Question

Neve Commercial Bank is the only bank in the town of​ York, Pennsylvania. On a typical​ Friday, an average of 8 customers per hour arrive at the bank to transact business. There is one teller at the​ bank, and the average time required to transact business is 3 minutes. It is assumed that service times may be described by the negative exponential distribution. A single line would be​ used, and the customer at the front of the line would go to the first available bank teller. If a single teller is​ used:

a) The average time in the line? = ______ minutes?.

b) The average number in the line? = _______customers?.

c) The average time in the system? = _____minutes? .

d) The average number in the system? = ______customers?

e) The probability that the bank is empty? = ____?(round your response to two decimal? places).

f) CEO Benjamin Neve is considering adding a second teller? (who would work at the same rate as the? first) to reduce the waiting time for customers. He assumes that this will cut the waiting time in half.

g) If a second teller is? added, the average time a customer spends in the queue? = ____ minutes?

h) If a second teller is? added, the average number of customers in the queue? =??____ customers.

I) If a second teller is? added, the average time a customer spends in the system? = _____minutes

j) If a second teller is? added, the average number of customers in the system? =?? customers?

Answer 1

Answer)

a)

Given that, the mean arrival rate of customers = average number of customers arriving at the bank to transact business = λ = 8 per hour

Mean service rate = average number of customers served per unit of time = μ = 60 / 3 = 20 customers per hour (convert minutes into hours)

The average time in line =  λ / ( μ (μ -  λ)) = 8 / (20 (20 - 8)) = (8 / 240) = ( 8 x 60) / 240 = 2 minutes ( multiplying 8 with 60 to covert into minutes)

Therefore, average time in line = 2 minutes

b)

Average number of customers in the line = (λ^2) / (μ (μ -  λ)) = (8^2) / (20 (20 - 8)) = 64 / 240 = 0.26666

Therefore, Average number of customers in the line = 0.2666

c)

Average time in the system = 1 / (μ -  λ) = 1 / (20 - 8) = 1 / 12 = 60 / 12 = 5 minutes (multiplying by 60 to convert into minutes)

Therefore, Average time in the system = 5 minutes

d)

Average number of customers in the system = λ / (μ -  λ) = 8 / (20 - 8) = 0.6666

Therefore, Average number of customers in the system = 0.6666 customers

e)

Probability that the bank is empty (no customers waiting in the service system) = (μ -  λ) / μ = (20 - 8) / 20 =  0.6

Therefore, the probability that the bank is empty = 0.6