Online Scheduling of Bounded Length Jobs to Maximize Throughput
Résumé
We consider an online scheduling problem, motivated by the issues present at the joints of networks using ATM and TCP/IP. Namely, IP packets have to be broken down into small ATM cells and sent out before their deadlines, but cells corresponding to different packets can be interwoven. More formally, we consider the online scheduling problem with preemptions, where each job $j$ is revealed at release time $r_j$, and has processing time $p_j$, deadline $d_j$, and weight $w_j$. A preempted job can be resumed at any time. The goal is to maximize the total weight of all jobs completed on time. Our main results are as follows. Firstly, we prove that when the processing times of all jobs are at most $k$, the optimum deterministic competitive ratio is $\Theta(k /\log k)$. Secondly, we give a deterministic algorithm with competitive ratio depending on the ratio between the smallest and the largest processing time of all jobs. In particular, it attains competitive ratio 5 in the case when all jobs have identical processing times, for which we give a lower bound of 2.598. The latter upper bound also yields an $O(\log k)$-competitive randomized algorithm for the variant with processing times bounded by $k$.