Scheduling for Fading Channels Under Partial Channel Information Santanu Mondal and Vinod Sharma Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore – 560 012, India. Email: {santanu,vinod}@ece.iisc.ernet.in Abstract—We consider a scheduler for the downlink of a wireless channel when only partial channel-state information is available at the scheduler. We characterize the network stability region and provide two throughput-optimal scheduling policies. We also derive a deterministic bound on the mean packet delay in the network. Finally, we provide a throughput-optimal policy for the network under QoS constraints when real-time and rate- guaranteed data traffic may be present. Index Terms—Fading, scheduling, partial channel-state, throughput-optimal policies, delay bound, QoS. I. I NTRODUCTION Scheduling has always been an indispensable part of re- source allocation in wireless networks. The seminal work of Tassiulas et al. [20], and later [21], [23] considered the case where both channel states and queue lengths are fully available to the scheduler. It was shown that the MaxWeight algorithm, which serves the longest connected queue, is throughput- optimal. Subsequently, the MaxWeight algorithm was found to be throughput-optimal in many other settings as well ([1]- [12] and the references therein) using tools from Lyapunov optimization. Some other works (see [24], [25], [27]) also approach the scheduling problem using convex optimization and dual decomposition techniques. [5] even considers the role of imperfect queue length information on network throughput, showing that the stability region does not reduce. But, in all these cases, accurate information about channel-state is assumed as a modeling simplification. In a real-life network, e.g., Long Term Evolution (LTE) [19] or IEEE 802.16e WiMAX, the channel-state information fed back to the transmitter can have uncertainty. The primary rea- son being that although resource-allocation is done at the finer granularity of a Physical Resource Block (PRB), channel- state information is still fed back at the coarser granularity of a subband, which is a group of PRBs. This is done to reduce the feedback traffic from the users to the Base Station (BS). However, this averaging causes information loss and hence, the resulting uncertainty at the scheduler. Moreover, uncertainty might be present in the channel-estimates because of the very process of estimation. Some recent works have, hence, tried to model this un- certainty in the channel-estimate. In [7], the authors show that infrequent channel-state measurement, unlike infrequent queue length measurement, reduces the maximum attainable throughput. [13] considers the effect of inaccuracy of channel estimation on throughput, but does so assuming a specific probability distribution of the channel-state and does not study the stability of the data queues either. [12] attempts at modeling channel- and queue-state uncertainty by considering the case where only heterogeneously delayed information is available at the scheduler. They however assume knowledge of the channel-state transition probabilities. In [18], the authors study scheduling with rate adaptation in a single-hop network with a single channel under channel uncertainty. They consider cases when the channel estimates are inaccurate but complete or incomplete knowledge of the channel-estimator joint statis- tics is available at the scheduler. The authors, however, assume that the channel-estimates are independent across the channels for each user. Delay performance of various wireless systems has also been investigated by many researchers recently. Among pre- vious work in the area, the authors in [28] study the problem of opportunistic scheduling of a wireless channel while also trying to minimize the mean delay. Neely [17] has given a O(1/(1 − ρ)) delay bound in the case of ON/OFF channels and a O(N/(1 − ρ)) bound for multi-rate channels for the classical MaxWeight algorithm for a network of size N and any traffic input-rate vector within a ρ-scaled version of the stability region (where 0 <ρ< 1). Subsequent work in [16] established O(1/(1 − ρ)) and O(N/(1 − ρ)) delay bounds for the case of single-hop and multi-hop networks, respectively, for ON/OFF channels and under both i.i.d. and Markov modulated arrival traffic scenarios. [15] derives lower and upper bounds on the delay in a wireless system with single-hop traffic and general interference constraints. Our contributions of this paper are as follows: • Firstly, we model the channel-estimate inaccuracy and characterize the network stability region. Compared to [18], we allow the channel estimates to have dependence among themselves, which is a more realistic situation in a modern LTE or WiMax network. Besides, we study a multi-channel setup whereas they consider a single channel. • Secondly, we propose two simple MaxWeight based scheduling schemes that achieve any rate in the interior of the stability region. • Thirdly, we derive an O(N/(1 − ρ)) delay bound for our system under one of the throughput-optimal policies we propose.