52 research outputs found
Non-Equilibrium Statistical Physics of Currents in Queuing Networks
We consider a stable open queuing network as a steady non-equilibrium system
of interacting particles. The network is completely specified by its underlying
graphical structure, type of interaction at each node, and the Markovian
transition rates between nodes. For such systems, we ask the question ``What is
the most likely way for large currents to accumulate over time in a network
?'', where time is large compared to the system correlation time scale. We
identify two interesting regimes. In the first regime, in which the
accumulation of currents over time exceeds the expected value by a small to
moderate amount (moderate large deviation), we find that the large-deviation
distribution of currents is universal (independent of the interaction details),
and there is no long-time and averaged over time accumulation of particles
(condensation) at any nodes. In the second regime, in which the accumulation of
currents over time exceeds the expected value by a large amount (severe large
deviation), we find that the large-deviation current distribution is sensitive
to interaction details, and there is a long-time accumulation of particles
(condensation) at some nodes. The transition between the two regimes can be
described as a dynamical second order phase transition. We illustrate these
ideas using the simple, yet non-trivial, example of a single node with
feedback.Comment: 26 pages, 5 figure
Networks become navigable as nodes move and forget
We propose a dynamical process for network evolution, aiming at explaining
the emergence of the small world phenomenon, i.e., the statistical observation
that any pair of individuals are linked by a short chain of acquaintances
computable by a simple decentralized routing algorithm, known as greedy
routing. Previously proposed dynamical processes enabled to demonstrate
experimentally (by simulations) that the small world phenomenon can emerge from
local dynamics. However, the analysis of greedy routing using the probability
distributions arising from these dynamics is quite complex because of mutual
dependencies. In contrast, our process enables complete formal analysis. It is
based on the combination of two simple processes: a random walk process, and an
harmonic forgetting process. Both processes reflect natural behaviors of the
individuals, viewed as nodes in the network of inter-individual acquaintances.
We prove that, in k-dimensional lattices, the combination of these two
processes generates long-range links mutually independently distributed as a
k-harmonic distribution. We analyze the performances of greedy routing at the
stationary regime of our process, and prove that the expected number of steps
for routing from any source to any target in any multidimensional lattice is a
polylogarithmic function of the distance between the two nodes in the lattice.
Up to our knowledge, these results are the first formal proof that navigability
in small worlds can emerge from a dynamical process for network evolution. Our
dynamical process can find practical applications to the design of spatial
gossip and resource location protocols.Comment: 21 pages, 1 figur
A Simple Option Formula for General Jump-Diffusion and Other Exponential Levy Processes
Discrete time queueing networks with product form steady state. Availability and performance analysis in an integrated model
A manufacturing system with general stationary failure process: stability and ipa of hedging control policies
Rate of convergence to equilibrium of marked Hawkes processes
In this article we obtain rates of convergence to equilibrium of marked Hawkes processes in two situations. Firstly, the stationary process is the empty process, in which case we speak of the rate of extinction. Secondly, the stationary process is the unique stationary and nontrivial marked Hawkes process, in which case we speak of the rate of installation. The first situation models small epidemics, whereas the results in the second case are useful in deriving stopping rules for simulation algorithms of Hawkes processes with random marks
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