The Longest Queue Drop Policy for Shared-Memory Switches is   1.5-competitive
  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. 
Nicolaos Matsakis
cs.DS