A Monte-Carlo simulation approach for approximating multi-state two-terminal reliability
Introduction
The computation of two-terminal reliability (2TR) is a classical network reliability problem when considering binary and system states, i.e. both network and components can only be in fully working or fully failed states. Numerous approaches and methodologies have been proposed to solve this difficult problem [1], [2], [3], [4], [5], but these approaches operate under the assumption that a system and its components can only be in working or failed states. However, researchers have indicated that in some cases, binary state theory fails to characterize the actual system reliability behavior [6], [7], [8], [9], [10], which is multi-state. Misrepresenting system behavior as a binary event can be problematic since systems can have a range of intermediate states that are not accounted for in the reliability computation. For some systems, such erroneous appraisal of system reliability could translate into: (1) incorrect system modeling, (2) incorrect system reliability computation, and/or (3) incorrect conjectures regarding reliability-dependant measures.
Multi-state reliability has been proposed as a complementary theory to cope with the problem of analyzing systems where traditional binary reliability theory and models become insufficient [6], [7], [8], [9]. For systems such as water distribution systems, telecommunication systems, oil and gas supply systems, and power generation and transmission systems, an analysis of 2TR from a multi-state view may be the preferred approach [10], [27]. For these types of networked systems, it may be insufficient to consider a binary state behavior of the components. Generally, the components of these systems follow a degradation pattern that reduces the ability of the system to provide some required service.
Recently the multi-state extension of 2TR has received considerable attention [11], [12], [13], [14], [15], [16], [17]. For this extension, multi-state two-terminal reliability at demand level d (M2TRd) is defined as the probability that a demand of d units can be supplied from source to sink nodes through multi-state arcs. Most of these approaches solve the problem by providing multi-state minimal cut or path vectors (MMCV, MMPV), which are the multi-state equivalent of minimal path or cut sets in 2TR.
Previous research [13], [14], [15], [16] assumes or implies that once MMCV or MMPV are obtained, numerical reliability computation is straightforward or even trivial. The preferred approach used for obtaining M2TRd is the classical inclusion/exclusion formula. For small networks, this may be a simple approach. However, as the size of the network or the number of component states increase, using such formula may be computationally inefficient, and thus, not trivial at all. Therefore, alternative reliability computation approaches are needed.
Actual systems, where M2TRd is the most appropriate reliability metric, are complex. Reliability computation may be too costly to obtain through traditional techniques. Simulation has been effectively used for analyzing 2TR for relatively large systems [18]. However, simulation has been underutilized for approximating M2TRd. In this paper, a Monte-Carlo (MC) simulation methodology for estimating numerical M2TR is proposed based on identified MMCV [13], [14], [15], [16], [17]. The methodology consists of using MC simulation to generate system state vectors. Once a vector is obtained it is compared to the set of MMCV to decide if the capacity of such a vector satisfies the required demand.
The remainder of the paper is organized as follows: the rest of this section defines the M2TR problem. Section 2 illustrates different methodologies that have been developed for analyzing M2TRd and techniques used for approximating reliability. Section 3 presents the new methodology, and in Section 4, examples are used to illustrate and validate the methodology. These examples are used to compare the proposed methodology against other approximation methodologies. Finally, Section 5 presents conclusions.
Let G=(N, A) represent a stochastic capacitated network with a specified required demand d from source node s to sink node t. N represents the set of nodes, A={ai|1≤i≤|A|} represents the set of arcs. The current state (capacity) of arc ai, is represented by xi. The current state (capacity) of arc ai, represented by xi∈bi, takes values bi1=0,bi2,…,bil=Mi, where bij∈+ and bi represents arc ai state space vector. The vector pi represents the probability associated with each of the values taken by xi. The system state vector x=(x1,x2,…,x|A|) denotes the current state of all the arcs of the network. The function φ: |A|→+ maps the system state vector into a system state. That is, φ(x) is the available capacity from source to sink under system state vector x. M2TRd is understood as the probability that a demand of d units can be supplied from source to sink through the multi-state arcs, i.e. M2TRd=P(φ(x)≥d).
Definitions:
Vector dominance A vector y is said to be less than x, y<x, iff for every xi∈x and every yi∈y, yi≤xi and for some xk∈x, yk∈y, yk<xk. A vector y is said to be dominated by a vector x if y<x. Minimal path vector at level d A vector x is said to be a minimal path vector to level d if φ(x)≥d and for every other y<x, φ(y)<d [6]. Minimal cut vector at level d A vector x is said to be a minimal cut vector to level d if φ(x)<d and for every other y>x, φ(y)≥d [6].
Assumptions:
- (1)
Component states are statistically independent.
- (2)
The structure function φ(x) is coherent. That is, improvement of component states cannot damage the system.
- (3)
Component states and associated probabilities are known.
Acronyms:
- MMCV
multi-state minimal cut vector
- MMPV
multi-state minimal path vector
- M2TRd
multi-state two-terminal reliability
- MC
Monte-Carlo
Section snippets
Multi-state two-terminal reliability
For systems where binary state analysis is insufficient, incorrect reliability assessment can lead to faulty decision-making regarding network performance. Unnecessary expenditures, incorrect maintenance scheduling and reduction of safety standards are problems that can potentially be related to unsatisfactory reliability assessments. Thus, there is a real need to incorporate a more realistic view of system performance concerning multi-state behavior.
A first approach to solve this problem has
Monte-Carlo simulation for M2TRd
As previously discussed, most other methodologies [13], [14], [15], [16] assume the reliability computation for the M2TRd is straightforward once MMCV (or MMPV) have been determined. For large systems with even a relatively small number of states, this may not be the case. Although bounding methodologies have been proposed for this problem [19], numerical examples show that, in some cases, the approximation to the actual M2TRd may not be sufficiently accurate. Even when the bounds do provide
Experimental results and comparison of existing methodologies
In this section, three examples are used to obtain experimental results using the proposed simulation approach. The value of M2TRd obtained through the simulation is compared to other approximation methodologies available in the literature [19]. The network in Example 1 is used to validate the methodology. 4.2 Example 2, 4.3 Example 3 consider medium-sized and relatively large networks where the accuracy of traditional computational approaches may not be sufficiently accurate. The results
Performance and comparisons
The results obtained for the three examples presented in Section 4 have been analyzed considering two perspectives: computational effort and quality of results. For each of these perspectives, figures of merit are presented considering the approximation methodologies and the MC simulation approach. These analyses intend to clarify and illustrate when simulation-based approaches can be preferable to analytical approximation formulae.
Conclusions
A Monte-Carlo simulation approach has been developed for estimating M2TRd. M2TRd is defined as the probability that a demand of d units can be supplied from source to sink through multi-state arcs. The new methodology effectively estimates M2TRd if MMCV or MMPV is known. Additionally, it has been discussed that most methodologies [13], [14], [15], [16] assume the reliability computation for the M2TRd based on MMCV (MMPV) is straightforward. For large systems, even with a relatively small number
References (29)
- et al.
Reliability evaluation of flow networks considering multistate modeling of network elements
Microelectron Reliab
(1993) - et al.
Evaluation of probability mass function of flow in a communication network considering a multistate model of network links
Microelectron Reliab
(1996) A simple algorithm for reliability evaluation of a stochastic-flow network with node failure
Comput Oper Res
(2001)Using minimal cuts to evaluate the system reliability of a stochastic-flow network with failures at nodes and arcs
Reliab Eng Syst Saf
(2002)A simple MC-based algorithm for evaluating reliability of a stochastic-flow network with unreliable nodes
Reliab Eng Syst Saf
(2004)- et al.
Network reliability estimates using linear and quadratic unreliability of minimal cuts
Reliab Eng Syst Saf
(2003) Edge-packings of graphs and network reliability
Discrete Math
(1988)A fast recursive algorithm to calculate the reliability of a communication network
IEEE Trans Commun
(1972)- et al.
A new analysis technique for probabilistic graphs
IEEE Trans Circuits Syst
(1979) Calculation of node-pair reliability in large networks with unreliable nodes
IEEE Trans Reliab
(1994)
Determining terminal pair reliability based on edge expansion diagrams using OBDD
IEEE Trans Reliab
Multistate reliability theory—a case study
Adv Appl Probab
Customer-driven reliability models for multistate coherent systems
IEEE Trans Reliab
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