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This paper focuses upon a particular conservative algorithm for parallel simulation, the Time of Next Event (TNE) suite of algorithms [13]. TNE relies upon a shortest path algorithm which is independently executed on each processor in order to unblock LPs in the processor and to increase the parallelism of the simulation. TNE differs fundamentally from other conservative approaches in that it takes advantage of having several LPs assigned to each processor, and does not rely upon message passing to provide lookahead. Instead, it relies upon a shortest path algorithm executed independently in each processor. A deadlock resolution algorithm is employed for inter-processor deadlocks. We describe an empirical investigation of the performance of TNE on the iPSC/i860 hypercube multiprocessor. Several factors which play an important role in TNE's behavior are identified, and the speedup relative to a fast uniprocessor-based event list algorithm is reported. Our results indicate that TNE yields good speedups and out-performs an optimized version of the Chandy&Misra-null message (CMB) algorithm. TNE was 2-5 times as fast as the CM approach for less than 10 processors (and 1.5-3 times as fast when more than 10 processors were used for the same population of processes.)