arXiv subjects

Malgorzata M. O'Reilly

Publications and source records attributed to Malgorzata M. O'Reilly.

9 recordsLinked to original sources

arXiv2026

Level-dependent quasi-birth-and-death processes: Application to cost analysis of multi-server systems

Analysing costs is crucial for optimising the operational efficiency and resource allocation in systems evolving under uncertainty. In this paper, we study the distribution of costs associated with the evolution of level-dependent quasi-birth-and-death (LD-QBD) processes, which are useful in modelling many multi-server systems. We derive analytical expressions for the Laplace-Stieltjes transforms (LSTs) of the distribution of total costs accumulated during the times the LD-QBD processes spend in a specified set of levels. We present algorithms for the numerical evaluation of these LSTs. We also give memory efficient versions of the algorithms and discuss their algorithmic complexity. To assess the robustness of the distribution of costs with respect to model parameters, we develop algorithms for the sensitivity analysis of the corresponding LSTs. To illustrate the application potential of our results, we construct LD-QBD example models for a finite capacity multi-server queueing systems with admissions policies including redirection, preemptive transfer, and guard-channel threshold. The analysis is based on a large dataset obtained from a tertiary referral hospital in Australia. We compute the long-run performance measures, the distribution of time until some number of beds become available following congestion, and the distribution of the associated costs. We present valuable insights into how the system behaves under the various policies. We also perform the sensitivity analysis of the distribution of costs with respect to model parameters.

math.PR↗

arXiv2026

Waiting time analysis in a finite-capacity multi-server systems with dynamic priorities, dynamically evolving customer types, and abandonment

In many service systems, an estimation of customers' waiting times for the service can assist in decision making focused on enhancing the operational efficiency, improving the customers' experience, and ensuring efficient resource allocation. In this paper, we study the customers' waiting times in a finite-capacity service system with a finite number of parallel servers and a shared waiting area. We consider two customer types, Type 1 and Type 2, with dynamic admission priorities, dynamically evolving customer type, and abandonment. We model the system under such assumptions using a continuous-time Markov chain (CTMC) and develop a methodology based on Krylov subspace approximation methods to evaluate the conditional waiting time distributions of Type 1 and Type 2 customers in the system. This methodology (CTMC-Krylov) offers a scalable computational approach that is well suited for analysing large complex systems. Next, we model this system using a quasi-birth-and-death (QBD) process and derive analytical expressions building on matrix-analytic methods to evaluate the conditional and long-run waiting time distributions using recursion. We illustrate the practical applicability of our models in a hospital system through a suite of numerical examples based on a large dataset obtained from a tertiary referral hospital in Australia, considering two types of patients, complex (Type 1) and other (Type 2).

stat.AP↗

arXiv2026

Quasi-birth-and-death processes evolving within trees: Applications to comparative phylogenetics

We consider a quasi-birth-and-process (QBD) that duplicates itself at some fixed times within a tree that contains information about duplication times and potentially partially observed states. We analyse a continuous trait by discretising it to obtain the QBD level variable. Then, the phase variable is used to model the dynamics of the underlying environment. Here, we extend the framework of Soewongsono et al. to enable a more general analysis. We develop an efficient recursive algorithm for computing the likelihood of an observed tree under this model and construct several numerical examples to illustrate its application potential. Through our synthetic data examples, we show a range of potential behaviours that could be modelled with this approach. Further, we apply the framework to two empirical examples from comparative phylogenetics (the evolution of range area and body size traits across a phylogeny of 49 mammals) to gain different insights into the evolution of these continuous traits. In this setting duplication of the QBD represents speciation and continuous trait evolution is modelled in a discretised state space. In our empirical examples, we explore the impact of different parameter choices on the corresponding likelihood of observing a given phylogenetic tree and the observed levels at its tips.

q-bio.PE↗

arXiv2024

Markov Decision Process and Approximate Dynamic Programming for a Patient Assignment Scheduling problem

We study the Patient Assignment Scheduling (PAS) problem in a random environment that arises in the management of patient flow in the hospital systems, due to the stochastic nature of the arrivals as well as the Length of Stay distribution. We develop a Markov Decision Process (MDP) which aims to assign the newly arrived patients in an optimal way so as to minimise the total expected long-run cost per unit time over an infinite horizon. We assume Poisson arrival rates that depend on patient types, and Length of Stay distributions that depend on whether patients stay in their primary wards or not. Since the instances of realistic size of this problem are not easy to solve, we develop numerical methods based on Approximate Dynamic Programming. We illustrate the theory with numerical examples with parameters obtained by fitting to data from a tertiary referral hospital in Australia, and demonstrate the application potential of our methodology under practical considerations.

math.OC↗

arXiv2023

Matrix-analytic methods for the evolution of species trees, gene trees, and their reconciliation

We consider the reconciliation problem, in which the task is to find a mapping of a gene tree into a species tree, so as to maximize the likelihood of such fitting, given the available data. We describe a model for the evolution of the species tree, a subfunctionalisation model for the evolution of the gene tree, and provide an algorithm to compute the likelihood of the reconciliation. We derive our results using the theory of matrix-analytic methods and describe efficient algorithms for the computation of a range of useful metrics. We illustrate the theory with examples and provide the physical interpretations of the discussed quantities, with a focus on the practical applications of the theory to incomplete data.

q-bio.PE↗

arXiv2023

A Stochastic Fluid Model Approach to the Stationary Distribution of the Maximum Priority Process

In traditional priority queues, we assume that every customer upon arrival has a fixed, class-dependent priority, and that a customer may not commence service if a customer with a higher priority is present in the queue. However, in situations where a performance target in terms of the tails of the class-dependent waiting time distributions has to be met, such models of priority queueing may not be satisfactory. In fact, there could be situations where high priority classes easily meet their performance target for the maximum waiting time, while lower classes do not. Here, we are interested in the stationary distribution at the times of commencement of service of this maximum priority process. Until now, there has been no explicit expression for this distribution. We construct a mapping of the maximum priority process to a tandem fluid queue, which enables us to find expressions for this stationary distribution. We derive the results for the stationary distribution of the maximum priority process at the times of the commencement of service.

math.PR↗

arXiv2023

Stochastic niche-based models for the evolution of species

There have been many studies to examine whether one trait is correlated with another trait across a group of present-day species (for example, do species with larger brains tend to have longer gestation times. Since the introduction of the phylogenetic comparative method some authors have argued that it is necessary to have a biologically realistic model to generate evolutionary trees that incorporates information about the ecological niche occupied by species. Price presented a simple model along these lines in 1997. He defined a two-dimensional niche space formed by two continuous-valued traits, in which new niches arise with trait values drawn from a bivariate normal distribution. When a new niche arises, it is occupied by a descendant species of whichever current species is closest in ecological niche space. In sequence, more species are then evolved from already-existing species to which they are ecologically closest. Here we explore ways of extending Price's adaptive radiation model. One extension is to increase the dimensionality of the niche space by considering more than two continuous traits. A second extension is to allow both extinction of species (which may leave unoccupied niches) and removal of niches (which causes species occupying them to go extinct). To model this problem, we consider a continuous-time stochastic process which implicitly defines a phylogeny. To explore if trees generated under such a model (or under different parametrizations of the model) are realistic we can compute a variety of summary statistics that can be compared to those of empirically observed phylogenies. For example, there are existing statistics that aim to measure: tree balance, the relative rate of diversification, and phylogenetic signal of traits.

q-bio.PE↗

arXiv2023

Modelling gene content across a phylogeny to determine when genes become associated

In this work, we develop a stochastic model of gene gain and loss with the aim of inferring when (if at all) in evolutionary history and association between two genes arises. The data we consider is a species tree along with information on the presence or absence of two genes in each of the species. The biological motivation for our model is that if two genes are involved in the same biochemical pathway, i.e. they are both required for some function, then the rate of gain or loss of one gene in the pathway should depend upon the presence or absence of the other gene in the pathway. However, if the two genes are not functionally linked, then the rate of gain or loss of one gene should be independent of the state of another gene. We simulate data under this model to determine under what conditions a shift from the independent rates class to the dependent rates class can be detected. For example, how large a tree is required and how large a shift in the rates is needed before Akaike information criterion (AIC) supports a model with two rate classes over a simpler model with just one rate class? If a model with two rate classes is preferred, can it correctly detect where on the evolutionary tree the shift occurred?

q-bio.PE↗

arXiv2019

A discontinuous Galerkin method for approximating the stationary distribution of stochastic fluid-fluid processes

Introduced by Bean and O'Reilly (2014), a stochastic fluid-fluid process is a Markov processes $\{X_t, Y_t, φ_t\}_{t \geq 0}$, where the first fluid $X_t$ is driven by the Markov chain $φ_t$, and the second fluid $Y_t$ is driven by $φ_t$ as well as by $X_t$. That paper derived a closed-form expression for the joint stationary distribution, given in terms of operators acting on measures, which does not lend itself easily to numerical computations. Here, we construct a discontinuous Galerkin method for approximating this stationary distribution, and illustrate the methodology using an on-off bandwidth sharing system, which is a special case of a stochastic fluid-fluid process.

math.PR↗