Stability of a Subcritical Fluid Model for Fair Bandwidth Sharing with General File Size Distributions

Abstract

This work concerns the asymptotic behavior of solutions to a (strictly) subcritical fluid model for a data communication network, where file sizes are generally distributed and the network operates under a fair bandwidth-sharing policy. Here we consider fair bandwidth-sharing policies that are a slight generalization of the [Formula: see text]-fair policies introduced by Mo and Walrand [Mo J, Walrand J (2000) Fair end-to-end window-based congestion control. IEEE/ACM Trans. Networks 8(5):556–567.] Since the year 2000, it has been a standing problem to prove stability of the data communications network model of Massoulié and Roberts [Massoulié L, Roberts J (2000) Bandwidth sharing and admission control for elastic traffic. Telecommunication Systems 15(1):185–201.], with general file sizes and operating under fair bandwidth sharing policies, when the offered load is less than capacity (subcritical conditions). A crucial step in an approach to this problem is to prove stability of subcritical fluid model solutions. In 2012, Paganini et al. [Paganini F, Tang A, Ferragut A, Andrew LLH (2012) Network stability under alpha fair bandwidth allocation with general file size distribution. IEEE Trans. Automatic Control 57(3):579–591.] introduced a Lyapunov function for this purpose and gave an argument, assuming that fluid model solutions are sufficiently smooth in time and space that they are strong solutions of a partial differential equation and assuming that no fluid level on any route touches zero before all route levels reach zero. The aim of the current paper is to prove stability of the subcritical fluid model without these strong assumptions. Starting with a slight generalization of the Lyapunov function proposed by Paganini et al., assuming that each component of the initial state of a measure-valued fluid model solution, as well as the file size distributions, have no atoms and have finite first moments, we prove absolute continuity in time of the composition of the Lyapunov function with any subcritical fluid model solution and describe the associated density. We use this to prove that the Lyapunov function composed with such a subcritical fluid model solution converges to zero as time goes to infinity. This implies that each component of the measure-valued fluid model solution converges vaguely on [Formula: see text] to the zero measure as time goes to infinity. Under the further assumption that the file size distributions have finite pth moments for some p > 1 and that each component of the initial state of the fluid model solution has finite pth moment, it is proved that the fluid model solution reaches the measure with all components equal to the zero measure in finite time and that the time to reach this zero state has a uniform bound for all fluid model solutions having a uniform bound on the initial total mass and the pth moment of each component of the initial state. In contrast to the analysis of Paganini et al., we do not need their strong smoothness assumptions on fluid model solutions and we rigorously treat the realistic, but singular situation, where the fluid level on some routes becomes zero, whereas other route levels remain positive.

Large-time limit of nonlinearly coupled measure-valued equations that model many-server queues with reneging

“…For all 0 ≤ t ≤ t * , applying Lemma 2.7 with T = τ and (2. 19) with ψ = h s , and using the fact that (2.20) implies K(τ + t) − K(τ ) = λt for t ∈ (0, t * ), the identity g s = h s Ḡs , the upper bound on h s from Assumption 3.1( 1) and (4.33), we obtain…”

“…Using the definition of D from (2.6), setting ψ = h s in (2. 19), interchanging the order of integration, using integration by parts and the fact G s (0+) = 0, we obtain…”

“…To establish the remaining limits, we begin with the representation for the age measure ν t given in (2. 19), which shows that ν t = θ t + µ t , where θ t , µ t ∈ M F [0, ∞), are defined by…”

“…Now, recalling B(•) = 1, ν • from (4.36) and combining Lemma 2.7 and Theorem 2.6, it follows that (2. 19) holds with ψ = 1, and ν t and K t replaced with ν T1+t , and K T1+t − K T1 , respectively, or in other words, for each t ≥ 0,…”

“…In the case that service time distribution is class-independent and satisfies the same conditions as above, reneging times are exponential, but possibly class-dependent, and the fluid equations have a unique invariant state, we establish (in Theorem 5.2) uniqueness of the random fixed points and analogous long-time convergence results in the supercritical regime. It should be mentioned here that in the queueing context, other works that have studied long-time behavior of measure-valued fluid equations using Lyapunov functionals include [35,19,38]. All of these works focus on the dynamics of residual times for jobs in bandwidth sharing and processor sharing models, which have a different structure from the measure-valued equations arising from our fluid equations.…”