6,270 research outputs found
Efficient robust routing for single commodity network flows
We study single commodity network flows with suitable robustness and efficiency specs. An original use of a maximum entropy problem for distributions on the paths of the graph turns this problem into a steering problem for Markov chains with prescribed initial and final marginals. From a computational standpoint, viewing scheduling this way is especially attractive in light of the existence of an iterative algorithm to compute the solution. The present paper builds on [13] by introducing an index of efficiency of a transportation plan and points, accordingly, to efficient-robust transport policies. In developing the theory, we establish two new invariance properties of the solution (called bridge) \u2013 an iterated bridge invariance property and the invariance of the most probable paths. These properties, which were tangentially mentioned in our previous work, are fully developed here. We also show that the distribution on paths of the optimal transport policy, which depends on a \u201ctemperature\u201d parameter, tends to the solution of the \u201cmost economical\u201d but possibly less robust optimal mass transport problem as the temperature goes to zero. The relevance of all of these properties for transport over networks is illustrated in an example
Knudsen gas in a finite random tube: transport diffusion and first passage properties
We consider transport diffusion in a stochastic billiard in a random tube
which is elongated in the direction of the first coordinate (the tube axis).
Inside the random tube, which is stationary and ergodic, non-interacting
particles move straight with constant speed. Upon hitting the tube walls, they
are reflected randomly, according to the cosine law: the density of the
outgoing direction is proportional to the cosine of the angle between this
direction and the normal vector. Steady state transport is studied by
introducing an open tube segment as follows: We cut out a large finite segment
of the tube with segment boundaries perpendicular to the tube axis. Particles
which leave this piece through the segment boundaries disappear from the
system. Through stationary injection of particles at one boundary of the
segment a steady state with non-vanishing stationary particle current is
maintained. We prove (i) that in the thermodynamic limit of an infinite open
piece the coarse-grained density profile inside the segment is linear, and (ii)
that the transport diffusion coefficient obtained from the ratio of stationary
current and effective boundary density gradient equals the diffusion
coefficient of a tagged particle in an infinite tube. Thus we prove Fick's law
and equality of transport diffusion and self-diffusion coefficients for quite
generic rough (random) tubes. We also study some properties of the crossing
time and compute the Milne extrapolation length in dependence on the shape of
the random tube.Comment: 51 pages, 3 figure
Diffusive propagation of wave packets in a fluctuating periodic potential
We consider the evolution of a tight binding wave packet propagating in a
fluctuating periodic potential. If the fluctuations stem from a stationary
Markov process satisfying certain technical criteria, we show that the square
amplitude of the wave packet after diffusive rescaling converges to a
superposition of solutions of a heat equation.Comment: 13 pages (v2: added a paragraph on the history of the problem, added
some references, correct a few typos; v3 minor corrections, added keywords
and subject classes
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
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