1,146 research outputs found
Supermarket Model on Graphs
We consider a variation of the supermarket model in which the servers can
communicate with their neighbors and where the neighborhood relationships are
described in terms of a suitable graph. Tasks with unit-exponential service
time distributions arrive at each vertex as independent Poisson processes with
rate , and each task is irrevocably assigned to the shortest queue
among the one it first appears and its randomly selected neighbors. This
model has been extensively studied when the underlying graph is a clique in
which case it reduces to the well known power-of- scheme. In particular,
results of Mitzenmacher (1996) and Vvedenskaya et al. (1996) show that as the
size of the clique gets large, the occupancy process associated with the
queue-lengths at the various servers converges to a deterministic limit
described by an infinite system of ordinary differential equations (ODE). In
this work, we consider settings where the underlying graph need not be a clique
and is allowed to be suitably sparse. We show that if the minimum degree
approaches infinity (however slowly) as the number of servers approaches
infinity, and the ratio between the maximum degree and the minimum degree in
each connected component approaches 1 uniformly, the occupancy process
converges to the same system of ODE as the classical supermarket model. In
particular, the asymptotic behavior of the occupancy process is insensitive to
the precise network topology. We also study the case where the graph sequence
is random, with the -th graph given as an Erd\H{o}s-R\'enyi random graph on
vertices with average degree . Annealed convergence of the occupancy
process to the same deterministic limit is established under the condition
, and under a stronger condition ,
convergence (in probability) is shown for almost every realization of the
random graph.Comment: 32 page
Nonlinear Markov Processes in Big Networks
Big networks express various large-scale networks in many practical areas
such as computer networks, internet of things, cloud computation, manufacturing
systems, transportation networks, and healthcare systems. This paper analyzes
such big networks, and applies the mean-field theory and the nonlinear Markov
processes to set up a broad class of nonlinear continuous-time block-structured
Markov processes, which can be applied to deal with many practical stochastic
systems. Firstly, a nonlinear Markov process is derived from a large number of
interacting big networks with symmetric interactions, each of which is
described as a continuous-time block-structured Markov process. Secondly, some
effective algorithms are given for computing the fixed points of the nonlinear
Markov process by means of the UL-type RG-factorization. Finally, the Birkhoff
center, the Lyapunov functions and the relative entropy are used to analyze
stability or metastability of the big network, and several interesting open
problems are proposed with detailed interpretation. We believe that the results
given in this paper can be useful and effective in the study of big networks.Comment: 28 pages in Special Matrices; 201
Traffic at the Edge of Chaos
We use a very simple description of human driving behavior to simulate
traffic. The regime of maximum vehicle flow in a closed system shows
near-critical behavior, and as a result a sharp decrease of the predictability
of travel time. Since Advanced Traffic Management Systems (ATMSs) tend to drive
larger parts of the transportation system towards this regime of maximum flow,
we argue that in consequence the traffic system as a whole will be driven
closer to criticality, thus making predictions much harder. A simulation of a
simplified transportation network supports our argument.Comment: Postscript version including most of the figures available from
http://studguppy.tsasa.lanl.gov/research_team/. Paper has been published in
Brooks RA, Maes P, Artifical Life IV: ..., MIT Press, 199
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