Distributed simulation often requires a large amount of overhead that can severely affect performance of the simulator. In this paper we study the distributed simulation of a special class of systems, namely acyclic systems, which can be simulated via distributed simulation with low overhead. These acyclic systems are any arbitrary feedforward networks of processes that communicate solely by exchanging messages asynchronously. Such systems include acyclic logic circuits, acyclic queuing networks, etc. We define a low overhead distributed simulation scheme specifically tailored for these systems and prove correctness of this scheme. The scheme is simple to implement and the number of overhead messages is nearly zero. By isolating the class of acyclic systems and designing a distributed simulation algorithm specifically geared for it, we have been able to avoid almost all overhead messages. By contrast, other distributed simulation algorithms are usually designed for more general systems and they usually require a significant number of overhead messages, not only for more general systems, but also for these acyclic systems.