Asymptotic behavior of a discrete-time queue with long range dependent input

In this paper we derive an expression for the asymptotics of the buffer length distribution of a discrete-time infinite capacity single server queue with deterministic service time and its input process belonging to a class of long range dependent discrete-time M/G/∞ processes. This class of arrival process is defined as follows. At each time slot sequences of back-to back customers are generated according to a Poisson distribution with parameter X. The length of such a sequence is assumed to asymptotically behave like a Pareto distribution with parameter s, i.e. the probability that a sequence consists of k customers is given by ck^-3 for k→∞, with c>0 and 2<s<3. Due to the heavy tail of the members of this class of distributions, the presented class of M/G/∞ processes has the long range dependence property (i.e., the autocorrelation function decays as a power of the lag time). We show that in this case the asymptotic behavior of the tail probabilities of the stationary distribution of the buffer occupancy is given by (λcp^s-2/(s-2)(s-1)(1-ρ))n^2-s for n→∞, with ρ representing the load of the system. This result is obtained using a generating function approach and the Tauberian theorem for a power series. Furthermore, an application towards traffic management and simulation results are presented