We prove the optimality of routing policies with limited state information that are employed in queueing systems with finite capacities and customers arriving at arbitrary instants. Using sample path arguments we show that, on one hand, when service times are exponential and capacities are equal, the round robin policy minimizes the expected number of losses by any time t and at the same time, maximizes the expected number of departures by t, over all policies that use no state information. On the other hand, when service times are deterministic, a modified round robin policy that makes use of limited state information outperforms stochastically all dynamic policies that have richer state information, in terms of the number of losses and the number of departures by t.