Online packet scheduling with bounded delay and lookahead

We study the online bounded-delay packet scheduling problem (Packet Scheduling), 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 ofφ≈1.618 [11, 2, 6].In the variant of Packet Scheduling with s-bounded instances, each packet can be scheduled in at most s consecutive slots, starting at its release time. The lower bound of φ applies even to the special case of 2-bounded instances, and a φ-competitive algorithm for 3-boundedinstances was given in [5]. Improving that result, and addressing a question posed by Goldwasser [9], we present a φ-competitive algorithm for 4-boundedinstances. We also study a variant of Packet Scheduling where an online algorithm has the additional power of1-lookahead, knowing at time t which packets will arrive at time t+ 1. For Packet Scheduling with 1-lookahead restricted to 2-bounded instances, we present an online algorithm with competitive ratio12(√13−1)≈1.303 and we prove a nearly tight lower boundof14(1 +√17)≈1.281. In fact, our lower bound result is more general: using only 2-boundedinstances, for any integer`≥0 we prove a lower bound of12(`+1)(1 +√5 + 8`+ 4`2) for online algorithms with`-look ahead, i.e., algorithms that at time t can see all packets arriving by time t+`. Finally, for non-restricted instances we show a lower bound of 1.25 for randomized algorithms with`-lookahead, for any`≥0