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File Fragmentation over an Unreliable Channel
It has been recently discovered that heavy-tailed
file completion time can result from protocol interaction even
when file sizes are light-tailed. A key to this phenomenon is
the RESTART feature where if a file transfer is interrupted
before it is completed, the transfer needs to restart from the
beginning. In this paper, we show that independent or bounded
fragmentation guarantees light-tailed file completion time as long
as the file size is light-tailed, i.e., in this case, heavy-tailed file
completion time can only originate from heavy-tailed file sizes.
If the file size is heavy-tailed, then the file completion time is
necessarily heavy-tailed. For this case, we show that when the
file size distribution is regularly varying, then under independent
or bounded fragmentation, the completion time tail distribution
function is asymptotically upper bounded by that of the original
file size stretched by a constant factor. We then prove that if the
failure distribution has non-decreasing failure rate, the expected
completion time is minimized by dividing the file into equal sized
fragments; this optimal fragment size is unique but depends on
the file size. We also present a simple blind fragmentation policy
where the fragment sizes are constant and independent of the
file size and prove that it is asymptotically optimal. Finally, we
bound the error in expected completion time due to error in
modeling of the failure process
Universal performance bounds of restart
As has long been known to computer scientists, the performance of
probabilistic algorithms characterized by relatively large runtime fluctuations
can be improved by applying a restart, i.e., episodic interruption of a
randomized computational procedure followed by initialization of its new
statistically independent realization. A similar effect of restart-induced
process acceleration could potentially be possible in the context of enzymatic
reactions, where dissociation of the enzyme-substrate intermediate corresponds
to restarting the catalytic step of the reaction. To date, a significant number
of analytical results have been obtained in physics and computer science
regarding the effect of restart on the completion time statistics in various
model problems, however, the fundamental limits of restart efficiency remain
unknown. Here we derive a range of universal statistical inequalities that
offer constraints on the effect that restart could impose on the completion
time of a generic stochastic process. The corresponding bounds are expressed
via simple statistical metrics of the original process such as harmonic mean
, median value and mode , and, thus, are remarkably practical. We
test our analytical predictions with multiple numerical examples, discuss
implications arising from them and important avenues of future work.Comment: 12 pages, 2 figure
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