State-Dependent Benes Buffer Model with Fast Loading and Output Rates

Abstract

We consider a state-dependent generalization of the exponential Benes model of single-source buffer system in which the source process consists of alternating transmission and idle periods. Martingale methods are applied for analyzing limit nonstationary behavior of the buffer content process, when the buffer is loaded and depleted, with rates proportional to a large parameter $N$. Depending on traffic conditions, defined by parameters of the model, different types of approximations are established for the buffer content. We show that in heavy traffic the buffer content grows linearly in $N$, whereas the deviations of the order $\sqrt{N}$ from the deterministic limit are approximated by the Gaussian diffusion process. In moderate traffic the buffer content grows as $\sqrt{N}$, and the normalized buffer content is approximated by a diffusion process with reflection at zero. In the case of normal traffic, we show that the buffer utilization tends to the ratio of "the input-to-output rate." Moreover, we show that the main contribution to the utilization comes from arbitrary small buffer content.