Dynamic reliability via computational solution of generalized state-transition equations for entry-time processes

doi:10.1016/j.ress.2006.08.005

Reliability Engineering & System Safety

Volume 92, Issue 9, September 2007, Pages 1281-1293

https://doi.org/10.1016/j.ress.2006.08.005 Get rights and content

Abstract

Entry-time processes are finite-state continuous-time jump processes with transition rates depending only on the two states involved, the calendar time, and the most recent arrival time (entry time). Entry-time processes are transformed into Markov processes via the standard technique of incorporating entry time into the state variables. It is shown that the associated state-transition (Chapman–Kolmogorov) equations can be written as a coupled pair of integrodifferential equations. A finite-difference approximation to these equations is developed. This computational approach is verified, and some of its properties delineated, via two hypothetical examples. One of these examples admits a semi-analytic solution, while simulations provide the base of comparison for the other.

Introduction

An entry-time process is heuristically a finite-state continuous-time jump process such that the transition rate between states depends only on (the two states and) the calendar time and the most recent arrival time, termed as entry time. (See the following section for a more formal definition.) An entry-time process is not a finite-state Markov process, because of the dependence of the transition rates on the entry time (generically denoted here asτ), which is no later than the calendar time (t). If one augments the state space by the entry time, there results a Markov process, with associated Chapman–Kolmogorov equations, termed here as generalized state-transition equations; see [1] for elaborations on this well-known idea.

The objectives of this article are:

(i)
To demonstrate explicitly (Section 3) the generalized state-transition equations as a coupled set of integro/partial-differential equations on the two-dimensional region $0 ⩽ τ ⩽ t$ .
(ii)
To describe (Section 4) a discrete algorithm for direct computational approximation to these generalized state-transition equations.
(iii)
To verify (5 A semianalytically solvable example, 6 A simplified computational example, with comparison to simulation) this algorithm, by demonstrating its accuracy for two somewhat contrived examples, which were selected largely in order to permit alternative solutions for comparison, but also to expose the potential applicability of entry-time processes to reliability engineering.

The remainder of this introductory section is a brief review of prior related work. The following Section 2 contains a mathematical development of the essentials of entry-time processes.

The work described here has been motivated by the potential application to optimization of preventive maintenance activities within risk-informed asset management (RIAM) in nuclear power plants; e.g., [2]. Computational solution of the systems of ordinary differential equations that comprise the state-transition equations for finite-state continuous-time Markov models has been demonstrated to comprise a viable tool for analysis of RIAM activities in nuclear power plants [3]. The present approach is a natural extension, to permit inclusion of the aging (“wearout”) and “wearin” effects that are central to the modern study of reliability issues [4], [5], and that appear likely to become increasingly important within the nuclear energy industry (e.g., [6]).

Such concerns are by no means unique to nuclear power [7], [8], [9]. Nonetheless, an approach via computational solution of integrodifferential equations is natural to nuclear energy, because this approach already is widespread there, as deterministic (e.g., multigroup discrete-ordinate) methods for computational approximation to the neutron transport equation. Indeed DeVooght and Smidts [10], [11] have previously suggested such an approach as a generalization of the traditional discrete fault tree approach to risk analysis in nuclear power plants. These researchers employed only Monte Carlo methods for the solution of their prototypical state-transition equations; the present work can be considered a natural counterpart of that prior work, in that we instead apply direct (“deterministic”) computational methods.

Earlier work most directly related to the present is that of Becker et al. [12]. Entry-time processes are precisely semi-Markov processes viewed, as in [12], to be defined by transition rates that depend upon calendar time and entry time, and hence upon sojourn time=difference between the two. As in [12] we believe in applications to reliability the transition rates are a more natural starting point than the nonhomogeneous semi-Markov kernel, and computation of the state probabilities corresponding to an a priori known initial state probability is the typical objective, in contrast to the focus upon transition probabilities that is traditional in probability theory (e.g., [13]). We defer to [12] for a discussion of connections between the two views and limitations of entry-time processes in their application to reliability, and to [13] and references cited therein for computational approaches to non-homogeneous semi-Markov processes from the traditional viewpoint.

The primary difference between the present work and that of Ref. [12] is our focus upon detailed description, verification and delineation of properties of computational methods for the direct determination of the state probabilities. Our use of integrodifferential state-transition equations, as opposed to the mathematically equivalent integral form in [12], is a key enabler of our focus upon computational techniques, in that it permits ready adaptation of the ideas of the extensive existing methodology for deterministic solution of (e.g., neutron) transport equations. We do make some effort to extend the framework of [12] to incorporate deterministic transitions, as that seems useful for applications (e.g., scheduled maintenance).

This work also shares with [12] that it can be considered a specialization, to the case that the physical process dynamics is negligible and the only dynamics stems from the (physically artificial but mathematically and computationally important) incorporation of entry time into the state space, of the significant ongoing work being conducted under the general heading of “dynamic reliability” (cf. [14], and references cited therein). The focus in these works upon the integral form of the state-transition equations has led to employment of Monte Carlo methodology as the predominant tool for obtaining quantitative results. Our intention here is to illustrate there may be direct computational methods (e.g., finite differences) that conceivably could be competitive with (e.g., Monte Carlo) simulation.

Section snippets

Entry-time processes

In this section we basically follow the terminology, notation and conventions of the monograph of Sigman [15]; however, we find it convenient to denote the sequence of arrival times of a simple marked point process as ${τ_{n}}_{n = 0}^{\infty}$ (i.e., by “τ” rather than “t”). Given a simple marked point process, the entry time associated to calendar time t is defined as the most recent arrival time; symbolically $τ (t) = τ_{N (t)}$ , where N is the associated counting process. The mark values will commonly be referred to

The generalized state-transition equations

The central point of entry-time processes is that their (marginal) distributions are defined by (presumed a priori known) associated transition rates $λ_{ij} = λ_{ij} (τ, t)$ . These are defined so that if at calendar time t the system has entry time $τ ⩽ t$ and associated mark j, the probability it has subsequent entry time $τ^{'} > t$ with associated mark i is $λ_{ij} (τ, t) (τ^{'} - t) + o (τ^{'} - t)$ . These transition rates generalize, to multistate processes, the familiar concept of failure rates. (That is, if the only allowable

A finite-difference methodology

Fig. 1 depicts the first quadrant of the (τ,t)-plane. The P_i(τ,t) are sought in the relevant region $t > τ$ , overlaid by a mesh of square cells of dimension Δt×Δt. We use m and n, respectively, as discrete surrogates for τ and t, $τ_{m} = m Δ t$ and $t_{n} = n Δ t$ . The objective is to obtain computational approximations $P_{i}^{(m, n)} \approx P_{i} (m Δ t, n Δ t)$ . We use the notations $C_{mn} = {(τ, t) : τ_{m} ⩽ τ ⩽ τ_{m + 1}, t_{n} ⩽ t ⩽ t_{n + 1}}$ for m⩽n, and $T_{n} = C_{nn} ⋂ {(τ, t) : t ⩾ τ}$ .

We begin by evaluating Eq. (6) at τ_m=mΔt, and then integrating the resulting equation on t,

A semianalytically solvable example

The purpose of this section is to verify the computational method developed in the preceding section, by comparing its performance to the (nearly) analytic solution of a simple three-state problem contrived to lend itself to such a solution. The example and its semianalytic solution are formulated in Subsection 5.1. Computational results via the above finite-difference method are described in Subsection 5.2.

A simplified computational example, with comparison to simulation

This section is devoted to a hypothetical three-state example that permits state reentry. Computational results from the above methodology are compared to simulations.

The three-state example is formulated in Subsection 6.1. In Subsection 6.2 we apply to the three-state example the MatLab finite-difference code used to obtain the results described in the preceding section, and we verify this application by comparison against (differential and integral) results obtained from simulation.

Conclusions

Entry-time processes were introduced as finite-state continuous-time jump processes such that the transition rate between two states depends only on the two states, the calendar time and the entry time (most recent arrival time). Such processes were shown to have (generalized) state-transition (Chapman–Kolmogorov) equations having the form of a coupled pair of two-dimensional integrodifferential equations. A simple finite-difference method for the solutions of these equations was introduced,

Acknowledgements

This material is based upon work partially supported by the Department of Energy under Grant no. DE-FG07-05ID14694, “Application of Entry-time Processes in Asset Management for Nuclear Power Plants.” The authors gratefully acknowledge the encouragement and advice of Mr. Ernest J. Kee and Ms. Alice Y. Sun, of the South Texas Nuclear Project.

References (16)

K.N. Fleming
Markov models for evaluating risk-informed in-service inspection strategies for nuclear power plant piping systems
Reliab Eng Syst Saf
(2004)
J. DeVooght et al.
Probabilistic dynamics as a tool for dynamic PSA
Rel Eng Syst Saf
(1996)
G. Becker et al.
A semi-Markovian model allowing for inhomogenities with respect to process time
Rel Eng Syst Saf
(2000)
G. Becker et al.
Dynamic reliability under random shocks
Rel Eng Syst Saf
(2002)
D.R. Cox
The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables
Proc Camb Philos Soc
(1955)
J.K. Liming et al.
Risk-informed asset management (RIAM) for nuclear power plants
Trans Am Nucl Soc
(2003)
G.B. Gertsbakh
Models of preventive maintenance
(1977)
G.B. Gertsbakh
Reliability theory
(2000)

There are more references available in the full text version of this article.

Cited by (8)

A parameter estimation method for a condition-monitored device under multi-state deterioration
2012, Reliability Engineering and System Safety
Citation Excerpt :
With this type of transition, the associated degradation process from state i to state j starts only at the time the equipment reached state i. This type of transition rate has been also used in [56,57]. To be able to describe more general cases, we assume that any of the above mentioned degradation transition types may exist for multi-state equipment.
The overall performance of a mechanical device under random shocks, fatigue, and gradual degradation may continuously deteriorate over time, leading to multi-state health conditions. This deterioration can be represented by a continuous-time degradation process – with multiple discrete states – that reflects the relative degree of deterioration. This paper focuses on a condition-monitored device with multi-state deterioration, where its degradation state is not directly observable and only incomplete information is available through condition monitoring. After modeling this multi-state device, an unsupervised parameter estimation method is developed, which employs historical condition monitoring information to estimate the unknown characteristic parameters of the degradation process and the observation process. The results are evaluated through numerical experiments.
Spatially constrained forest cover dynamics using Markovian random processes
2012, Forest Policy and Economics
Essential to analyses of forest-cover change is application of geospatial empirical or semi-empirical models of transition potentials based on the likelihood that forest land would change to non-forested land or vice versa depending on prevailing conditions of land-use change. Modeling land-cover as a function of land-use aids in understanding pertinent land-cover dynamics. This can enable forecasting of ramifications of current conversion processes on land designated for agriculture or development. National Agricultural Statistics Service (NASS) grid data published by USDA for years 1997–2002 were used as preliminary inputs. Two prevalent transition probabilities were derived: probability of a pixel changing (a) from forested to non-forested, P_fnf and (b) from non-forested to forested, P_nff. These probabilities were determined for the years (i) 1999–2000, (ii) 2000–2001, (iii) 1999–2001, and (iv) 1999–2009 using decade stratification. The maximal transition probability ranges for forest to non-forest transition were higher for 1999–2000 and 1999–2001 transiting periods compared to 2000–2001. The maximal transition probability ranges for non-forest to forest transition were lower for 1999–2000 transiting period compared to 2000–2001, and 1999–2001 transiting periods. The analysis provided a glimpse on areas deemed prone to forest conversion and those that would immensely benefit from federally funded programs.
Dynamic reliability of digital-based transmitters
2011, Reliability Engineering and System Safety
Citation Excerpt :
These considerations further increase when data and deviation variables are added and, for the latter variables, it is also not always obvious to know how to a priori define efficiently ranges of evolution. Discussions about numerical methods which can be applied to these topics can be found in the literature (e.g. [71–73]). Since transition rates and differential equations are used to describe the system evolution (cf. Eqs. (1)–(5)), the numerical method adopted in the present paper is based on Taylor series and finite differences.
Dynamic reliability explicitly handles the interactions between the stochastic behaviour of system components and the deterministic behaviour of process variables. While dynamic reliability provides a more efficient and realistic way to perform probabilistic risk assessment than “static” approaches, its industrial level applications are still limited. Factors contributing to this situation are the inherent complexity of the theory and the lack of a generic platform. More recently the increased use of digital-based systems has also introduced additional modelling challenges related to specific interactions between system components. Typical examples are the “intelligent transmitters” which are able to exchange information, and to perform internal data processing and advanced functionalities. To make a contribution to solving these challenges, the mathematical framework of dynamic reliability is extended to handle the data and information which are processed and exchanged between systems components. Stochastic deviations that may affect system properties are also introduced to enhance the modelling of failures. A formalized Petri net approach is then presented to perform the corresponding reliability analyses using numerical methods. Following this formalism, a versatile model for the dynamic reliability modelling of digital-based transmitters is proposed. Finally the framework's flexibility and effectiveness is demonstrated on a substantial case study involving a simplified model of a nuclear fast reactor.
Mathematical formulation and numerical treatment based on transition frequency densities and quadrature methods for non-homogeneous semi-Markov processes
2009, Reliability Engineering and System Safety
Non-homogeneous semi-Markov processes (NHSMP) are important stochastic tools for modeling reliability metrics over time for systems where the future behavior depends on the current and next states as well as on sojourn and process times. The classical method to solve the interval transition probabilities of NHSMPs consists of directly applying any general quadrature method to some non-convolution integral equations. However, this approach has a considerable computational effort. Namely, N²-coupled integral equations with two variables must be solved, where N is the number of states. Therefore, this article proposes a more efficient mathematical formulation and numerical treatment, which are based on transition frequency densities and general quadrature methods respectively, for NHSMPs. The approach consists of only solving N-coupled integral equations with one variable and N straightforward integrations. Two examples in the context of reliability are also presented. The first one addresses a case where a semi-analytical solution is available. Then an example of application concerning pressure–temperature optical monitoring systems for oil wells is discussed. In both cases, the proposed approach is validated via the comparison against the results obtained from the semi-analytical solution (for the first example) as well as from both the classic and the Monte Carlo methods.
Analyze the core damage scenario via the Rismc-based Metatheory and reliability theory
2014, International Conference on Nuclear Engineering, Proceedings, ICONE
Leveraging a spatio-temporal drought severity and coverage index with crop yield modelled as a stochastic process
2012, International Journal of Hydrology Science and Technology

View all citing articles on Scopus

View full text

Dynamic reliability via computational solution of generalized state-transition equations for entry-time processes

Abstract

Introduction

Section snippets

Entry-time processes

The generalized state-transition equations

A finite-difference methodology

A semianalytically solvable example

A simplified computational example, with comparison to simulation

Conclusions

Acknowledgements

Reliab Eng Syst Saf

Rel Eng Syst Saf

Rel Eng Syst Saf

Rel Eng Syst Saf

The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables

Proc Camb Philos Soc

Risk-informed asset management (RIAM) for nuclear power plants

Trans Am Nucl Soc

Models of preventive maintenance

Reliability theory