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Markov and Semi-markov Chains, Processes, Systems and Emerging Related Fields
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Markov and Semi-markov Chains, Processes, Systems and Emerging Related Fields

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

Deadline for manuscript submissions: closed (30 June 2021) | Viewed by 31636

Printed Edition Available!
A printed edition of this Special Issue is available here.

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

E-Mail Website
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,

Modern Probability has a face with two different sides. One side is the closest possible to being a branch of pure mathematics, derived from certain axioms in classical areas of algebra. On the other side, however, probability is an applied mathematics discipline, most commonly known as applied probability, although the usual distinction between pure and applied mathematics is, in our opinion, merely artificial. This side, without being less demanding mathematically or stochastically, produces valuable models for studying everyday phenomena in the real world. Stochastic processes are by now well established as an extension of probability theory. In the area of stochastic processes, Markov and semi-Markov processes play a vital role as an independent area of study, generating important and novel applications and new mathematical results. Additionally, they play a vital part in emerging new areas of research by blending with dynamic developing areas of statistical research and large-scale data manipulation.

We invite our colleagues to submit papers to areas related to the following keywords and also in any closed linked field of study not mentioned specifically here.

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

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

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.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Markov chains and Markov processes
  • semi-Markov chains and processes
  • Markov renewal processes
  • matrix analysis in stochastic processes and applications
  • nonhomogeneous Markov and semi-Markov systems
  • discrete and continuous time Markov chains
  • stochastic analysis for finance
  • stochastic processes in social sciences
  • Martingales and related fields
  • first step analysis and random walks
  • stochastic stability and asymptotic analysis
  • perturbation and sensitivity analysis in Markov processes
  • queuing networks and Markov chains
  • Markovian modeling and applications in biology, epidemiology and healthcare
  • inference in Markov, semi-Markov and diffusion processes
  • Markov reward models
  • reversible Markov chains
  • Monte Carlo methods and simulation-related fields
  • Markov chains and computer science
  • optimization techniques and efficiency evaluation
  • hidden Markov chains and applications
  • modeling and applications in classification and machine learning

Published Papers (15 papers)

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Editorial

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6 pages, 206 KiB  
Editorial
Markov and Semi-Markov Chains, Processes, Systems, and Emerging Related Fields
by P.-C.G. Vassiliou and Andreas C. Georgiou
Mathematics 2021, 9(19), 2490; https://doi.org/10.3390/math9192490 - 04 Oct 2021
Cited by 5 | Viewed by 2149
Abstract
Probability resembles the ancient Roman God Janus since, like Janus, probability also has a face with two different sides, which correspond to the metaphorical gateways and transitions between the past and the future [...] Full article
(This article belongs to the Special Issue Markov and Semi-markov Chains, Processes, Systems and Emerging Related Fields)

Research

Jump to: Editorial

18 pages, 368 KiB  
Article
Sequential Interval Reliability for Discrete-Time Homogeneous Semi-Markov Repairable Systems
by Vlad Stefan Barbu, Guglielmo D’Amico and Thomas Gkelsinis
Mathematics 2021, 9(16), 1997; https://doi.org/10.3390/math9161997 - 20 Aug 2021
Cited by 6 | Viewed by 1632
Abstract
In this paper, a new reliability measure, named sequential interval reliability, is introduced for homogeneous semi-Markov repairable systems in discrete time. This measure is the probability that the system is working in a given sequence of non-overlapping time intervals. Many reliability measures are [...] Read more.
In this paper, a new reliability measure, named sequential interval reliability, is introduced for homogeneous semi-Markov repairable systems in discrete time. This measure is the probability that the system is working in a given sequence of non-overlapping time intervals. Many reliability measures are particular cases of this new reliability measure that we propose; this is the case for the interval reliability, the reliability function and the availability function. A recurrent-type formula is established for the calculation in the transient case and an asymptotic result determines its limiting behaviour. The results are illustrated by means of a numerical example which illustrates the possible application of the measure to real systems. Full article
(This article belongs to the Special Issue Markov and Semi-markov Chains, Processes, Systems and Emerging Related Fields)
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13 pages, 907 KiB  
Article
Evaluating the Efficiency of Off-Ball Screens in Elite Basketball Teams via Second-Order Markov Modelling
by Nikolaos Stavropoulos, Alexandra Papadopoulou and Pavlos Kolias
Mathematics 2021, 9(16), 1991; https://doi.org/10.3390/math9161991 - 20 Aug 2021
Cited by 5 | Viewed by 3202
Abstract
In basketball, the offensive movements on both strong and weak sides and tactical behavior play major roles in the effectiveness of a team’s offense. In the literature, studies are mostly focused on offensive actions, such as ball screens on the strong side. In [...] Read more.
In basketball, the offensive movements on both strong and weak sides and tactical behavior play major roles in the effectiveness of a team’s offense. In the literature, studies are mostly focused on offensive actions, such as ball screens on the strong side. In the present paper, for the first time a second-order Markov model is defined to evaluate players’ interactions on the weak side, particularly for exploring the effectiveness of tactical structures and off-ball screens regarding the final outcome. The sample consisted of 1170 possessions of the FIBA Basketball Champions League 2018–2019. The variables of interest were the type of screen on the weak side, the finishing move, and the outcome of the shot. The model incorporates partial non-homogeneity according to the time of the execution (0–24″) and the quarter of playtime, and it is conditioned on the off-ball screen type. Regarding the overall performance, the results indicated that the outcome of each possession was influenced not only by the type of the executed shot, but also by the specific type of screen that took place earlier on the weak side of the offense. Thus, the proposed model could operate as an advisory tool for the coach’s strategic plans. Full article
(This article belongs to the Special Issue Markov and Semi-markov Chains, Processes, Systems and Emerging Related Fields)
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13 pages, 307 KiB  
Article
State Space Modeling with Non-Negativity Constraints Using Quadratic Forms
by Ourania Theodosiadou and George Tsaklidis
Mathematics 2021, 9(16), 1908; https://doi.org/10.3390/math9161908 - 10 Aug 2021
Cited by 5 | Viewed by 1467
Abstract
State space model representation is widely used for the estimation of nonobservable (hidden) random variables when noisy observations of the associated stochastic process are available. In case the state vector is subject to constraints, the standard Kalman filtering algorithm can no longer be [...] Read more.
State space model representation is widely used for the estimation of nonobservable (hidden) random variables when noisy observations of the associated stochastic process are available. In case the state vector is subject to constraints, the standard Kalman filtering algorithm can no longer be used in the estimation procedure, since it assumes the linearity of the model. This kind of issue is considered in what follows for the case of hidden variables that have to be non-negative. This restriction, which is common in many real applications, can be faced by describing the dynamic system of the hidden variables through non-negative definite quadratic forms. Such a model could describe any process where a positive component represents “gain”, while the negative one represents “loss”; the observation is derived from the difference between the two components, which stands for the “surplus”. Here, a thorough analysis of the conditions that have to be satisfied regarding the existence of non-negative estimations of the hidden variables is presented via the use of the Karush–Kuhn–Tucker conditions. Full article
(This article belongs to the Special Issue Markov and Semi-markov Chains, Processes, Systems and Emerging Related Fields)
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17 pages, 430 KiB  
Article
On State Occupancies, First Passage Times and Duration in Non-Homogeneous Semi-Markov Chains
by Andreas C. Georgiou, Alexandra Papadopoulou, Pavlos Kolias, Haris Palikrousis and Evanthia Farmakioti
Mathematics 2021, 9(15), 1745; https://doi.org/10.3390/math9151745 - 24 Jul 2021
Cited by 3 | Viewed by 1928
Abstract
Semi-Markov processes generalize the Markov chains framework by utilizing abstract sojourn time distributions. They are widely known for offering enhanced accuracy in modeling stochastic phenomena. The aim of this paper is to provide closed analytic forms for three types of probabilities which describe [...] Read more.
Semi-Markov processes generalize the Markov chains framework by utilizing abstract sojourn time distributions. They are widely known for offering enhanced accuracy in modeling stochastic phenomena. The aim of this paper is to provide closed analytic forms for three types of probabilities which describe attributes of considerable research interest in semi-Markov modeling: (a) the number of transitions to a state through time (Occupancy), (b) the number of transitions or the amount of time required to observe the first passage to a state (First passage time) and (c) the number of transitions or the amount of time required after a state is entered before the first real transition is made to another state (Duration). The non-homogeneous in time recursive relations of the above probabilities are developed and a description of the corresponding geometric transforms is produced. By applying appropriate properties, the closed analytic forms of the above probabilities are provided. Finally, data from human DNA sequences are used to illustrate the theoretical results of the paper. Full article
(This article belongs to the Special Issue Markov and Semi-markov Chains, Processes, Systems and Emerging Related Fields)
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13 pages, 308 KiB  
Article
Discrete Time Hybrid Semi-Markov Models in Manpower Planning
by Brecht Verbeken and Marie-Anne Guerry
Mathematics 2021, 9(14), 1681; https://doi.org/10.3390/math9141681 - 16 Jul 2021
Cited by 5 | Viewed by 2536
Abstract
Discrete time Markov models are used in a wide variety of social sciences. However, these models possess the memoryless property, which makes them less suitable for certain applications. Semi-Markov models allow for more flexible sojourn time distributions, which can accommodate for duration of [...] Read more.
Discrete time Markov models are used in a wide variety of social sciences. However, these models possess the memoryless property, which makes them less suitable for certain applications. Semi-Markov models allow for more flexible sojourn time distributions, which can accommodate for duration of stay effects. An overview of differences and possible obstacles regarding the use of Markov and semi-Markov models in manpower planning was first given by Valliant and Milkovich (1977). We further elaborate on their insights and introduce hybrid semi-Markov models for open systems with transition-dependent sojourn time distributions. Hybrid semi-Markov models aim to reduce model complexity in terms of the number of parameters to be estimated by only taking into account duration of stay effects for those transitions for which it is useful. Prediction equations for the stock vector are derived and discussed. Furthermore, the insights are illustrated and discussed based on a real world personnel dataset. The hybrid semi-Markov model is compared with the Markov and the semi-Markov models by diverse model selection criteria. Full article
(This article belongs to the Special Issue Markov and Semi-markov Chains, Processes, Systems and Emerging Related Fields)
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31 pages, 6058 KiB  
Article
Partial Diffusion Markov Model of Heterogeneous TCP Link: Optimization with Incomplete Information
by Andrey Borisov, Alexey Bosov, Gregory Miller and Igor Sokolov
Mathematics 2021, 9(14), 1632; https://doi.org/10.3390/math9141632 - 10 Jul 2021
Cited by 5 | Viewed by 2303
Abstract
The paper presents a new mathematical model of TCP (Transmission Control Protocol) link functioning in a heterogeneous (wired/wireless) channel. It represents a controllable, partially observable stochastic dynamic system. The system state describes the status of the modeled TCP link and expresses it via [...] Read more.
The paper presents a new mathematical model of TCP (Transmission Control Protocol) link functioning in a heterogeneous (wired/wireless) channel. It represents a controllable, partially observable stochastic dynamic system. The system state describes the status of the modeled TCP link and expresses it via an unobservable controllable MJP (Markov jump process) with finite-state space. Observations are formed by low-frequency counting processes of packet losses and timeouts and a high-frequency compound Poisson process of packet acknowledgments. The information transmission through the TCP-equipped channel is considered a stochastic control problem with incomplete information. The main idea to solve it is to impose the separation principle on the problem. The paper proposes a mathematical framework and algorithmic support to implement the solution. It includes a solution to the stochastic control problem with complete information, a diffusion approximation of the high-frequency observations, a solution to the MJP state filtering problem given the observations with multiplicative noises, and a numerical scheme of the filtering algorithm. The paper also contains the results of a comparative study of the proposed state-based congestion control algorithm with the contemporary TCP versions: Illinois, CUBIC, Compound, and BBR (Bottleneck Bandwidth and RTT). Full article
(This article belongs to the Special Issue Markov and Semi-markov Chains, Processes, Systems and Emerging Related Fields)
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29 pages, 1859 KiB  
Article
Open Markov Type Population Models: From Discrete to Continuous Time
by Manuel L. Esquível, Nadezhda P. Krasii and Gracinda R. Guerreiro
Mathematics 2021, 9(13), 1496; https://doi.org/10.3390/math9131496 - 25 Jun 2021
Cited by 4 | Viewed by 2223
Abstract
We address the problem of finding a natural continuous time Markov type process—in open populations—that best captures the information provided by an open Markov chain in discrete time which is usually the sole possible observation from data. Given the open discrete time Markov [...] Read more.
We address the problem of finding a natural continuous time Markov type process—in open populations—that best captures the information provided by an open Markov chain in discrete time which is usually the sole possible observation from data. Given the open discrete time Markov chain, we single out two main approaches: In the first one, we consider a calibration procedure of a continuous time Markov process using a transition matrix of a discrete time Markov chain and we show that, when the discrete time transition matrix is embeddable in a continuous time one, the calibration problem has optimal solutions. In the second approach, we consider semi-Markov processes—and open Markov schemes—and we propose a direct extension from the discrete time theory to the continuous time one by using a known structure representation result for semi-Markov processes that decomposes the process as a sum of terms given by the products of the random variables of a discrete time Markov chain by time functions built from an adequate increasing sequence of stopping times. Full article
(This article belongs to the Special Issue Markov and Semi-markov Chains, Processes, Systems and Emerging Related Fields)
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16 pages, 1686 KiB  
Article
Particle Filtering: A Priori Estimation of Observational Errors of a State-Space Model with Linear Observation Equation
by Rodi Lykou and George Tsaklidis
Mathematics 2021, 9(12), 1445; https://doi.org/10.3390/math9121445 - 21 Jun 2021
Cited by 4 | Viewed by 1790
Abstract
Observational errors of Particle Filtering are studied over the case of a state-space model with a linear observation equation. In this study, the observational errors are estimated prior to the upcoming observations. This action is added to the basic algorithm of the filter [...] Read more.
Observational errors of Particle Filtering are studied over the case of a state-space model with a linear observation equation. In this study, the observational errors are estimated prior to the upcoming observations. This action is added to the basic algorithm of the filter as a new step for the acquisition of the state estimations. This intervention is useful in the presence of missing data problems mainly, as well as sample tracking for impoverishment issues. It applies theory of Homogeneous and Non-Homogeneous closed Markov Systems to the study of particle distribution over the state domain and, thus, lays the foundations for the employment of stochastic control against impoverishment. A simulating example is quoted to demonstrate the effectiveness of the proposed method in comparison with existing ones, showing that the proposed method is able to combine satisfactory precision of results with a low computational cost and provide an example to achieve impoverishment prediction and tracking. Full article
(This article belongs to the Special Issue Markov and Semi-markov Chains, Processes, Systems and Emerging Related Fields)
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12 pages, 1732 KiB  
Article
Using Markov Models to Characterize and Predict Process Target Compliance
by Sally McClean
Mathematics 2021, 9(11), 1187; https://doi.org/10.3390/math9111187 - 24 May 2021
Cited by 3 | Viewed by 1598
Abstract
Processes are everywhere, covering disparate fields such as business, industry, telecommunications, and healthcare. They have previously been analyzed and modelled with the aim of improving understanding and efficiency as well as predicting future events and outcomes. In recent years, process mining has appeared [...] Read more.
Processes are everywhere, covering disparate fields such as business, industry, telecommunications, and healthcare. They have previously been analyzed and modelled with the aim of improving understanding and efficiency as well as predicting future events and outcomes. In recent years, process mining has appeared with the aim of uncovering, observing, and improving processes, often based on data obtained from logs. This typically requires task identification, predicting future pathways, or identifying anomalies. We here concentrate on using Markov processes to assess compliance with completion targets or, inversely, we can determine appropriate targets for satisfactory performance. Previous work is extended to processes where there are a number of possible exit options, with potentially different target completion times. In particular, we look at distributions of the number of patients failing to meet targets, through time. The formulae are illustrated using data from a stroke patient unit, where there are multiple discharge destinations for patients, namely death, private nursing home, or the patient’s own home, where different discharge destinations may require disparate targets. Key performance indicators (KPIs) of this sort are commonplace in healthcare, business, and industrial processes. Markov models, or their extensions, have an important role to play in this work where the approach can be extended to include more expressive assumptions, with the aim of assessing compliance in complex scenarios. Full article
(This article belongs to the Special Issue Markov and Semi-markov Chains, Processes, Systems and Emerging Related Fields)
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29 pages, 7604 KiB  
Article
Optimizing a Multi-State Cold-Standby System with Multiple Vacations in the Repair and Loss of Units
by Juan Eloy Ruiz-Castro
Mathematics 2021, 9(8), 913; https://doi.org/10.3390/math9080913 - 20 Apr 2021
Cited by 6 | Viewed by 1616
Abstract
A complex multi-state redundant system with preventive maintenance subject to multiple events is considered. The online unit can undergo several types of failure: both internal and those provoked by external shocks. Multiple degradation levels are assumed as both internal and external. Degradation levels [...] Read more.
A complex multi-state redundant system with preventive maintenance subject to multiple events is considered. The online unit can undergo several types of failure: both internal and those provoked by external shocks. Multiple degradation levels are assumed as both internal and external. Degradation levels are observed by random inspections and, if they are major, the unit goes to a repair facility where preventive maintenance is carried out. This repair facility is composed of a single repairperson governed by a multiple vacation policy. This policy is set up according to the operational number of units. Two types of task can be performed by the repairperson, corrective repair and preventive maintenance. The times embedded in the system are phase type distributed and the model is built by using Markovian Arrival Processes with marked arrivals. Multiple performance measures besides the transient and stationary distribution are worked out through matrix-analytic methods. This methodology enables us to express the main results and the global development in a matrix-algorithmic form. To optimize the model, costs and rewards are included. A numerical example shows the versatility of the model. Full article
(This article belongs to the Special Issue Markov and Semi-markov Chains, Processes, Systems and Emerging Related Fields)
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10 pages, 240 KiB  
Article
Period-Life of a Branching Process with Migration and Continuous Time
by Khrystyna Prysyazhnyk, Iryna Bazylevych, Ludmila Mitkova and Iryna Ivanochko
Mathematics 2021, 9(8), 868; https://doi.org/10.3390/math9080868 - 15 Apr 2021
Cited by 1 | Viewed by 1572
Abstract
The homogeneous branching process with migration and continuous time is considered. We investigated the distribution of the period-life τ, i.e., the length of the time interval between the moment when the process is initiated by a positive number of particles and the [...] Read more.
The homogeneous branching process with migration and continuous time is considered. We investigated the distribution of the period-life τ, i.e., the length of the time interval between the moment when the process is initiated by a positive number of particles and the moment when there are no individuals in the population for the first time. The probability generating function of the random process, which describes the behavior of the process within the period-life, was obtained. The boundary theorem for the period-life of the subcritical or critical branching process with migration was found. Full article
(This article belongs to the Special Issue Markov and Semi-markov Chains, Processes, Systems and Emerging Related Fields)
26 pages, 441 KiB  
Article
Tails of the Moments for Sums with Dominatedly Varying Random Summands
by Mantas Dirma, Saulius Paukštys and Jonas Šiaulys
Mathematics 2021, 9(8), 824; https://doi.org/10.3390/math9080824 - 10 Apr 2021
Cited by 7 | Viewed by 2079
Abstract
The asymptotic behaviour of the tail expectation ?E(Snξ)α?{Snξ>x} is investigated, where exponent α is a nonnegative real number and [...] Read more.
The asymptotic behaviour of the tail expectation ?E(Snξ)α?{Snξ>x} is investigated, where exponent α is a nonnegative real number and Snξ=ξ1+…+ξn is a sum of dominatedly varying and not necessarily identically distributed random summands, following a specific dependence structure. It turns out that the tail expectation of such a sum can be asymptotically bounded from above and below by the sums of expectations ?Eξiα?{ξi>x} with correcting constants. The obtained results are extended to the case of randomly weighted sums, where collections of random weights and primary random variables are independent. For illustration of the results obtained, some particular examples are given, where dependence between random variables is modelled in copulas framework. Full article
(This article belongs to the Special Issue Markov and Semi-markov Chains, Processes, Systems and Emerging Related Fields)
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25 pages, 382 KiB  
Article
Non-Homogeneous Markov Set Systems
by P.-C.G. Vassiliou
Mathematics 2021, 9(5), 471; https://doi.org/10.3390/math9050471 - 25 Feb 2021
Cited by 7 | Viewed by 1645
Abstract
A more realistic way to describe a model is the use of intervals which contain the required values of the parameters. In practice we estimate the parameters from a set of data and it is natural that they will be in confidence intervals. [...] Read more.
A more realistic way to describe a model is the use of intervals which contain the required values of the parameters. In practice we estimate the parameters from a set of data and it is natural that they will be in confidence intervals. In the present study, we study Non-Homogeneous Markov Systems (NHMS) processes for which the required basic parameters are in intervals. We call such processes Non-Homogeneous Markov Set Systems (NHMSS). First we study the set of the relative expected population structure of memberships and we prove that under certain conditions of convexity of the intervals of the parameters the set is compact and convex. Next, we establish that if the NHMSS starts with two different initial distributions sets and allocation probability sets under certain conditions, asymptotically the two expected relative population structures coincide geometrically fast. We continue proving a series of theorems on the asymptotic behavior of the expected relative population structure of a NHMSS and the properties of their limit set. Finally, we present an application for geriatric and stroke patients in a hospital and through it we solve problems that surface in an application. Full article
(This article belongs to the Special Issue Markov and Semi-markov Chains, Processes, Systems and Emerging Related Fields)
14 pages, 343 KiB  
Article
Geometric Ergodicity of the Random Walk Metropolis with Position-Dependent Proposal Covariance
by Samuel Livingstone
Mathematics 2021, 9(4), 341; https://doi.org/10.3390/math9040341 - 08 Feb 2021
Cited by 4 | Viewed by 1764
Abstract
We consider a Metropolis–Hastings method with proposal N(x,hG(x)−1), where x is the current state, and study its ergodicity properties. We show that suitable choices of G(x) can change [...] Read more.
We consider a Metropolis–Hastings method with proposal N(x,hG(x)−1), where x is the current state, and study its ergodicity properties. We show that suitable choices of G(x) can change these ergodicity properties compared to the Random Walk Metropolis case N(x,hΣ), either for better or worse. We find that if the proposal variance is allowed to grow unboundedly in the tails of the distribution then geometric ergodicity can be established when the target distribution for the algorithm has tails that are heavier than exponential, in contrast to the Random Walk Metropolis case, but that the growth rate must be carefully controlled to prevent the rejection rate approaching unity. We also illustrate that a judicious choice of G(x) can result in a geometrically ergodic chain when probability concentrates on an ever narrower ridge in the tails, something that is again not true for the Random Walk Metropolis. Full article
(This article belongs to the Special Issue Markov and Semi-markov Chains, Processes, Systems and Emerging Related Fields)
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