  We consider the problem of managing the buffer of a shared-memory switch that
transmits packets of unit value. A shared-memory switch consists of an input
port, a number of output ports, and a buffer with a specific capacity. In each
time step, an arbitrary number of packets arrive at the input port, each packet
designated for one output port. Each packet is added to the queue of the
respective output port. If the total number of packets exceeds the capacity of
the buffer, some packets have to be irrevocably evicted. At the end of each
time step, each output port transmits a packet in its queue and the goal is to
maximize the number of transmitted packets.
  The Longest Queue Drop (LQD) online algorithm accepts any arriving packet to
the buffer. However, if this results in the buffer exceeding its memory
capacity, then LQD drops a packet from whichever queue is currently the
longest, breaking ties arbitrarily. The LQD algorithm was first introduced in
1991, and is known to be $2$-competitive since 2001. Although LQD remains the
best known online algorithm for the problem and is of practical interest,
determining its true competitiveness is a long-standing open problem. We show
that LQD is 1.6918-competitive, establishing the first $(2-\varepsilon)$ upper
bound for the competitive ratio of LQD, for a constant $\varepsilon>0$.
