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In this paper we study an optimal server allocation problem, where a single server is shared among multiple queues based on the queue backlog information. Due to the physical nature of the system this information is delayed, in that when the allocation decision is made, the server only has the backlog information from an earlier time. Queues have different arrival processes as well as different buffering/holding costs. The objective is to minimize the expected total discounted holding cost over a finite or infinite horizon. We introduce an index policy where the index of a queue is a function of the state of the queue. Our primary interest is to characterize conditions under which this index policy is optimal. We present a fairly general method bounding the reward of serving one queue instead of another. Using this result, sufficient conditions on the optimality of the index policy can be derived for a variety of arrival processes and packet holding costs. These conditions are in general in the form of sufficient separation among indices, and they characterize the part of the state space where the index policy is optimal. We provide examples and derive the indices and illustrate the region where the index policy is optimal.