The Harmonic Policy for Online Buffer Sharing is (2 + ln n)-Competitive: A Simple Proof

The problem of online buffer sharing is expressed as follows. A switch with $n$ output ports receives a stream of incoming packets. When an incoming packet is accepted by the switch, it is stored in a shared buffer of capacity $B$ common to all packets and awaits its transmission through its corresponding output port determined by its destination. Each output port transmits one packet per time unit. The problem is to find an algorithm for the switch to accept or reject a packet upon its arrival in order to maximize the total number of transmitted packets. Building on the work of Kesselman et al. (STOC 2001) on split buffer sharing, Kesselman and Mansour (TCS 2004) considered the problem of online buffer sharing which models most deployed internet switches. In their work, they presented the Harmonic policy and proved that it is $(2 + \ln n)$-competitive, which is the best known competitive ratio for this problem. The Harmonic policy unfortunately saw less practical relevance as it performs $n$ threshold checks per packets which is deemed costly in practice, especially on network switches processing multiple terabits of packets per second. While the Harmonic policy is elegant, the original proof is also rather complex and involves a lengthy matching routine along with multiple intermediary results. This note presents a simplified Harmonic policy, both in terms of implementation and proof. First, we show that the Harmonic policy can be implemented with a constant number of threshold checks per packet, matching the widely deployed \emph{Dynamic Threshold} policy. Second, we present a simple proof that shows the Harmonic policy is $(2 + \ln n)$-competitive. In contrast to the original proof, the current proof is direct and relies on a 3-partitioning of the packets.