Malgorzata M. O'REilly

Markov modulated (MM) models in the theory of matrix-analytic methods (MAMs) are a class of probabilistic models whose key feature is an underlying Markov environment that drives the evolution of some complex and unpredictable system. Our goal in this paper is to emphasise their great potential for understanding evolution and the need for close multidisciplinary collaborations between probabilists and evolutionary biologists to advance the field. We give an overview of the fundamental MM models in MAMs, namely quasi-birth-and-death processes (QBDs) and stochastic fluid models (SFMs), in which the fluctuations in the level variable are modulated by fluctuations in the phase variable. We illustrate their application potential in evolution through an example of phylogenetic inference, state a general tree likelihood formula applicable to a wide range of MM models, and discuss future directions required to advance the theory of MM models as well as in the applications of the existing theory to evolutionary contexts.

We establish results for the first sensitivity analysis of the stochastic fluid models (SFMs). We derive expressions for the sensitivity analysis of the key stationary and transient (time-dependent) quantities of this class of models. We also construct numerical examples to demonstrate the application potential of our methodology in queueing systems, such as deteriorating systems and insurance risk processes. This work forms foundation for the sensitivity analysis of other Markovian modulated models, which are generalisations of the SFMs, and have widespread applications.

Fog computing providers have started providing services in closer proximity to the users by leasing the unused computational resources of users' devices. The leased devices are increasingly used for many time-sensitive and IoT applications. Despite the advantages of Fog, due to the highly mobile and dynamic nature of Fog devices, the resources or devices may fail at any time or may not always be available for the processing of the applications, which leads to degradation of service quality and an increase in application processing time. Hence, effective selection of the resources in the Fog computing environment by considering mobility, heterogeneity, and failure of devices is a complex task. Traditional resource selection techniques may not be applied directly in Fog computing environments due to their dynamic and unique resource characteristics. Therefore, this article proposes a Markov chain-based resource selection model to improve the quality of service by minimizing disruptions and managing device failures. The effectiveness of the proposed algorithm is evaluated using simulations, which take failure traces, current resource usage, and mobility as input from a dataset archive. Our results demonstrate the effectiveness of the proposed algorithm in terms of average disruption rates, average latency, and average overutilization. Our analysis shows significant improvements in average latency reduction of approximately 11.83%, and an average overutilization improvement by 16.17%.

Emergency Department overcrowding is a critical issue that compromises patient safety and operational efficiency, necessitating accurate demand forecasting for effective resource allocation. This study evaluates and compares three distinct predictive models: Seasonal AutoRegressive Integrated Moving Average with eXogenous regressors (SARIMAX), EXtreme Gradient Boosting (XGBoost) and Long Short-Term Memory (LSTM) networks for forecasting daily ED arrivals over a seven-day horizon. Utilizing data from an Australian tertiary referral hospital spanning January 2017 to December 2021, this research distinguishes itself by decomposing demand into eight specific ward categories and stratifying patients by clinical complexity. To address data distortions caused by the COVID-19 pandemic, the study employs the Prophet model to generate synthetic counterfactual values for the anomalous period. Experimental results demonstrate that all three proposed models consistently outperform a seasonal naive baseline. XGBoost demonstrated the highest accuracy for predicting total daily admissions with a Mean Absolute Error of 6.63, while the statistical SARIMAX model proved marginally superior for forecasting major complexity cases with an MAE of 3.77. The study concludes that while these techniques successfully reproduce regular day-to-day patterns, they share a common limitation in underestimating sudden, infrequent surges in patient volume.

We study the patient assignment scheduling (PAS) problem in a random environment that arises in the management of patient flow in hospital systems, due to the stochastic nature of the arrivals as well as the length of stay (LoS) distribution. At the start of each time period, emergency patients in the waiting area of a hospital system need to be admitted to relevant wards. Decisions may involve allocation to less suitable wards, or transfers of the existing inpatients to accommodate higher priority cases when wards are at full capacity. However, the LoS for patients in non-primary wards may increase, potentially leading to long-term congestion. To assist with decision-making in this PAS problem, we construct a discrete-time Markov decision process over an infinite horizon, with multiple patient types and multiple wards. Since the instances of realistic size of this problem are not easy to solve, we develop numerical methods based on approximate dynamic programming. We demonstrate the application potential of our methodology under practical considerations with numerical examples, using parameters obtained from data at a tertiary referral hospital in Australia. We gain valuable insights, such as the number of patients in non-primary wards, the number of transferred patients, and the number of patients redirected to other facilities, under different policies that enhance the system’s performance. This approach allows for more realistic assumptions and can also help determine the appropriate size of wards for different patient types within the hospital system.

We consider the reconciliation problem, in which the task is to find a mapping of a gene tree into a species tree. In this paper we present a method, where for a given choice of parameters, we are able to compute the likelihood for alternative reconciliations. We describe a Markovian binary tree (MBT) model for the evolution of species trees, a quasi-birth-and-death (QBD) model for the evolution of gene trees, and provide a recursive algorithm to compute the likelihood of a given reconciliation between a species tree and a gene tree. We derive our results using the theory of matrix-analytic methods, prove them using rigorous mathematics together with decomposition of sample path arguments, and describe 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.

Background: Poor patient flow or patient movement through a healthcare organisation can lead to adverse outcomes for patients and organisational inefficiency. Many hospitals have addressed suboptimal patient flow by increasing resources, such as bed stock and staffing; however, this is an unsustainable approach. In determining the nature of poor patient flow issues, it is important to collect data from healthcare professionals who manage patient flow daily. Doing so provides insights into the current state of patient flow management in its entirety, whilst also helping with the development of sustainable solutions. Methods: Thirteen semi-structured interviews were conducted with healthcare professionals who were directly involved with patient flow at a referring hospital in Tasmania, Australia. Results: Using a thematic analysis method, four major themes were developed. The first major theme was: ‘managing patient flow’ which centered around known and unknown demands on bed availability. The second theme, ‘communication for decisions’, highlighted the essential role of communication in maintaining patient flow. The third theme, ‘tools as enablers and barriers, relates to the number of software programs which can both help and hinder patient flow. The final theme of ‘increasing complexity’ is related to an apparent trend towards greater numbers of patients requiring care of an increasingly specialised nature. Conclusions: The findings of this study provide great insights into patient flow issues, with potential solutions identified to address them.