Server behaviours in healthcare queueing systems

In the classical queueing theory literature, a server is commonly assumed to work at a constant speed. Motivated by observations from healthcare applications, a study is made to explore the nature of the relationship between service times and workload in order to assess and quantify any workforce (server) behaviours. Consequently, an initial analytical queueing model is considered with switching thresholds to allow for two-speed service. In this model service time depends on queue length, which for example captures the congestion in the waiting room and the resulting change in speed of the workforce to try and cope with the backlog of patients. Furthermore, related behavioural characteristics resulting from workload fatigue and service breakdown are considered. A developed analytical model with ‘catastrophic’ service failure is proposed to examine the consequences on patient service levels. The research helps to demonstrate the importance of more accurately capturing server behaviours in workload-dependent environment

Hotelling Games on Networks: Efficiency of Equilibria

URL des Documents de travail : http://centredeconomiesorbonne.univ-paris1.fr/Documents de travail du Centre d'Economie de la Sorbonne 2014.33 - ISSN : 1955-611XWe consider a Hotelling game where a finite number of retailers choose a location, given that their potential customers are distributed on a network. Retailers do not compete on price but only on location, therefore each consumer shops at the closest store. We show that when the number of retailers is large enough, the game admits a pure Nash equilibrium and we construct it. We then compare the equilibrium cost bore by the consumers with the cost that could be achieved if the retailers followed the dictate of a benevolent planner. We perform this comparison in term of the induced price of anarchy, i.e., the ratio of the worst equilibrium cost and the optimal cost, and the induced price of stability, i.e., the ratio of the best equilibrium cost and the optimal cost. We show that, asymptotically in the number of retailers, these ratios are two and one, respectively.On considère un jeu à la Hotelling où un nombre fini de magasins doivent choisir un emplacement sachant que leurs clients potentiels sont situés sur un réseau donné. Les magasins ne sont pas en compétition sur les prix, mais seulement sur les emplacements. Nous montrons de manière constructive que lorsque le nombre de magasins est suffisamment grand ce jeu admet un équilibre de Nash en stratégies pures. Ensuite, nous comparons le coût de déplacement des consommateurs à l'équilibre avec le coût engendré par la situation optimale qui aurait été décidée par un planificateur extérieur. Pour cela, nous calculons le prix de l'anarchie induit, c'est-à-dire le ratio entre le pire coût à l'équilibre et le coût à l'optimum. Nous regardons aussi le prix de la stabilité induit, le ratio entre le meilleur coût à l'équilibre et le coût à l'optimum. Nous montrons que lorsque le nombre de vendeurs devient grand ces ratios tendent respectivement vers 2 et 1

A game theoretic model of the behavioural gaming that takes place at the EMS - ED interface

This research describes the development and application of a 3-player game theoretic model between two queueing systems and a service that distributes individuals to them. The resultant model is used to explore dynamics between all players. The first aspect of this work is the development of a queueing system with two consecutive waiting spaces where the strategic managerial behaviour corresponds to how individuals use these waiting spaces. Two modelling techniques are deployed: discrete event simulation and Markov chains. The state probabilities of the Markov chain system are used to extract the performance measures of the queueing model (e.g. mean time in each waiting room, mean number of individuals in each room, etc.). A 3-player game theoretic model is subsequently proposed between the two queueing systems and the service that distributes individuals to them. In particular this can be viewed as a 2-player normal-form game where the utilities are determined by a third player with its own strategies and objectives. A backwards induction technique is used to get the utilities of the normal-form game between the two queueing systems. This particular system has many applications, including those in healthcare where it captures the emergent behaviour between the Emergency Medical Service (EMS) and the Emergency Department (ED). The impact of time-target measures on patient well-being is explored in this paper

Evacuation planning under selfish evacuation routing

In case of an evacuation a large number of evacuees must be routed through a street network to let them leave the endangered area and reach safe places. In such a situation a lot of evacuees use the street network in a short time span and so the network capacity will be insufficient. With an evacuation plan the traffic could be guided through the network for a better use of network capacity. But to implement the solution planned by a central decision maker, optimal routes must be communicated to all network users, which lead to a high communication effort. Furthermore, it must be ensured that the evacuees take the given routes. But a lot of people do not follow the instructions from authorities in a panic situation. They do what they assume is best for themselves. Such selfish behaviour leads to a suboptimal distribution of traffic and results in congestion. In this thesis we present a concept to guide the evacuees through the network without determining optimal routes for all network users. With the blockage of street sections we force the evacuees to use other routes than the preferred ones but give them the possibility to choose their routes on their own. The thesis presents different mathematical model formulations and heuristic for the described problem. In a comprehensive computational study, with real world examples, the functionality of the presented concept and methods are tested

Efficiency and resilience of heterogeneous networks

Many systems such as traffic or electrical flow can be described as flows following paths of least resistance in networks. The efficiency and resilience of these networks define the system’s ability to function effectively. Research into network efficiency and resilience often focuses on the role of network topology, with the aim of uncovering optimal network structures that boost system performance. However, little attention has been paid to the role of node behaviour. This thesis bridges that gap by analysing the efficiency and resilience of networks whose nodes have heterogeneous behaviour. The nodes may variably be sources or sinks of the flow. The nodes may also be equipped with the ability to adjust their behaviour in response to the state of the network. The efficiency and resilience of networks are evaluated as a function of their composition of node types and behaviours. The primary motivation for this is the proliferation of renewable sources of electrical power in energy grids. The resulting electrical networks have highly dynamic and heterogeneous nodes. This thesis provides a framework in which to analyse the behaviour of these systems. A variety of mathematical methods are utilised throughout this thesis. The efficiency of network flows is analysed using a measurement from game theory called the Price of Anarchy, from which an equivalency between least resistance network flows and Nash equilibria is also identified. The average variation of efficiency with node composition is found to be approximately invariant across different network structures. The highest inefficiencies are found to always occur when there are an equal number of source and sink nodes. Resilience is investigated using models of cascading network failures. Both a steady state and a dynamical model are employed. Analytical results for cascades on simple lattices are derived, while for complex networks it is shown that resilience can often be improved by increasing the numbers of source and sink nodes. This analysis is employed on a test case of electrical networks, constructed using real household power consumption and photovoltaic generation data. The impact of the dynamic variability of these data-driven networks on resilience is analysed. Lowest resiliences are found during times when high numbers of photo-voltaic source nodes are active

Efficiency and resilience of heterogeneous networks

Many systems such as traffic or electrical flow can be described as flows following paths of least resistance in networks. The efficiency and resilience of these networks define the system’s ability to function effectively. Research into network efficiency and resilience often focuses on the role of network topology, with the aim of uncovering optimal network structures that boost system performance. However, little attention has been paid to the role of node behaviour. This thesis bridges that gap by analysing the efficiency and resilience of networks whose nodes have heterogeneous behaviour. The nodes may variably be sources or sinks of the flow. The nodes may also be equipped with the ability to adjust their behaviour in response to the state of the network. The efficiency and resilience of networks are evaluated as a function of their composition of node types and behaviours. The primary motivation for this is the proliferation of renewable sources of electrical power in energy grids. The resulting electrical networks have highly dynamic and heterogeneous nodes. This thesis provides a framework in which to analyse the behaviour of these systems. A variety of mathematical methods are utilised throughout this thesis. The efficiency of network flows is analysed using a measurement from game theory called the Price of Anarchy, from which an equivalency between least resistance network flows and Nash equilibria is also identified. The average variation of efficiency with node composition is found to be approximately invariant across different network structures. The highest inefficiencies are found to always occur when there are an equal number of source and sink nodes. Resilience is investigated using models of cascading network failures. Both a steady state and a dynamical model are employed. Analytical results for cascades on simple lattices are derived, while for complex networks it is shown that resilience can often be improved by increasing the numbers of source and sink nodes. This analysis is employed on a test case of electrical networks, constructed using real household power consumption and photovoltaic generation data. The impact of the dynamic variability of these data-driven networks on resilience is analysed. Lowest resiliences are found during times when high numbers of photo-voltaic source nodes are active

Modelling critical care unit activities through queueing theory

Critical Care Units (CCUs) are one of the most complex and expensive of all medical resources and hospital managers are challenged to meet the demand for critical care services with adequate capacity. The pressure on critical care beds is continuously increasing as new medical equipment provides the opportunity to save more patients lives. It is therefore crucial that beds are managed well and used efficiently. This thesis describes two major projects, the first undertaken in conjunction with the CCU at the University Hospital of Wales in Cardiff (UHW); and the second with two CCUs from the Aneurin Bevan Health Board. In the first project data has been analysed to determine the flow of patients through the Unit. Admissions to CCUs were categorised under two headings: emergency, and elective. The length of stay in the CCU is heavily dependent on the admission category. In this thesis, both computer simulation and theoretical queueing models have been considered, which show how improvements in bed management may be achieved by considering these two categories of patients separately. The vast majority of previous literature in this field is concerned only with steady-state conditions, whereas in reality the processes are time-dependent. This thesis goes some way to addressing this deficiency. The second project relates to work undertaken with managers from the Royal Gwent Hospital in Newport and at the Nevill Hall Hospital in Abergavenny. Data from both hospitals have been analysed to define arrival and service processes. A state-dependent theoretical queueing model has been considered which has been used to investigate the significance of combining the two units. The model has been also utilised to advise on the number of beds the new combined unit should have in order to satisfy targets quoted by the hospital managers. In the final part of the thesis, consideration has been given to the impact of collaboration, or lack thereof, between hospitals using a game theoretical approach. The effect of patient diversion has been studied. To formally investigate the impact of patients transfers, a Markov chain model of the two CCUs has been set-up, each admitting two arrival streams: namely, their own patients and transfers from other hospital. Four different models were considered and for each model the effect of targets, demand and capacity were studied. The efficiency of a system which degrades due to selfish behaviour of its agents has been measured in terms of Price of Anarchy

Optimal control of queueing systems with multiple heterogeneous facilities

This thesis discusses queueing systems in which decisions are made when customers arrive, either by individual customers themselves or by a central controller. Decisions are made concerning whether or not customers should be admitted to the system (admission control) and, if they are to be admitted, where they should go to receive service (routing control). An important objective is to compare the effects of "selfish" decision-making, in which customers make decisions aimed solely at optimising their own outcomes, with those of "socially optimal" control policies, which optimise the economic performance of the system as a whole. The problems considered are intended to be quite general in nature, and the resulting findings are therefore broad in scope. Initially, M/M/1 queueing systems are considered, and the results presented establish novel connections between two distinct areas of the literature. Subsequently, a more complicated problem is considered, involving routing control in a system which consists of heterogeneous, multiple-server facilities arranged in parallel. It is shown that the multiple-facility system can be formulated mathematically as a Markov Decision Process (MDP), and this enables a fundamental relationship to be proved between individually optimal and socially optimal policies which is of great theoretical and practical importance. Structural properties of socially optimal policies are analysed rigorously, and it is found that 'simple' characterisations of socially optimal policies are usually unattainable in systems with heterogeneous facilities. Finally, the feasibility of finding 'near-optimal' policies for large scale systems by using heuristics and simulation-based methods is considered