arXiv 1604.048742016-04-17math.PR

测度值路上的Skorokhod映射及其在优先排队中的应用

A Skorokhod Map on Measure-Valued Paths with Applications to Priority Queues

Rami Atar, Anup Biswas, Haya Kaspi, Kavita Ramanan

备注：: 51 pages, 1 figure

摘要：

半直线上的Skorokhod映射已被证明是研究非负约束过程的一个有用工具。在这项工作中，我们介绍了一个测量值模拟这个地图，转换的每个元素$ζ$的某类càdlàg路径，采取的空间中的带号测度在半线上的值的càdlàg路径，采取的空间中的非负措施$[0,\infty)$在这样的方式，对于每个$x > 0$，路径$t \mapsto ζ_t[0,x]$是通过Skorokhod映射变换半线上，且不同$x > 0$的调节功能耦合。我们建立了这个地图的规律性，并表明该地图提供了一个方便的工具，用于研究任务的优先级，根据一个连续的参数的调度系统。三个著名的模型是最早截止日期优先，最短作业优先和最短剩余处理时间的调度策略。对于这些应用程序，我们展示了如何映射提供了一个统一的框架内，形成流体模型方程，证明这些方程的解的唯一性，并建立收敛的缩放状态过程的流体模型。特别是，对于这些模型，我们得到了新的收敛结果在时间不均匀的设置，这似乎属于现有方法的范围之外。

英文摘要：

The Skorokhod map on the half-line has proved to be a useful tool for studying processes with non-negativity constraints. In this work we introduce a measure-valued analog of this map that transforms each element $ζ$ of a certain class of càdlàg paths that take values in the space of signed measures on the half-line to a càdlàg path that takes values in the space of non-negative measures on $[0,\infty)$ in such a way that for each $x > 0$, the path $t \mapsto ζ_t[0,x]$ is transformed via a Skorokhod map on the half-line, and the regulating functions for different $x > 0$ are coupled. We establish regularity properties of this map and show that the map provides a convenient tool for studying queueing systems in which tasks are prioritized according to a continuous parameter. Three such well known models are the earliest-deadline-first, the shortest-job-first and the shortest-remaining-processing-time scheduling policies. For these applications, we show how the map provides a unified framework within which to form fluid model equations, prove uniqueness of solutions to these equations and establish convergence of scaled state processes to the fluid model. In particular, for these models, we obtain new convergence results in time-inhomogeneous settings, which appear to fall outside the purview of existing approaches.

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摘要：

英文摘要：

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