  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.
