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Optimal data placements for triple replication in distributed storage systems
Advances in Mathematics of Communications ( IF 0.9 ) Pub Date : 2022-01-01 , DOI: 10.3934/amc.2022037 Ruijing Liu 1 , Junling Zhou 1
Advances in Mathematics of Communications ( IF 0.9 ) Pub Date : 2022-01-01 , DOI: 10.3934/amc.2022037 Ruijing Liu 1 , Junling Zhou 1
Affiliation
<p style='text-indent:20px;'>In distributed storage systems it is essential to store files (data) in replication to ensure reliability and fault-tolerance. Given a set <inline-formula><tex-math id="M1">\begin{document}$ V $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M2">\begin{document}$ v $\end{document}</tex-math></inline-formula> servers along with <inline-formula><tex-math id="M3">\begin{document}$ b $\end{document}</tex-math></inline-formula> files, each file is replicated (placed) on exactly <inline-formula><tex-math id="M4">\begin{document}$ k $\end{document}</tex-math></inline-formula> servers and thus a file can be represented by a set of <inline-formula><tex-math id="M5">\begin{document}$ k $\end{document}</tex-math></inline-formula> servers. Then we produce a data placement consisting of <inline-formula><tex-math id="M6">\begin{document}$ b $\end{document}</tex-math></inline-formula> subsets of <inline-formula><tex-math id="M7">\begin{document}$ V $\end{document}</tex-math></inline-formula> called blocks, each of size <inline-formula><tex-math id="M8">\begin{document}$ k $\end{document}</tex-math></inline-formula>. Each server has some probability to fail and we want to find an optimal data placement that minimizes the variance of the number of available files for any value of the probability of failure. An optimal data placement with <inline-formula><tex-math id="M9">\begin{document}$ b $\end{document}</tex-math></inline-formula> blocks of size three on a <inline-formula><tex-math id="M10">\begin{document}$ v $\end{document}</tex-math></inline-formula>-set was proved to exist by Wei et al. [J. Combin. Des. 24 (2016) 77-100] if <inline-formula><tex-math id="M11">\begin{document}$ v $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M12">\begin{document}$ b $\end{document}</tex-math></inline-formula> meet some conditions. We observe that there is as yet no complete solution to the existence problem of optimal data placements for triple replication. Hence this article concentrates on those missing parameters by former research and we characterize the combinatorial properties of the corresponding optimal data placements. Nearly well-balanced triple systems (NWBTSs) are defined to produce optimal data placements. Many constructions for NWBTSs are developed, mainly by constructing candelabra systems with various desirable partitions. The main result of this article is that there always exist optimal data placements for triple replication with <inline-formula><tex-math id="M13">\begin{document}$ b $\end{document}</tex-math></inline-formula> blocks on a <inline-formula><tex-math id="M14">\begin{document}$ v $\end{document}</tex-math></inline-formula>-set for all positive integers <inline-formula><tex-math id="M15">\begin{document}$ v, b $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M16">\begin{document}$ v \equiv 0, 1, 3, 5 $\end{document}</tex-math></inline-formula> (mod 6).</p>
更新日期：2022-01-01
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