Numerical solution of reliability models described by stochastic automata networks
Introduction
Markov chains are one of the most often applied stochastic processes in system reliability models [1], [2]. Application of Markov chains in reliability modelling include such diverse examples as analysis of software reliability [3], maintenance of road networks [4], and evaluation of reliability of small electronic components [5] or electrical power transmission systems [6]. As compared to the methods based on the independence of system components (e.g. fault tree analysis or reliability block diagrams), Markov models can be applied to systems with more complex behaviour, such as dependent failures or repairable items [7]. Moreover, unlike simulation-based techniques (e.g. a Monte Carlo method), Markov chain modelling is a stochastic analytic method which, in theory, can provide an exact solution. However, all the aforementioned techniques can be cumbersome and require many computational resources, especially if large systems and rare events are considered [8], [9], [10], [11]. This is also true for Markov chains, which are rarely applied in reliability modelling of large systems due to an exponential growth of state-space. Therefore, most authors apply Markov chains for relatively small models, which can be tracked analytically [12], [13], [14]. In more complex cases, special software [15], [16] is also applied for model creation and numerical solution [17], [18]. However, for many real-life problems, the state-space of a Markov chain model can be estimated in millions. In such a case, the creation and storage of the transition matrix, as well as the calculation of steady-state probabilities require special attention.
Thus, efficient application of numerical methods is a crucial part of Markov chain modelling of large systems. However, it is still rarely addressed in reliability modelling studies, and only in a few exceptions can one find examples of steady-state solutions of reliability models whose state-space exceeds a million [19]. Thus, it seems that well-established methods of numerical solution of Markov chain models [20] have not yet been extensively applied in reliability modelling studies. So far, the most often used approaches in dealing with large Markov models are various state-space reduction strategies [21]. Among them, we can mention an exclusion of redundant system states and truncation of states with low probabilities [22], [23], state aggregation and lumping methods [24], [25] or some mixture of both these techniques [26]. Another similar approach is to decompose a large system into smaller, nearly independent components, and to solve the smaller models separately [27], [28]. However, these techniques are by definition approximations and exhibit inherent solution errors. Only the solution of the complete Markov model can provide an exact (as far as computational precision allows) solution, thus evaluating all possible scenarios, including all rare events.
In this study, we address a numerical solution of complete Markov chain models, described by the Stochastic Automata Networks (SANs) formalism [29]. The SAN formalism applies Kronecker algebra operations to store a transition matrix of the Markov chain model in a compact form, thus mitigating the problem of dimensionality. However, a steady-state solution remains a serious problem because Kronecker's algebra approach requires a more sophisticated application of standard numerical methods [30].
So far, most examples of SANs applications considered queuing system models [31], [32], [33]. The use of SANs in system reliability modelling is still relatively rare [34], [35] and, to our knowledge, the published studies did not consider a steady-state probability calculation. In our previous paper [36], we presented a methodology for power system reliability specification using the SAN formalism. We demonstrated that various reliability scenarios could be described using arrowhead matrices and functional transition rates. As a result, the transition rate matrices of the created Markov models have a distinctive structure and we proposed that it could be advantageous for an efficient estimation of steady-state probabilities. However, we did not address this problem in the previous study because for the relatively small models presented in [36] it was possible to generate the full transition matrix from the SAN descriptor and to estimate steady-state probabilities using standard direct algorithms, such as Gaussian elimination. For larger models, such an approach would not be feasible due to operating memory constraints. In that case, one must apply various iterative techniques for calculation of steady-state probabilities, and we address this problem in this study.
Our research showed that the distinctive structure of the created reliability models allows us to apply a block Gauss–Seidel method very efficiently. That is, due to the high fidelity of system components, steady-state solution of power systems described by arrowhead matrices converges very rapidly when using the Gauss–Seidel algorithm. In addition, the specific block structure arising from the SAN formalism is very suitable for block-iterative methods. Moreover, the analytical research showed that inner iteration of the block Gauss–Seidel method can be solved very efficiently. Numerical experiments supported the analytical results and the block Gauss–Seidel method outperformed other standard methods in solving steady-state probabilities of a representative Markov model of a 3/2 substation configuration with nearly two million states.
The structure of this paper is as follows. An introduction to the steady-state solution of Markov chain models is presented in Section 2. In Section 3, we explain the choice of the block Gauss–Seidel method for the steady-state solution, based on the structural properties of the model transition matrix. In Section 4, we present the efficient implementation of the block Gauss–Seidel method specially modified for models described by the SAN formalism and arrowhead matrices. In Section 5, we present the SAN specification of the 3/2 substation configuration using the methodology proposed in [36]. This model is later used in Section 6 as a case study to illustrate the efficiency of the proposed implementation of the block Gauss–Seidel method.
Section snippets
Steady-state solution of Markov chain models
In this section, we briefly introduce the theoretical background and numerical solution methods for steady-state calculations of Markov chain models. For more information, we refer the reader to [20], [37].
Motivation
In the previous study [36], we presented the methodology for specification of system reliability by the SAN formalism. We demonstrated that a variety of reliability scenarios could be described using arrowhead matrices and functional transition rates. The basic idea was to divide a system into smaller linear branches of components, which are disconnected together under repair in case any of them fails. Such linear branch of components can then be modelled by a single automaton, which has the
Efficient application of the block Gauss–Seidel method for reliability models described by the SAN formalism
Based on the motivation that is provided in the previous section, we decided to analyse the use of the block Gauss–Seidel method for reliability models specified by the SAN formalism and arrowhead matrices in detail. Our research showed that the intrinsic structure of the transition rate matrix of SAN reliability models allows for a very efficient implementation of the block Gauss–Seidel method. More specifically, in our implementation, the solution of linear systems of equations during the
SAN reliability model of the 3/2 substation configuration
To illustrate the efficiency of the proposed methodology for steady-state probabilities calculations, we consider a reliability model of the standard 3/2 substation configuration (see Fig. 1).
Results and discussion
The created 3/2 substation configuration model was used to test the efficiency of the proposed block Gauss–Seidel modification. Failure and repair rates of system components were evaluated from the data collected by the Lithuanian Energy Institute (see Table 4).
It is easy to see that the repair rates are much higher than the failure rates (||M|| > > ||Λ||); therefore, one can expect a fast convergence of the block Gauss–Seidel method, as we showed in Section 2. The convergence speed of an
Acknowledgement
This research was funded by a grant (ATE-No. 04/2012) from the Research Council of Lithuania.
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