Minimizing Large Deviation Paths for a Family of Long-Range Dependent Processes and Their Fractional Brownian Approximations

Abstract

The long-range dependent cumulative traffic processes generated by the superposition of constant rate fluid sources having exponentially distributed inter-start times and Pareto-distributed durations with finite mean and infinite variance satisfy a large deviation principle when the session intensity is increased and the processes are centered and appropriately scaled. We show that close to its zero point the rate function in this large deviation principle behaves approximately like the one appearing in a sample path large deviation principle for fractional Brownian motion. This allows for an approximation of large deviation rates and minimizing paths of the traffic processes by those of fractional Brownian motions. We explicitly calculate certain minimizing paths representing the asymptotically most probable behavior of the traffic processes leading to a fixed positive or negative increase and compare them to the corresponding behavior of fractional Brownian motions.

We develop analogous results for the minimizing paths of the queue length processes obtained by sending the traffic through a buffer with infinite capacity and constant service rate. These paths are approximated by minimizing paths of reflected fractional Brownian motion.

Keywords:

• Most probable path
• Queue length process
• Rate function convergence
• Reflected fractional Brownian motion

Mathematics Subject Classification:

• Primary 60F10
• Secondary 60J65, 60K25, 68M20