Dagstuhl Seminar Proceedings, Volume 7461
Dagstuhl Seminar Proceedings
DagSemProc
https://www.dagstuhl.de/dagpub/1862-4405
https://dblp.org/db/series/dagstuhl
1862-4405
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
7461
2008
https://drops.dagstuhl.de/entities/volume/DagSemProc-volume-7461
07461 Abstracts Collection – Numerical Methods for Structured Markov Chains
From 11.11. to 14.11.07, the Dagstuhl Seminar 07461 ``Numerical Methods for Structured Markov Chains'' was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available.
Matrix analytic methods
markov processes
queuing theory
numerical methods
structured matrices
telecommunication modeling
performance evaluation
1-0
Regular Paper
Dario A.
Bini
Dario A. Bini
Beatrice
Meini
Beatrice Meini
Vaidyanathan
Ramaswami
Vaidyanathan Ramaswami
Marie-Ange
Remiche
Marie-Ange Remiche
Peter
Taylor
Peter Taylor
10.4230/DagSemProc.07461.1
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07461 Executive Summary – Numerical Methods for Structured Markov Chains
This Dagstuhl seminar has brought together leaders and young researchers
in the fields of analysis of numerical algorithms, applied stochastic modeling and
statistical inference, with the result of stimulating exchange of methodologies
and experiences and generating synergetic collaborations.
This has favored a better communication between these worlds where problems from the applications feed the theoretical research and where advanced
numerical tools can be utilized in applications with reciprocal advantages.
Matrix analytic methods
Markov processes
queuing theory
numerical methods
structured matrices
telecommunication modeling
performance evaluation.
1-2
Regular Paper
Dario A.
Bini
Dario A. Bini
Beatrice
Meini
Beatrice Meini
Vaidyanathan
Ramaswami
Vaidyanathan Ramaswami
Marie-Ange
Remiche
Marie-Ange Remiche
Peter
Taylor
Peter Taylor
10.4230/DagSemProc.07461.2
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A policy iteration algorithm for Markov decision processes skip-free in one direction
In this paper we present a new algorithm for policy iteration for Markov decision processes (MDP) skip-free in one direction. This algorithm, which is based on matrix analytic methods, is in the same spirit as the algorithm of White (Stochastic Models, 21:785-797, 2005) which was limited to matrices that are skip-free in both directions.
Optimization problems that can be solved using Markov decision processes arise in the domain of optical buffers, when trying to improve loss rates of fibre delay line (FDL) buffers. Based on the analysis of such an FDL buffer we present a comparative study between the different techniques available to solve an MDP. The results illustrate that the exploitation of the structure of the transition matrices places us in a position to deal with larger systems, while reducing the computation times.
Markov Decision Process
Policy Evaluation
Skip-Free
Optical buffers
Fibre Delay Lines
1-3
Regular Paper
Joke
Lambert
Joke Lambert
Benny
van Houdt
Benny van Houdt
Chris
Blondia
Chris Blondia
10.4230/DagSemProc.07461.3
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Characterizing Coxian Distributions of Algebraic Degree q and Triangular Order p
In this research note we present a procedure to characterize the set of all Coxian distributions of algebraic degree q that have Coxian representations of order p where p > q.
Phase-type distribution
Coxian distribution
algebraic degree
triangular order
rational Laplace-Stieltjes transform
1-10
Regular Paper
Mark
Fackrell
Mark Fackrell
10.4230/DagSemProc.07461.4
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Current results and open questions on PH and MAP characterization
Stochastic processes with matrix exponential kernels have a wide range
of applications due to the availability of efficient matrix analytic methods.
The characterization of these processes is in progress in recent years.
Basic questions like
the flexibility,
the degree of freedom,
the most efficient (canonical) representation of these models
are under study.
The presentation collects a set of available results and related open questions.
PH distribution
ME distribution
MAP
MEP
1-6
Regular Paper
Levente
Bodrog
Levente Bodrog
Armin
Heindl
Armin Heindl
Gábor
Horváth
Gábor Horváth
Miklós
Telek
Miklós Telek
András
Horváth
András Horváth
10.4230/DagSemProc.07461.5
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Erlangian Approximation to Finite Time Ruin Probabilities in Perturbed Risk Models
In this work-in-progress, we consider perturbed risk processes that have an underlying Markovian structure, including Markovian risk processes, and Sparre-Andersen risk processes when both inter claim times and claim sizes are phase-type. We apply the Erlangization method to this risk process in order to obtain an accurate approximation of the finite time ruin probability. In addition, we recognize a repeating structure in the probability matrices we work with. This is the key element in developing more efficent algorithms for the computation of the ruin probabilities. Several numerical examples are present to illustrate the model.
Perturbed risk processes
finite-time ruin probability
phase-type distribution
fluid flow models
Erlangization
1-15
Regular Paper
Kaiqi
Yu
Kaiqi Yu
David A.
Stanford
David A. Stanford
Jiandong
Ren
Jiandong Ren
10.4230/DagSemProc.07461.6
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From Algebraic Riccati equations to unilateral quadratic matrix equations: old and new algorithms
The problem of reducing an algebraic Riccati equation $XCX-AX-XD+B=0$ to a unilateral quadratic matrix equation (UQME) of the
kind $PX^2+QX+R$ is analyzed. New reductions are introduced
which enable one to prove some theoretical and computational properties.
In particular we show that the structure preserving doubling algorithm
of B.D.O. Anderson [Internat. J. Control, 1978] is nothing else but the
cyclic reduction algorithm applied to a suitable UQME. A new algorithm
obtained by complementing our reductions with the shrink-and-shift tech-
nique of Ramaswami is presented. Finally, faster algorithms which require
some non-singularity conditions, are designed. The non-singularity re-
striction is relaxed by introducing a suitable similarity transformation of
the Hamiltonian.
Algebraic Riccati Equation
Matrix Equation
Cyclic Reduction
Structured doubling algorithm
1-28
Regular Paper
Dario A.
Bini
Dario A. Bini
Beatrice
Meini
Beatrice Meini
Federico
Poloni
Federico Poloni
10.4230/DagSemProc.07461.7
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Interarrival Times Characterization and Fitting for Markovian Traffic Analysis
We propose a traffic fitting algorithm for Markovian Arrival Processes (MAPs) that can capture statistics of any order of interarrival times. By studying real traffic traces, we show that matching higher order properties, in addition to first and second order descriptors, results in increased queueing prediction accuracy with respect to other algorithms that only match the mean, coefficient of variation, and autocorrelations. The result promotes the idea of modeling traffic traces using the interarrival time process instead of the counting process that is more frequently employed in previous work, but for which higher order moments are difficult to manipulate.
We proceed by first characterizing the general properties of MAPs using a spectral approach. Based on this characterization, we show how different MAP processes can be combined together using Kronecker products to define a larger MAP with predefined properties of interarrival times. We then devise an algorithm that is based on this Kronecker composition and can accurately fit traffic traces. The algorithm employs nonlinear optimization programs that can be customized to fit an arbitrary number of moments and to meet the desired cost-accuracy tradeoff.
Numerical results of the fitting algorithm on real HTTP and TCP traffic data, such as the Bellcore Aug89 trace, indicate that the proposed fitting methods achieve increased prediction accuracy with respect to other state-of-the-art fitting methods.
MAP fitting
interarrival time process
higher-order moments
1-8
Regular Paper
Giuliano
Casale
Giuliano Casale
Eddy Z.
Zhang
Eddy Z. Zhang
Evgenia
Smirni
Evgenia Smirni
10.4230/DagSemProc.07461.8
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Matrix Analytic Methods in Branching processes
We examine the question of solving the extinction
probability of a particular class of continuous-time multi-type
branching processes, named Markovian binary trees (MBT). The
extinction probability is the minimal nonnegative solution of a
fixed point equation that turns out to be quadratic, which makes its
resolution particularly clear.
We analyze first two linear algorithms to compute the extinction
probability of an MBT, of which one is new, and, we propose a
quadratic algorithm arising from Newton's iteration method for
fixed-point equations.
Finally, we add a catastrophe process to the
initial MBT, and we analyze the resulting system. The extinction
probability turns out to be much more difficult to compute; we use a
$G/M/1$-type Markovian process approach to approximate this
probability.
Branching Processes
Matrix Analytic Methods
Extinction Probability
Catastrophe Process
1-3
Regular Paper
Sophie
Hautphenne
Sophie Hautphenne
Guy
Latouche
Guy Latouche
Marie-Ange
Remiche
Marie-Ange Remiche
10.4230/DagSemProc.07461.9
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Multivariate matrix-exponential distributions
We review what is currently known about one-dimensional distributions on
the non-negative reals with rational Laplace transform, also known as
matrix-exponential distributions. In particular we discuss a flow
interpreation which enables one to mimic certain probabilisticly
inspired arguments which are known from the theory of phase-type distributions.
We then move on to present ongoing research for higher dimensions.
We discuss a characterization result, some closure properties, and
a number of examples. Finally we present open problems and future
perspectives.
Multivariate matrix-exponential distributions
multivariate phase-type distributions
rational Laplace transform
1-13
Regular Paper
Mogens
Bladt
Mogens Bladt
Bo Friis
Nielsen
Bo Friis Nielsen
10.4230/DagSemProc.07461.10
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Nonsymmetric algebraic Riccati equations associated with an M-matrix: recent advances and algorithms
We survey on theoretical properties and algorithms concerning the problem of solving a nonsymmetric algebraic Riccati equation, and we report on some known methods and new algorithmic advances. In particular, some results on the number of positive solutions are proved and a careful convergence analysis of Newton's iteration is carried out in the cases of interest where some singularity conditions are encountered. From this analysis we determine initial approximations which still guarantee the quadratic convergence.
Nonsymmetric algebraic Riccati equations
matrix equation
M-matrices
Newton method
quadratically convergent algorithms
cyclic reduction
doubling
1-31
Regular Paper
Dario A.
Bini
Dario A. Bini
Bruno
Iannazzo
Bruno Iannazzo
Beatrice
Meini
Beatrice Meini
Federico
Poloni
Federico Poloni
10.4230/DagSemProc.07461.11
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On the Properties of Moments of Matrix Exponential Distributions and Matrix Exponential Processes
In this paper we provide properties of moments of matrix exponential
distributions and joint moments of matrix exponential processes. Based on
the provided properties, an algorithm is presented to compute any finite
dimensional moments of these processes based on a set of required (low
order) moments. This algorithm does not require the computation of any
representation of the given process. We present some related examples to
demonstrate the potential use of the properties of moments.
Matrix exponential process
Markov arrival process
Matrix exponential distribution
phase type distribution
1-12
Regular Paper
Levente
Bodrog
Levente Bodrog
András
Horváth
András Horváth
Miklós
Telek
Miklós Telek
10.4230/DagSemProc.07461.12
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On the tail decay of M/G/1-type Markov renewal processes
The tail decay of M/G/1-type Markov renewal processes is studied. The Markov
renewal process is transformed into a Markov chain so that the problem of
tail decay is reformulated in terms of the decay of the coefficients of a
suitable power series. The latter problem is reduced to analyze the
analyticity domain of the power series.
Renewal processes
tail decay
M/G/1-type Markov chains
1-7
Regular Paper
Dario A.
Bini
Dario A. Bini
Beatrice
Meini
Beatrice Meini
Vaidyanathan
Ramaswami
Vaidyanathan Ramaswami
10.4230/DagSemProc.07461.13
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QBD processes and matrix orthogonal polynomilas: somw new explicit examples
In the case of birth-and-death processes
there are a few exactly solvable
situations where the n-step transition
matrix can be written down using the
Karlin-McGregor formula. A few of these
come from group representation theory.
I plan to show how this can be extended
to some instances of QBD processes with an
arbitrary finite number of phases. The group
involved is the set of all unitary
matrices of size N. For a fixed N one gets
examples where the number of phases is
a free parameter, and there are a few extra
parameters to play with. By tunning
these parameters
one can exhibit examples where states
are recurrent or transient.
The rather surprising fact that for these
examples one can compute everything
explicitly raises the issue of finding
a possible network application
for this piece of
mathematics that involves matrix
valued orthogonal polynomials.
I will give an ab-initio discussion of
the examples starting with the case of one phase.
QBD
orthogonal polynomials
Karlin-McGregor formula
representation theory
0-0
Regular Paper
Alberto F.
Grünbaum
Alberto F. Grünbaum
10.4230/DagSemProc.07461.14
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Structured Markov Chains Arising from Finite-Source Retrial Queues with Orital Search
We consider retrial queueing systems with a finite number of
homogeneous sources of calls, a single reliable server, and the
search for orbiting customers by the server after job completion.
During this investigation, the infinitesimal generator of the
underlying (finite) continuous-time Markov chain takes a
(level-dependent) QBD-like form.
After solving for the steady state probabilities using the MOSEL-2
tool, the results show a surprising maximum of the mean response
time. This maximum was already discovered by other researchers
dealing with finite-source retrial queues. However, to our best
knowledge, no thorough investigation was done yet why this
maximum exists and in which way it depends on the system
parameters.
In the talk, after introducing the backgrounds of finite-source
retrial queues with orbital search, a generalized stochastic
Petri net is used to derive the underlying continuous-time
Markov chain and its generator. Finally, using the seminar,
we can hopefully bring forward discussions how to make more
general statements on the parameter-dependent behavior of
the response time’s maximum.
Structured Markov chain
finite source
retrial queues
orbital search
performance measures
performance tool
1-4
Regular Paper
Patrick
Wüchner
Patrick Wüchner
János
Sztrik
János Sztrik
Hermann
de Meer
Hermann de Meer
10.4230/DagSemProc.07461.15
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