Dynamic reliability via computational solution of generalized state-transition equations for entry-time processes
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 .
- (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 (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 , 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 . These are defined so that if at calendar time t the system has entry time and associated mark j, the probability it has subsequent entry time with associated mark i is . 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 Pi(τ,t) are sought in the relevant region , overlaid by a mesh of square cells of dimension Δt×Δt. We use m and n, respectively, as discrete surrogates for τ and t, and . The objective is to obtain computational approximations . We use the notationsfor m⩽n, and .
We begin by evaluating Eq. (6) at τm=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.
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