Abstract
We study the online bounded delay packet scheduling problem (PacketScheduling), where packets of unit size arrive at a router over time and need to be transmitted over a network link. Each packet has two attributes: a non-negative weight and a deadline for its transmission.
The objective is to maximize the total weight of the transmitted packets. This problem has been well studied in the literature; yet currently the best published upper bound is 1.828 [8], still quite far from the best lower bound of phi approximate to 1.618 [11,2,6].
In the variant of PacketScheduling with s-bounded instances, each packet can be scheduled in at most s consecutive slots, starting at its release time. The lower bound of phi applies even to the special case of 2-bounded instances, and a phi-competitive algorithm for 3-bounded instances was given in [5].
Improving that result, and addressing a question posed by Goldwasser [9], we present a phi-competitive algorithm for 4 bounded instances. We also study a variant of PacketScheduling where an online algorithm has the additional power of 1- lookahead, knowing at time t which packets will arrive at time t + 1.
For PacketScheduling with 1-lookahead restricted to 2-bounded instances, we present an online algorithm with competitive ratio 1/2 (root 13-1) approximate to 1.303 and we prove a nearly tight lower bound of 1/4 (1 + root 17) approximate to 1.281. In fact, our lower bound result is more general: using only 2-bounded instances, for any integer l >= 0 we prove a lower bound of 1/2(l+1) (1+root 5+8l+4l(2)) for online algorithms with l-lookahead, i.e., algorithms that at time t can see all packets arriving by time t + l.
Finally, for non-restricted instances we show a lower bound of 1.25 for randomized algorithms with l-lookahead, for any l >= 0.