(Sennott, 1989a) Consider an infinite-horizon countable-state version of the service rate control model of Sec. 3.7.2, in which the service rate distribution is Bernoulli with probability pb, where b is chosen from a finite set E and p+= maXbeB pb < 1 and p-= minbeB pb > 0. This means that the probability of a service completion in a single time slot equals pb when action 6 is chosen. The cost of serving at rate p6 for one period equals ~(6). There is no cost associated with changing the service rate. The revenue R = 0 and the holding cost h(s) is linear in s. The objective is to determine a service rate policy which minimizes the long-run average expected cost.
a. Give the optimality equations for the discounted model.
b. Using Theorem 8.10.7, show that, if the second moment of the arrival distribution is finite and the mean arrival rate p is less than p', there exists a lim inf average optimal stationary policy.
c. Show that (8.10.9) holds with equality if, in addition, p
d. Provide conditions for the existence of a stationary lim inf average optimal policy and for the optimality equation to hold under the assumption that h(s) is quadratic in s.
e. Determine the structure of an optimal policy under the linear holding cost assumption.
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