8.3. (This problem is taken from Probability and Statistical Inference for Engineers by Derman and Klein, Oxford University Press, London, 1959.) The number of oil tankers, say N, arriving at a certain refinery each day has a Poisson distribution with parameter = 2. Present port facilities can service three tankers a day. If more than three tankers arrive in a day, the tankers in excess of three must be sent to another port.

(a) On a given day, what is the probability of having to send tankers away?

(b) How much must present facilities be increased to permit handling all tankers on approximately 90 percent of the days?

(c) What is the expected number of tankers arriving per day?

(d) What is the most probable number of tankers arriving daily?

(e) What is the expected number of tankers serviced daily?

(f) What is the expected number of tankers turned away daily?