Breaking the Barrier of 2 for the Competitiveness of Longest Queue Drop

TL;DR

This paper improves the known upper bound on the competitiveness of the Longest Queue Drop (LQD) algorithm for buffer management in shared-memory switches, showing it is at most approximately 1.69-competitive, better than the longstanding 2 bound.

Contribution

The paper establishes the first explicit upper bound below 2 for LQD's competitive ratio, advancing understanding of its performance.

Findings

LQD is proven to be at most 1.6918-competitive.

This is the first bound below 2 for LQD's competitiveness.

The result narrows the gap towards the optimal performance of LQD.

Abstract

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 $\varepsilon>0$.