303 research outputs found
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
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
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
On the Generating Functionals of a Class of Random Packing Point Processes
Consider a symmetrical conflict relationship between the points of a point
process. The Mat\'ern type constructions provide a generic way of selecting a
subset of this point process which is conflict-free. The simplest one consists
in keeping only conflict-free points. There is however a wide class of Mat\'ern
type processes based on more elaborate selection rules and providing larger
sets of selected points. The general idea being that if a point is discarded
because of a given conflict, there is no need to discard other points with
which it is also in conflict. The ultimate selection rule within this class is
the so called Random Sequential Adsorption, where the cardinality of the
sequence of conflicts allowing one to decide whether a given point is selected
is not bounded. The present paper provides a sufficient condition on the span
of the conflict relationship under which all the above point processes are well
defined when the initial point process is Poisson. It then establishes, still
in the Poisson case, a set of differential equations satisfied by the
probability generating functionals of these Mat\'ern type point processes.
Integral equations are also given for the Palm distributions
Interference Queueing Networks on Grids
Consider a countably infinite collection of interacting queues, with a queue
located at each point of the -dimensional integer grid, having independent
Poisson arrivals, but dependent service rates. The service discipline is of the
processor sharing type,with the service rate in each queue slowed down, when
the neighboring queues have a larger workload. The interactions are translation
invariant in space and is neither of the Jackson Networks type, nor of the
mean-field type. Coupling and percolation techniques are first used to show
that this dynamics has well defined trajectories. Coupling from the past
techniques are then proposed to build its minimal stationary regime. The rate
conservation principle of Palm calculus is then used to identify the stability
condition of this system, where the notion of stability is appropriately
defined for an infinite dimensional process. We show that the identified
condition is also necessary in certain special cases and conjecture it to be
true in all cases. Remarkably, the rate conservation principle also provides a
closed form expression for the mean queue size. When the stability condition
holds, this minimal solution is the unique translation invariant stationary
regime. In addition, there exists a range of small initial conditions for which
the dynamics is attracted to the minimal regime. Nevertheless, there exists
another range of larger though finite initial conditions for which the dynamics
diverges, even though stability criterion holds.Comment: Minor Spell Change
- …