可索引和不可索引不安分强盗的渐近最优优先策略
Asymptotically optimal priority policies for indexable and nonindexable restless bandits
摘要：
研究了多类不安分土匪的渐近最优控制。一个不安分的强盗是一个可控的随机过程，其状态演变取决于强盗是否被激活。由于寻找最优控制通常是棘手的，我们提出了一类优先级的政策，证明是渐近最优的全局吸引子的性质和技术条件。我们既考虑了固定的土匪人口，也考虑了土匪可以离开和到达的动态人口。作为一个例子，一个动态人口的土匪，我们分析了多类$\mathit{M/M/S+M}$队列，我们表现出渐近最优的索引政策。我们结合流体缩放技术与线性规划的结果证明，当土匪是可索引的，惠特尔的索引政策包括在我们的类的优先级政策。从而推广了Weber和Weiss [J. Appl. Probab. 27（1990）637-648]关于惠特尔指数策略对（i）几类强盗，（ii）新强盗的到来和（iii）多个行动的设置的渐近最优性。我们的结果并不需要土匪的指数化。对于不可索引的土匪，我们描述了如何选择优先级的政策，从类的渐近最优的政策和目前的数值证据表明，在渐近制度之外，我们提出的优先级政策的性能是接近最优的。
英文摘要：
We study the asymptotic optimal control of multi-class restless bandits. A restless bandit is a controllable stochastic process whose state evolution depends on whether or not the bandit is made active. Since finding the optimal control is typically intractable, we propose a class of priority policies that are proved to be asymptotically optimal under a global attractor property and a technical condition. We consider both a fixed population of bandits as well as a dynamic population where bandits can depart and arrive. As an example of a dynamic population of bandits, we analyze a multi-class $\mathit{M/M/S+M}$ queue for which we show asymptotic optimality of an index policy. We combine fluid-scaling techniques with linear programming results to prove that when bandits are indexable, Whittle's index policy is included in our class of priority policies. We thereby generalize a result of Weber and Weiss [J. Appl. Probab. 27 (1990) 637-648] about asymptotic optimality of Whittle's index policy to settings with (i) several classes of bandits, (ii) arrivals of new bandits and (iii) multiple actions. Indexability of the bandits is not required for our results to hold. For nonindexable bandits, we describe how to select priority policies from the class of asymptotically optimal policies and present numerical evidence that, outside the asymptotic regime, the performance of our proposed priority policies is nearly optimal.
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