Scheduling tasks with precedence constraints on multiple servers

We consider the problem of scheduling jobs which are modeled by directed acyclic graphs (DAG). In such graphs, nodes represent tasks of a job and edges represent precedence constraints in processing these tasks. The DAG scheduling problem, also known as scheduling in fork-join processing networks, is motivated by examples such as job scheduling in data centers and cloud computing, patient flow scheduling in health systems and many other applications. We consider a flexible system, in which servers may process different, possibly overlapping, sets of task types. In this paper, we first discuss the difficulties in designing provably efficient policies for DAG scheduling, which arise due to interactions between the flexibility of the processing environment and the precedence constraints in the system. A major difficulty is the classical synchronization issue, which is further complicated in the presence of system flexibility. Then, we propose two queueing networks to model the scheduling problem that overcome this difficulty. These are virtual queues that enable us to design provably efficient scheduling policies. We show that the well-known Max-Weight policy for these queueing networks is throughput-optimal. Finally, to compare the delay performance of the two queueing networks, we consider a simplified model in which tasks and servers are identical. We characterize their delay performances under a simple first-come-first-serve policy, via a novel coupling argument.