32 research outputs found
Analysis of a One-Dimensional Continuous Delay-Tolerant Network Model
The packet speed and transmission cost are examined, for a single packet traveling along a simple one-dimensional, continuous-time network, using a combination of wireless transmissions and physical transports. We assume that the network consists of two nodes moving at constant speed on a circle, and changing their direction of travel after independent exponential times. The packet wishes to travel in the clockwise direction as fast and as far as possible. It travels either by being physically transported on a node’s buffer, or by being wirelessly transmitted to the other node when the two are in the same location. We derive exact, explicit expressions for the long-term average packet speed (in the clockwise direction), and also for the average wireless transmission cost. These results can be viewed as initial steps towards the development of analogous exact expressions for the speed and cost, in more realistic, two-dimensional wireless delay-tolerant network models
A simple network of nodes moving on the circle
Two simple Markov processes are examined, one in discrete and one in
continuous time, arising from idealized versions of a transmission protocol for
mobile, delay-tolerant networks. We consider two independent walkers moving
with constant speed on either the discrete or continuous circle, and changing
directions at independent geometric (respectively, exponential) times. One of
the walkers carries a message that wishes to travel as far and as fast as
possible in the clockwise direction. The message stays with its current carrier
unless the two walkers meet, the carrier is moving counter-clockwise, and the
other walker is moving clockwise. In that case, the message jumps to the other
walker. The long-term average clockwise speed of the message is computed. An
explicit expression is derived via the solution of an associated boundary value
problem in terms of the generator of the underlying Markov process. The average
transmission cost is also similarly computed, measured as the long-term number
of jumps the message makes per unit time. The tradeoff between speed and cost
is examined, as a function of the underlying problem parameters
Condensation in randomly perturbed zero-range processes
The zero-range process is a stochastic interacting particle system that
exhibits a condensation transition under certain conditions on the dynamics. It
has recently been found that a small perturbation of a generic class of jump
rates leads to a drastic change of the phase diagram and prevents condensation
in an extended parameter range. We complement this study with rigorous results
on a finite critical density and quenched free energy in the thermodynamic
limit, as well as quantitative heuristic results for small and large noise
which are supported by detailed simulation data. While our new results support
the initial findings, they also shed new light on the actual (limited)
relevance in large finite systems, which we discuss via fundamental diagrams
obtained from exact numerics for finite systems.Comment: 18 pages, 6 figure
Thermodynamic Limit for the Invariant Measures in Supercritical Zero Range Processes
We prove a strong form of the equivalence of ensembles for the invariant
measures of zero range processes conditioned to a supercritical density of
particles. It is known that in this case there is a single site that
accomodates a macroscopically large number of the particles in the system. We
show that in the thermodynamic limit the rest of the sites have joint
distribution equal to the grand canonical measure at the critical density. This
improves the result of Gro\ss kinsky, Sch\"{u}tz and Spohn, where convergence
is obtained for the finite dimensional marginals. We obtain as corollaries
limit theorems for the order statistics of the components and for the
fluctuations of the bulk