5. Consider a queuing situation involving a single server. Service time is exponentially distributed with an expected value of: 6 min/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 min when there are four in the system; and 2 minutes/customer when there are five in the system.

Suppose arrivals occur in Poisson fashion at an average rate of 15/hour when no more than one customer is in the system. If one customer is waiting for service, then the arrival rates drop to a level of 12/hour. If two are waiting, then arrivals occur at a rate of 9/hour; if three are waiting then the arrival rate is 6/hour; If four are waiting, then arrival is no longer occur.

a) Determine the probability of an idle server.

b) If the probability of an idle server is 0.12 for the arrival and service rates given above:

i) What is the probability of exactly one customer in the system?

ii) What is the probability of exactly two customers in the system?

iii) What is the probability of less than four customers in the system?

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