Consider an \(M / M / 1 / N\) queue whose arrival and service rates are equal. Suppose that customers who join the queue receive a reward \(R\) at the end of service, but pay at a rate of \(C\) per unit time while they are waiting. Find inequalities that characterize the optimal waiting room size \(N\) that maximizes the long-run expected profit per customer, per unit time:

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AR lim P[X,