Capacity of 1-to-K Broadcast Packet Erasure Channels with Channel Output Feedback

This paper focuses on the 1-to-K broadcast packet erasure channel (PEC), which is a generalization of the broadcast binary erasure channel from the binary symbol to that of arbitrary finite fields GF(q) with sufficiently large q. We consider the setting in which the source node has instant feedback of the channel outputs of the K receivers after each transmission. Such a setting directly models network coded packet transmission in the downlink direction with integrated feedback mechanisms (such as Automatic Repeat reQuest (ARQ)). The main results of this paper are: (i) The capacity region for general 1-to-3 broadcast PECs, and (ii) The capacity region for two classes of 1-to-K broadcast PECs: the symmetric PECs, and the spatially independent PECs with one-sided fairness constraints. This paper also develops (iii) A pair of outer and inner bounds of the capacity region for arbitrary 1-to-K broadcast PECs, which can be evaluated by any linear programming solver. For most practical scenarios, the outer and inner bounds meet and thus jointly characterize the capacity.Comment: 8 pages, 2 figures. Published in Allerton 2010. The journal version of this work was submitted to IEEE Trans IT in May, 201

When and By How Much Can Helper Node Selection Improve Regenerating Codes?

Regenerating codes (RCs) can significantly reduce the repair-bandwidth of distributed storage networks. Initially, the analysis of RCs was based on the assumption that during the repair process, the newcomer does not distinguish (among all surviving nodes) which nodes to access, i.e., the newcomer is oblivious to the set of helpers being used. Such a scheme is termed the blind repair (BR) scheme. Nonetheless, it is intuitive in practice that the newcomer should choose to access only those "good" helpers. In this paper, a new characterization of the effect of choosing the helper nodes in terms of the storage-bandwidth tradeoff is given. Specifically, answers to the following fundamental questions are given: Under what conditions does proactively choosing the helper nodes improve the storage-bandwidth tradeoff? Can this improvement be analytically quantified? This paper answers the former question by providing a necessary and sufficient condition under which optimally choosing good helpers strictly improves the storage-bandwidth tradeoff. To answer the latter question, a low-complexity helper selection solution, termed the family repair (FR) scheme, is proposed and the corresponding storage/repair-bandwidth curve is characterized. For example, consider a distributed storage network with 60 total number of nodes and the network is resilient against 50 node failures. If the number of helper nodes is 10, then the FR scheme and its variant demonstrate 27% reduction in the repair-bandwidth when compared to the BR solution. This paper also proves that under some design parameters, the FR scheme is indeed optimal among all helper selection schemes. An explicit construction of an exact-repair code is also proposed that can achieve the minimum-bandwidth-regenerating point of the FR scheme. The new exact-repair code can be viewed as a generalization of the existing fractional repetition code.Comment: 35 pages, 10 figures, submitted to IEEE Transactions on Information Theory on September 04, 201

Robust And Optimal Opportunistic Scheduling For Downlink 2-Flow Network Coding With Varying Channel Quality and Rate Adaptation

This paper considers the downlink traffic from a base station to two different clients. When assuming infinite backlog, it is known that inter-session network coding (INC) can significantly increase the throughput of each flow. However, the corresponding scheduling solution (when assuming dynamic arrivals instead and requiring bounded delay) is still nascent. For the 2-flow downlink scenario, we propose the first opportunistic INC + scheduling solution that is provably optimal for time-varying channels, i.e., the corresponding stability region matches the optimal Shannon capacity. Specifically, we first introduce a new binary INC operation, which is distinctly different from the traditional wisdom of XORing two overheard packets. We then develop a queue-length-based scheduling scheme, which, with the help of the new INC operation, can robustly and optimally adapt to time-varying channel quality. We then show that the proposed algorithm can be easily extended for rate adaptation and it again robustly achieves the optimal throughput. A byproduct of our results is a scheduling scheme for stochastic processing networks (SPNs) with random departure, which relaxes the assumption of deterministic departure in the existing results. The new SPN scheduler could thus further broaden the applications of SPN scheduling to other real-world scenarios