A major impediment towards the industrial adoption of decentralized distributed systems comes from the difficulty to theoretically prove that these systems exhibit the required behavior. In this paper, we use probability theory to analyze a decentralized auto-scaling algorithm in which each node probabilistically decides to scale in or out. We prove that, in the context of dynamic workloads, the average load of the system is maintained within a variation interval with a given probability, provided that the number of nodes and the variation interval length are higher than certain bounds. The paper also proposes numerical algorithms for approximating these minimum bounds.
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