  In the area of discrete event simulation (DES), event simultaneity occurs
when any two events are scheduled to happen at the same point in simulated
time. Simulation determinism is the expectation that the same semantically
configured simulation will be guaranteed to repeatedly reproduce identical
results. Since events in DES are the sole mechanism for state change, ensuring
consistent real-time event processing order is crucial to maintaining
determinism. This is synonymous with finding a consistent total ordering of
events.
  In this work, we extend the concept of virtual time to utilize an
arbitrary-length series of tie-breaking values to preserve determinism in
parallel, optimistically executed simulations without imposing additional bias
influencing the ordering of otherwise incomparable events. Furthermore, by
changing the core pseudo-random number generator seed at initialization,
different orderings of events incomparable by standard virtual time can be
observed, allowing for fair probing of other potential simulation outcomes. We
implement and evaluate this extended definition of virtual time in the
Rensselaer Optimistic Simulation System (ROSS) with three simulation models and
discuss the importance of deterministic event ordering given the existence of
event ties.
