1,729 research outputs found
A Computational Framework for the Mixing Times in the QBD Processes with Infinitely-Many Levels
In this paper, we develop some matrix Poisson's equations satisfied by the
mean and variance of the mixing time in an irreducible positive-recurrent
discrete-time Markov chain with infinitely-many levels, and provide a
computational framework for the solution to the matrix Poisson's equations by
means of the UL-type of -factorization as well as the generalized inverses.
In an important special case: the level-dependent QBD processes, we provide a
detailed computation for the mean and variance of the mixing time. Based on
this, we give new highlight on computation of the mixing time in the
block-structured Markov chains with infinitely-many levels through the
matrix-analytic method
Nonlinear Markov Processes in Big Networks
Big networks express various large-scale networks in many practical areas
such as computer networks, internet of things, cloud computation, manufacturing
systems, transportation networks, and healthcare systems. This paper analyzes
such big networks, and applies the mean-field theory and the nonlinear Markov
processes to set up a broad class of nonlinear continuous-time block-structured
Markov processes, which can be applied to deal with many practical stochastic
systems. Firstly, a nonlinear Markov process is derived from a large number of
interacting big networks with symmetric interactions, each of which is
described as a continuous-time block-structured Markov process. Secondly, some
effective algorithms are given for computing the fixed points of the nonlinear
Markov process by means of the UL-type RG-factorization. Finally, the Birkhoff
center, the Lyapunov functions and the relative entropy are used to analyze
stability or metastability of the big network, and several interesting open
problems are proposed with detailed interpretation. We believe that the results
given in this paper can be useful and effective in the study of big networks.Comment: 28 pages in Special Matrices; 201
Error bounds for last-column-block-augmented truncations of block-structured Markov chains
This paper discusses the error estimation of the last-column-block-augmented
northwest-corner truncation (LC-block-augmented truncation, for short) of
block-structured Markov chains (BSMCs) in continuous time. We first derive
upper bounds for the absolute difference between the time-averaged functionals
of a BSMC and its LC-block-augmented truncation, under the assumption that the
BSMC satisfies the general -modulated drift condition. We then establish
computable bounds for a special case where the BSMC is exponentially ergodic.
To derive such computable bounds for the general case, we propose a method that
reduces BSMCs to be exponentially ergodic. We also apply the obtained bounds to
level-dependent quasi-birth-and-death processes (LD-QBDs), and discuss the
properties of the bounds through the numerical results on an M/M/ retrial
queue, which is a representative example of LD-QBDs. Finally, we present
computable perturbation bounds for the stationary distribution vectors of
BSMCs.Comment: This version has fixed the bugs for the positions of Figures 1
through
Block SOR for Kronecker structured representations
Cataloged from PDF version of article.The Kronecker structure of a hierarchical Markovian model (HMM) induces nested block
partitionings in the transition matrix of its underlying Markov chain. This paper shows how
sparse real Schur factors of certain diagonal blocks of a given partitioning induced by the
Kronecker structure can be constructed from smaller component matrices and their real Schur
factors. Furthermore, it shows how the column approximate minimum degree (COLAMD)
ordering algorithm can be used to reduce fill-in of the remaining diagonal blocks that are
sparse LU factorized. Combining these ideas, the paper proposes three-level block successive
over-relaxation (BSOR) as a competitive steady state solver for HMMs. Finally, on a set of
numerical experiments it demonstrates how these ideas reduce storage required by the factors
of the diagonal blocks and improve solution time compared to an all LU factorization implementation
of the BSOR solver.
© 2004 Elsevier Inc. All rights reserved
Finding equilibrium probabilities of QBD processes by spectral methods when eigenvalues vanish
AbstractIn this paper, we discuss the use of spectral or eigenvalue methods for finding the equilibrium probabilities of quasi-birth–death processes for the case where some eigenvalues are zero. Since this leads to multiple eigenvalues at zero, a difficult problem to analyze, we suggest to eliminate such eigenvalues. To accomplish this, the dimension of the largest Jordan block must be established, and some initial equations must be eliminated. The method is demonstrated by two examples, one dealing with a tandem queue, the other one with a shorter queue problem
On the numerical solution of Kronecker-based infinite level-dependent QBD processes
Cataloged from PDF version of article.Infinite level-dependent quasi-birth-and-death (LDQBD) processes can be used to model Markovian systems with countably infinite multidimensional state spaces. Recently it has been shown that sums of Kronecker products can be used to represent the nonzero blocks of the transition rate matrix underlying an LDQBD process for models from stochastic chemical kinetics. This paper extends the form of the transition rates used recently so that a larger class of models including those of call centers can be analyzed for their steady-state. The challenge in the matrix analytic solution then is to compute conditional expected sojourn time matrices of the LDQBD model under low memory and time requirements after truncating its countably infinite state space judiciously. Results of numerical experiments are presented using a Kronecker-based matrix-analytic solution on models with two or more countably infinite dimensions and rules of thumb regarding better implementations are derived. In doing this, a more recent approach that reduces memory requirements further by enabling the computation of steady-state expectations without having to obtain the steady-state distribution is also considered. (C) 2013 Elsevier B.V. All rights reserved
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