4,606 research outputs found
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
Entropy and Time
The emergence of a direction of time in statistical mechanics from an
underlying time-reversal-invariant dynamics is explained by examining a simple
model. The manner in which time-reversal symmetry is preserved and the role of
initial conditions are emphasized. An extension of the model to finite
temperatures is also discussed.Comment: 9 pages, 8eps figures. To appear in the theme issue of the American
Journal of Physics on Statistical Physic
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