Service requests arrive to an input buffer in a digital system. The buffer capacity and work discipline are to be chosen such that the maximum rate at which tasks are processed through the buffer while still meeting delay criteria is not above some maximum level. This input buffer is intended to be a control valve to prevent internal system overload. A processor visits the buffer at random time epochs, serves up to a given maximum number of tasks, leaves, and repeats this process. The equilibrium probability of buffer overflow, and the equilibrium waiting time distribution of a task until it is processed are studied as a function of model parameters.