We consider the problem of task scheduling for dispersed computing in which arriving jobs are modeled as chains, where the nodes represent the smaller tasks of a job, and edges represent precedence constraints among tasks. In our model, motivated by significant communication cost in large scale computing environments, the communication times are taken into account. We consider a network where servers are capable of serving all task types and sending the output of processed tasks to other servers. In this paper, we first characterize the capacity region of the network and then propose a virtual queueing network encoding the state of the network. We show that the Max-Weight policy is throughput-optimal for the network. Moreover, we extend our work to a more general case where jobs are modeled by directed acyclic graph (DAG).