# ATM Queuing

Topics: Arithmetic mean, Week-day names, Queueing theory Pages: 10 (1332 words) Published: September 25, 2013
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Case Study for Bank ATM Queuing Model

Bhavin Patel1 and Pravin Bhathawala2
1Assistant Professor, Humanities Department, Sankalchand Patel College of Engineering, Visnagar, Gujarat, India; 2 Professor & Head, Department of Mathematics, BIT, Baroda, Gujarat, India

Abstract:
Bank ATMs would avoid losing their
customers due to a long wait on the line. The bank initially provides one ATM. However, one ATM would not serve a purpose when customers withdraw to use ATM and try to use other bank ATM. Thus, to maintain the customers, the service time needs to be improved. This paper shows that the queuing theory may be used to solve this problem. We obtained the data from a bank ATM in a city. We used Little’s Theorem and M/M/1 queuing model. The arrival rate at a bank ATM on Monday during banking time is

customer per minute (cpm) while the service rate
is ? ? cpm. The average number of customers
in the ATM is ? ?and the utilization period is
? ?. We discuss the benefits of applying
queuing theory to a busy ATM in conclusion.
Keywords: Bank ATM, Little’s Theorem, M/M/1
queuing model, Queue, Waiting lines
Here, ?is the average customer arrival rate and 𝑇 is the average service time for a customer.

B. ATM Model (M/M/1 queuing model)
M/M/1 queuing model means that the arrival and service time are exponentially distributed (Poisson process ). For the analysis of the ATM M/M/1 queuing model, the following variables will be investigated:  ? The mean customers arrival rate

 ? The mean service rate
 𝜌: ?: utilization factor

 Probability of zero customers in the ATM:
𝑃0 = 1 − 𝜌 (2)
 𝑃?: The probability of having ? customers
in the ATM:
𝑃? = 𝑃0 𝜌? = (1 − 𝜌)𝜌? (3)  𝐿: The average number of customers in the
ATM:
I. Introduction
𝐿 = 𝜌 = ?
(4)
This paper uses queuing theory to study the
1−𝜌
?−

waiting lines in Bank ATM in a city. The bank provides one ATM in the main branch.  𝐿?: The average number of customers in
the queue:
𝜌 2
𝜌
In ATM, bank customers arrive randomly
and the service time i.e. the time customer takes to
𝐿? = 𝐿 × 𝜌 = 1−𝜌 = ?−? (5)
 𝑊?: The average waiting time in the queue:
do transaction in ATM, is also random. We use
𝐿 ?
𝜌
M/M/1 queuing model to derive the arrival rate, service rate, utilization rate, waiting time in the queue and the average number of customers in the 𝑊? = ? = ?−? (6)

 𝑊: The average time spent in the ATM,
including the waiting time:
queue. On average, 500 customers are served on
𝑊 = 𝐿 = 1
(7)
weekdays ( monday to Friday ) and 300 customers
are served on weekends ( Saturday and Sunday )
monthly. Generally, on Mondays, there are more customers coming to ATM, during 10 𝑎. ? ? 5 ? ?.
II. Queuing theory
A. Little’s Theorem
Little’s Theorem describes the relationship between throughput rate (i.e. arrival and service rate), cycle time and work in process (i.e. number of customers/jobs in the system). The theorem states that the expected number of customers (𝑁) for a

system in steady state can be determined using the
following equation:
𝐿 = ?? (1) ? ?−
III. Observation and Discussion
We have collected the one month daily
customer data by observation during banking time, as shown in Table-1.

Table-1 Monthly Customer counts

Sun
Mon
Tue
Wed
Thu
Fri
Sat
1st week
70
139
128
116
119
112
138
2nd week
71
155
140
108
72
78
75
3rd week
70
110
111
83
94
119
113
4th week
40
96
90
87
70
60
70
Total
251
500
469
394
355
369
396

180

160

140

120

100

80

60

40

20

0

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