12. A supermarket has two exponential checkout counters, each operating at rate . Arrivals are Poisson at

12. A supermarket has two exponential checkout counters, each operating at rate μ.

Arrivals are Poisson at rate λ. The counters operate in the following way:

(i) One queue feeds both counters.

(ii) One counter is operated by a permanent checker and the other by a stock clerk who instantaneously begins checking whenever there are two or more customers in the system. The clerk returns to stocking whenever he completes a service, and there are fewer than two customers in the system.

(a) Find Pn, proportion of time there are n in the system.

(b) At what rate does the number in the system go from 0 to 1? From 2 to 1?

(c) What proportion of time is the stock clerk checking?

Hint: Be a little careful when there is one in the system.