Analysis of a Stochastic Model of Replication in Large Distributed Storage Systems: A Mean-Field Approach

Distributed storage systems such as Hadoop File System or Google File System (GFS) ensure data availability and durability using replication. This paper is focused on the analysis of the efficiency of replication mechanism that determines the location of the copies of a given file at some server. The variability of the loads of the nodes of the network is investigated for several policies. Three replication mechanisms are tested against simulations in the context of a real implementation of a such a system: Random, Least Loaded and Power of Choice. The simulations show that some of these policies may lead to quite unbalanced situations: if $\beta$ is the average number of copies per node it turns out that, at equilibrium, the load of the nodes may exhibit a high variability. It is shown in this paper that a simple variant of a power of choice type algorithm has a striking effect on the loads of the nodes: at equilibrium, the distribution of the load of a node has a bounded support, most of nodes have a load less than $2\beta$ which is an interesting property for the design of the storage space of these systems. Mathematical models are introduced and investigated to explain this interesting phenomenon. The analysis of these systems turns out to be quite complicated mainly because of the large dimensionality of the state spaces involved. Our study relies on probabilistic methods, mean-field analysis, to analyze the asymptotic behavior of an arbitrary node of the network when the total number of nodes gets large. An additional ingredient is the use of stochastic calculus with marked Poisson point processes to establish some of our results.