In the online packet scheduling problem with deadlines (PacketSchD, for short), the goal is to schedule transmissions of packets that arrive over time in a network switch and need to be sent across a link. Each packet has a deadline, representing its urgency, and a non-negative weight, that represents its priority. Only one packet can be transmitted in any time slot, so if the system is overloaded, some packets will inevitably miss their deadlines and be dropped. In this scenario, the natural objective is to compute a transmission schedule that maximizes the total weight of packets that are successfully transmitted. The problem is inherently online, with the scheduling decisions made without the knowledge of future packet arrivals. The central problem concerning PacketSchD, that has been a subject of intensive study since 2001, is to determine the optimal competitive ratio of online algorithms, namely the worst-case ratio between the optimum total weight of a schedule (computed by an offline algorithm) and the weight of a schedule computed by a (deterministic) online algorithm. We solve this open problem by presenting a $\phi$-competitive online algorithm for PacketSchD (where $\phi\approx 1.618$ is the golden ratio), matching the previously established lower bound.

翻译：针对带截止时间的在线数据包调度问题（简称PacketSchD），其目标是在网络交换机中调度随时间到达的数据包传输，并使其通过链路发送。每个数据包都有一个表示紧急程度的截止时间，以及一个代表优先级的非负权重。在任何时隙中只能传输一个数据包，因此当系统过载时，部分数据包将不可避免地错过截止时间而被丢弃。在此场景下，自然目标是计算一个传输调度方案，使得成功传输的数据包总权重最大化。该问题本质上是在线的，调度决策需在不了解未来数据包到达信息的情况下做出。自2001年起，关于PacketSchD的核心问题一直是确定在线算法的最优竞争比，即最优调度方案（由离线算法计算得出）的总权重与（确定性）在线算法计算的调度方案权重之间的最坏情况比值。我们通过提出一个具有$\phi$竞争比的PacketSchD在线算法（其中$\phi\approx 1.618$为黄金分割比）解决了这一开放问题，该结果与先前确立的下界相匹配。