3,698 research outputs found
Matrix geometric approach for random walks: stability condition and equilibrium distribution
In this paper, we analyse a sub-class of two-dimensional homogeneous nearest
neighbour (simple) random walk restricted on the lattice using the matrix
geometric approach. In particular, we first present an alternative approach for
the calculation of the stability condition, extending the result of Neuts drift
conditions [30] and connecting it with the result of Fayolle et al. which is
based on Lyapunov functions [13]. Furthermore, we consider the sub-class of
random walks with equilibrium distributions given as series of product-forms
and, for this class of random walks, we calculate the eigenvalues and the
corresponding eigenvectors of the infinite matrix appearing in the
matrix geometric approach. This result is obtained by connecting and extending
three existing approaches available for such an analysis: the matrix geometric
approach, the compensation approach and the boundary value problem method. In
this paper, we also present the spectral properties of the infinite matrix
Spectral properties of the tandem Jackson network, seen as a quasi-birth-and-death process
Quasi-birth-and-death (QBD) processes with infinite ``phase spaces'' can
exhibit unusual and interesting behavior. One of the simplest examples of such
a process is the two-node tandem Jackson network, with the ``phase'' giving the
state of the first queue and the ``level'' giving the state of the second
queue. In this paper, we undertake an extensive analysis of the properties of
this QBD. In particular, we investigate the spectral properties of Neuts's
R-matrix and show that the decay rate of the stationary distribution of the
``level'' process is not always equal to the convergence norm of R. In fact, we
show that we can obtain any decay rate from a certain range by controlling only
the transition structure at level zero, which is independent of R. We also
consider the sequence of tandem queues that is constructed by restricting the
waiting room of the first queue to some finite capacity, and then allowing this
capacity to increase to infinity. We show that the decay rates for the finite
truncations converge to a value, which is not necessarily the decay rate in the
infinite waiting room case. Finally, we show that the probability that the
process hits level n before level 0 given that it starts in level 1 decays at a
rate which is not necessarily the same as the decay rate for the stationary
distribution.Comment: Published at http://dx.doi.org/10.1214/105051604000000477 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Quantum Interference in Superconducting Wire Networks and Josephson Junction Arrays: Analytical Approach based on Multiple-Loop Aharonov-Bohm Feynman Path-Integrals
We investigate analytically and numerically the mean-field
superconducting-normal phase boundaries of two-dimensional superconducting wire
networks and Josephson junction arrays immersed in a transverse magnetic field.
The geometries we consider include square, honeycomb, triangular, and kagome'
lattices. Our approach is based on an analytical study of multiple-loop
Aharonov-Bohm effects: the quantum interference between different electron
closed paths where each one of them encloses a net magnetic flux. Specifically,
we compute exactly the sums of magnetic phase factors, i.e., the lattice path
integrals, on all closed lattice paths of different lengths. A very large
number, e.g., up to for the square lattice, exact lattice path
integrals are obtained. Analytic results of these lattice path integrals then
enable us to obtain the resistive transition temperature as a continuous
function of the field. In particular, we can analyze measurable effects on the
superconducting transition temperature, , as a function of the magnetic
filed , originating from electron trajectories over loops of various
lengths. In addition to systematically deriving previously observed features,
and understanding the physical origin of the dips in as a result of
multiple-loop quantum interference effects, we also find novel results. In
particular, we explicitly derive the self-similarity in the phase diagram of
square networks. Our approach allows us to analyze the complex structure
present in the phase boundaries from the viewpoint of quantum interference
effects due to the electron motion on the underlying lattices.Comment: 18 PRB-type pages, plus 8 large figure
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
Achievable Performance in Product-Form Networks
We characterize the achievable range of performance measures in product-form
networks where one or more system parameters can be freely set by a network
operator. Given a product-form network and a set of configurable parameters, we
identify which performance measures can be controlled and which target values
can be attained. We also discuss an online optimization algorithm, which allows
a network operator to set the system parameters so as to achieve target
performance metrics. In some cases, the algorithm can be implemented in a
distributed fashion, of which we give several examples. Finally, we give
conditions that guarantee convergence of the algorithm, under the assumption
that the target performance metrics are within the achievable range.Comment: 50th Annual Allerton Conference on Communication, Control and
Computing - 201
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