A Monte-Carlo simulation approach for approximating multi-state two-terminal reliability

Abstract

This paper describes a Monte-Carlo (MC) simulation methodology for estimating the reliability of a multi-state network. The problem under consideration involves multi-state two-terminal reliability (M2TR) computation. Previous approaches have relied on enumeration or on the computation of multi-state minimal cut vectors (MMCV) and the application of inclusion/exclusion formulae. This paper discusses issues related to the reliability calculation process based on MMCV. For large systems with even a relatively small number of component states, reliability computation can become prohibitive or inaccurate using current methods. The major focus of this paper is to present and compare a new MC simulation approach that obtains accurate approximations to the actual M2TR. The methodology uses MC to generate system state vectors. Once a vector is obtained, it is compared to the set of MMCV to determine whether the capacity of the vector satisfies the required demand. Examples are used to illustrate and validate the methodology. The estimates of the simulation approach are compared to exact and approximation procedures from solution quality and computational effort perspectives. Results obtained from the simulation approach show that for relatively large networks, the maximum absolute relative error between the simulation and the actual M2TR is less than 0.9%, yet when considering approximation formulae, this error can be as large as 18.97%. Finally, the paper discusses that the MC approach consistently yields accurate results while the accuracy of the bounding methodologies can be dependant on components that have considerable impact on the system design.

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 (M2TR_d) 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 M2TR_d 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 M2TR_d 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 M2TR_d. 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 M2TR_d 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={a_i|1≤i≤|A|} represents the set of arcs. The current state (capacity) of arc a_i, is represented by x_i. The current state (capacity) of arc a_i, represented by x_i∈b_i, takes values b_i1=0,b_i2,…,b_il=M_i, where b_ij∈$\mathfrak{R}$⁺ and b_i represents arc a_i state space vector. The vector p_i represents the probability associated with each of the values taken by x_i. The system state vector x=(x₁,x₂,…,x_|A|) denotes the current state of all the arcs of the network. The function φ: $\mathfrak{R}$^|A|→$\mathfrak{R}$⁺ 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. M2TR_d 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. M2TR_d=P(φ(x)≥d).

Definitions:

Vector dominance

A vector y is said to be less than x, y<x, iff for every x_i∈x and every y_i∈y, y_i≤x_i and for some x_k∈x, y_k∈y, y_k<x_k. 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:

• Component states are statistically independent.

• The structure function φ(x) is coherent. That is, improvement of component states cannot damage the system.

• Component states and associated probabilities are known.

Acronyms:

MMCV

multi-state minimal cut vector

MMPV

multi-state minimal path vector

M2TR_d

multi-state two-terminal reliability

MC

Monte-Carlo