# Maximizing throughput in flow shop real-time scheduling

## Abstract

We consider scheduling real-time jobs in the classic flow shop model. The input is a set of n jobs, each consisting of m segments to be processed on m machines in the specified order, such that segment Ii of a job can start processing on machine Mi only after segment Ii−1 of the same job completed processing on machine Mi−1, for 2 ≤ i ≤ m. Each job also has a release time, a due date, and a weight. The objective is to maximize the throughput (or, profit) of the n jobs, i.e., to find a subset of the jobs that have the maximum total weight and can complete processing on the m machines within their time windows. This problem has numerous real-life applications ranging from manufacturing to cloud and embedded computing platforms, already in the special case where m = 2. Previous work in the flow shop model has focused on makespan, flow time, or tardiness objectives. However, little is known for the flow shop model in the real-time setting. In this work, we give the first nontrivial results for this problem and present a pseudo-polynomial time (2m + 1)-approximation algorithm for the problem on m ≥ 2 machines, where m is a constant. This ratio is essentially tight due to a hardness result of Ω(m/logm) for the approximation ratio. We further give a polynomial-time algorithm for the two-machine case, with an approximation ratio of (9 + ε) where ε = O(1/n). We obtain better bounds for some restricted subclasses of inputs with two machines. To the best of our knowledge, this fundamental problem of throughput maximization in the flow shop scheduling model is studied here for the first time.