Mean-Field Analysis of Loss Models with Mixed-Erlang Distributions under Power-of-d Routing

In this paper, we study the behavior of a large number N of parallel loss servers operating under a randomized power-of-d routing scheme for the arrivals. Such models are of importance in several cloud architectures that offer IaaS (Infrastructure as a Service). The probabilistic behavior of such modelshas previously been analyzed for jobs with exponential holding times. However, in most realistic applications, the assumption of exponential holding times does not holdand therefore it is of importance to understand the performance of the power-of-d routing scheme under more general holding time distributions. In this paper, we analyze the dynamics of the system under mixed-Erlang service time distributions since any distribution on [0,∞) can be approximated by the mixed-Erlangdistribution with arbitrary accuracy. We focus on the limiting regime when N → ∞. This leads to a mean-field dynamicsthat are significantly more difficult to analyze than the exponential case since the stateof each server is multi-dimensional with no monotonicity properties. In particular we show that that the mean-field equation (MFE) has a unique fixed-point that corresponds to the fixed-point obtained with exponential assumptions on the holding times showing that the fixed-point is insensitive to the parameters of the mixed-Erlang distribution and only depends on the mean. This has important implications for practical systems.