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Distributed simulation often involves a large number of overhead messages. This may lead to severe performance degradation of the simulator as pointed out in several previous works. In this work special classes of systems which can be simulated with low overhead-in most systems identified here the total number of overhead messages on any given communication line is at most one, irrespective of the time up to which the physical system is to be simulated-are presented. This includes acyclic systems, some closed queuing networks, some open cyclic queuing networks, and other special types of cyclic networks. Distributed simulation algorithms specially tailored to these systems are given. By isolating these classes of systems and designing distributed simulation algorithms specifically geared for them, a large number of overhead messages can be avoided. By contrast, other distributed simulation algorithms are usually designed for more general systems and they usually require too many overhead messages, even in simulating these special classes of systems.