Małgorzata M. O'Reilly

ALGORITHMS FOR RETURN PROBABILITIES FOR STOCHASTIC FLUID FLOWS

Abstract We consider several known algorithms and introduce some new algorithms that can be used to calculate the probability of return to the initial level in the Markov stochastic fluid flow model. We give the physical interpretations of these algorithms within the fluid flow environment. The rates of convergence are explained in terms of the physical properties of the fluid flow processes. We compare these algorithms with respect to the number of iterations required and their complexity. The performance of the algorithms depends on the nature of the process considered in the analysis. We illustrate this with examples and give appropriate recommendations.

Hitting probabilities and hitting times for stochastic fluid flows: The bounded model

We consider a Markovian stochastic fluid flow model in which the fluid level has a lower bound zero and a positive upper bound. The behavior of the process at the boundaries is modeled by parameters that are different than those in the interior and allow for modeling a range of desired behaviors at the boundaries. We illustrate this with examples. We establish formulas for several time-dependent performance measures of significance to a number of applied probability models. These results are achieved with techniques applied within the fluid flow model directly. This leads to useful physical interpretations, which are presented.

A Stochastic Fluid Flow Model of the Operation and Maintenance of Power Generation Systems

A tool to inform strategic decision making on electricity market bidding prices, based on prediction of long-term system operation, degradation, and maintenance, is described. The operation and maintenance of a hydro-power generation system is modeled using a bounded stochastic fluid flow model with special behavior at the boundaries. The stationary distribution for the model and useful time-dependent performance measures are derived. The application potential of the model is illustrated through a practical industry-derived example modeling the operation of a hydro-power generator, in which a number of operation strategies are compared using several performance measures.

A stochastic two-dimensional fluid model

We introduce a Stochastic Two-Dimensional Fluid Model that consists of two stochastic fluid flows driven by the same underlying Markov chain, where one of the fluids is unconstrained. We develop the theoretical and numerical framework for the transient analysis of the model. We derive the important generator matrix of a particular Laplace-Stieltjes transform of the model, which is the foundation of our analysis. We use it to develop expressions for the Laplace-Stieltjes transforms of various performance measures for the transient analysis of the model and construct powerful algorithms for their numerical evaluation. An example of an application in a queueing environment is given.

On mechanistic modeling of gene content evolution: birth-death models and mechanisms of gene birth and gene retention

Characterizing the mechanisms of duplicate gene retention using phylogenetic methods requires models that are consistent with different biological processes. The interplay between complex biological processes and necessarily simpler statistical models leads to a complex modeling problem. A discussion of the relationship between biological processes, existing models for duplicate gene retention and data is presented. Existing models are then extended in deriving two new birth/death models for phylogenetic application in a gene tree/species tree reconciliation framework to enable probabilistic inference of the mechanisms from model parameterization. The goal of this work is to synthesize a detailed discussion of modeling duplicate genes to address biological questions, moving from previous work to future trajectories with the aim of generating better models and better inference.

Loss rates for stochastic fluid models

We introduce loss rates, a novel class of performance measures for Markovian stochastic fluid models and discuss their applications potential. We derive analytical expressions for loss rates and describe efficient methods for their evaluation. Further, we study interesting asymptotic properties of loss rates for large size of the buffer, which are crucial for identifying the Quality of Service requirements guaranteed for each user. We illustrate the theory with a numerical example.

A Relative Density Ratio-Based Framework for Detection of Land Cover Changes in MODIS NDVI Time Series

To improve statistical approaches for near real-time land cover change detection in nonGaussian time-series data, we propose a supervised land cover change detection framework in which a MODIS NDVI time series is modeled as a triply modulated cosine function using the extended Kalman filter and the trend parameter of the triply modulated cosine function is used to derive repeated sequential probability ratio test (RSPRT) statistics. The statistics are based on relative density ratios estimated directly from the training set by a relative unconstrained least squares importance Fitting (RULSIF) algorithm, unlike traditional likelihood ratio-based test statistics. We test the framework on simulated, synthetic, and real-world beetle infestation datasets, and show that using estimated relative density ratios, instead of assuming the individual density functions to be Gaussian or approximating them with Gaussian Kernels, in the RSPRT statistics achieves better performance in terms of accuracy and detection delay. We verify the efficiency of the proposed approach by comparing its performance with three existing methods on all the three datasets under consideration in this study. We also propose a simple heuristic technique that tunes the threshold efficiently in difficult cases of near real-time change detection, when we need to take three performance indices, namely, false positives, false negatives, and mean detection delay, into account simultaneously.

A stochastic fluid model driven by an uncountable-state process, which is a stochastic fluid model itself: the stochastic fluid-fluid model

Stochastic fluid models (SFMs) are stochastic models with a two dimensional state-space, consisting of a continuous level variable X(t) and a finite phase variable φ(t) ∈ S, where S is some finite set. The phase variable φ(t) is often used to describe the state of the environment at time t. The model assumes that the transitions between phases occur according to a continuous-time Markov chain. At time t, when φ(t) = i, the rate of increase of the level X(·) is given by the constant ci, which may be positive, negative or zero. We say that the underlying Markov chain drives the fluid level X(·). A SFM is therefore a two-dimensional Markov process, {(φ(t), X(t)), t ≥ 0}. In this paper, we are interested in the following model where, besides the level X(t) at time t, we would like to model some other continuous performance measure Y (t), such as the net profit at time t for example. To illustrate this, consider a hydro-power generator, which can be operated in various modes of operation. The generator must be periodically maintained, in order to improve its performance and prolong its lifespan. An important problem is the evaluation of maintenance strategies and the impact of maintenance on reliability. This problem can be modelled as a bounded SFM {(φ(t), X(t)), t ≥ 0} with S consisting of operating phases and deterioration level X(t) ∈ [0, 1] [1]. Now, consider the continuous revenue process {Y (t), t ≥ 0} associated with the process {(X(t), φ(t)), t ≥ 0}. Clearly the rate at which the revenue is increasing/decreasing at time t, will depend on the operating mode φ(t). However, the rate of earning revenue at time t may also depend on the deterioration level X(t). For example, when the generator is new, it will operate very efficiently, produce more energy and require less-costly maintenance. Hence, the rates at which the revenue level Y (t) is increasing/decreasing may depend both on the operating mode i ∈ S and the deterioration level X(t) = x at time t. We denote these rates by ri(x). In this paper, we propose a new class of fluid models, denoted by {(φ(t), X(t), Y (t)), t ≥ 0}. We refer to this class as the stochastic fluid-fluid model, since it is a SFM with level Y (t) driven by the uncountable state-space process, {(φ(t), X(t)), t ≥ 0}, which is also a SFM itself. Another example is the process of growth and bleaching of coral reefs [3, 2]. To model this process, a classical SFM can be used [2], in which X(t) is interpreted as the density of symbiotic zooxanthellae at time t; positive rates ci correspond to the growth of the zooxanthellae, while negative rates correspond to bleaching. If the density drops below some given level x, the coral does not receive sufficient energy and has to rely on stored lipids. It cannot start to restore those lipids until the density reaches x again. If the coral runs out of stored lipids then the coral dies. The important question here is what is the survival distribution of the coral, that is, what is the probability that the level of stored lipids will not reach 0 before time t. If we let Y (t) represent the amount of stored lipids, then a doublefluid model {(φ(t), X(t), Y (t)), t ≥ 0} can be applied. So we see that the modelling potential of the stochastic fluid-fluid model is quite extensive. In this paper we focus on developing the theoretical framework for the stochastic fluid-fluid model in the form of analytical expressions for various performance measures. We assume that Y (t) ≥ 0, so buffer Y is bounded below by zero. The idea that we apply here is to study the evolution of the process {Y (t)} in terms of the uncountable-state process {(φ(t), X(t))}, and develop appropriate operator expressions corresponding to relevant sample paths in {Y (t)}. We consider the standard SFM {(φ(t), X(t))} and derive the generator of its dynamics, with respect to time. These preliminary results are used to derive expressions for the Laplce-Stieltjes Transforms (LSTs) with respect to level in the process {Y (t)}. We derive expressions for the important operator Ψ(s), which records the LSTs of the time to first return to the original level zero in the process {Y (t)}. The importance of Ψ(s) lies in the fact that, just like in the onedimensional case, other useful performance measures can be derived from it, such as the stationary distribution of the process {(φ(t), X(t), Y (t)), t ≥ 0}. Whilst the analysis in this paper is focused on the development of the theoretical framework, the numerical treatment of the model is also briefly discussed.

Spatially-coherent uniformization of a stochastic fluid model to a Quasi-Birth-and-Death process

We derive a uniformization of a stochastic fluid model (SFM) to a Quasi-Birth-and-Death process (QBD) that is spatially-coherent since the continuous level in the SFM has a natural correspondence to the discrete level in the QBD. As a consequence of this, the QBD can be used as a direct approximation of the original SFM, in those situations where a discrete state space is an advantage. We treat the unbounded as well as the bounded cases and illustrate the theory with a numerical example. The key fluid generator, Q, and matrix @J for the SFMs emerge from the QBD calculations in the natural limit.

Stochastic model for maintenance in continuously deteriorating systems

We construct a stochastic model for maintenance suitable for the analysis of real-life systems which deteriorate over time before they eventually fail and are replaced. The model uses a continuous deterioration level, where the rate of change depends on the current operating mode as well as the current level of deterioration. We demonstrate how to construct a model in which the uncertainty about the state of deterioration, when the system is not continuously observed, is accurately represented. This feature addresses some drawbacks of previous work that is known to cause modelling errors. The key performance measures for this model can be evaluated efficiently using existing algorithms. The theory is illustrated using numerical examples, in which we discuss how this model can be used in a practical evaluation of different maintenance strategies.