We study a variant of the k -server problem, the infinite server problem, in which infinitely many servers reside initially at a particular point of the metric space and serve a sequence of requests. In the framework of competitive analysis, we show a surprisingly tight connection between this problem and the resource augmentation version of the k -server problem, also known as the (h,k) -server problem, in which an online algorithm with k servers competes against an offline algorithm with h servers. Specifically, we show that the infinite server problem has bounded competitive ratio if and only if the (h,k) -server problem has bounded competitive ratio for some k = O ( h ). We give a lower bound of 3.146 for the competitive ratio of the infinite server problem, which holds even for the line and some simple weighted stars. It implies the same lower bound for the (h,k) -server problem on the line, even when k/h → ∞, improving on the previous known bounds of 2 for the line and 2.4 for general metrics. For weighted trees and layered graphs, we obtain upper bounds, although they depend on the depth. Of particular interest is the infinite server problem on the line, which we show to be equivalent to the seemingly easier case in which all requests are in a fixed bounded interval. This is a special case of a more general reduction from arbitrary metric spaces to bounded subspaces. Unfortunately, classical approaches (double coverage and generalizations, work function algorithm, balancing algorithms) fail even for this special case.

中文翻译：

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无限服务器问题

我们研究了一个变种ķ-服务器问题，无限服务器问题，其中无限多的服务器最初驻留在度量空间的特定点并服务于一系列请求。在竞争分析的框架中，我们展示了这个问题与资源增强版本之间惊人的紧密联系。ķ-服务器问题，也称为(h,k)-服务器问题，其中一个在线算法与ķ服务器与离线算法竞争H服务器。具体来说，我们证明了无限服务器问题具有有限的竞争比率当且仅当(h,k)-服务器问题对某些人来说具有有限的竞争力ķ=○(H）。我们为无限服务器问题的竞争比率给出了 3.146 的下限，即使对于直线和一些简单的加权星也适用。这意味着相同的下限(h,k)-服务器问题就行了，即使当千/小时→ ∞，改进了之前已知的直线边界 2 和一般度量的 2.4。对于加权树和分层图，我们获得了上限，尽管它们取决于深度。特别感兴趣的是在线上的无限服务器问题，我们证明它等同于所有请求都在固定有界区间内的看似更简单的情况。这是从任意度量空间到有界子空间的更一般化简的特殊情况。不幸的是，即使对于这种特殊情况，经典方法（双重覆盖和泛化、功函数算法、平衡算法）也会失败。