7.1T Consider a person providing a certain type of service to customers. The person receives at the beginning of each time period with probability pi a proposal by a customer of type t', where e' : 1, 2, . . . , n, who offers an amount of mionie}r Ma We assume that 2;, p; = 1. The person ma}r reject the offer, in which case the customer leaves: and the person remains idle during that period, or the person ma}.F accept the offer in which case the person spends some random amount of time with that customer. In particular, we assume that the probability that the type 1' customer will leave after t: periods [in = 1, 2, . . J, given that the customer has already stayed with the person for k r 1 periods is a given scalar ,8;- E {II}, 1}. The problem is to determine an acceptancerejection policy that maximizes 1 . lim {Expected payment over N periods}. Noo N {a} Formulate the person's problem as an average cost per stage problem, and show that the optimal cost is independent of the initial state. {b} Show that there exists a scalar A and an optimal policyr that accepts the offer of a type 11 customer if and only if e:_... A_, where T; is the expected time spent with the type i customer