We consider the Longest Queue Drop memory management policy in shared-memory switches consisting of $N$ output ports. The shared memory of size $M\geq N$ may have an arbitrary number of input ports. Each packet may be admitted by any incoming port, but must be destined to a specific output port and each output port may be used by only one queue. The Longest Queue Drop policy is a natural online strategy used in directing the packet flow in buffering problems. According to this policy and assuming unit packet values and cost of transmission, every incoming packet is accepted, whereas if the shared memory becomes full, one or more packets belonging to the longest queue are preempted, in order to make space for the newly arrived packets. It was proved in 2001 [Hahne et al., SPAA '01] that the Longest Queue Drop policy is 2-competitive and at least $\sqrt{2}$-competitive. It remained an open question whether a (2-\epsilon) upper bound for the competitive ratio of this policy could be shown, for any positive constant \epsilon. We show that the Longest Queue Drop online policy is 1.5-competitive.

翻译：我们考虑由$N$个输出端口组成的共享内存交换机中的最长队列丢弃内存管理策略。大小为$M\geq N$的共享内存可拥有任意数量的输入端口。每个数据包可由任意入端口接收，但必须发往指定输出端口，且每个输出端口仅能由一个队列使用。最长队列丢弃策略是一种用于缓冲问题中数据包流导向的自然在线策略。根据该策略，在假设数据包单位价值与传输成本的前提下，每个进入数据包均被接受；当共享内存满时，将抢占属于最长队列的一个或多个数据包，以释放空间容纳新到达数据包。2001年[Hahne等人, SPAA '01]已证明该策略的竞争比为2且至少为$\sqrt{2}$。该策略是否存在(2-\epsilon)的上界（对任意正常数\epsilon）一直是未解问题。我们证明最长队列丢弃在线策略的竞争比为1.5。