Question

Management Science Bank is the only bank in the small town of Sto Thomas. On a typical Friday, an average of 10 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 4 minutes. It is assumed that service times may be described by the 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, find:
a.) The average time in the line.
b.) The average number in the line.
c.) The average time in the system.
d.) The average number in the system.
e.) The probability that the bank is empty.
f.) Management Science Bank is considering adding a second teller (who would work at the same rate as the first) to reduce the waiting time for customers. She assumes that this will cut the waiting time in half. If a second teller is added, find the new answers to parts (a) to (e).

Answer 1

Incoming Customers arrival rate = L = 10 per hour. Teller processes each customer in a single transaction in 4 minutes. Hence Processing rate = 60/4 = u = 15 per hour. Using these parameters we can apply queuing theory principles to find out various measures as required.

a,c) Utilization p = L/u = 0.67. Average Time in System = W = 1/(u - L) = 1/5 hours = 12 minutes. Hence average time in line = p*W = 8 minutes

b,d) Average number of customers in the system C = L/(u-L) = 10/5 = 2. Hence Average number of customers in queue Q = p*C = 1.33

e.) Probability that bank is empty is that there is nobody in service. = (1 - p)*p0 as n = 0; i.e = 1 - p = 0.33

f) if a second teller is added then we have s = number of stations = 2. Hence effective processing rate = s*u = 30 per hour. The new utilization p = L/(s*u) 10/30 = 0.33. Other parameters can be calculated as per queuing theory principles for a multi server model.