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Niko Limnios

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Sandra (A.C.C.) van Wijk

American University of Sharjah

Jeffrey Townsend

Yale University

Younès EL AMRANI

University Mohamed V, Faculté Des Sciences Rabat Agdal

Yashwant Malaiya

Colorado State University

Islam AbdulMaksoud

Alexandria University

Tampere University

Umberto Villano

Università degli Studi del Sannio

Valentina Casola

Università degli Studi di Napoli "Federico II"

Kishor S Trivedi

Duke University

International University of east africa

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Papers by Niko Limnios

Lévy Approximation of Increment Processes with Markov Switching

Stochastics and Stochastics Reports, 2004

We study weak convergence of increment processes with embedded Markov chain switching in a series... more We study weak convergence of increment processes with embedded Markov chain switching in a series scheme. The limit process is a Lévy process where the jump part is a compound Poisson process. A result concerning the rate of convergence is also given. This study is motivated by risk theory and its applications.

Reliability of Semi-Markov Systems in Discrete Time: Modeling and Estimation

Handbook of Performability Engineering

Reliability Measures of Semi-Markov Systems with General State Space

Methodology and Computing in Applied Probability, 2011

The aim of this paper is to present a systematic modeling of reliability and related measures: av... more The aim of this paper is to present a systematic modeling of reliability and related measures: availability, maintainability, failure rate, rate of occurrence of failures, mean times, etc., known in the literature under the term dependability. This model includes the continuous and discrete time semi-Markov processes with general state space. This is one of the most general models in reliability theory since it includes as particular cases the Markov and renewal processes. Keywords Semi-Markov process • Reliability • Availability • Failure rate • Rate of occurrence of failure • Mean time to failure • Survival function

Average and diffusion approximation of stochastic evolutionary systems in an asymptotic split state space

The Annals of Applied Probability, 2004

Stochastic evolutionary systems of additive functional type, described by processes with locally ... more Stochastic evolutionary systems of additive functional type, described by processes with locally independent increments, are considered with Markov switching in an asymptotic split state space having a stoppage state. The average and diffusion approximation limit theorems are established in both single and double merging. The proofs of these results are obtained using a singular perturbation approach of linear reducible-invertible operators and the tightness of processes. Particular cases of these systems including integral functionals, dynamic systems, storage processes and compound Poisson processes are also considered. The application of limit theorems in reliability and reward problems is discussed.

The rate of occurrence of failures for semi-Markov processes and estimation

Statistics & Probability Letters, 2002

This paper investigates the rate of occurrence of failures (ROCOF) of ÿnite state semi-Markov sys... more This paper investigates the rate of occurrence of failures (ROCOF) of ÿnite state semi-Markov systems. Firstly, a formula for evaluating the ROCOF for semi-Markov systems is derived. As a consequence of this result, we derive the ROCOF of Markov systems as well as the ROCOF of the alternating renewal processes and we give their asymptotic values. From the semi-Markov ROCOF formula, we propose a statistical estimator of this function. Finally, the uniform strong consistency and the asymptotic normality of this estimator are obtained.

Statistical and Probabilistic Models in Reliability

Technometrics, 2000

Statistica! and probabilistic models in reliability / D.C. Ionescu, N. Limnios, editors. p. cm. I... more

Markov Decision Processes with a Constraint on the Asymptotic Failure Rate

Methodology And Computing In Applied Probability

In this paper, we introduce a Markov decision model with absorbing states and a constraint on the... more In this paper, we introduce a Markov decision model with absorbing states and a constraint on the asymptotic failure rate. The objective is to find a stationary policy which minimizes the infinite horizon expected average cost, given that the system never fails. Using Perron-Frobenius theory of non-negative matrices and spectral analysis, we show that the problem can be reduced to a linear programming problem. Finally, we apply this method to a real problem for an aeronautical system.

Advances in degradation modeling

Lévy approximation of an impulse recurrent process with Markov switching

Theory of Probability and Mathematical Statistics, 2010

The weak convergence of an impulse recurrent process with Markov switching is proved in this pape... more The weak convergence of an impulse recurrent process with Markov switching is proved in this paper for the scheme of the Lévy approximation. A modified Liptser semimartingale method is applied to the proof of the relative compactness of the process under consideration. The modification of the Liptser method used in the paper relies upon a solution of a singular perturbation problem instead of an ergodic theorem as used in the original Liptser method.

Mathematical Methods in Survival Analysis, Reliability and Quality of Life

Discrete Time Semi-Markov Processes for Reliability and Survival Analysis — A Nonparametric Estimation Approach

Statistics for Industry and Technology, 2004

In this work we focus on multi-state systems modeled by means of a particular class of non-homoge... more In this work we focus on multi-state systems modeled by means of a particular class of non-homogeneous Markov processes introduced in [37], called drifting Markov processes. The main idea behind this type of processes is to consider a non-homogeneity that is "smooth", of a known shape. More precisely, the Markov transition matrix is assumed to be a linear (polynomial) function of two (several) Markov transition matrices. For this class of systems, we first obtain explicit expressions for reliability/survival indicators of drifting Markov models, like reliability, availability, maintainability and failure rates. Then, under different statistical settings, we estimate the parameters of the model, obtain plug-in estimators of the associated reliability/survival indicators and investigate the consistency of the estimators. The quality of the proposed estimators and the model validation is illustrated through numerical experiments. Keywords Drifting Markov chains • Multi-state systems • Reliability theory • Survival analysis • Estimation • Asymptotic properties Mathematics Subject Classification (2000) 60J10 • 60K15 • 90B25 • 62N02 • 62F12

First integrals in the diffusion approximation scheme

Applied Mathematics Letters, 2008

This work deals with the diffusion approximation of the first integral of a stochastic dynamic sy... more

Diffusion approximation of evolutionary systems with equilibrium in asymptotic split phase space

Theory of Probability and Mathematical Statistics, 2005

In this paper we consider an additive functional of a Markov process with locally independent inc... more In this paper we consider an additive functional of a Markov process with locally independent increments switched by a Markov process. For this functional, we obtain nonhomogeneous diffusion approximation results without balance condition on the drift parameter. A more general diffusion approximation result is obtained in the case of an asymptotic split phase space of the switching Markov process.

Nonparametric estimation of performance and performability for semi-markov processes

International Journal of …, 2006

In this paper a nonparametric estimator of the performability for semi-Markov systems is proposed... more

Applied Nonparametric Statistics in Reliability

Springer Series in Reliability Engineering, 2011

Nonparametric statistics has probably become the leading methodology for researchers performing d... more Nonparametric statistics has probably become the leading methodology for researchers performing data analysis. It is nevertheless true that, whereas these methods have already proved highly effective in other applied areas of knowledge such as biostatistics or social sciences, nonparametric analyses in reliability currently form an interesting area of study that has not yet been fully explored. Applied Nonparametric Statistics in Reliability is focused on the use of modern statistical methods for the estimation of dependability measures of reliability systems that operate under different conditions. The scope of the book includes: smooth estimation of the reliability function and hazard rate of non-repairable systems; study of stochastic processes for modelling the time evolution of systems when imperfect repairs are performed; nonparametric analysis of discrete and continuous time semi-Markov processes; isotonic regression analysis of the structure function of a reliability system, and lifetime regression analysis. Besides the explanation of the mathematical background, several numerical computations or simulations are presented as illustrative examples. The corresponding computer-based methods have been implemented using R and MATLAB®. A concrete modelling scheme is chosen for each practical situation and, in consequence, a nonparametric inference procedure is conducted. Applied Nonparametric Statistics in Reliability will serve the practical needs of scientists (statisticians and engineers) working on applied reliability subjects

Poisson Approximation of Processes with Locally Independent Increments and Semi-Markov Switching – Toward Application in Reliability

In this chapter, the weak convergence of additive functionals of processes with locally independe... more In this chapter, the weak convergence of additive functionals of processes with locally independent increments and with semi-Markov switching in the scheme of Poisson approximation is investigated. Singular perturbation problem for the compensating operator of the extended Markov renewal process is used to prove the relative compactness. This approach can be used in applications and especially in shock and degradation in random environment arising in reliability. Keywords and phrases Poisson approximation - semimartingale - semi-Markov process - locally independent increments process - piecewise deterministic Markov process - weak convergence - singular perturbation - degradation - reliability

Semi-Markov Models for Seismic Hazard Assessment in Certain Areas of Greece

The long-term probabilistic seismic hazard is studied through the application of semi-Markov mode... more The long-term probabilistic seismic hazard is studied through the application of semi-Markov model. In this model a sequence of earthquakes is considered as a Markov process and the waiting time distributions depend only on the type of the last and the next event. The principal hypothesis of the model is the property of one-step memory, according to which the probability of moving to any future state depends only on the current state. The model under consideration defines a continuous-time, discrete-state stationary process in which successive state occupancies are governed by the transition probabilities of the Markov process. The space of states is considered to be finite and the process started far in the past has achieved stationarity. Firstly, a non-parametric method is applied in order to determine the waiting times. Then, the waiting times derived by means of the exponential and Weibull distributions will be compared to each other, as well as with the actual waiting times. Th...

Stochastic Stability of Vector Sde with Applications to a Stochastic Epidemic Model

International Journal of Pure and Apllied Mathematics

In this paper we consider stochastic stability, namely, asymptotic stability in mean, asymptotic ... more In this paper we consider stochastic stability, namely, asymptotic stability in mean, asymptotic mean-square stability, asymptotic stability a.s., exponential p-stability and stability with stochastic Lyapunov fucntion for vector stochastic differential equations. We apply these results to the stochastic epidemic models induced by bacteriophages in the marine bacteria populations. The novelty of the paper consists of new stability results for vector SDE and their applications to a new stochastic epidemic model.

Diffusion Approximation with Equilibrium for Evolutionary Systems Switched by Semi-Markov Processes

Ukrainian Mathematical Journal, 2005

We consider an evolutionary system switched by a semi-Markov process. For this system we obtain a... more We consider an evolutionary system switched by a semi-Markov process. For this system we obtain a nonhomogeneous diffusion approximation results where the initial process is compensated by the averaging function in the average approximation scheme. Dlq system, wo peremykagt\sq napivmarkovs\kymy procesamy, oderΩano rezul\taty pro neodnoridnu dyfuzijnu aproksymacig, de vysxidnyj proces kompensu[t\sq userednenog funkci[g v aproksymacijnij sxemi userednennq.

Diffusion approximation of evolutionary systems with equilibrium in asymptotic split phase space

In this paper we consider an additive functional of a Markov process with locally independent inc... more In this paper we consider an additive functional of a Markov process with locally independent increments switched by a Markov process. For this functional, we obtain nonhomogeneous diffusion approximation results without balance condition on the drift parameter. A more general diffusion approximation result is obtained in the case of an asymptotic split phase space of the switching Markov process.

Lévy Approximation of Increment Processes with Markov Switching

Stochastics and Stochastics Reports, 2004

We study weak convergence of increment processes with embedded Markov chain switching in a series... more We study weak convergence of increment processes with embedded Markov chain switching in a series scheme. The limit process is a Lévy process where the jump part is a compound Poisson process. A result concerning the rate of convergence is also given. This study is motivated by risk theory and its applications.

Reliability of Semi-Markov Systems in Discrete Time: Modeling and Estimation

Handbook of Performability Engineering

Reliability Measures of Semi-Markov Systems with General State Space

Methodology and Computing in Applied Probability, 2011

The aim of this paper is to present a systematic modeling of reliability and related measures: av... more The aim of this paper is to present a systematic modeling of reliability and related measures: availability, maintainability, failure rate, rate of occurrence of failures, mean times, etc., known in the literature under the term dependability. This model includes the continuous and discrete time semi-Markov processes with general state space. This is one of the most general models in reliability theory since it includes as particular cases the Markov and renewal processes. Keywords Semi-Markov process • Reliability • Availability • Failure rate • Rate of occurrence of failure • Mean time to failure • Survival function

Average and diffusion approximation of stochastic evolutionary systems in an asymptotic split state space

The Annals of Applied Probability, 2004

Stochastic evolutionary systems of additive functional type, described by processes with locally ... more Stochastic evolutionary systems of additive functional type, described by processes with locally independent increments, are considered with Markov switching in an asymptotic split state space having a stoppage state. The average and diffusion approximation limit theorems are established in both single and double merging. The proofs of these results are obtained using a singular perturbation approach of linear reducible-invertible operators and the tightness of processes. Particular cases of these systems including integral functionals, dynamic systems, storage processes and compound Poisson processes are also considered. The application of limit theorems in reliability and reward problems is discussed.

The rate of occurrence of failures for semi-Markov processes and estimation

Statistics & Probability Letters, 2002

This paper investigates the rate of occurrence of failures (ROCOF) of ÿnite state semi-Markov sys... more This paper investigates the rate of occurrence of failures (ROCOF) of ÿnite state semi-Markov systems. Firstly, a formula for evaluating the ROCOF for semi-Markov systems is derived. As a consequence of this result, we derive the ROCOF of Markov systems as well as the ROCOF of the alternating renewal processes and we give their asymptotic values. From the semi-Markov ROCOF formula, we propose a statistical estimator of this function. Finally, the uniform strong consistency and the asymptotic normality of this estimator are obtained.

Statistical and Probabilistic Models in Reliability

Technometrics, 2000

Statistica! and probabilistic models in reliability / D.C. Ionescu, N. Limnios, editors. p. cm. I... more

Markov Decision Processes with a Constraint on the Asymptotic Failure Rate

Methodology And Computing In Applied Probability

In this paper, we introduce a Markov decision model with absorbing states and a constraint on the... more In this paper, we introduce a Markov decision model with absorbing states and a constraint on the asymptotic failure rate. The objective is to find a stationary policy which minimizes the infinite horizon expected average cost, given that the system never fails. Using Perron-Frobenius theory of non-negative matrices and spectral analysis, we show that the problem can be reduced to a linear programming problem. Finally, we apply this method to a real problem for an aeronautical system.

Advances in degradation modeling

Lévy approximation of an impulse recurrent process with Markov switching

Theory of Probability and Mathematical Statistics, 2010

The weak convergence of an impulse recurrent process with Markov switching is proved in this pape... more The weak convergence of an impulse recurrent process with Markov switching is proved in this paper for the scheme of the Lévy approximation. A modified Liptser semimartingale method is applied to the proof of the relative compactness of the process under consideration. The modification of the Liptser method used in the paper relies upon a solution of a singular perturbation problem instead of an ergodic theorem as used in the original Liptser method.

Mathematical Methods in Survival Analysis, Reliability and Quality of Life

Discrete Time Semi-Markov Processes for Reliability and Survival Analysis — A Nonparametric Estimation Approach

Statistics for Industry and Technology, 2004

In this work we focus on multi-state systems modeled by means of a particular class of non-homoge... more In this work we focus on multi-state systems modeled by means of a particular class of non-homogeneous Markov processes introduced in [37], called drifting Markov processes. The main idea behind this type of processes is to consider a non-homogeneity that is "smooth", of a known shape. More precisely, the Markov transition matrix is assumed to be a linear (polynomial) function of two (several) Markov transition matrices. For this class of systems, we first obtain explicit expressions for reliability/survival indicators of drifting Markov models, like reliability, availability, maintainability and failure rates. Then, under different statistical settings, we estimate the parameters of the model, obtain plug-in estimators of the associated reliability/survival indicators and investigate the consistency of the estimators. The quality of the proposed estimators and the model validation is illustrated through numerical experiments. Keywords Drifting Markov chains • Multi-state systems • Reliability theory • Survival analysis • Estimation • Asymptotic properties Mathematics Subject Classification (2000) 60J10 • 60K15 • 90B25 • 62N02 • 62F12

First integrals in the diffusion approximation scheme

Applied Mathematics Letters, 2008

This work deals with the diffusion approximation of the first integral of a stochastic dynamic sy... more

Diffusion approximation of evolutionary systems with equilibrium in asymptotic split phase space

Theory of Probability and Mathematical Statistics, 2005

In this paper we consider an additive functional of a Markov process with locally independent inc... more In this paper we consider an additive functional of a Markov process with locally independent increments switched by a Markov process. For this functional, we obtain nonhomogeneous diffusion approximation results without balance condition on the drift parameter. A more general diffusion approximation result is obtained in the case of an asymptotic split phase space of the switching Markov process.

Nonparametric estimation of performance and performability for semi-markov processes

International Journal of …, 2006

In this paper a nonparametric estimator of the performability for semi-Markov systems is proposed... more

Applied Nonparametric Statistics in Reliability

Springer Series in Reliability Engineering, 2011

Nonparametric statistics has probably become the leading methodology for researchers performing d... more Nonparametric statistics has probably become the leading methodology for researchers performing data analysis. It is nevertheless true that, whereas these methods have already proved highly effective in other applied areas of knowledge such as biostatistics or social sciences, nonparametric analyses in reliability currently form an interesting area of study that has not yet been fully explored. Applied Nonparametric Statistics in Reliability is focused on the use of modern statistical methods for the estimation of dependability measures of reliability systems that operate under different conditions. The scope of the book includes: smooth estimation of the reliability function and hazard rate of non-repairable systems; study of stochastic processes for modelling the time evolution of systems when imperfect repairs are performed; nonparametric analysis of discrete and continuous time semi-Markov processes; isotonic regression analysis of the structure function of a reliability system, and lifetime regression analysis. Besides the explanation of the mathematical background, several numerical computations or simulations are presented as illustrative examples. The corresponding computer-based methods have been implemented using R and MATLAB®. A concrete modelling scheme is chosen for each practical situation and, in consequence, a nonparametric inference procedure is conducted. Applied Nonparametric Statistics in Reliability will serve the practical needs of scientists (statisticians and engineers) working on applied reliability subjects

Poisson Approximation of Processes with Locally Independent Increments and Semi-Markov Switching – Toward Application in Reliability

In this chapter, the weak convergence of additive functionals of processes with locally independe... more In this chapter, the weak convergence of additive functionals of processes with locally independent increments and with semi-Markov switching in the scheme of Poisson approximation is investigated. Singular perturbation problem for the compensating operator of the extended Markov renewal process is used to prove the relative compactness. This approach can be used in applications and especially in shock and degradation in random environment arising in reliability. Keywords and phrases Poisson approximation - semimartingale - semi-Markov process - locally independent increments process - piecewise deterministic Markov process - weak convergence - singular perturbation - degradation - reliability

Semi-Markov Models for Seismic Hazard Assessment in Certain Areas of Greece

The long-term probabilistic seismic hazard is studied through the application of semi-Markov mode... more The long-term probabilistic seismic hazard is studied through the application of semi-Markov model. In this model a sequence of earthquakes is considered as a Markov process and the waiting time distributions depend only on the type of the last and the next event. The principal hypothesis of the model is the property of one-step memory, according to which the probability of moving to any future state depends only on the current state. The model under consideration defines a continuous-time, discrete-state stationary process in which successive state occupancies are governed by the transition probabilities of the Markov process. The space of states is considered to be finite and the process started far in the past has achieved stationarity. Firstly, a non-parametric method is applied in order to determine the waiting times. Then, the waiting times derived by means of the exponential and Weibull distributions will be compared to each other, as well as with the actual waiting times. Th...

Stochastic Stability of Vector Sde with Applications to a Stochastic Epidemic Model

International Journal of Pure and Apllied Mathematics

In this paper we consider stochastic stability, namely, asymptotic stability in mean, asymptotic ... more In this paper we consider stochastic stability, namely, asymptotic stability in mean, asymptotic mean-square stability, asymptotic stability a.s., exponential p-stability and stability with stochastic Lyapunov fucntion for vector stochastic differential equations. We apply these results to the stochastic epidemic models induced by bacteriophages in the marine bacteria populations. The novelty of the paper consists of new stability results for vector SDE and their applications to a new stochastic epidemic model.

Diffusion Approximation with Equilibrium for Evolutionary Systems Switched by Semi-Markov Processes

Ukrainian Mathematical Journal, 2005

We consider an evolutionary system switched by a semi-Markov process. For this system we obtain a... more We consider an evolutionary system switched by a semi-Markov process. For this system we obtain a nonhomogeneous diffusion approximation results where the initial process is compensated by the averaging function in the average approximation scheme. Dlq system, wo peremykagt\sq napivmarkovs\kymy procesamy, oderΩano rezul\taty pro neodnoridnu dyfuzijnu aproksymacig, de vysxidnyj proces kompensu[t\sq userednenog funkci[g v aproksymacijnij sxemi userednennq.

Diffusion approximation of evolutionary systems with equilibrium in asymptotic split phase space

In this paper we consider an additive functional of a Markov process with locally independent inc... more In this paper we consider an additive functional of a Markov process with locally independent increments switched by a Markov process. For this functional, we obtain nonhomogeneous diffusion approximation results without balance condition on the drift parameter. A more general diffusion approximation result is obtained in the case of an asymptotic split phase space of the switching Markov process.