Mean field analysis of loss models with mixed-Erlang distributions under Power-of-d routing
29th International Teletraffic Congress (ITC 29), Genoa, Italy, (2017)Abstract
In this paper, we study a model for cloud systems consisting of $N$ heterogeneous parallel servers of finite capacities, where jobs arrive according to a Poisson process with rate \$N\$ and each arriving job is routed to the server with the maximum vacancy amongst $d$ randomly selected servers. The performance of the above routing scheme, known as the Power-of-$d$ routing scheme, has previously been analyzed for such systems but only under the assumption of exponential holding times of the jobs at the servers. However, in most realistic applications, the assumption of exponential holding times does not hold and 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,$\backslash$infty)\$ can be approximated by the mixed-Erlang distribution with arbitrary accuracy. We focus on the limiting regime where \$N $\backslash$to $\backslash$infty\$. This leads to a mean-field dynamics, that are significantly more difficult to analyze than the exponential case since the state of each server now becomes multi-dimensional. We derive the mean field dynamics of the system and show that the mean field 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.
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