Opportunistic Splitting for Scheduling via Stochastic Approximation Vinay Joseph, Vinod Sharma and Utpal Mukherji Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore, India. Email: vinayjoseph@mail.utexas.edu, {vinod,utpal}@ece.iisc.ernet.in Abstract—We consider the problem of scheduling a wireless channel among multiple users. A slot is given to a user with a highest metric (e.g., channel gain) in that slot. The scheduler may not know the channel states of all the users at the beginning of each slot. In this scenario opportunistic splitting is an attractive solution. However this algorithm requires that the metrics of different users form independent, identically distributed (iid) sequences with same distribution and that their distribution and number be known to the scheduler. This limits the usefulness of opportunistic splitting. In this paper we develop a paramet- ric version of this algorithm. The optimal parameters of the algorithm are learnt online through a stochastic approximation scheme. Our algorithm does not require the metrics of different users to have the same distribution. The statistics of these metrics and the number of users can be unknown and also vary with time. We prove the convergence of the algorithm and show its utility by scheduling the channel to maximize its throughput while satisfying some fairness and/or quality of service constraints. Keywords: Quality of Service, Opportunistic Scheduling, Opportunistic Splitting, Stochastic Approximation, Multiple access channel. I. I NTRODUCTION In a wireless network often the optimal policy chooses a node with the highest metric. The nodes may be distributed in space and the value of the metric for a node may be available only with the node. For instance, in a wireless Multiple Access Channel (MAC), scheduling a node with the maximum channel gain for transmission maximizes the throughput (provided we perform proper power control) when the nodes have infinite backlogs ([1]). The challenge lies in finding the node quickly using minimum resources. This problem can be seen as one of decentralized contention resolution (CR) where, in each slot, several nodes want to transmit to a central node and we have to resolve the contention by opportunistically scheduling a node with the maximum value for some metric. A simple scheme is to let all the nodes feed back their metric to the central node following which the central node selects the node with the highest metric. But, when the number of nodes is large, the overhead required for the transmission of the control information from each node can become prohibitively large. An early work that addressed this problem was [2] in which a scheme was proposed which uses the idea of opportunistic splitting. The use of splitting for CR is not new ([3]). In [2], opportunistic splitting is used to split the set of contending nodes based on their metrics. We refer to the splitting algorithm given in [2] for a homogeneous setting with iid metrics as ALGO-QB. In this setting, the channel gains of the nodes have the same distribution and for each node, the channel gains are iid from slot to slot. For this algorithm, it is shown in [2] that, independent of the fading distribution and the number of nodes, the average number of mini-slots required to find the best node is upper bounded by 2.51 which is close to an asymptotic lower bound. In a later work ([4]), the same authors have considered a heterogeneous setting where different nodes’ channel gains have different distributions although they are still independent from slot to slot. But, they consider only one kind of metric that ensures temporal fairness. In the rest of the paper, the terms homogeneous setting and heterogeneous setting are used with the same meaning as described above. Some other works on opportunistic CR are discussed next. In [5], a scheme called channel-aware Aloha is studied where users randomly transmit packets with probabilities that in- crease with their channel gain. Similar opportunistic exten- sions of ALOHA are also considered in [6] and [7]. In [8], a random access based feedback protocol which uses a fixed number of feedback slots for CR is presented. [9] also consid- ers CR, and improves on and analyzes the policies presented in [2]. Further, [9] extends the algorithms for multiple node selection. However, [8] and [9] restrict themselves to the case in which the channel gains of all nodes are assumed to be iid. [10] presents algorithms for reducing feedback required for CR for a heterogeneous (and homogeneous) setting using static splitting. All these algorithms require the schedular to know the distributions of the channel gains and the number of users. In this paper, we propose decentralized opportunistic split- ting algorithms whose parameters are learnt using stochastic approximation algorithms. The algorithms do not need the knowledge of distributions or the number of users, and perform well in heterogeneous settings too. Using these algorithms we will also provide certain Quality of Service (QoS) guarantees. Also, the algorithms can track the parameters if they change slowly with time. The tracking ability makes our algorithms very useful in dynamic systems. We prove the convergence of our stochastic algorithms used in learning the parameters, and illustrate this using simulations. The paper is organized as follows. Section II explains the problem scenario. In Section III we present our opportunis- 978-1-4244-6385-5/10/$26.00 ©2010 IEEE