In many on-demand online platforms such as ride-sharing, grocery delivery, or shipping, some arriving agents are patient and willing to wait a short amount of time for the resource or service as long as there is an upfront guarantee that service will be ultimately provided within a certain delay. Motivated by this, we present a setting with patient and impatient agents who seek a resource or service that replenishes periodically. Impatient agents demand the resource immediately upon arrival while patient agents are willing to wait a short period conditioned on an upfront commitment to receive the resource. We study this setting under adversarial arrival models using a relaxed notion of competitive ratio. We present a class of POLYtope-based Resource Allocation (POLYRA) algorithms that achieve optimal or near-optimal competitive ratios. Such POLYRA algorithms work by consulting a particular polytope and only making decisions that guarantee the algorithm's state remains feasible in this polytope. When the number of agent types is either two or three, POLYRA algorithms can obtain the optimal competitive ratio. To design these polytopes, we construct an upper bound on the competitive ratio of any algorithm, which is characterized via a linear program (LP) that considers a collection of overlapping worst-case input sequences. Our designed POLYRA algorithms then mimic the optimal solution of this upper bound LP via its polytope's definition, obtaining the optimal competitive ratio. When there are more than three types, our overlapping worst-case input sequences do not necessarily result in an attainable competitive ratio, and so we present a class of simple and interpretable POLYRA algorithm which achieves at least 80% of the optimal competitive ratio. We complement our theoretical studies with numerical analysis which shows the efficiency of our algorithms beyond adversarial arrivals

翻译：在许多按需在线平台（如拼车、杂货配送或物流运输）中，部分到达的代理具有耐心，只要事先保证在一定延迟内最终提供服务，他们就愿意等待短时间。受此启发，我们提出一个包含耐心和急躁代理的场景，这些代理寻求周期性补充的资源或服务。急躁代理在到达时立即要求资源，而耐心代理则愿意在获得提前承诺的前提下等待短时间。我们在对抗性到达模型下，利用竞争比的松弛概念研究该场景。我们提出一类基于多面体的资源分配（POLYRA）算法，可实现最优或接近最优的竞争比。此类POLYRA算法通过参考特定多面体并仅做出确保算法状态在该多面体中可行的决策来运行。当代理类型数为二或三时，POLYRA算法可获得最优竞争比。为设计这些多面体，我们构建了任意算法竞争比的上界，该上界通过一个线性规划（LP）刻画，该线性规划考虑了一组重叠的最坏情况输入序列。随后，我们设计的POLYRA算法通过其多面体定义模仿此上界LP的最优解，从而获得最优竞争比。当类型数超过三种时，我们的重叠最坏情况输入序列不一定能得到可达的竞争比，因此我们提出一类简单且可解释的POLYRA算法，该算法至少能达到最优竞争比的80%。我们通过数值分析补充理论研究，展示了算法在对抗性到达场景之外的效率。