具有线性减速的处理机共享系统的调度
Scheduling for a Processor Sharing System with Linear Slowdown
摘要：
我们考虑的问题调度到达的拥塞系统与有限数量的用户具有相同的确定性需求大小。拥塞是处理器共享类型的，在这个意义上，在任何给定的时间，系统中的所有用户都同时得到服务。然而，与经典的处理器共享拥塞模型相比，处理速度的减慢与系统中的用户数量成正比。也就是说，所有用户所体验的服务速率随着用户的数量而线性下降。对于每个用户都有一个理想的出发时间（截止日期）。一个集中式的调度目标，然后选择到达时间，以便最小化由于偏离理想的时间加权逗留时间的总惩罚。假设每个偏差是二次的，或者更一般地是凸的。但由于系统的动态性，调度目标函数是非凸的。具体地，系统目标函数是非光滑分段凸函数。尽管如此，我们能够利用问题的结构，推导出一个算法，找到全局最优值在一个（大，但）有限数量的步骤，每个涉及的解决方案的约束凸计划。在此基础上，提出了几种新的解决方案。第一种方法是遍历相邻约束凸规划问题，保证到达集中问题的一个局部极小值。这是一种“局部搜索”的形式，我们以一种新颖的方式使用问题结构。第二个是一个坐标的“全局搜索”，用于坐标枢轴迭代。然后，我们将这两个启发式合并成一个统一的“局部-全局”启发式，并数值说明了这种启发式的有效性。
英文摘要：
We consider the problem of scheduling arrivals to a congestion system with a finite number of users having identical deterministic demand sizes. The congestion is of the processor sharing type in the sense that all users in the system at any given time are served simultaneously. However, in contrast to classical processor sharing congestion models, the processing slowdown is proportional to the number of users in the system at any time. That is, the rate of service experienced by all users is linearly decreasing with the number of users. For each user there is an ideal departure time (due date). A centralized scheduling goal is then to select arrival times so as to minimize the total penalty due to deviations from ideal times weighted with sojourn times. Each deviation is assumed quadratic, or more generally convex. But due to the dynamics of the system, the scheduling objective function is non-convex. Specifically, the system objective function is a non-smooth piecewise convex function. Nevertheless, we are able to leverage the structure of the problem to derive an algorithm that finds the global optimum in a (large but) finite number of steps, each involving the solution of a constrained convex program. Further, we put forward several heuristics. The first is the traversal of neighbouring constrained convex programming problems, that is guaranteed to reach a local minimum of the centralized problem. This is a form of a "local search", where we use the problem structure in a novel manner. The second is a one-coordinate "global search", used in coordinate pivot iteration. We then merge these two heuristics into a unified "local-global" heuristic, and numerically illustrate the effectiveness of this heuristic.
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