3 At Hot Dog Queen Restaurant, the following sequence of events occurs during each minute:

a With probability p, a customer arrives and waits in line.

b Hot Dog Queen determines the rate s at which customers are served. If any customers are in the restaurant, then with probability s, one of the customers completes service and leaves the restaurant. It costs c(s) dollars per period to serve customers at a rate s. Each customer spends R dollars, and the customer’s food costs Hot Dog Queen R  1 dollars to prepare.

c For each customer in line at the end of the minute, a cost of h dollars is assessed (because of customer inconvenience).

d The next minute begins.

Formulate a recursion that could be used to maximize expected revenues less costs (including customer inconvenience costs) incurred during the next T minutes.

Assume that initially there are no customers present.