2,212 research outputs found
Using imprecise continuous time Markov chains for assessing the reliability of power networks with common cause failure and non-immediate repair.
We explore how imprecise continuous time Markov
chains can improve traditional reliability models based
on precise continuous time Markov chains. Specifically,
we analyse the reliability of power networks under very
weak statistical assumptions, explicitly accounting for
non-stationary failure and repair rates and the limited
accuracy by which common cause failure rates can be
estimated. Bounds on typical quantities of interest
are derived, namely the expected time spent in system
failure state, as well as the expected number of
transitions to that state. A worked numerical example
demonstrates the theoretical techniques described.
Interestingly, the number of iterations required for
convergence is observed to be much lower than current
theoretical bounds
Efficient computation of updated lower expectations for imprecise continuous-time hidden Markov chains
We consider the problem of performing inference with imprecise
continuous-time hidden Markov chains, that is, imprecise continuous-time Markov
chains that are augmented with random output variables whose distribution
depends on the hidden state of the chain. The prefix `imprecise' refers to the
fact that we do not consider a classical continuous-time Markov chain, but
replace it with a robust extension that allows us to represent various types of
model uncertainty, using the theory of imprecise probabilities. The inference
problem amounts to computing lower expectations of functions on the state-space
of the chain, given observations of the output variables. We develop and
investigate this problem with very few assumptions on the output variables; in
particular, they can be chosen to be either discrete or continuous random
variables. Our main result is a polynomial runtime algorithm to compute the
lower expectation of functions on the state-space at any given time-point,
given a collection of observations of the output variables
Hitting times and probabilities for imprecise Markov chains
We consider the problem of characterising expected hitting times and hitting probabilities for imprecise Markov chains. To this end, we consider three distinct ways in which imprecise Markov chains have been defined in the literature: as sets of homogeneous Markov chains, as sets of more general stochastic processes, and as game-theoretic probability models. Our first contribution is that all these different types of imprecise Markov chains have the same lower and upper expected hitting times, and similarly the hitting probabilities are the same for these three types. Moreover, we provide a characterisation of these quantities that directly generalises a similar characterisation for precise, homogeneous Markov chains
Imprecise continuous-time Markov chains : efficient computational methods with guaranteed error bounds
Imprecise continuous-time Markov chains are a robust type of continuous-time Markov chains that allow for partially specified time-dependent parameters. Computing inferences for them requires the solution of a non-linear differential equation. As there is no general analytical expression for this solution, efficient numerical approximation methods are essential to the applicability of this model. We here improve the uniform approximation method of Krak et al. (2016) in two ways and propose a novel and more efficient adaptive approximation method. For ergodic chains, we also provide a method that allows us to approximate stationary distributions up to any desired maximal error
Imprecise Continuous-Time Markov Chains
Continuous-time Markov chains are mathematical models that are used to
describe the state-evolution of dynamical systems under stochastic uncertainty,
and have found widespread applications in various fields. In order to make
these models computationally tractable, they rely on a number of assumptions
that may not be realistic for the domain of application; in particular, the
ability to provide exact numerical parameter assessments, and the applicability
of time-homogeneity and the eponymous Markov property. In this work, we extend
these models to imprecise continuous-time Markov chains (ICTMC's), which are a
robust generalisation that relaxes these assumptions while remaining
computationally tractable.
More technically, an ICTMC is a set of "precise" continuous-time finite-state
stochastic processes, and rather than computing expected values of functions,
we seek to compute lower expectations, which are tight lower bounds on the
expectations that correspond to such a set of "precise" models. Note that, in
contrast to e.g. Bayesian methods, all the elements of such a set are treated
on equal grounds; we do not consider a distribution over this set.
The first part of this paper develops a formalism for describing
continuous-time finite-state stochastic processes that does not require the
aforementioned simplifying assumptions. Next, this formalism is used to
characterise ICTMC's and to investigate their properties. The concept of lower
expectation is then given an alternative operator-theoretic characterisation,
by means of a lower transition operator, and the properties of this operator
are investigated as well. Finally, we use this lower transition operator to
derive tractable algorithms (with polynomial runtime complexity w.r.t. the
maximum numerical error) for computing the lower expectation of functions that
depend on the state at any finite number of time points
Imprecise Markov Models for Scalable and Robust Performance Evaluation of Flexi-Grid Spectrum Allocation Policies
The possibility of flexibly assigning spectrum resources with channels of
different sizes greatly improves the spectral efficiency of optical networks,
but can also lead to unwanted spectrum fragmentation.We study this problem in a
scenario where traffic demands are categorised in two types (low or high
bit-rate) by assessing the performance of three allocation policies. Our first
contribution consists of exact Markov chain models for these allocation
policies, which allow us to numerically compute the relevant performance
measures. However, these exact models do not scale to large systems, in the
sense that the computations required to determine the blocking
probabilities---which measure the performance of the allocation
policies---become intractable. In order to address this, we first extend an
approximate reduced-state Markov chain model that is available in the
literature to the three considered allocation policies. These reduced-state
Markov chain models allow us to tractably compute approximations of the
blocking probabilities, but the accuracy of these approximations cannot be
easily verified. Our main contribution then is the introduction of
reduced-state imprecise Markov chain models that allow us to derive guaranteed
lower and upper bounds on blocking probabilities, for the three allocation
policies separately or for all possible allocation policies simultaneously.Comment: 16 pages, 7 figures, 3 table
Hitting Times and Probabilities for Imprecise Markov Chains
We consider the problem of characterising expected hitting times and hitting
probabilities for imprecise Markov chains. To this end, we consider three
distinct ways in which imprecise Markov chains have been defined in the
literature: as sets of homogeneous Markov chains, as sets of more general
stochastic processes, and as game-theoretic probability models. Our first
contribution is that all these different types of imprecise Markov chains have
the same lower and upper expected hitting times, and similarly the hitting
probabilities are the same for these three types. Moreover, we provide a
characterisation of these quantities that directly generalises a similar
characterisation for precise, homogeneous Markov chains
Computing Inferences for Large-Scale Continuous-Time Markov Chains by Combining Lumping with Imprecision
If the state space of a homogeneous continuous-time Markov chain is too
large, making inferences - here limited to determining marginal or limit
expectations - becomes computationally infeasible. Fortunately, the state space
of such a chain is usually too detailed for the inferences we are interested
in, in the sense that a less detailed - smaller - state space suffices to
unambiguously formalise the inference. However, in general this so-called
lumped state space inhibits computing exact inferences because the
corresponding dynamics are unknown and/or intractable to obtain. We address
this issue by considering an imprecise continuous-time Markov chain. In this
way, we are able to provide guaranteed lower and upper bounds for the
inferences of interest, without suffering from the curse of dimensionality.Comment: 9th International Conference on Soft Methods in Probability and
Statistics (SMPS 2018
A Recursive Algorithm for Computing Inferences in Imprecise Markov Chains
We present an algorithm that can efficiently compute a broad class of
inferences for discrete-time imprecise Markov chains, a generalised type of
Markov chains that allows one to take into account partially specified
probabilities and other types of model uncertainty. The class of inferences
that we consider contains, as special cases, tight lower and upper bounds on
expected hitting times, on hitting probabilities and on expectations of
functions that are a sum or product of simpler ones. Our algorithm exploits the
specific structure that is inherent in all these inferences: they admit a
general recursive decomposition. This allows us to achieve a computational
complexity that scales linearly in the number of time points on which the
inference depends, instead of the exponential scaling that is typical for a
naive approach
Bounding inferences for large-scale continuous-time Markov chains : a new approach based on lumping and imprecise Markov chains
If the state space of a homogeneous continuous-time Markov chain is too large, making inferences becomes computationally infeasible. Fortunately, the state space of such a chain is usually too detailed for the inferences we are interested in, in the sense that a less detailedâsmallerâstate space suffices to unambiguously formalise the inference. However, in general this so-called lumped state space inhibits computing exact inferences because the corresponding dynamics are unknown and/or intractable to obtain. We address this issue by considering an imprecise continuous-time Markov chain. In this way, we are able to provide guaranteed lower and upper bounds for the inferences of interest, without suffering from the curse of dimensionality
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