15 research outputs found
Efficient cyclic reduction for QBDs with rank structured blocks
We provide effective algorithms for solving block tridiagonal block Toeplitz
systems with quasiseparable blocks, as well as quadratic matrix
equations with quasiseparable coefficients, based on cyclic
reduction and on the technology of rank-structured matrices. The algorithms
rely on the exponential decay of the singular values of the off-diagonal
submatrices generated by cyclic reduction. We provide a formal proof of this
decay in the Markovian framework. The results of the numerical experiments that
we report confirm a significant speed up over the general algorithms, already
starting with the moderately small size
Occupation densities in solving exit problems for Markov additive processes and their reflections
This paper solves exit problems for spectrally negative Markov additive
processes and their reflections. A so-called scale matrix, which is a
generalization of the scale function of a spectrally negative \levy process,
plays a central role in the study of exit problems. Existence of the scale
matrix was shown in Thm. 3 of Kyprianou and Palmowski (2008). We provide a
probabilistic construction of the scale matrix, and identify the transform. In
addition, we generalize to the MAP setting the relation between the scale
function and the excursion (height) measure. The main technique is based on the
occupation density formula and even in the context of fluctuations of
spectrally negative L\'{e}vy processes this idea seems to be new. Our
representation of the scale matrix W(x)=e^{-\Lambda x}\eL(x) in terms of nice
probabilistic objects opens up possibilities for further investigation of its
properties
A Linear Programming Approach to Error Bounds for Random Walks in the Quarter-plane
We consider the approximation of the performance of random walks in the
quarter-plane. The approximation is in terms of a random walk with a
product-form stationary distribution, which is obtained by perturbing the
transition probabilities along the boundaries of the state space. A Markov
reward approach is used to bound the approximation error. The main contribution
of the work is the formulation of a linear program that provides the
approximation error
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
A computational framework for two-dimensional random walks with restarts
The treatment of two-dimensional random walks in the quarter plane leads to
Markov processes which involve semi-infinite matrices having Toeplitz or block
Toeplitz structure plus a low-rank correction. Finding the steady state
probability distribution of the process requires to perform operations
involving these structured matrices. We propose an extension of the framework
of [5] which allows to deal with more general situations such as processes
involving restart events. This is motivated by the need for modeling processes
that can incur in unexpected failures like computer system reboots.
Algebraically, this gives rise to corrections with infinite support that cannot
be treated using the tools currently available in the literature. We present a
theoretical analysis of an enriched Banach algebra that, combined with
appropriate algorithms, enables the numerical treatment of these problems. The
results are applied to the solution of bidimensional Quasi-Birth-Death
processes with infinitely many phases which model random walks in the quarter
plane, relying on the matrix analytic approach. This methodology reduces the
problem to solving a quadratic matrix equation with coefficients of infinite
size. We provide conditions on the transition probabilities which ensure that
the solution of interest of the matrix equation belongs to the enriched
algebra. The reliability of our approach is confirmed by extensive numerical
experimentation on some case studies