Flow shop for dual CPUs in dynamic voltage scaling

We study the following flow shop scheduling problem on two processors. We are given n jobs with a common deadline D, where each job j has workload p_i,j on processor i and a set of processors which can vary their speed dynamically. Job j can be executed on the second processor if the execution of job j is completed on the first processor. Our objective is to find a feasible schedule such that all jobs are completed by the common deadline D with minimized energy consumption. For this model, we present a linear program for the discrete speed case, where the processor can only run at specific speeds in S={s₁,s₂,⋯,s_q} and the job execution order is fixed. We also provide a m^α−1-approximation algorithm for the arbitrary order case and for continuous speed model where m is the number of processors and α is a parameter of the processor. 

We then introduce a new variant of flow shop scheduling problem called sense-and-aggregate model motivated by data aggregation in wireless sensor networks where the base station needs to receive data from sensors and then compute a single aggregate result. In this model, the first processor will receive unit size data from sensors and the second processor is responsible for calculating the aggregate result. The second processor can decide when to aggregate and the workload that needs to be done to aggregate x data will be f(x) and another unit size data will be generated as the result of the partial aggregation which will then be used in the next round aggregation. Our objective is to find a schedule such that all data are received and aggregated by the deadline with minimum energy consumption. We present an O(n⁵) dynamic programming algorithm when f(x)=x and a greedy algorithm when f(x)=x−1. 

Finally, we investigate the performance of the flowshop problem when the order of jobs is fixed by comparing it to the approximation algorithm with an arbitrary order. We show experimentally that the approximation ratio is close to 1 when there are few machines and when there are more jobs.