Arrival rate of telephone calls at a telephone both are according to a Poisson distribution with an average time of 9 minutes between two consecutive arrivals. The length of the telephone call is assumed to be exponentially distributed with mean 3 minutes.
a) Determine the probability that the person arriving at the booth will have to wait.
b) Find the average queue length that is formed from time to time c) The telephone company will install a second booth when convinced that an arrival would expect to have to wait at least four minutes for the phone. Find the increase in flow rate of arrivals which will justify a second booth.
d) What is the probability that an arrival will have to wait for more than 10 minutes before the phone is free?
e) What is the probability that he will have to wait for more than 10 minutes before the phone is available and the call is also complete?
f) Find the fraction of a day that the phone will be in use