Speed Scaling with Tandem Servers

Speed scaling for a tandem server setting is considered, where there is a series of servers, and each job has to be processed by each of the servers in sequence. Servers have a variable speed, their power consumption being a convex increasing function of the speed. We consider the worst case setting as well as the stochastic setting. In the worst case setting, the jobs are assumed to be of unit size with arbitrary (possibly adversarially determined) arrival instants. For this problem, we devise an online speed scaling algorithm that is constant competitive with respect to the optimal offline algorithm that has non-causal information. The proposed algorithm, at all times, uses the same speed on all active servers, such that the total power consumption equals the number of outstanding jobs. In the stochastic setting, we consider a more general tandem network, with a parallel bank of servers at each stage. In this setting, we show that random routing with a simple gated static speed selection is constant competitive. In both cases, the competitive ratio depends only on the power functions, and is independent of the workload and the number of servers.