Packet-mode policies for input-queued switches

This paper considers the problem of packet-mode scheduling of input queued switches. Packets have variable lengths, and are divided into cells of unit length. Each packet arrives to the switch with a given deadline by which it must traverse the switch. A packet successfully passes the switch if the sequence of cells comprising it is contiguously transmitted out of the switch before the packet's deadline expires. A packet transmission may be preempted and restarted from the beginning later. The scheduling policy has to decide at each time step which packets to serve. The problem is online in nature, and thus we use competitive analysis to measure the performance of our scheduling policies.First we consider the case where the goal of the switch policy is to maximize the total number of successfully transmitted packets. We derive two algorithms achieving the competitive ratios of (2^{2√log L} +1) and N+ 1, respectively, where L is the ratio between the longest and the shortest packet lengths and N is the number of input/output ports. We also show that any deterministic online algorithm has a competitive ratio of at least (⌊log L⌋ + 1, N).Then we study the general case in which each packet has an intrinsic value representing its priority, and the goal is to maximize the total value of successfully transmitted packets. We derive an algorithm which achieves a competitive ratio of 2κ+2√κ+1/2+ (2κ+√κ+1/2+1) (√κ+1/2+3), where κ is the ratio between the maximum and the minimum value per cell. We note that [4] gives a lower bound of Ω(κ) on the performance of any deterministic online algorithm. In particular, our algorithm achieves a competitive ratio of approximately 11.123 for κ=1, which improves upon the previous best-known upper bound for this problem [17].We complement our results by studying the offline version of the problem, which is NP-hard We give a pseudo-polynomial 3-approximation algorithm for the general case and a polynomial 3-approximation algorithm for the case of unit value packets.