Differentiated bandwidth sharing with disparate flow sizes

Abstract

Abstract-We consider a multi-class queueing system operating under the Discriminatory Processor-Sharing (DPS) discipline. The DPS discipline provides a natural approach for modeling the flow-level performance of differentiated bandwidth-sharing mechanisms. Motivated by the extreme diversity in flow sizes observed in the Internet, we examine the system performance in an asymptotic regime where the flow dynamics of the various classes occur on separate time scales. Specifically, from the perspective of a given class, the arrival and service completions of some of the competing classes (called mice) evolve on an extremely fast time scale. In contrast, the flow dynamics of the remaining classes (referred to as elephants) occur on a comparatively slow time scale. Assuming a strict separation of time scales, we obtain simple explicit expressions for various performance measures of interest, such as the distribution of the numbers of flows, mean delays, and flow throughputs. In particular, the latter performance measures are insensitive, in the sense that they only depend on the service requirement distributions through their first moments. Numerical experiments show that the limiting results provide remarkably accurate approximations in certain cases 1 .

Comparison of bandwidth-sharing policies in a linear network

“…Another interesting observation from the numerical section is that the total mean number of users is monotone in μ0 for given load ρ0, when the other parameters are kept fixed, see Figure 4 b). Similar monotonicity results have been discussed for a single-server queue in [3,11], but to the best of our knowledge there does not exist any proof. There is no hope that this monotonicity property can be proved using sample-path arguments, since this requires the same realizations for the service requirements.…”

Monotonicity Properties for Multi-Class Queueing Systems

“…When α > 1, the quasi-stationary regime (µ 0 → ∞) is a lower bound on the total mean number of users and the fluid regime (µ 0 → 0) an upper bound on the total mean number of users, and when α < 1 vice versa. A similar observation was made in [18] for a DPS queue.…”

“…Hence it seems plausible that giving relatively more preference to classes with a high c i µ i will decrease the weighted mean number of users. For a single-server system with only two classes of users (K = 2) we can indeed prove this, see (18) and (19) in Proposition 6.1. A single server with two classes is equivalent to a linear network with one node (L = 1).…”

“…In [5] the authors introduce the so-called quasi-stationary and fluid limit regimes (see also [18]). In these regimes, a complete separation of time scales takes place which can greatly simplify the analysis.…”

Beyond processor sharing

“…In the limit, a strict separation of time scales occurs, allowing for exact analysis of the queue length distribution. Under the technical assumption of phase-type distributions [43], this yields the following result for K = 2. (All limits in this subsection hold as r → ∞, which we omit for compactness of the presentation.…”