Reliability of Erasure Coded Storage Systems: A Geometric Approach

We consider the probability of data loss, or equivalently, the reliability function for an erasure coded distributed data storage system under worst case conditions. Data loss in an erasure coded system depends on probability distributions for the disk repair duration and the disk failure duration. In previous works, the data loss probability of such systems has been studied under the assumption of exponentially distributed disk failure and disk repair durations, using well-known analytic methods from the theory of Markov processes. These methods lead to an estimate of the integral of the reliability function. Here, we address the problem of directly calculating the data loss probability for general repair and failure duration distributions. A closed limiting form is developed for the probability of data loss and it is shown that the probability of the event that a repair duration exceeds a failure duration is sufficient for characterizing the data loss probability. For the case of constant repair duration, we develop an expression for the conditional data loss probability given the number of failures experienced by a each node in a given time window. We do so by developing a geometric approach that relies on the computation of volumes of a family of polytopes that are related to the code. An exact calculation is provided and an upper bound on the data loss probability is obtained by posing the problem as a set avoidance problem. Theoretical calculations are compared to simulation results.Comment: 28 pages. 8 figures. Presented in part at IEEE International Conference on BigData 2013, Santa Clara, CA, Oct. 2013 and to be presented in part at 2014 IEEE Information Theory Workshop, Tasmania, Australia, Nov. 2014. New analysis added May 2015. Further Update Aug. 201

A Zador-Like Formula for Quantizers Based on Periodic Tilings

We consider Zador's asymptotic formula for the distortion-rate function for a variable-rate vector quantizer in the high-rate case. This formula involves the differential entropy of the source, the rate of the quantizer in bits per sample, and a coefficient G which depends on the geometry of the quantizer but is independent of the source. We give an explicit formula for G in the case when the quantizing regions form a periodic tiling of n-dimensional space, in terms of the volumes and second moments of the Voronoi cells. As an application we show, extending earlier work of Kashyap and Neuhoff, that even a variable-rate three-dimensional quantizer based on the ``A15'' structure is still inferior to a quantizer based on the body-centered cubic lattice. We also determine the smallest covering radius of such a structure.Comment: 8 page

Multiple Description Vector Quantization with Lattice Codebooks: Design and Analysis

The problem of designing a multiple description vector quantizer with lattice codebook Lambda is considered. A general solution is given to a labeling problem which plays a crucial role in the design of such quantizers. Numerical performance results are obtained for quantizers based on the lattices A_2 and Z^i, i=1,2,4,8, that make use of this labeling algorithm. The high-rate squared-error distortions for this family of L-dimensional vector quantizers are then analyzed for a memoryless source with probability density function p and differential entropy h(p) < infty. For any a in (0,1) and rate pair (R,R), it is shown that the two-channel distortion d_0 and the channel 1 (or channel 2) distortions d_s satisfy lim_{R -> infty} d_0 2^(2R(1+a)) = (1/4) G(Lambda) 2^{2h(p)} and lim_{R -> infty} d_s 2^(2R(1-a)) = G(S_L) 2^2h(p), where G(Lambda) is the normalized second moment of a Voronoi cell of the lattice Lambda and G(S_L) is the normalized second moment of a sphere in L dimensions.Comment: 46 pages, 14 figure

Lattice Erasure Codes of Low Rank with Noise Margins

We consider the following generalization of an $(n,k)$ MDS code for application to an erasure channel with additive noise. Like an MDS code, our code is required to be decodable from any $k$ received symbols, in the absence of noise. In addition, we require that the noise margin for every allowable erasure pattern be as large as possible and that the code satisfy a power constraint. In this paper we derive performance bounds and present a few designs for low rank lattice codes for an additive noise channel with erasures

Constructive spherical codes on layers of flat tori

A new class of spherical codes is constructed by selecting a finite subset of flat tori from a foliation of the unit sphere S^{2L-1} of R^{2L} and designing a structured codebook on each torus layer. The resulting spherical code can be the image of a lattice restricted to a specific hyperbox in R^L in each layer. Group structure and homogeneity, useful for efficient storage and decoding, are inherited from the underlying lattice codebook. A systematic method for constructing such codes are presented and, as an example, the Leech lattice is used to construct a spherical code in R^{48}. Upper and lower bounds on the performance, the asymptotic packing density and a method for decoding are derived.Comment: 9 pages, 5 figures, submitted to IEEE Transactions on Information Theor