8,228 research outputs found
State space collapse and diffusion approximation for a network operating under a fair bandwidth sharing policy
We consider a connection-level model of Internet congestion control,
introduced by Massouli\'{e} and Roberts [Telecommunication Systems 15 (2000)
185--201], that represents the randomly varying number of flows present in a
network. Here, bandwidth is shared fairly among elastic document transfers
according to a weighted -fair bandwidth sharing policy introduced by Mo
and Walrand [IEEE/ACM Transactions on Networking 8 (2000) 556--567] []. Assuming Poisson arrivals and exponentially distributed document
sizes, we focus on the heavy traffic regime in which the average load placed on
each resource is approximately equal to its capacity. A fluid model (or
functional law of large numbers approximation) for this stochastic model was
derived and analyzed in a prior work [Ann. Appl. Probab. 14 (2004) 1055--1083]
by two of the authors. Here, we use the long-time behavior of the solutions of
the fluid model established in that paper to derive a property called
multiplicative state space collapse, which, loosely speaking, shows that in
diffusion scale, the flow count process for the stochastic model can be
approximately recovered as a continuous lifting of the workload process.Comment: Published in at http://dx.doi.org/10.1214/08-AAP591 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Robustness of Regularity for the D Convective Brinkman-Forchheimer Equations
We prove a robustness of regularity result for the D convective
Brinkman-Forchheimer equations \partial_tu -\mu\Delta u + (u \cdot \nabla)u
+ \nabla p + \alpha u + \beta\abs{u}^{r - 1}u = f, for the range of the
absorption exponent (for there exist global-in-time
regular solutions), i.e. we show that strong solutions of these equations
remain strong under small enough changes of the initial condition and forcing
function. We provide a smallness condition which is similar to the robustness
conditions given for the D incompressible Navier-Stokes equations by
Chernyshenko et al. (2007) and Dashti & Robinson (2008).Comment: 22 page
Multilevel Monte Carlo for Random Degenerate Scalar Convection Diffusion Equation
We consider the numerical solution of scalar, nonlinear degenerate
convection-diffusion problems with random diffusion coefficient and with random
flux functions. Building on recent results on the existence, uniqueness and
continuous dependence of weak solutions on data in the deterministic case, we
develop a definition of random entropy solution. We establish existence,
uniqueness, measurability and integrability results for these random entropy
solutions, generalizing \cite{Mishr478,MishSch10a} to possibly degenerate
hyperbolic-parabolic problems with random data. We next address the numerical
approximation of random entropy solutions, specifically the approximation of
the deterministic first and second order statistics. To this end, we consider
explicit and implicit time discretization and Finite Difference methods in
space, and single as well as Multi-Level Monte-Carlo methods to sample the
statistics. We establish convergence rate estimates with respect to the
discretization parameters, as well as with respect to the overall work,
indicating substantial gains in efficiency are afforded under realistic
regularity assumptions by the use of the Multi-Level Monte-Carlo method.
Numerical experiments are presented which confirm the theoretical convergence
estimates.Comment: 24 Page
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