The operating characteristics of a typical single-node hierarchical element in a multiprocessor network such as might be used for process control are described, and the class of scheduling problems arising from the use of such a network is discussed and related to similar problems which have been reported in the literature. Two measures of system performance are then defined: the "settling time," defined as the maximum of the finishing times of all jobs when a set of jobs is initiated once in the network, and the "steady-state service," a related performance index for the case where jobs are initiated on a periodic basis. Finally, a simulation model is used to evaluate the operating characteristics of the hierarchical network using the two metrics in conjunction with several synthetic workloads; a general optimization procedure is shown to lead to values of scheduling parameters which are close to optimal even for network workloads with many degrees of freedom (i.e., when many computers are initiating jobs).