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Stochastic Processes and Their Applications: In Honor of Prof. Sally...
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Stochastic Processes and Their Applications: In Honor of Prof. Sally McClean

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Probability and Statistics".

Deadline for manuscript submissions: closed (15 November 2024) | Viewed by 15250

Special Issue Editors

Prof. Dr. Panagiotis-Christos Vassiliou

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Guest Editor
Department of Statistical Science, University College London, Gower st, London WC1E 6BT, UK
Interests: probability; stochastic processes; stochastic modeling; applied probability; financial mathematics; stochastic analysis; mathematical finance; Markov chains; Markov processes; option pricing
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Andreas C. Georgiou

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Guest Editor
Quantitative Methods and Decision Analytics Lab, Department of Business Administration, University of Macedonia, 54636 Thessaloniki, Greece
Interests: Markovian processes; stochastic modeling; manpower planning; multiple objective optimization; simulation; performance evaluation; business analytics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The Journal of Mathematics is publishing a Special Issue under the above title to honor Prof. Sally McClean in the occasion of her semi-retirement and in recognition of her important contributions in research.

Sally Ida McClean was born in Belfast and took her first degree, an M.A. in mathematics, from the University of Oxford in 1970. She earned an M.Sc. in Mathematical Statistics and Operations Research from Cardiff University in 1972, and completed her Ph.D. in 1976 at the Ulster University at Coleraine. Her contribution to mathematical modeling in health care planning is huge, and, in particular, her studies for improving the wellbeing of the elderly is greatly respected amongst her peers.  She is currently Professor of Mathematics at Ulster University. Her main research interests are in Stochastic Modelling and Optimization for Healthcare Planning and Computer Science, and, more specifically, in Databases, Internet of Things, Sensor Technology, and Telecommunications. She has published more than 300 papers on the subjects, with an impressive record of citations. Prof. McClean is a recipient of Ulster University's Senior Distinguished Research Fellowship.

Stochastic processes are probably the most important tool in many areas of science, such as biology, operational research, social sciences, stochastic finance, and others.  Important characteristics in these areas evolve with time in a more or less a random way, and since stochastic processes are mainly sequences or families of random variables, in which their index represents time, they are the natural tool to use. The theory and applications of stochastic processes emerged in the genesis of one of the richest one, that is, the Brownian motion. That was rather unexpected since Brownian motion is a beautiful object which is at the same time a martingale, a Gaussian process, a diffusion, a Levy process, a Markov process, etc.; concepts that were discovered quite latter in the evolution of time. We cordially invite researchers and colleagues to submit their papers following the journal’s guidelines and the topics of interest listed below.

Prof. Dr. Panagiotis-Christos Vassiliou
Prof. Dr. Andreas C. Georgiou
Guest Editors

Manuscript Submission Information

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

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Keywords

  • anomaly detection and Markov processes
  • applications of artificial intelligence in medical diagnosis processes
  • business processes and Markov chains
  • conditional phase-type distributions
  • discrete q-distributions
  • epidemic models
  • hidden Markov chains and applications
  • inference in Markov, semi-Markov and diffusion processes
  • machine learning lingo for a stochastic environment
  • Markov and semi-Markov chains and processes,
  • Markov chains in computing science
  • Markov decision processes and Markov processes in reinforcement learning
  • Markov models and applications in biology, epidemiology and healthcare
  • Markov models and efficiency evaluation
  • Markov Populations
  • Markov renewal processes
  • Markovian Arrival Processes (MAP)
  • martingales
  • mathematical finance
  • matrix analysis
  • migration process in credit risk
  • modeling and applications in supervised and unsupervised machine learning
  • nonhomogeneous Markov and semi-Markov systems
  • optimization techniques and efficiency evaluation
  • perturbation and sensitivity analysis in Markov processes
  • phase type survival trees
  • point process
  • process mining techniques
  • queuing networks and Markov chains
  • random walks
  • reversible Markov chains
  • stochastic analysis
  • stochastic biological models
  • stochastic environments
  • stochastic processes in finance and social sciences
  • stochastic stability and asymptotic analysis

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Published Papers (11 papers)

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Research

18 pages, 341 KiB  
Article
On a Mixed Transient–Asymptotic Result for the Sequential Interval Reliability for Semi-Markov Chains
by Guglielmo D’Amico and Thomas Gkelsinis
Mathematics 2024, 12(12), 1842; https://doi.org/10.3390/math12121842 - 13 Jun 2024
Cited by 1 | Viewed by 555
Abstract
In this paper, we are concerned with the study of sequential interval reliability, a measure recently introduced in the literature. This measure represents the probability of the system working during a sequence of nonoverlapping time intervals. In the cited work, the authors proposed [...] Read more.
In this paper, we are concerned with the study of sequential interval reliability, a measure recently introduced in the literature. This measure represents the probability of the system working during a sequence of nonoverlapping time intervals. In the cited work, the authors proposed a recurrent-type formula for computing this indicator in the transient case and investigated the asymptotic behavior as all the time intervals go to infinity. The purpose of the present work is to further explore the asymptotic behavior when only some of the time intervals are allowed to go to infinity while the remaining ones are not. In this way, we provide a unique indicator that is able to describe the process evolution in the transient and asymptotic cases as well. It is important to mention that this is not a straightforward result since, in order to achieve it, we need to develop several mathematical ingredients that generalize the classical renewal and Markov renewal frameworks. A numerical example illustrates our theoretical results. Full article
(This article belongs to the Special Issue Stochastic Processes and Their Applications: In Honor of Prof. Sally McClean)
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15 pages, 342 KiB  
Article
The Arsenal of Perturbation Bounds for Finite Continuous-Time Markov Chains: A Perspective
by Alexander Y. Mitrophanov
Mathematics 2024, 12(11), 1608; https://doi.org/10.3390/math12111608 - 21 May 2024
Cited by 1 | Viewed by 3399
Abstract
Perturbation bounds are powerful tools for investigating the phenomenon of insensitivity to perturbations, also referred to as stability, for stochastic and deterministic systems. This perspective article presents a focused account of some of the main concepts and results in inequality-based perturbation theory for [...] Read more.
Perturbation bounds are powerful tools for investigating the phenomenon of insensitivity to perturbations, also referred to as stability, for stochastic and deterministic systems. This perspective article presents a focused account of some of the main concepts and results in inequality-based perturbation theory for finite state-space, time-homogeneous, continuous-time Markov chains. The diversity of perturbation bounds and the logical relationships between them highlight the essential stability properties and factors for this class of stochastic processes. We discuss the linear time dependence of general perturbation bounds for Markov chains, as well as time-independent (i.e., time-uniform) perturbation bounds for chains whose stationary distribution is unique. Moreover, we prove some new results characterizing the absolute and relative tightness of time-uniform perturbation bounds. Specifically, we show that, in some of them, an equality is achieved. Furthermore, we analytically compare two types of time-uniform bounds known from the literature. Possibilities for generalizing Markov-chain stability results, as well as connections with stability analysis for other systems and processes, are also discussed. Full article
(This article belongs to the Special Issue Stochastic Processes and Their Applications: In Honor of Prof. Sally McClean)
15 pages, 2094 KiB  
Article
Cost Evaluation for Capacity Planning Based on Patients’ Pathways via Semi-Markov Reward Modelling
by Christina Chatzimichail, Pavlos Kolias and Alexandra Papadopoulou
Mathematics 2024, 12(10), 1430; https://doi.org/10.3390/math12101430 - 7 May 2024
Viewed by 900
Abstract
In the present paper, we develop a non-homogeneous semi-Markov reward model, deriving expressions for a healthcare system’s expected structure along with the expected costs generated by medical services and patients’ holding times in the states. We provide a novel definition and investigation for [...] Read more.
In the present paper, we develop a non-homogeneous semi-Markov reward model, deriving expressions for a healthcare system’s expected structure along with the expected costs generated by medical services and patients’ holding times in the states. We provide a novel definition and investigation for states’ availability, which is critical for capacity planning based on service demand in an environment of limited resources. The study is based on patients’ mobility through hospital care, where each patient spends an amount of time in every state of the hospital (emergency room, short-term acute care, hospitalization, surgery room, and intensive care unit). Multiple outcomes, such as discharge or death, can also be taken into account. We envisage a situation where any discharges are immediately replaced by a number of new admissions that carry on the pathways of the patients who exit. By assuming an expanding system, the new idea of states’ inflows is considered due to new patients who create pathways through hospital care, along with internal entrances. The theoretical results are illustrated numerically with simulated hospital data informed by aggregated public data of the Greek public health sector. The framework can be used for both strategic planning and cost evaluation purposes for hospital resources. Full article
(This article belongs to the Special Issue Stochastic Processes and Their Applications: In Honor of Prof. Sally McClean)
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14 pages, 325 KiB  
Article
Attainability for Markov and Semi-Markov Chains
by Brecht Verbeken and Marie-Anne Guerry
Mathematics 2024, 12(8), 1227; https://doi.org/10.3390/math12081227 - 19 Apr 2024
Viewed by 907
Abstract
When studying Markov chain models and semi-Markov chain models, it is useful to know which state vectors n, where each component ni represents the number of entities in the state Si, can be maintained or attained. This question leads [...] Read more.
When studying Markov chain models and semi-Markov chain models, it is useful to know which state vectors n, where each component ni represents the number of entities in the state Si, can be maintained or attained. This question leads to the definitions of maintainability and attainability for (time-homogeneous) Markov chain models. Recently, the definition of maintainability was extended to the concept of state reunion maintainability (SR-maintainability) for semi-Markov chains. Within the framework of semi-Markov chains, the states are subdivided further into seniority-based states. State reunion maintainability assesses the maintainability of the distribution across states. Following this idea, we introduce the concept of state reunion attainability, which encompasses the potential of a system to attain a specific distribution across the states after uniting the seniority-based states into the underlying states. In this paper, we start by extending the concept of attainability for constant-sized Markov chain models to systems that are subject to growth or contraction. Afterwards, we introduce the concepts of attainability and state reunion attainability for semi-Markov chain models, using SR-maintainability as a starting point. The attainable region, as well as the state reunion attainable region, are described as the convex hull of their respective vertices, and properties of these regions are investigated. Full article
(This article belongs to the Special Issue Stochastic Processes and Their Applications: In Honor of Prof. Sally McClean)
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21 pages, 2661 KiB  
Article
Educational Status as a Mediator of Intergenerational Social Mobility in Europe: A Positional Analysis Approach
by Glykeria Stamatopoulou, Eva Tsouparopoulou and Maria Symeonaki
Mathematics 2024, 12(7), 966; https://doi.org/10.3390/math12070966 - 25 Mar 2024
Viewed by 1256
Abstract
This paper investigates the transmission of educational attainment from parents to offspring as a mediator of intergenerational class mobility in Europe. The study covers the last two decades with data drawn from a cross-national large-scale sample survey, namely the European Social Survey (ESS), [...] Read more.
This paper investigates the transmission of educational attainment from parents to offspring as a mediator of intergenerational class mobility in Europe. The study covers the last two decades with data drawn from a cross-national large-scale sample survey, namely the European Social Survey (ESS), for the years 2002–2018. Interest has focused on the question of the persistence of inequality of educational opportunities by examining the attainment of nominal levels of education and the association between the educational attainment of the parent with the highest level of education and their descendants. The study also covers new trends in social mobility that consider education as a “positional good”, and a novel method of incorporating educational expansion into the transition probabilities is proposed, providing answers to whether the rising accessibility of educational qualifications attenuates the association between social origin and educational attainment. Therefore, the concept of positionality is taken into account in the estimation of intergenerational transition probabilities, and to complement the analysis, mobility measures are provided for both methods, nominal and positional. The proposed positional method is validated through a correlation analysis between the upward mobility scores (nominal and positional) with the Education Expansion Index (EEI) for the respective years. The upward mobility scores estimated via the positional method are more highly correlated with the EEI for all years, indicating a better alignment with the broader trends in educational participation and achievement. Full article
(This article belongs to the Special Issue Stochastic Processes and Their Applications: In Honor of Prof. Sally McClean)
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14 pages, 260 KiB  
Article
A Class of Power Series q-Distributions
by Charalambos A. Charalambides
Mathematics 2024, 12(5), 712; https://doi.org/10.3390/math12050712 - 28 Feb 2024
Viewed by 696
Abstract
A class of power series q-distributions, generated by considering a q-Taylor expansion of a parametric function into powers of the parameter, is discussed. Its q-factorial moments are obtained in terms of q-derivatives of its series (parametric) function. Also, it [...] Read more.
A class of power series q-distributions, generated by considering a q-Taylor expansion of a parametric function into powers of the parameter, is discussed. Its q-factorial moments are obtained in terms of q-derivatives of its series (parametric) function. Also, it is shown that the convolution of power series q-distributions is also a power series q-distribution. Furthermore, the q-Poisson (Heine and Euler), q-binomial of the first kind, negative q-binomial of the second kind, and q-logarithmic distributions are shown to be members of this class of distributions and their q-factorial moments are deduced. In addition, the convolution properties of these distributions are examined. Full article
(This article belongs to the Special Issue Stochastic Processes and Their Applications: In Honor of Prof. Sally McClean)
21 pages, 1138 KiB  
Article
Estimation–Calibration of Continuous-Time Non-Homogeneous Markov Chains with Finite State Space
by Manuel L. Esquível, Nadezhda P. Krasii and Gracinda R. Guerreiro
Mathematics 2024, 12(5), 668; https://doi.org/10.3390/math12050668 - 24 Feb 2024
Cited by 1 | Viewed by 1049
Abstract
We propose a method for fitting transition intensities to a sufficiently large set of trajectories of a continuous-time non-homogeneous Markov chain with a finite state space. Starting with simulated data computed with Gompertz–Makeham transition intensities, we apply the proposed method to fit piecewise [...] Read more.
We propose a method for fitting transition intensities to a sufficiently large set of trajectories of a continuous-time non-homogeneous Markov chain with a finite state space. Starting with simulated data computed with Gompertz–Makeham transition intensities, we apply the proposed method to fit piecewise linear intensities and then compare the transition probabilities corresponding to both the Gompertz–Makeham transition intensities and the fitted piecewise linear intensities; the main comparison result is that the order of magnitude of the average fitting error per unit time—chosen as a year—is always less than 1%, thus validating the methodology proposed. Full article
(This article belongs to the Special Issue Stochastic Processes and Their Applications: In Honor of Prof. Sally McClean)
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16 pages, 296 KiB  
Article
Strong Ergodicity in Nonhomogeneous Markov Systems with Chronological Order
by P.-C.G. Vassiliou
Mathematics 2024, 12(5), 660; https://doi.org/10.3390/math12050660 - 23 Feb 2024
Viewed by 652
Abstract
In the present, we study the problem of strong ergodicity in nonhomogeneous Markov systems. In the first basic theorem, we relax the fundamental assumption present in all studies of asymptotic behavior. That is, the assumption that the inherent inhomogeneous Markov chain converges to [...] Read more.
In the present, we study the problem of strong ergodicity in nonhomogeneous Markov systems. In the first basic theorem, we relax the fundamental assumption present in all studies of asymptotic behavior. That is, the assumption that the inherent inhomogeneous Markov chain converges to a homogeneous Markov chain with a regular transition probability matrix. In addition, we study the practically important problem of the rate of convergence to strong ergodicity for a nonhomogeneous Markov system (NHMS). In a second basic theorem, we provide conditions under which the rate of convergence to strong ergodicity is geometric. With these conditions, we in fact relax the basic assumption present in all previous studies, that is, that the inherent inhomogeneous Markov chain converges to a homogeneous Markov chain with a regular transition probability matrix geometrically fast. Finally, we provide an illustrative application from the area of manpower planning. Full article
(This article belongs to the Special Issue Stochastic Processes and Their Applications: In Honor of Prof. Sally McClean)
21 pages, 447 KiB  
Article
A Bilevel DEA Model for Efficiency Evaluation and Target Setting with Stochastic Conditions
by Andreas C. Georgiou, Konstantinos Kaparis, Eleni-Maria Vretta, Kyriakos Bitsis and George Paltayian
Mathematics 2024, 12(4), 529; https://doi.org/10.3390/math12040529 - 8 Feb 2024
Viewed by 1361
Abstract
The effective allocation of limited resources and the establishment of targeted goals play a pivotal role in enhancing the overall efficiency of large enterprises and organizations. To achieve optimal organizational efficiency, managers seek dynamic strategies that adapt to the constraints of limited and [...] Read more.
The effective allocation of limited resources and the establishment of targeted goals play a pivotal role in enhancing the overall efficiency of large enterprises and organizations. To achieve optimal organizational efficiency, managers seek dynamic strategies that adapt to the constraints of limited and uncertain historical data. This paper introduces an evaluation of organizational efficiency through a stochastic framework, employing a bilevel data envelopment analysis (DEA) approach. This decision-making process is centralized within a decision-making unit (DMU) overseeing subordinate decision-making units (subDMUs). Discrete scenarios, each associated with a realization probability, define the uncertain parameters in the bilevel DEA-based model. This stochastic approach allows for recourse actions upon scenario realization leading to an enhanced overall organizational strategy. Decision-makers acting within uncertain and dynamic environments can benefit from this research since it allows the investigation of efficiency assessment under alternative scenarios in the presence of volatility and risk. The potential impact of applying this methodology varies depending on the specific domain. Although, the context of this paper focuses on banking, in general, enhancing resource allocation and target setting under stochasticity, contributes to advancing sustainability across all its three dimensions (economic, environmental, social). As mentioned earlier, the practical application of our approach is demonstrated via a case study in the banking sector. Full article
(This article belongs to the Special Issue Stochastic Processes and Their Applications: In Honor of Prof. Sally McClean)
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12 pages, 300 KiB  
Article
Analyzing the Asymptotic Behavior of an Extended SEIR Model with Vaccination for COVID-19
by Vasileios E. Papageorgiou, Georgios Vasiliadis and George Tsaklidis
Mathematics 2024, 12(1), 55; https://doi.org/10.3390/math12010055 - 23 Dec 2023
Cited by 7 | Viewed by 1387
Abstract
Several research papers have attempted to describe the dynamics of COVID-19 based on systems of differential equations. These systems have taken into account quarantined or isolated cases, vaccinations, control measures, and demographic parameters, presenting propositions regarding theoretical results that often investigate the asymptotic [...] Read more.
Several research papers have attempted to describe the dynamics of COVID-19 based on systems of differential equations. These systems have taken into account quarantined or isolated cases, vaccinations, control measures, and demographic parameters, presenting propositions regarding theoretical results that often investigate the asymptotic behavior of the system. In this paper, we discuss issues that concern the theoretical results proposed in the paper “An Extended SEIR Model with Vaccination for Forecasting the COVID-19 Pandemic in Saudi Arabia Using an Ensemble Kalman Filter”. We propose detailed explanations regarding the resolution of these issues. Additionally, this paper focuses on extending the local stability analysis of the disease-free equilibrium, as presented in the aforementioned paper, while emphasizing the derivation of theorems that validate the global stability of both epidemic equilibria. Emphasis is placed on the basic reproduction number R0, which determines the asymptotic behavior of the system. This index represents the expected number of secondary infections that are generated from an already infected case in a population where almost all individuals are susceptible. The derived propositions can inform health authorities about the long-term behavior of the phenomenon, potentially leading to more precise and efficient public measures. Finally, it is worth noting that the examined paper still presents an interesting epidemiological scheme, and the utilization of the Kalman filtering approach remains one of the state-of-the-art methods for modeling epidemic phenomena. Full article
(This article belongs to the Special Issue Stochastic Processes and Their Applications: In Honor of Prof. Sally McClean)
16 pages, 2530 KiB  
Article
Semi-Markov Models for Process Mining in Smart Homes
by Sally McClean and Lingkai Yang
Mathematics 2023, 11(24), 5001; https://doi.org/10.3390/math11245001 - 18 Dec 2023
Cited by 2 | Viewed by 1042
Abstract
Generally, these days people live longer but often with increased impairment and disabilities; therefore, they can benefit from assistive technologies. In this paper, we focus on the completion of activities of daily living (ADLs) by such patients, using so-called Smart Homes and Sensor [...] Read more.
Generally, these days people live longer but often with increased impairment and disabilities; therefore, they can benefit from assistive technologies. In this paper, we focus on the completion of activities of daily living (ADLs) by such patients, using so-called Smart Homes and Sensor Technology to collect data, and provide a suitable analysis to support the management of these conditions. The activities here are cast as states of a Markov-type process, while changes of state are indicated by sensor activations. This facilitates the extraction of key performance indicators (KPIs) in Smart Homes, e.g., the duration of an important activity, as well as the identification of anomalies in such transitions and durations. The use of semi-Markov models for such a scenario is described, where the state durations are represented by mixed gamma models. This approach is illustrated and evaluated using a publicly available Smart Home dataset comprising an event log of sensor activations, together with an annotated record of the actual activities. Results indicate that the methodology is well-suited to such scenarios. Full article
(This article belongs to the Special Issue Stochastic Processes and Their Applications: In Honor of Prof. Sally McClean)
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