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 2kB⌊B/k⌋+k for any B≥k, where B is the size of the buffer. They also gave a lower bound of B⌊2B/k⌋ 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(k2) by Kesselman et al. to 5B+⌊B/k⌋−4⌊B/2k⌋=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 2B⌊B/(k−1)⌋+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 B⌊2B/k⌋ 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.