Consider a queueing situation involving a single server. Service time is exponentially distributed with an expected value of 6 minutes per customer when there is one customer in the system, 5 minutes when there are two in the system, 4 minutes when there are three in the system, 3 minutes when there are four in the system, and 2 minutes when there are five in the system.

Suppose arrivals occur in a Poisson fashion at an average rate of 20 per hour when no more than one customer is in the system. If one customer is waiting for service, then the arrival rate drops to a level of 15 per hour. If two are waiting, then arrivals occur at a rate of 10 per hour; if three are waiting, then the arrival rate is 5 per hour;

if four are waiting, then arrivals no longer occur.

a. Determine the probability of an idle server.

b. If the probability of an idle server equals 0.1, for the arrival and service rates given above, what is the probability of exactly one customer in the system?