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Admission Control for Multidimensional Workload with Heavy Tails and Fractional Ornstein-Uhlenbeck Process

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 $H>1/2$.