is usually specified by the service time of the system (time required by a server to serve a customer). The duration of service might depend on the number of customers already in the service facility or it might be independent of the state of the system. The service time may be deterministic or it may be a random variable whose probability distribution is assumed to be exponentially distributed or not. It is also important to know whether customers are attended to completely by one server or by a sequence of servers. This study assumes that customers in the defined queuing system are completely serviced by one single server.

**2.2.4 Service Mechanism**

This represents the configuration of the service facility. The service mechanism consists of one or more service facilities, each of which contains one or more parallel service channels, called servers. If there is more than one service facility, the customer may receive service from a sequence of these service channels in series or enter one of the parallel service channels and is completely serviced by that server. A queueing model must specify the arrangement of the facilities and the number of servers (parallel channels). The time elapsed from the commencement of service to its completion for a customer at a service facility is referred to as the service time (or holding time). The service time distribution that is most frequently assumed in practice is the exponential distribution (Hillier and Lieberman, 2015). There are five (5) basic service structures, which includes:

- Single server with a single queue.
- Single server with multiple queues.
- Multiple servers with a single queue.
- Multiple servers with a multiple queues.
- Multiple servers in series with a single queue.

**2.2.5 System Capacity**

(Taha, 2007), states that the capacity of a system is the maximum number of customers in service and in the queue that are permitted into the service facility. The system is said to have an infinite capacity if there is no limit on the number of customers permitted into the service facility while a finite capacity system is said to have a limit on the units of customers allowed into the service facility. It is also important to note the physical arrangement of the service facility which could be a single channel or multiple channel service facility.

**Material and Methods**

The purpose of this study is to examine the performance characteristics (measures) of the EcoBank plc, Okigwe ATM service unit during banking peak hours of the day. The characteristics of interest in this research that would be examined includes: the utilization factor of the system, the arrival rate (The number of customers arriving to the ATM service unit at a given time), the service time (The time it takes the system to attend to a customer), the expected number of customers in the system and in the queue, the waiting time in the system and in the queue including building a model to determine the probability of having a certain number of customers in the system.

**3.1.1 Data collection**

The method of data collection adopted for this study is the primary source and it is limited to information gathered from EcoBank plc, ATM service unit located in Okigwe, Imo state. The data was collected by direct personal observation in which the number of customers arriving at the ATM service facility was recorded as well as each customer’s arrival and service time respectively. The period of data collection was during the peak hours of 8:00 am to 2:00 pm for a period of ten (10) working days. The M/G/1 queuing model was used to analyze the collected data based on the arrival and service patterns as well as taking cognizance of the assumptions of queuing theory with the help of Minitab statistical analysis tool.

**3.1.2 Statistical analysis technique**

The statistical methods used are: the simple frequency table for finding the mean and variance of the arrival and service rates. The Chi-square goodness of fit test is used to test the validity of the hypotheses concerning the arrival and service rates in the study using the observed and theoretically expected frequencies.

(3)

**3.1.3 Pollaczek-Khintchine theory**

The theory states that queues which the arrival process follow the Poisson distribution (with arrival rate) and no restriction on the service distribution are complex with the service time represented by any probability distribution with mean and variance.

The derivation of the Pollaczek-Khintchine formula follows from the results obtained by John, D.C. Little in his proof that:

(4)

(5a)

(5b)

Where:

Delay in queue of a customer arriving at a random instant of time.

Expected values of A.

Time required to complete the service of a customer in service at a random instant of time.

Expected value of B.

Time to service the customers in queue at a random instant of time.

Expected values of C.

According to (Cox, 1962) ………

(6)

(7)

Substituting (6) and (7) into (5b), solving for (Waiting time in the queue).

(8)

(9)

This equation (9) is commonly referred to as the Pollaczek-Khintchine formula, named after two pioneers who derived the formula in the early 1930’s. The other measures of performance are estimated from.

**3.2 Queuing models**

Queuing models are used to approximate a real queuing situation, so that the queuing behaviour can be analyzed mathematically. Although, there are several queuing models, (Taha, 2007) highlighted the following models:

The single server queuing system is the model considered in this study.

**3.2.1 Assumptions made on the system**

In order to make the problem in this study mathematically tractable, the following queuing system assumptions are expressed:

- The arrival is completely random and occurred in a single form, neither balking nor reneging occurred.
- The source population is infinite.
- The arrival process (inter-arrival times) and the service times are independent and identically distributed.
- The First Come First Serve (FCFS) queue discipline was employed.
- The single server channel and the capacity of the service system is infinite.

**3.3 ATM model (M/G/1 Queuing model)**

The queuing system model assumes that the queuing system has a single server, FCFS queuing discipline, infinite source population, infinite system capacity and a Poisson input with a fixed mean arrival rate () as well as customers having independent service times with the same probability distribution. However, no restrictions are imposed on what the service time distribution should be (Hillier and Lieberman, 2015). The only basic requirement is to know (or estimate) the mean and variance of this distribution. For an M/G/1 queuing system, the utilization level (traffic intensity) is defined as, (Steady-state condition).

**3.3.1 Measures of performance**

For the queuing system model, the Pollaczek-Khintchine formula will be used to find the basic measures of effectiveness of the system.

- The mean arrival rate
- The mean service rate
- Traffic intensity: (10)
- Probability of an idle system.

(11)

- Expected number of customers in the system.

(12)

- Expected number of customers in the queue.

(13)

- The average waiting time in the system (Includes service time).

(14)

- The average waiting time in the queue (Excludes service time).

(15)

- Probability of having customers in the system.

(16)

**Results and discussion**

The tables below shows a summary of frequencies for the number of arrivals and the service time from the collected data as depicted in

**appendix A**. The distribution of the number of customers (arrivals) in Table 1 and the service time (duration of service) in Table 2.**4.1 The arrival rate**

The expected arrival rate of customers is given by

Where: = 646, = 60.

Thus, the average arrival rate of customers per hour is

The chi-square goodness of fit test confirmed that the number of arrivals follow the Poisson distribution at 5% sig. level with parameter

**Figure 1: Arrival rate of customers.**

**4.2 Service rate**

The mean service time of customers is given by

Where, . Hence,

Thus, the average service rate of customers per hour is

The chi-square goodness of fit test confirmed that the service time did not follow the exponential distribution at 5% sig. level with parameter

**Figure 2: The service time of customers.**

**4.3 Implementation of formulas**

The utilization level and the measures of effectiveness of the ATM service unit is calculated using the derived formulas in section 3.3.1.

- Traffic-intensity (Utilization level):

- Probability of the idle system (ATM).
- The average number of customers in the system.
- The average number of customers in the queue.

- The waiting time in the system (ATM).

- The waiting time in the queue.
- The probability of having a certain number of customers in the system can be obtained, using this model:

The Kolmogorov-Smirnov (K-S) goodness of fit test was used to confirm the adequacy of the model in predicting the observed probabilities in the queuing system.

The above results revealed the following from the study analysis:

- The service time does not follow the exponential distribution (General Service time) while the number of arrivals followed the Poisson distribution. This suggest that the model with Poisson arrival rates, general service time and a single service channel (one server system) would best describe the ATM queuing system.
- It was determined that the average number of customers (arrival rate) was 10.767 per hour and the average service time (service rate) of customers to be 20.683 per hour.
- The traffic intensity () of the system is 0.521. Otherwise, known as the utilization level to be less than one i.e. (), which suggest that the system operates under the steady-state condition. Hence, it implies that 52.1% of the time during the data collection period that the system was busy as against the idle time of 47.9%.
- The measures of effectiveness for the results, shows the average number of customers in the system to be 1.086 customers (i.e. approx. one (1)) while the average number of customers in the queue to be 0.565 customers (i.e. approx. one (1)). The average waiting time of customers in the system and in the queue was estimated to be 0.101 hours and 0.0526 hours respectively, that is, customers are expected to spend an average of 6.06 minutes before he/she is completely serviced by the ATM system while a customer is likely to spend an average of 3.16 minutes waiting in the queue.

**Conclusion**

This research paper discussed the use of the number of arrivals as against the inter-arrival time to compute the average arrival rate and the use of Pollaczek-Khintchine formula to compute the measures of performance of a system in wake of a non-exponential service time. (Atkinson, 2000), highlighted in his study that an average of one customer is not so uncommon per queue. The study results revealed that though there were practically no long queue in the ATM service facility but the ATM system lacked the efficiency to offer service at a faster rate as shown by the average amount of time customers are expected to spend before being serviced by the ATM system. Hence, the need for the bank management to adjust the mechanism of the ATM system to enable it respond to customers service request at a much more faster speed during banking peak periods.

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