Age Replacement and Service Rate Control of Stochastically Degrading Queues

This thesis considers the problem of optimally selecting a periodic replacement time for a multiserver queueing system in which each server is subject to degradation as a function of the mean service rate and a stochastic and dynamic environment. Also considered is the problem of optimal service rate selection for such a system. In both cases, the performance metric is the long-run average cost rate. Analytical expressions are obtained, in terms of Laplace transforms, for the nonlinear objective functions, necessitating the use of numerical Laplace transform inversion to evaluate candidate solutions in conjunction with standard numerical algorithms. Due to the convexity of the objective function, the optimal replacement time is computed using a hybrid bisection-secant method which yields globally optimal solutions. The optimal service rates are obtained via gradient search methods but are only guaranteed to provide locally optimal solutions. The analytical results are implemented on three notional examples that demonstrate the benefits of dynamically adjusting service rates under the described maintenance policy

MoMA Algorithm: A Bottom-Up Modeling Procedure for a Modular System under Environmental Conditions

The functioning of complex systems relies on subsystems (modules) that in turn are composed of multiple units. In this paper, we focus on modular systems that might fail due to wear on their units or environmental conditions (shocks). The lifetimes of the units follow a phase-type distribution, while shocks follow a Markovian Arrival Process. The use of Matrix-Analytic methods and a bottom-up approach for constructing the system generator is proposed. The use of modular structures, as well as its implementation by the Modular Matrix-Analytic (MoMA) algorithm, make our methodology flexible in adapting to physical changes in the system, e.g., incorporation of new modules into the current model. After the model for the system is built, the modules are seen as a ‘black box’, i.e., only the contribution of the module as a whole to system performance is considered. However, if required, our method is able to keep track of the events within the module, making it possible to identify the state of individual units. Compact expressions for different reliability measures are obtained with the proposed description, optimal maintenance strategies based on critical operative states are suggested, and a numerical application based on a k-out-of-n structure is developed.Spanish Ministry of Science and Innovation-State Research Agency PID2020-120217RB-I00 PID2021-123737NB-I00Junta de Andalucia B-FQM-284-UGR20 CEX2020-001105-/AEI/10.13039/50110001103

Optimal data pooling for shared learning in maintenance operations

This paper addresses the benefits of pooling data for shared learning in maintenance operations. We consider a set of systems subject to Poisson degradation that are coupled through an a-priori unknown rate. Decision problems involving these systems are high-dimensional Markov decision processes (MDPs). We present a decomposition result that reduces such an MDP to two-dimensional MDPs, enabling structural analyses and computations. We leverage this decomposition to demonstrate that pooling data can lead to significant cost reductions compared to not pooling

(Batch) Markovian arrival processes: the identifiability issue and other applied aspects

Mención Internacional en el título de doctorThis dissertation is mainly motivated by the problem of statistical modeling via a specific point process, namely, the Batch Markovian arrival processes. Point processes arise in a wide range of situations of our daily activities, such as people arriving to a bank, claims of an insurance company or failures in a system. They are defined by the occurrence of an event at a specific time, where the event occurrences may be understood from different perspectives, either by the arrival of a person or group of people in a waiting line, the different claims to the insurance companies or failures occurring in a system. Point processes are defined in terms of one or several stochastic processes which implies more versatility than mere single random variables, for modeling purposes. A traditional assumption when dealing with the analysis of point processes is that the occurrence of events are independent and identically distributed, which considerably simplifies the theoretical calculations and computational complexity, and again because of simplicity, the Poisson process has been widely considered in stochastic modelling. However, the independence and exponentiability assumptions become unrealistic and restrictive in practice. For example, in teletraffic or insurance contexts it is usual to encounter dependence amongst observations, high variability, arrivals occurring in batches, and therefore, there is a need of more realistic models to fit the data. In particular, in this dissertation we investigate new theoretical and applied properties concerning the (batch) Markovian arrival processes, or (B)MAP, which is well known to be a versatile class of point process that allows for dependent and non-exponentially distributed inter-event times as well as correlated batches. They inherit the tractability of the Poisson processes, and turn out suitable models to fit data with statistical features that differ from the classical Poisson assumptions. In addition, in spite of the large amount of works considering the BMAP, still there are a number of open problems which are of interest and which shall be considered in this dissertation. This dissertation is organized as follows. In Chapter 1, we present a brief theoretical background that introduces the most important concepts and properties that are needed to carry out our analyses. We give a theoretical background of point processes and describe them from a probabilistic point of view. We introduce the Markovian point processes and its main properties, and also provide some point process estimation backdrop with a review of recent works. An important problem to consider when the statistical inference for any model is to be developed is the uniqueness of its representation, the identifiability problem. In Chapter 2 we analyze the identifiability of the non-stationary two-state MAP. We prove that, when the sample information is given by the inter-event times, then, the usual parametrization of the process is redundant, that is, the process is nonidentifiable. We present a methodology to build an equivalent non-stationary two-state MAPs from any fixed one. Also, we provide a canonical and unique parametrization of the process so that the redundant versions of the same process can be reduced to its canonical version. In Chapter 3 we study an estimation approach for the parameters of the non-stationary version of the MAP under a specific observed information. The framework to be considered is the modelling of the failures of N electrical components that are identically distributed, but for which it is not reasonable to assume that the operational times related to each component are independent and identically distributed. We propose a moments matching estimation approach to fit the data to the non-stationary two-state MAP. A simulated and a real data set provided by the Spanish electrical group Iberdrola are used to illustrate the approach. Unlike Chapters 2 and 3, which are devoted to the Markovian arrival process, Chapters 4 and 5 focus on its arrivals-in-batches counterpart, the BMAP. The capability of modeling non-exponentially distributed and dependent inter-event times as well as correlated batches makes the BMAP suitable in different real-life settings as teletraffic, queueing theory or actuarial contexts, to name a few. In Chapter 4 we analyze the identifiability issue of the BMAP. Specifically, we explore the identifiability of the stationary two-state BMAP noted as BMAP2(k), where k is the maximum batch arrival size, under the assumptions that both the inter-event times and batches sizes are observed. It is proven that for k ≥ 2 the process cannot be identified. The proof is based on the construction of an equivalent BMAP2(k) to a given one, and on the decomposition of a BMAP2(k) into k BMAP2(2)s. In Chapter 5 we study the auto-correlation functions of the inter-event times and batch sizes of the BMAP. This chapter examines the characterization of both auto-correlation functions for the stationary BMAP2(k), for k ≥ 2, where four behavior patterns are identified for both functions for the BMAP2(2). It is proven that both auto-correlation functions decrease geometrically as the time lag increases. Also, the characterization of the autocorrelation functions has been extended for the general BMAPm(k) case, m ≥ 3. To conclude, Chapter 6 summarizes the most significant contributions of this dissertation, and also give a short description of possible research lines.Esta tesis está motivada por el problema de modelización estadística mediante un tipo específico de procesos puntuales, los procesos de llegada Markovianos en tandas. Los procesos puntuales surgen en una gran variedad de situaciones de la vida real, como las personas que llegan a un banco, reclamaciones en compañías de seguro o fallos en un sistema. Los procesos puntuales se definen como la ocurrencia de eventos en diferentes instantes temporales, donde las ocurrencias de eventos se pueden entender desde diferentes perspectivas, llegadas de personas o un grupo de personas a una cola, las distintas reclamaciones en una compañía de seguros o los fallos que ocurren en un sistema. Los procesos puntuales se definen en términos de uno o varios procesos estocásticos lo que implica más versatilidad, en términos de modelización, que la que se obtiene mediante variables aleatorias que no consideren la dimensión temporal. Una suposición tradicional en la literatura al estudiar y analizar procesos puntuales es que los tiempos entre la ocurrencia de eventos son independientes e idénticamente distribuidos, lo que simplifica considerablemente los cálculos teóricos y la complejidad computacional. Adicionalmente, por simplicidad, el proceso de Poisson ha sido ampliamente considerado en modelización estocástica. Sin embargo, las suposiciones de independencia y exponenciabilidad son poco realistas en la práctica. Por ejemplo, en el contexto teletráfico o de seguros es usual encontrar dependencia entre las observaciones, alta variabilidad, llegadas que ocurren en tandas, por lo que hay una necesidad de ajustar los datos a modelos más reales. En particular, en esta tesis investigamos nuevas propiedades teóricas y aplicadas sobre los procesos de llegada Markovianos (en tanda), denotados (B)MAP, que son conocidos por ser procesos puntuales versátiles que permiten la dependencia y no-exponenciabilidad de los tiempos entre eventos, así como la correlación entre las tandas. Ya que heredan la manejabilidad de los procesos de Poisson, son procesos adecuados para ajustar datos con características estadísticas que difieren de los supuestos clásicos de Poisson. Además, a pesar de la gran cantidad de trabajos que consideran los BMAP, todavía hay una serie de problemas abiertos que son de interés y que serán considerados en esta tesis. La estructura de esta tesis es la siguiente. En el Capítulo 1, se presenta una breve revisión teórica que introduce las definiciones y propiedades más importantes necesarias para el desarrollo de nuestros análisis. Se definen los procesos puntuales y se describen desde un punto de vista probabilístico. Se introducen los procesos puntuales Markovianos y sus propiedades principales, además se proporciona una revisión de la literatura sobre la estimación de los procesos puntuales. Un problema importante a considerar cuando se quieren desarrollar métodos de inferencia sobre cualquier modelo es la unicidad de su parametrización, o alternativamente, el problema de identificabilidad. En el Capítulo 2 estudiamos el problema de identificabilidad del MAP no estacionario con dos estados. Se demuestra que, cuando la información muestral está dada por los tiempos entre eventos, entonces, la parametrización usual del proceso es redundante, esto es, el proceso es no-identificable. Se presenta un procedimiento para construir un MAP no estacionario con dos estados equivalente a uno fijo. Además, se proporciona una parametrización canónica y única del proceso, de manera que las versiones redundantes o equivalentes de un mismo proceso se pueden reducir a su versión canónica. En el Capítulo 3 se estudia un método de estimación para los parámetros del MAP no estacionario con dos estados. El esquema que se considerará es la modelización de los fallos de N componentes eléctricos que son idénticamente distribuidos, pero que no es razonable considerar que los tiempos operacionales asociados a cada componente son independientes ni idénticamente distribuidos. Se propone un método de igualdad de momentos para ajustar datos a un MAP no estacionarios con dos estados. Se presenta un ejemplo simulado y un ejemplo con datos reales proporcionados por la compañía eléctrica Iberdrola para ilustrar la metodología propuesta. A diferencia de los capítulos 2 y 3, que están dedicados a los procesos de llegada Markovianos, los capítulos 4 y 5 se centran en su generalización para considerar llegadas en tandas, el BMAP. La capacidad de modelar tiempos entre eventos dependientes y no-exponenciales, así como llegadas en tandas correladas, hace que los BMAP sean modelos apropiados en problemas de la vida real, como en contextos teletráficos, de teoría de colas o actuariales, entre otros. En el Capítulo 4 se explora la identificabilidad para el BMAP estacionario de 2 estados, BMAP2(k), donde k es el tamaño máximo de las tandas, bajo la suposición de que los tiempos entre eventos y los tamaños de las tandas son los datos observados. Se demuestra que para k ≥ 2 el proceso no es único. La demostración se basa en la construcción de un BMAP2(k) equivalente a uno fijo, y en la descomposición de un BMAP2(k) en k BMAP2(2)s. En el Capítulo 5 se estudia las funciones de autocorrelación para los tiempos entre-eventos y las llegadas en tanda del BMAP. Además, también se examina la caracterización de ambas funciones de autocorrelación para el BMAP2(k), k ≥ 2, estacionario, donde se identifican cuatro patrones para el BMAP2(2). Se demuestra que ambas funciones de autocorrelación decrecen geométricamente. Finalmente, se extiende la caracterización de las funciones de autocorrelación para el caso general BMAPm(k), m ≥ 3. Finalmente, en el Capítulo 6 se resumen las contribuciones más importantes de esta tesis y futuras líneas de investigación.Programa Oficial de Doctorado en Ingeniería MatemáticaPresidente: Rafael Pérez Ocón.- Secretario: D Auria , Bernardo.- Vocal: Mogens Blad

Generalized models of repairable systems: A survey via stochastic processes formalism

In this article, we survey the developments in the generalised models of repairable systems reliability during 1990s, particularly the last five years. In this field, we notice the sharp fundamental problem that voluminous complex models were developed but there is an absence of sufficient data of interest for justifying the success in tackling the real engineering problems. Instead of following the myth of using simple models to face the complex reality, we select and review some practical models, particularly the stochastic processes behind them. The Models in three quick growth areas: age models, condition monitoring technique related models, say, proportional intensity and their extensions, and shock and wearing models, including the delay-time models are reviewed. With the belief that only those stochastic processes reflecting the instinct nature of the actual physical processes of repairable systems, without excessive assumptions, may have a better chance to meet the demands of engineers and managers

Utilizing prediction analytics in the optimal design and control of healthcare systems

In recent years, increasing availability of data and advances in predictive analytics present new opportunities and challenges to healthcare management. Predictive models are developed to evaluate various aspects of healthcare systems, such as patient demand, patient pathways, and patient outcomes. While these predictions potentially provide valuable information to improve healthcare delivery, there are still many open questions considering how to integrate these forecasts into operational decisions. In this context, this dissertation develops methodologies to combine predictive analytics with the design of healthcare delivery systems. The first part of dissertation considers how to schedule proactive care in the presence of patient deterioration. Healthcare systems are typically limited resource environments where scarce capacity is reserved for the most urgent patients. However, there has been a growing interest in the use of proactive care when a less urgent patient is predicted to become urgent while waiting. On one hand, providing care for patients when they are less critical could mean that fewer resources are needed to fulfill their treatment requirement. On the other hand, due to prediction errors, the moderate patients who are predicted to deteriorate in the future may self cure on their own and never need the treatment. Hence, allocating limited resource for these patients takes the capacity away from other more urgent ones who need it now. To understand this tension, we propose a multi-server queueing model with two patient classes: moderate and urgent. We allow patients to transition classes while waiting. In this setting, we characterize how moderate and urgent patients should be prioritized for treatment when proactive care for moderate patients is an option. The second part of the dissertation focuses on the nurse staffing decisions in the emergency departments (ED). Optimizing ED nurse staffing decisions to balance the quality of service and staffing cost can be extremely challenging, especially when there is a high level of uncertainty in patient demand. Increasing data availability and continuing advancements in predictive analytics provide an opportunity to mitigate demand uncertainty by utilizing demand forecasts. In the second part of the dissertation, we study a two-stage prediction-driven staffing framework where the prediction models are integrated with the base (made weeks in advance) and surge (made nearly real-time) staffing decisions in the ED. We quantify the benefit of having the ability to use the more expensive surge staffing. We also propose a near-optimal two-stage staffing policy that is straightforward to interpret and implement. Lastly, we develop a unified framework that combines parameter estimation, real-time demand forecasts, and capacity sizing in the ED. High-fidelity simulation experiments for the ED demonstrate that the proposed framework can reduce annual staffing costs by 11%-16% ($2 M-$3 M) while guaranteeing timely access to care

An Application of Matrix Analytic Methods to Queueing Models with Polling

We review what it means to model a queueing system, and highlight several components of interest which govern the behaviour of customers, as well as the server(s) who tend to them. Our primary focus is on polling systems, which involve one or more servers who must serve multiple queues of customers according to their service policy, which is made up of an overall polling order, and a service discipline defined at each queue. The most common polling orders and service disciplines are discussed, and some examples are given to demonstrate their use. Classic matrix analytic method theory is built up and illustrated on models of increasing complexity, to provide context for the analyses of later chapters. The original research contained within this thesis is divided into two halves, finite population maintenance models and infinite population cyclic polling models. In the first half, we investigate a 2-class maintenance system with a single server, expressed as a polling model. In Chapter 2, the model we study considers a total of C machines which are at risk of failing when working. Depending on the failure that a machine experiences, it is sorted into either the class-1 or class-2 queue where it awaits service among other machines suffering from similar failures. The possible service policies that are considered include exhaustive, non-preemptive priority, and preemptive resume priority. In Chapter 3, this model is generalized to allow for a maintenance float of f spare machines that can be turned on to replace a failed machine. Additionally, the possible server behaviours are greatly generalized. In both chapters, among other topics, we discuss the optimization of server behaviour as well as the limiting number of working machines as we let C go to infinity. As these are systems with a finite population (for a given C and f), their steady-state distributions can be solved for using the algorithm for level-dependent quasi-birth-and-death processes without loss of accuracy. When a class of customers are impatient, the algorithms covered in this thesis require their queue length to be truncated in order for us to approximate the steady-state distribution for all but the simplest model. In Chapter 4, we model a 2-queue polling system with impatient customers and k_i-limited service disciplines. Finite buffers are assumed for both queues, such that if a customer arrives to find their queue full then they are blocked and lost forever. Finite buffers are a way to interpret a necessary truncation level, since we can simply assume that it is impossible to observe the removed states. However, if we are interested in approximating an infinite buffer system, this inconsistency will bias the steady-state probabilities if blocking probabilities are not negligible. In Chapter 5, we introduce the Unobserved Waiting Customer approximation as a way to reduce this natural biasing that is incurred when approximating an infinite buffer system. Among the queues considered within this chapter is a N-queue system with exhaustive service and customers who may or may not be impatient. In Chapter 6, we extend this approximation to allow for reneging rates that depend on a customer's place in their queue. This is applied to a N-queue polling system which generalizes the model of Chapter 4