Homogeneous semi-Markov Process Research Articles

Reliability analysis of multi-component systems subjected to dependent degradation processes and random shocks in dynamic environments

This article presents reliability models for multi-component systems subjected to multiple internally dependent degradation processes and external shocks in dynamic environments. We use the homogeneous semi-Markov process (HSMP) to describe the evolution of the environment state and the non-homogeneous Poisson process (NHPP) to model the arrival of shocks. The degradation processes of the system components are modeled using either general path models (GPM) or stochastic processes. The models highlight the shock-induced degradation acceleration, the degradation-shock dependence, and the influence of dynamic environments on the degradation processes and shock damage. The integrated process evolution (i.e., system degradations, shock arrivals, and environment shifts) follows a non-homogeneous Markov renewal process (NHMRP) in a multidimensional space. The NHMRP semi-Markov kernel and theoretical formulation for system reliability are derived. We provide a simulation algorithm to estimate system reliability. Finally, we demonstrate the theoretical correctness and applicability of the model using a micro-electro-mechanical system (MEMS).

On Unattainable Boundary of a Diffusion Process Range: Semi-Markov Approach

One-dimensional homogeneous semi-Markov processes of diffusion type are considered. The transition function of such a process satisfies an ordinary second-order differential equation. It is supposed that the process does not break and has no intervals of constancy. Under these conditions, the Dirichlet problem has a solution on any finite interval. This solution is presented in explicit form in terms of solutions having values 1 and 0 at the boundaries of the interval. A criterion for the left-hand boundary of the interval to be unattainable is derived, and for the corresponding values 0 and 1, a criterion for the right-hand boundary of the interval to be unattainable is derived. This criterion applied to a diffusion process follows from known formulas which are derived by considerably complex methods of the stochastic differential equations theory.

Aggregation of Markov flows I: theory.

A Markov flow is a stationary measure, with the associated flows and mean first passage times, for a continuous-time regular jump homogeneous semi-Markov process on a discrete state space. Nodes in the state space can be eliminated to produce a smaller Markov flow which is a factor of the original one. Some improvements to the elimination methods of Wales are given. The main contribution of the paper is to present an alternative, namely a method to aggregate groups of nodes to produce a factor. The method can be iterated to make hierarchical aggregation schemes. The potential benefits are efficient computation, including recomputation to take into account local changes, and insights into the macroscopic behaviour.This article is part of the theme issue 'Hilbert's sixth problem'.

On semi-Markov processes and their Kolmogorov's integro-differential equations

Semi-Markov processes are a generalization of Markov processes since the exponential distribution of time intervals is replaced with an arbitrary distribution. This paper provides an integro-differential form of the Kolmogorov's backward equations for a large class of homogeneous semi-Markov processes, having the form of an abstract Volterra integro-differential equation. An equivalent evolutionary (differential) form of the equations is also provided. Fractional equations in the time variable are a particular case of our analysis. Weak limits of semi-Markov processes are also considered and their corresponding integro-differential Kolmogorov's equations are identified.

Minimum divergence estimators for the Radon–Nikodym derivatives of the semi-Markov kernel

In this paper we face the problem of estimating the elements that define a homogeneous semi-Markov process. Given an observed sample path of the process in a finite interval , non-parametric estimators of the entries of the semi-Markov kernel and its derivatives can be constructed. These estimators exhibit good properties as consistency and asymptotic normality. On the other hand, in our approach we consider that we have some prior information about the underlying process. This information concerns the mean sojourn times in the different states of the process and has to be taken into account in the estimation procedure. So we start from the smoothed estimators and construct minimum divergence estimators (maximum entropy) under some constraints for the associated sojourn moments.

Wind Time Series Simulation with Underlying Semi-Markov Model: An Application to Weather Derivatives

Forecasting wind speed and direction is a challenging issue in the field of meteorological research. This topic appears to be of great importance in other research areas. For instance, quantitative finance may be involved due to the recent development of wind insurance risk contracts.In this paper we set up a stochastic model able to determine the dynamic evolution of wind direction and speed at a given location. The wind direction is modeled through a non-parametric homogeneous semi-Markov process where the states are the eight main sectors of the wind rose. The wind speed is then modeled for each state by an empirical distribution.The main features of the model are estimated thanks to a database containing 30 years’ wind characteristics for Boston city on an hourly basis. The model is then tested with a Monte Carlo simulation and we compare then the results with the empirical data. Finally, we use the results for the pricing of a wind risk contract issued for a wind farm.

Semi-Markov Disability Insurance Models

In this article, we present a stochastic model for disability insurance contracts. The model is based on a discrete time non homogeneous semi-Markov process (DTNHSMP) to which the backward recurrence time process is introduced. This permits a more exhaustive study of disability evolution and a more efficient approach to the duration problem. The use of semi-Markov reward processes facilitates the possibility of deriving equations of the prospective and retrospective mathematical reserves. The model is applied to a sample of contracts drawn at random from a mutual insurance company.

Smoothed Estimation of a 3-State Semi-Markov Reliability Model

In this paper, we obtain results involving the estimation of the characteristics of functioning of a reliability system modeled by a homogeneous semi-Markov process ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SMP</i> ). We use smoothing techniques for estimating the semi-Markov kernel matrix of a finite state homogeneous <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SMP</i> . In particular, we consider a system with two up or functioning states, and one down or failed state. We assume that transitions between the two up states are allowed, and that the elements of the state space are ordered according to the level of performance of the system that they are representing. We obtain an estimate of the semi-Markov matrix by estimating the transition probability matrix of the embedded Markov chain, and the distribution functions of the transition times between states. The uniformly strong consistency of the estimator is explored. We also construct bootstrap confidence intervals for the elements of the semi-Markov matrix based on the percentile method. Finally, we propose a simplified method to obtain an estimation of the reliability function. The methods presented are easily generalized to more complex situations.

A semi-Markov model for price returns

We study the high frequency price dynamics of traded stocks by a model of returns using a semi-Markov approach. More precisely we assume that the intraday returns are described by a discrete time homogeneous semi-Markov process and the overnight returns are modeled by a Markov chain. Based on this assumptions we derived the equations for the first passage time distribution and the volatility autocorrelation function. Theoretical results have been compared with empirical findings from real data. In particular we analyzed high frequency data from the Italian stock market from 1 January 2007 until the end of December 2010. The semi-Markov hypothesis is also tested through a nonparametric test of hypothesis.

Reliability of manufacturing equipment in complex environments

We present two stochastic failure models for the reliability evaluation of manufacturing equipment that degrades due to its complex operating environment. The first model examines the case when the environment is a temporally nonhomogeneous continuous-time Markov chain, and the second assumes the environment is a temporally homogeneous semi-Markov process on a finite space. Derived are transform expressions for the lifetime distributions. A few examples are provided to illustrate the main results.

Numerical Approach for Assessing System Dynamic Availability Via Continuous Time Homogeneous Semi-Markov Processes

Continuous-time homogeneous semi-Markov processes (CTHSMP) are important stochastic tools to model reliability measures for systems whose future behavior is dependent on the current and next states occupied by the process as well as on sojourn times in these states. A method to solve the interval transition probabilities of CTHSMP consists of directly applying any general quadrature method to the N2 coupled integral equations which describe the future behavior of a CTHSMP, where N is the number of states. However, the major drawback of this approach is its considerable computational effort. In this work, it is proposed a new more efficient numerical approach for CTHSMPs described through either transition probabilities or transition rates. Rather than N2 coupled integral equations, the approach consists of solving only N coupled integral equations and N straightforward integrations. Two examples in the context of availability assessment are presented in order to validate the effectiveness of this method against the comparison with the results provided by the classical and Monte Carlo approaches. From these examples, it is shown that the proposed approach is significantly less time-consuming and has accuracy comparable to the method of N2 computational effort.

A Stochastic Model for the HIV/AIDS Dynamic Evolution

This paper analyses the HIV/AIDS dynamic evolution as defined by CD4 levels, from a macroscopic point of view, by means of homogeneous semi-Markov stochastic processes. A large number of results have been obtained including the following conditional probabilities: an infected patient will be in statejafter a timetgiven that she/he entered at time 0 (starting time) in statei; that she/he will survive up to a timet, given the starting state; that she/he will continue to remain in the starting state up to timet; that she/he reach stagejof the disease in the next transition, if the previous state wasiand no state change occurred up to timet. The immunological states considered are based on CD4 counts and our data refer to patients selected from a series of 766 HIV-positive intravenous drug users.

Future pricing through homogeneous semi-Markov processes

An Erratum for this article has been published in Applied Stochastic Models in Business and Industry 2005; (in press) This paper presents a future pricing model based on the discrete time homogeneous semi-Markov process (DTHSMP). The model is adapted to the real data of the Italian primary future stock index. After showing the pricing model, the DTHSMP solution is given. The solution of the semi-Markov process gives, for each period of the considered horizon time, and for each starting state, the probability distribution of the future price. Copyright © 2005 John Wiley & Sons, Ltd.

Numerical Treatment of Homogeneous Semi-Markov Processes in Transient Case–a Straightforward Approach

This paper presents the numerical solution of the process evolution equation of a homogeneous semi-Markov process (HSMP) with a general quadrature method. Furthermore, results that justify this approach proving that the numerical solution tends to the evolution equation of the continuous time HSMP are given. The results obtained generalize classical results on integral equation numerical solutions applying them to particular kinds of integral equation systems. A method for obtaining the discrete time HSMP is shown by applying a very particular quadrature formula for the discretization. Following that, the problem of obtaining the continuous time HSMP from the discrete one is considered. In addition, the discrete time HSMP in matrix form is presented and the fact that the solution of the evolution equation of this process always exists is proved. Afterwards, an algorithm for solving the discrete time HSMP is given. Finally, a simple application of the HSMP is given for a real data social security example.

Numerical Treatment of Homogeneous and Non-homogeneous Semi-Markov Reliability Models#

In this paper, we extend to our knowledge, for the first case, some reliability results using homogeneous semi-Markov processes to the case of homogeneous modeling semi-Markov processes. Moreover, we apply some of our preceding results to give the numerical solutions and so the possibility to treat real life problems for which non-homogeneity in time is important.

The first exit of almost strongly recurrent semi-Markov processes

Let $\stackrelnX(·)$, n ∈ N, be a sequence of homogeneous semi-Markov processes (HSMP) on a countable set K, all with the same initial p.d. concentrated on a non-empty proper subset J. The subrenewal kernels which are restrictions of the corresponding ren

Rapprochement of consolidated stochastic systems and nonhomogeneous semi-Markov systems

The problem of asymptotic consolidation has been considered in a large number of investigations; they can be divided into three directions. In [I-3] one develops an analytic approach, connected with the investigation of the behavior of the solutions of the Markov renewal equations, which operates effectively at the analysis of homogeneous Markov processes (MP) and semi-Markov processes (SMP). In [4, 5], by direct probabilistic methods one has obtained a rapproachement of the characteristics of a consoliated process and of a process of the SMP type for processes with a discrete set of states. In [6-9] one has developed a new approach to the problems of asymptotic consolidation, making use of a special class of processes, viz., switching processes, which is effectively applied in [~0-12] to the analysis of both homogeneous and nonhomogeneous MP and SMP.

A computational procedure for the asymptotic analysis of homogeneous semi-Markov processes

A new algorithm for classifying the states of a homogeneous Markov chain having finitely many states is presented, which enables the investigation of the asymptotic behavior of semi-Markov processes in which the Markov chain is embedded. An application of the algorithm to a social security problem is also presented.