Cloud storage systems generally add redundancy in storing content files such that K files are replicated or erasure coded and stored on N > K nodes. In addition to providing reliability against failures, the redundant copies can be used to serve a larger volume of content access requests. A request for one of the files can either be sent to a systematic node, or one of the repair groups. In this paper, we seek to maximize the service capacity region, that is, the set of request arrival rates for the K files that can be supported by a coded storage system. We explore two aspects of this problem: 1) for a given erasure code, how to optimally split incoming requests between systematic nodes and repair groups, and 2) choosing an underlying erasure code that maximizes the achievable service capacity region. In particular, we consider MDS and Simplex codes. Our analysis demonstrates that erasure coding makes the system more robust to skews in file popularity than simply replicating a file at multiple servers, and that coding and replication together can make the capacity region larger than either alone.

We consider storage systems in which K files are stored over N nodes. A node may be systematic for a particular file in the sense that access to it gives access to the file. Alternatively, a node may be coded, meaning that it gives access to a particular file only when combined with other nodes (which may be coded or systematic). Requests for file f k arrive at rate λ k , and we are interested in the rate that can be served by a particular system. In this paper, we determine the set of request arrival rates for the a 3-file coded storage system. We also provide an algorithm to maximize the rate of requests served for file K given λ1, . . . , λK−1 in a general K-file case.

Reed-Solomon and BCH codes were considered as kernels of polar codes by Mori and Tanaka (IEEE Information Theory Workshop, 2010, pp 1-5) and Korada et al. (IEEE Trans Inform Theory 56(12):6253-6264, 2010) to create polar codes with large exponents. Mori and Tanaka showed that Reed-Solomon codes over the finite field F q with q elements give the best possible exponent among all codes of length l ≤ q. They also stated that a Hermitian code over F 2 r with r ≥ 4, a simple algebraic geometric code, gives a larger exponent than the Reed-Solomon matrix over the same field. In this paper, we expand on these ideas by employing more general algebraic geometric (AG) codes to produce kernels of polar codes. Lower bounds on the exponents are given for kernels from general AG codes, Hermitian codes, and Suzuki codes. We demonstrate that both Hermitian and Suzuki kernels have larger exponents than Reed-Solomon codes over the same field, for q ≥ 3; however, the larger exponents are at the expense of larger kernel matrices. Comparing kernels of the same size, though over different fields, we see that Reed-Solomon kernels have larger exponents than both Hermitian and Suzuki kernels. These results indicate a tradeoff between the exponent, kernel matrix size, and field size.

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In 2009, Barrière, Dalfó, Fiol, and Mitjana introduced the generalized hierarchical product of graphs. This operation is a generalization of the Cartesian product of graphs. It is known that every connected graph has a unique prime factor decomposition with respect to the Cartesian product. We generalize this result to show that connected graphs indeed have a unique prime factor decomposition with respect to the generalized hierarchical product. We also give preliminary results on the domination number of generalized hierarchical products.

A graph is said to be well-edge-dominated if all its minimal edge dominating sets are minimum. It is known that every well-edge-dominated graph G is also equimatchable, meaning that every maximal matching in G is maximum. In this paper, we show that if G is a connected, triangle-free, nonbipartite, well-edge-dominated graph, then G is one of three graphs. We also characterize the well-edge-dominated split graphs and Cartesian products. In particular, we show that a connected Cartesian product G H is well-edge-dominated, where G and H have order at least 2, if and only if

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