2,374 research outputs found
Stationary flows and uniqueness of invariant measures
In this short paper, we consider a quadruple ,where is a -algebra of subsets of , and is
a measurable bijection from into itself that preserves the measure
. For each , we consider the measure obtained by taking
cycles (excursions) of iterates of from . We then derive a relation
for that involves the forward and backward hitting times of by the
trajectory at a point .
Although classical in appearance, its use in obtaining uniqueness of invariant
measures of various stochastic models seems to be new. We apply the concept to
countable Markov chains and Harris processes
A New Phase Transition for Local Delays in MANETs
We consider Mobile Ad-hoc Network (MANET) with transmitters located according
to a Poisson point in the Euclidean plane, slotted Aloha Medium Access (MAC)
protocol and the so-called outage scenario, where a successful transmission
requires a Signal-to-Interference-and-Noise (SINR) larger than some threshold.
We analyze the local delays in such a network, namely the number of times slots
required for nodes to transmit a packet to their prescribed next-hop receivers.
The analysis depends very much on the receiver scenario and on the variability
of the fading. In most cases, each node has finite-mean geometric random delay
and thus a positive next hop throughput. However, the spatial (or large
population) averaging of these individual finite mean-delays leads to infinite
values in several practical cases, including the Rayleigh fading and positive
thermal noise case. In some cases it exhibits an interesting phase transition
phenomenon where the spatial average is finite when certain model parameters
are below a threshold and infinite above. We call this phenomenon, contention
phase transition. We argue that the spatial average of the mean local delays is
infinite primarily because of the outage logic, where one transmits full
packets at time slots when the receiver is covered at the required SINR and
where one wastes all the other time slots. This results in the "RESTART"
mechanism, which in turn explains why we have infinite spatial average.
Adaptive coding offers a nice way of breaking the outage/RESTART logic. We show
examples where the average delays are finite in the adaptive coding case,
whereas they are infinite in the outage case.Comment: accepted for IEEE Infocom 201
Poisson Hail on a Hot Ground
We consider a queue where the server is the Euclidean space, and the
customers are random closed sets (RACS) of the Euclidean space. These RACS
arrive according to a Poisson rain and each of them has a random service time
(in the case of hail falling on the Euclidean plane, this is the height of the
hailstone, whereas the RACS is its footprint). The Euclidean space serves
customers at speed 1. The service discipline is a hard exclusion rule: no two
intersecting RACS can be served simultaneously and service is in the First In
First Out order: only the hailstones in contact with the ground melt at speed
1, whereas the other ones are queued; a tagged RACS waits until all RACS
arrived before it and intersecting it have fully melted before starting its own
melting. We give the evolution equations for this queue. We prove that it is
stable for a sufficiently small arrival intensity, provided the typical
diameter of the RACS and the typical service time have finite exponential
moments. We also discuss the percolation properties of the stationary regime of
the RACS in the queue.Comment: 26 page
The Boolean Model in the Shannon Regime: Three Thresholds and Related Asymptotics
Consider a family of Boolean models, indexed by integers , where the
-th model features a Poisson point process in of intensity
with as , and balls of
independent and identically distributed radii distributed like , with satisfying a large deviations principle. It is shown
that there exist three deterministic thresholds: the degree threshold;
the percolation threshold; and the volume fraction threshold;
such that asymptotically as tends to infinity, in a sense made precise in
the paper: (i) for , almost every point is isolated, namely its
ball intersects no other ball; (ii) for , almost every
ball intersects an infinite number of balls and nevertheless there is no
percolation; (iii) for , the volume fraction is 0 and
nevertheless percolation occurs; (iv) for , almost every
ball intersects an infinite number of balls and nevertheless the volume
fraction is 0; (v) for , the whole space covered. The analysis
of this asymptotic regime is motivated by related problems in information
theory, and may be of interest in other applications of stochastic geometry
Information-Theoretic Capacity and Error Exponents of Stationary Point Processes under Random Additive Displacements
This paper studies the Shannon regime for the random displacement of
stationary point processes. Let each point of some initial stationary point
process in give rise to one daughter point, the location of which is
obtained by adding a random vector to the coordinates of the mother point, with
all displacement vectors independently and identically distributed for all
points. The decoding problem is then the following one: the whole mother point
process is known as well as the coordinates of some daughter point; the
displacements are only known through their law; can one find the mother of this
daughter point? The Shannon regime is that where the dimension tends to
infinity and where the logarithm of the intensity of the point process is
proportional to . We show that this problem exhibits a sharp threshold: if
the sum of the proportionality factor and of the differential entropy rate of
the noise is positive, then the probability of finding the right mother point
tends to 0 with for all point processes and decoding strategies. If this
sum is negative, there exist mother point processes, for instance Poisson, and
decoding strategies, for instance maximum likelihood, for which the probability
of finding the right mother tends to 1 with . We then use large deviations
theory to show that in the latter case, if the entropy spectrum of the noise
satisfies a large deviation principle, then the error probability goes
exponentially fast to 0 with an exponent that is given in closed form in terms
of the rate function of the noise entropy spectrum. This is done for two
classes of mother point processes: Poisson and Mat\'ern. The practical interest
to information theory comes from the explicit connection that we also establish
between this problem and the estimation of error exponents in Shannon's
additive noise channel with power constraints on the codewords
On Scaling Limits of Power Law Shot-noise Fields
This article studies the scaling limit of a class of shot-noise fields
defined on an independently marked stationary Poisson point process and with a
power law response function. Under appropriate conditions, it is shown that the
shot-noise field can be scaled suitably to have a -stable limit,
intensity of the underlying point process goes to infinity. It is also shown
that the finite dimensional distributions of the limiting random field have
i.i.d. stable random components. We hence propose to call this limte the
- stable white noise field. Analogous results are also obtained for the
extremal shot-noise field which converges to a Fr\'{e}chet white noise field.
Finally, these results are applied to the analysis of wireless networks.Comment: 17 pages, Typos are correcte
The stochastic geometry of unconstrained one-bit data compression
A stationary stochastic geometric model is proposed for analyzing the data
compression method used in one-bit compressed sensing. The data set is an
unconstrained stationary set, for instance all of or a
stationary Poisson point process in . It is compressed using a
stationary and isotropic Poisson hyperplane tessellation, assumed independent
of the data. That is, each data point is compressed using one bit with respect
to each hyperplane, which is the side of the hyperplane it lies on. This model
allows one to determine how the intensity of the hyperplanes must scale with
the dimension to ensure sufficient separation of different data by the
hyperplanes as well as sufficient proximity of the data compressed together.
The results have direct implications in compressive sensing and in source
coding.Comment: 29 page
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