In the B&K model of Example 18.5-1, suppose that the interarrival time at the checkout area is exponential with mean 8 minutes and that the checkout time per customer is also exponential with mean 12 minutes. Suppose further that B&K will add a fourth counter. Counters 1, 2, and 3 will open based on increments of two customers and counter 4 will open when there are 7 or more in the store. Determine the following:

(a) The steady-state probabilities, pn for all n.

(b) The probability that a fourth counter will be needed.

(c) The average number of idle counters.