Consider a stable M/G/1 system in which, at time \(t=0\), there are exactly n customers with residual service times equal to \(v_1,v_2,\ldots ,v_n\). In addition, assume that there is an extra customer c who arrives at time \(t=0\) and has a service requirement of x. The externalities which are created by c are equal to the total waiting time that others will save if her service requirement is reduced to zero. In this work, we study the joint distribution (parameterized by \(n,v_1,v_2,\ldots ,v_n,x\)) of the externalities created by c when the underlying service distribution is either last-come, first-served with preemption or first-come, first-served. We start by proving a decomposition of the externalities under the above-mentioned service disciplines. Then, this decomposition is used to derive several other results regarding the externalities: moments, asymptotic approximations as \(x\rightarrow \infty \), asymptotics of the tail distribution, and a functional central limit theorem.

考虑一个稳定的 M/G/1 系统，其中在时间\(t=0\)时，正好有n 个客户的剩余服务时间等于\(v_1,v_2,\ldots ,v_n\)。此外，假设有一个额外的客户c在时间\(t=0\)到达并且有x的服务需求。由c创造的外部性等于如果她的服务需求减少到零，其他人将节省的总等待时间。在这项工作中，我们研究了由c创建的外部性的联合分布（由\(n,v_1,v_2,\ldots ,v_n,x\)参数化）当底层服务分配是后到先得或先到先得时。我们首先证明上述服务纪律下的外部性分解。然后，该分解用于推导出关于外部性的其他几个结果：矩、渐近近似为\(x\rightarrow \infty \)、尾部分布的渐近和函数中心极限定理。