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

Abstract

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 ≥ 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 ≥ 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 ≥ 2k. Note that this upper bound is tight up to a multiplicative constant factor since the lower bound given by Kesselman et al. is Ω(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 ≥ 2 and any B ≥ 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 ≥ 3 and any B. Since a deterministic algorithm, as mentioned above, achieves an upper bound of about 10k, this indicates that randomization does not help too much.