  This paper introduces a new asymptotic regime for simplifying stochastic
models having non-stationary effects, such as those that arise in the presence
of time-of-day effects. This regime describes an operating environment within
which the arrival process to a service system has an arrival intensity that is
fluctuating rapidly. We show that such a service system is well approximated by
the corresponding model in which the arrival process is Poisson with a constant
arrival rate. In addition to the basic weak convergence theorem, we also
establish a first order correction for the distribution of the cumulative
number of arrivals over $[0,t]$, as well as the number-in-system process for an
infinite-server queue fed by an arrival process having a rapidly changing
arrival rate. This new asymptotic regime provides a second regime within which
non-stationary stochastic models can be reasonably approximated by a process
with stationary dynamics, thereby complementing the previously studied setting
within which rates vary slowly in time.
