  We consider parallel simulations for asynchronous systems employing L
processing elements that are arranged on a ring. Processors communicate only
among the nearest neighbors and advance their local simulated time only if it
is guaranteed that this does not violate causality. In simulations with no
constraints, in the infinite L-limit the utilization scales (Korniss et al, PRL
84, 2000); but, the width of the virtual time horizon diverges (i.e., the
measurement phase of the algorithm does not scale). In this work, we introduce
a moving window global constraint, which modifies the algorithm so that the
measurement phase scales as well. We present results of systematic studies in
which the system size (i.e., L and the volume load per processor) as well as
the constraint are varied. The constraint eliminates the extreme fluctuations
in the virtual time horizon, provides a bound on its width, and controls the
average progress rate. The width of the window constraint can serve as a tuning
parameter that, for a given volume load per processor, could be adjusted to
optimize the utilization so as to maximize the efficiency. This result may find
numerous applications in modeling the evolution of general spatially extended
short-range interacting systems with asynchronous dynamics, including dynamic
Monte Carlo studies.
