The superposition of alternating on-off flows and a fluid model

Abstract

An on-off process is a 0-1 process $\xi_t$ in which consecutive 0-periods ${T_{0, n}}$ alternate with 1-periods ${T_{1, n}}(n = 1, 2,\dots)$. The on and off time sequences are independent, each consisting of i.i.d. r.v.s. By the superposed flow, we mean the process $Z_t = \Sigma_{\ell=1}^N r^{\ell}\xi_t^{\ell}$, where $r^{\ell} > 0$ and ${\xi_t^1}, \dots,{\xi_t^N}$ are independent on-off flows. The process $\xi_t^{\ell}$ is not Markovian; however, with the age component $\eta_t^{\ell}$, the process $w_t^{\ell} = (\xi_t^{\ell}, \eta_t^{\ell})$ is a piecewise deterministic Markov process. In this paper we study the buffer content process for which the tail of its steady-state distribution $\Psi (b)$ fulfills inequality $C_- e^{\gammab} \leq \Psi (b) \leq C_+ e^{-\gammab}, where $\gamma > 0$ is the solution of some basic nonlinear system of equations.