Steady-State Behavior of Some Load Balancing Mechanisms in Cloud Storage Systems

In large storage systems, files are often coded across several servers to improve reliability and retrieval speed. We consider a system of $n$ servers storing files using a Maximum Distance Separable code (cf. \cite{li2016mean}). Specifically, each file is stored in equally sized pieces across $L$ servers such that any $k$ pieces can reconstruct the original file. File requests are routed using the Batch Sampling routing scheme. I.e. when a request for a file is received, a centralized dispatcher routes the job into the $k$-shortest queues among the $L$ for which the corresponding server contains a piece of the file being requested. We study the long time behavior of this class of load balancing mechanisms. In particular, it is shown that the ODE system that describes the mean field limit of the occupancy measure process has a unique fixed point which is stable. This fixed point corresponds to a distribution on $\mathbb{N}_0$ of queue lengths with tails that decay super-exponentially. Upper and lower bounds on the decay rate are provided. Finally, we show that the unique invariant measure of the Markov occupancy measure process converges to the Dirac measure concentrated at the unique fixed point of the ODE system, establishing the interchangeability of the $t\to\infty$ and $n\to\infty$ limits.