Math / Data & Statistics

QuestionIn a Bank every 15 minutes one customer arrives for the cheque. The staff in the only payment counter takes 10 minutes for serving a customer on an average. Find (d) Average number of Customer in the bank (e) Average number of Customer in the queue
Average time that a customer spends in the bank

Studdy Solution

STEP 1

1. The inter-arrival time of customers is exponential with an average rate of $\lambda = 4$ customers per hour (since 1 customer arrives every 15 minutes).
2. The service time is exponential with an average rate of $\mu = 6$ customers per hour (since it takes 10 minutes to serve one customer).
3. The system can be modeled using the M/M/1 queue, which means a single server, exponential inter-arrival and service times, and a first-come-first-served discipline.

STEP 2

1. Calculate the utilization factor ( $\rho$ ) of the system.
2. Find the average number of customers in the system ( $L$ ).
3. Find the average number of customers in the queue ( $L_q$ ).
4. Calculate the average time a customer spends in the system ( $W$ ).

STEP 3

Calculate the utilization factor $\rho$ using $\rho = \frac{\lambda}{\mu}$ .
$\rho = \frac{4}{6} = \frac{2}{3}$

STEP 4

Find the average number of customers in the system $L$ using the formula for an M/M/1 queue: $L = \frac{\rho}{1 - \rho}$ .
$L = \frac{\frac{2}{3}}{1 - \frac{2}{3}} = \frac{\frac{2}{3}}{\frac{1}{3}} = 2$

STEP 5

Find the average number of customers in the queue $L_q$ using the formula: $L_q = \frac{\rho^2}{1 - \rho}$ .
$L_q = \frac{\left(\frac{2}{3}\right)^2}{1 - \frac{2}{3}} = \frac{\frac{4}{9}}{\frac{1}{3}} = \frac{4}{3}$

STEP 6

Calculate the average time a customer spends in the system $W$ using Little's Law: $W = \frac{L}{\lambda}$ .
$W = \frac{2}{4} = \frac{1}{2} \text{ hours} = 30 \text{ minutes}$
Solution: (d) The average number of customers in the bank is $L = 2$ . (e) The average number of customers in the queue is $L_q = \frac{4}{3}$ . The average time that a customer spends in the bank is $W = 30$ minutes.

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