  The infinite source Poisson arrival model with heavy-tailed workload
distributions has attracted much attention, especially in the modeling of data
packet traffic in communication networks. In particular, it is well known that
under suitable assumptions on the source arrival rate, the centered and scaled
cumulative workload process for the underlying processing system can be
approximated by fractional Brownian motion. In many applications one is
interested in the stabilization of the work inflow to the system by modifying
the net input rate, using an appropriate admission control policy. In this work
we study a natural family of admission control policies which keep the
associated scaled cumulative workload asymptotically close to a pre-specified
linear trajectory, uniformly over time. Under such admission control policies
and with natural assumptions on arrival distributions, suitably scaled and
centered cumulative workload processes are shown to converge weakly in the path
space to the solution of a $d$-dimensional stochastic differential equation
(SDE) driven by a Gaussian process. It is shown that the admission control
policy achieves moment stabilization in that the second moment of the solution
to the SDE (averaged over the $d$-stations) is bounded uniformly for all times.
In one special case of control policies, as time approaches infinity, we obtain
a fractional version of a stationary Ornstein-Uhlenbeck process that is driven
by fractional Brownian motion with Hurst parameter $H>1/2$.
