论文标题
队列中的外部性作为随机过程：FCFS M/g/1的情况
Externalities in queues as stochastic processes: The case of FCFS M/G/1
论文作者
论文摘要
外部性是公共资源的用户对他人施加的成本。例如，考虑一个FCFS M/g/1队列和服务需求为$ x \ geq0 $分钟的客户，当工作负载级别为$ v \ geq0 $分钟时，他们到达了系统。让$ e_v（x）$为如果客户放弃其服务需求，可以节省的总等待时间。在这项工作中，我们分析\ textit {外部性过程} $ e_v（\ cdot）= \ left \ {e_v（x）：x \ geq0 \ right \} $。结果表明，此过程可以用A（时间为$ v $分钟的时间移动）的积分来表示，并具有正离散跳转分布，因此$ e_v（\ cdot）$是凸的。此外，我们计算了$ e_v（\ cdot）$的有限维分布以及其平均值和自动协调功能的LST。我们还确定了条件，在这些条件下，一系列归一化外部性过程承认，在$ \ mathcal {d} [0，\ infty）$上的融合较弱，配备了统一度量的$（$ v $分钟）标准Wiener流程。最后，当$ v $是一个通用的非负随机变量时，我们还考虑扩展框架，该变量与到达过程和服务需求独立。这导致了先前的Haviv和Ritov（1998）作品的现有结果的概括。
Externalities are the costs that a user of a common resource imposes on others. For example, consider a FCFS M/G/1 queue and a customer with service demand of $x\geq0$ minutes who arrived into the system when the workload level was $v\geq0$ minutes. Let $E_v(x)$ be the total waiting time which could be saved if this customer gave up on his service demand. In this work, we analyse the \textit{externalities process} $E_v(\cdot)=\left\{E_v(x):x\geq0\right\}$. It is shown that this process can be represented by an integral of a (shifted in time by $v$ minutes) compound Poisson process with positive discrete jump distribution, so that $E_v(\cdot)$ is convex. Furthermore, we compute the LST of the finite-dimensional distributions of $E_v(\cdot)$ as well as its mean and auto-covariance functions. We also identify conditions under which, a sequence of normalized externalities processes admits a weak convergence on $\mathcal{D}[0,\infty)$ equipped with the uniform metric to an integral of a (shifted in time by $v$ minutes) standard Wiener process. Finally, we also consider the extended framework when $v$ is a general nonnegative random variable which is independent from the arrival process and the service demands. This leads to a generalization of an existing result from a previous work of Haviv and Ritov (1998).
