Large Buffer Asymptotics for Fluid Queues with Heterogeneous M/G/∞ Weibullian Inputs

Abstract

In this paper we consider a generalization of the so called M/G/∞ model where M types of sessions enter a buffer. The instantaneous rates of the sessions are functions of the occupancy of an M/G/∞ system with Weibullian G distributions. In particular we assume that a session of type i transmits r _i cells per unit time and lasts for a random time τ with a Weibull distribution given by $\Pr (\tau > x) \sim {\text{e}}^{{\text{ - }}\gamma ix^{\alpha i} }$, where 0<α _i <1, γ _i >0. We show that the complementary buffer occupancy distribution for large buffer size is Weibullian whose parameters can be determined as the solution of a deterministic nonlinear knapsack problem. For α _i <0.5, upper and lower bound factors are determined. When specialized to the homogeneous case, i.e., when all the sessions are identical, the result coincides with a lower bound reported in the literature.