In this paper we study three opportunistic schedulers for the problem of optimal multi-class flow-level scheduling in wireless downlink and uplink systems. For user channels we employ the Gilbert-Elliot model of good and bad channel condition with flow-level interpretation, and assume an automatic repeat query (ARQ) feedback, so that channel state information is available at the end of the slot only if the user was scheduled. The problem is essentially a Partially-Observable Markov Decision Process with a sample-path resource constraint. Given its complexity, we study two naive schedulers: the myopic rule and the belief-state rule. Further, realizing that the problem fits the multi-armed restless bandit framework, we consider the relaxation of the problem which instead of serving a given number of flows on sample-path allows for serving that number of flows only in expectation, and derive an optimal Whittle index policy in closed form. We further discuss the interpretation of the resulting novel Whittle-index-based heuristic scheduler and evaluate its performance against the two naive schedulers in simulations under the time-average criterion. According to the Whittle-index-based scheduler, the users whose last channel feedback gave good condition and those not served yet receive an absolute priority over those whose last channel feedback gave bad condition, which extends to this setting the property of channel-aware schedulers that are known to be maximally stable. In addition, we obtain tie-breaking index values for setting priorities among users in each of the two groups. In case of a single user class, the scheduler becomes independent of the problem parameters and equivalent to both the myopic and belief-state scheduler, and has a simple universal structure which can be represented by three first-in-first-out priority lists.
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