We consider the problem of selfish agents in discrete-time queuing systems, where competitive queues try to get their packets served. In this model, a queue gets to send a packet each step to one of the servers, which will attempt to serve the oldest arriving packet, and unprocessed packets are returned to each queue. We model this as a repeated game where queues compete for the capacity of the servers, but where the state of the game evolves as the length of each queue varies, resulting in a highly dependent random process. Earlier work by the authors [EC'20] shows that with no-regret learners, the system needs twice the capacity as would be required in the coordinated setting to ensure queue lengths remain stable despite the selfish behavior of the queues. In this paper, we demonstrate that this way of evaluating outcomes is myopic: if more patient queues choose strategies that selfishly maximize their long-run success rate, stability can be ensured with just $\frac{e}{e-1}\approx 1.58$ times extra capacity, better than what is possible assuming the no-regret property. As these systems induce highly dependent processes, our analysis draws heavily on techniques from probability theory. Though these systems are random under any fixed policies by the queues, we show that, surprisingly, these systems have deterministic and explicit asymptotic behavior. We show that the asymptotic growth rates of queues can be written as a ratio of a submodular and modular function, which provides significant game-theoretic properties. Our equilibrium analysis then relies on a novel deformation argument towards a more analyzable solution that differs significantly from previous price of anarchy results. While the intermediate points will not be equilibria, this analytic structure will ensure that this deformation is monotonic along this continuous path.

中文翻译：

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战略排队系统中的耐心美德

我们考虑离散时间排队系统中自私代理的问题，在这种系统中，竞争性队列试图使他们的数据包得到服务。在此模型中，队列将在每个步骤中将数据包发送到其中一个服务器，该服务器将尝试服务最早到达的数据包，并且未处理的数据包将返回到每个队列。我们将此模型模拟为一个重复的游戏，其中队列竞争服务器的容量，但是游戏状态随着每个队列长度的变化而演变，从而导致高度依赖的随机过程。作者[EC'20]的早期工作表明，对于无悔学习者，系统需要的容量是协调设置中所需容量的两倍，以确保尽管队列自私，队列长度仍保持稳定。在本文中，我们证明了这种评估结果的方法是近视的：如果更多的患者队列选择自私地最大化其长期成功率的策略，则只需$ \ frac {e} {e-1} \大约1.58 $倍的额外容量就可以确保稳定性，这要比假设没有容量的情况更好。后悔财产。由于这些系统引发高度依赖的过程，因此我们的分析大量借鉴了概率论中的技术。尽管这些系统在队列的任何固定策略下都是随机的，但令人惊讶的是，我们证明了这些系统具有确定性和显式渐近行为。我们表明，队列的渐近增长率可以表示为亚模块和模块函数的比率，这提供了重要的博弈论性质。然后，我们的均衡分析依赖于一种新颖的变形论证，该论证指向一种更具可分析性的解决方案，该解决方案与先前的无政府状态结果价格有显着差异。尽管中间点不会达到平衡，但这种分析结构将确保该变形沿此连续路径是单调的。