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We consider the following queuing system which arises as a model of a wireless link shared by multiple users+ There is a finite number N of input flows served by a server+ The system operates in discrete time t ϭ 0,1,2, + + + + Each input flow can be described as an irreducible countable Markov chain; waiting customers of each flow are placed in a queue+ The sequence of server states m~t !, t ϭ 0,1,2, + + + , is a Markov chain with finite number of states M+ When the server is in state m, it can serve µ i m customers of flow i~in one time slot!+ The scheduling discipline is a rule that in each time slot chooses the flow to serve based on the server state and the state of the queues+ Our main result is that a simple online scheduling discipline, Modified Largest Weighted Delay First, along with its generalizations, is throughput optimal; namely, it ensures that the queues are stable as long as the vector of average arrival rates is within the system maximum stability region+

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The relative delay tolerance of data applications, together with the bursty traffic characteristics, opens up the possibility for scheduling transmissions so as to optimize throughput. A particularly attractive approach, in fading environments, is to exploit the variations in the channel conditions, and transmit to the user with the currently 'best' channel. We show that the 'best' user may be identified as the maximum-rate user when the feasible rates are weighed with some appropriately determined coefficients. Interpreting the coefficients as shadow prices, or reward values, the optimal strategy may thus be viewed as a revenue-based policy, which always assigns the transmission slot to the user yielding the maximum revenue. Calculating the optimal revenue vector directly is a formidable task, requiring detailed information on the channel statistics. Instead, we present adaptive algorithms for determining the optimal revenue vector on-line in an iterative fashion, without the need for explicit knowledge of the channel behavior. Starting from an arbitrary initial vector, the algorithms iteratively adjust the reward values to compensate for observed deviations from the target throughput ratios. The algorithms are validated through extensive numerical experiments. Besides verifying long-run convergence, we also examine the transient performance, in particular the rate of convergence to the optimal revenue vector. The results show that the target throughput ratios are tightly maintained, and that the algorithms are well able to track sudden changes in the channel conditions or throughput targets.2000 Mathematics Subject Classification: 60K25 (primary), 68M20, 90B18, 90B22 (secondary).