Stability of a Subcritical Fluid Model for Fair Bandwidth Sharing with General File Size Distributions
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 more »
Authors:
;
Award ID(s):
Publication Date:
NSF-PAR ID:
10231281
Journal Name:
Stochastic Systems
Volume:
10
Issue:
3
Page Range or eLocation-ID:
251 to 273
ISSN:
1946-5238
2. We consider the well-known Lieb-Liniger (LL) model for \begin{document}$N$\end{document} bosons interacting pairwise on the line via the \begin{document}$\delta$\end{document} potential in the mean-field scaling regime. Assuming suitable asymptotic factorization of the initial wave functions and convergence of the microscopic energy per particle, we show that the time-dependent reduced density matrices of the system converge in trace norm to the pure states given by the solution to the one-dimensional cubic nonlinear Schrödinger equation (NLS) with an explict rate of convergence. In contrast to previous work [3] relying on the formalism of second quantization and coherent states and without an explicit rate, our proof is based on the counting method of Pickl [65,66,67] and Knowles and Pickl [44]. To overcome difficulties stemming from the singularity of the \begin{document}$\delta$\end{document} potential, we introduce a new short-range approximation argument that exploits the Hölder continuity of the \begin{document}$N$\end{document}-body wave function in a single particle variable. By further exploiting the \begin{document}$L^2$\end{document}-subcritical well-posedness theory for the 1D cubic NLS, we can prove mean-field convergence when the limiting solution to the NLS has finitemore »
3. The YBJ equation (Young & Ben Jelloul, J. Marine Res. , vol. 55, 1997, pp. 735–766) provides a phase-averaged description of the propagation of near-inertial waves (NIWs) through a geostrophic flow. YBJ is obtained via an asymptotic expansion based on the limit $\mathit{Bu}\rightarrow 0$ , where $\mathit{Bu}$ is the Burger number of the NIWs. Here we develop an improved version, the YBJ + equation. In common with an earlier improvement proposed by Thomas, Smith & Bühler ( J. Fluid Mech. , vol. 817, 2017, pp. 406–438), YBJ + has a dispersion relation that is second-order accurate in $\mathit{Bu}$ . (YBJ is first-order accurate.) Thus both improvements have the same formal justification. But the dispersion relation of YBJ + is a Padé approximant to the exact dispersion relation and with $\mathit{Bu}$ of order unity this is significantly more accurate than the power-series approximation of Thomas et al. (2017). Moreover, in the limit of high horizontal wavenumber $k\rightarrow \infty$ , the wave frequency of YBJ + asymptotes to twice the inertial frequency $2f$ . This enables solution of YBJ + with explicit time-stepping schemes using a time step determined by stable integration of oscillations with frequency $2f$ . Other phase-averaged equations have dispersion relations with frequency increasing as $k^{2}$more »
4. We study the focusing NLS equation in $R\mathbb{R}^N$ in the mass-supercritical and energy-subcritical (or intercritical ) regime, with $H^1$ data at the mass-energy threshold $\mathcal{ME}(u_0)=\mathcal{ME}(Q)$, where Q is the ground state. Previously, Duyckaerts–Merle studied the behavior of threshold solutions in the $H^1$-critical case, in dimensions $N = 3, 4, 5$, later generalized by Li–Zhang for higher dimensions. In the intercritical case, Duyckaerts–Roudenko studied the threshold problem for the 3d cubic NLS equation. In this paper, we generalize the results of Duyckaerts–Roudenko for any dimension and any power of the nonlinearity for the entire intercritical range. We show the existence of special solutions, $Q^\pm$, besides the standing wave $e^{it}Q$, which exponentially approach the standing wave in the positive time direction, but differ in its behavior for negative time. We classify solutions at the threshold level, showing either blow-up occurs in finite (positive and negative) time, or scattering in both time directions, or the solution is equal to one of the three special solutions above, up to symmetries. Our proof extends to the $H^1$-critical case, thus, giving an alternative proof of the Li–Zhang result and unifying the critical and intercritical cases. These results are obtained by studying the linearized equation around themore »