In this paper, we study a large system of $N$ servers each with capacity to process at most $C$ simultaneous jobs and an incoming job is routed to a server if it has the lowest occupancy amongst $d$ (out of N) randomly selected servers. A job that is routed to a server with no vacancy is assumed to be blocked and lost. Such randomized policies are referred to JSQ(d) (Join the Shortest Queue out of $d$) policies. Under the assumption that jobs arrive according to a Poisson process with rate $N\lambda^{(N)}$ where $\lambda^{(N)}=\sigma-\frac{\beta}{\sqrt{N}}$, $\sigma\in\mb{R}_+$ and $\beta\in\mb{R}$, we establish functional central limit theorems (FCLTs) for the fluctuation process in both the transient and stationary regimes when service time distributions are exponential. In particular, we show that the limit is an Ornstein-Uhlenbeck process whose mean and variance depend on the mean-field of the considered model. Using this, we obtain approximations to the blocking probabilities for large $N$, where we can precisely estimate the accuracy of first-order approximations.

中文翻译：

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带有随机路由的Erlang损失模型中均场波动的敏感性

在本文中，我们研究了一个由$ N $个服务器组成的大型系统，每个服务器最多可以处理$ C $个同时作业，如果传入的作业在$ d $（N个）中占用率最低，则将其路由到服务器随机选择的服务器。假定路由到无空缺的服务器的作业被阻止并丢失。此类随机策略称为JSQ（d）（从$ d $中加入最短队列）策略。假设作业按照泊松过程到达，速率为$ N \ lambda ^ {{N}} $，其中$ \ lambda ^ {{N}} = \ sigma- \ frac {\ beta} {\ sqrt {N} } $，$ \ sigma \ in \ mb {R} _ + $和$ \ beta \ in \ mb {R} $，我们建立了瞬态和平稳状态下波动过程的函数中心极限定理（FCLT）服务时间分布是指数的。特别是，我们证明极限是一个Ornstein-Uhlenbeck过程，其均值和方差取决于所考虑模型的均值场。使用此方法，我们可以获得大的$ N $的阻塞概率的近似值，在这里我们可以精确估算一阶近似值的准确性。