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Throughput optimal scheduling policies in general require the solution of a complex optimization problem. The past literature has shown that the complexity of this optimization problem can be greatly reduced, but at the expense of memory requirement that is exponential with the number of users. In this paper, we study the stability region of a class of linear-memory scheduling policies for time varying channels, and investigate how the channel memory impacts the supportable input rates. The set of scheduling policies in this paper covers a wide spectrum of resource allocation algorithms, which allows us to study policies with different complexity levels. In particular, we are able to model a class of low-complexity scheduling policies with linear memory, which are suitable for practical implementation.