In this paper we study a large system of N servers, each with capacity to process at most C simultaneous jobs; 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 $\lambda^{(N)}=\sigma-\frac{\beta}{\sqrt{N}\,}$ , $\sigma\in\mathbb{R}_+$ and $\beta\in\mathbb{R}$ , we establish functional central limit theorems 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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具有随机路由的 Erlang 损失模型中平均场波动的敏感性

在本文中，我们研究了一个大型系统ñ服务器，每个服务器最多可以处理C同时工作；如果传入作业的占用率最低，则将其路由到服务器d（在......之外ñ) 随机选择的服务器。假定路由到没有空缺的服务器的作业被阻止并丢失。这种随机化策略被称为 JSQ(d) (加入最短队列d) 政策。假设工作按照泊松过程到达，速率为$N\lambda^{(N)}$在哪里$\lambda^{(N)}=\sigma-\frac{\beta}{\sqrt{N}\,}$,$\sigma\in\mathbb{R}_+$和$\beta\in\mathbb{R}$，当服务时间分布为指数时，我们为瞬态和静止状态下的波动过程建立了功能中心极限定理。特别是，我们证明了极限是一个 Ornstein-Uhlenbeck 过程，其均值和方差取决于所考虑模型的均值场。使用这个，我们获得了大块阻塞概率的近似值ñ，我们可以精确估计一阶近似的准确性。