Better Bounds for Online $k$-Frame Throughput Maximization in Network Switches

We consider a variant of the online buffer management problem in network switches, called the $k$-frame throughput maximization problem ($k$-FTM). This problem models the situation where a large frame is fragmented into $k$ packets and transmitted through the Internet, and the receiver can reconstruct the frame only if he/she accepts all the $k$ packets. Kesselman et al.\ introduced this problem and showed that its competitive ratio is unbounded even when $k=2$. They also introduced an "order-respecting" variant of $k$-FTM, called $k$-OFTM, where inputs are restricted in some natural way. They proposed an online algorithm and showed that its competitive ratio is at most $\frac{2kB}{\lfloor B/k \rfloor} + k$ for any $B \ge k$, where $B$ is the size of the buffer. They also gave a lower bound of $\frac{B}{\lfloor 2B/k \rfloor}$ for deterministic online algorithms when $2B \geq k$ and $k$ is a power of 2. In this paper, we improve upper and lower bounds on the competitive ratio of $k$-OFTM. Our main result is to improve an upper bound of $O(k^{2})$ by Kesselman et al.\ to $\frac{5B + \lfloor B/k \rfloor - 4}{\lfloor B/2k \rfloor} = O(k)$ for $B\geq 2k$. Note that this upper bound is tight up to a multiplicative constant factor since the lower bound given by Kesselman et al.\ is $\Omega(k)$. We also give two lower bounds. First we give a lower bound of $\frac{2B}{\lfloor {B/(k-1)} \rfloor} + 1$ on the competitive ratio of deterministic online algorithms for any $k \geq 2$ and any $B \geq k-1$, which improves the previous lower bound of $\frac{B}{\lfloor 2B/k \rfloor}$ by a factor of almost four. Next, we present the first nontrivial lower bound on the competitive ratio of randomized algorithms. Specifically, we give a lower bound of $k-1$ against an oblivious adversary for any $k \geq 3$ and any $B$.