A New Monte Carlo Method for the Network Reliability Wei-Chang Yeh, Member, IEEE
Abstract-- A Hybrid Monte Carlo method (HMC) method is proposed in this article for estimating the system reliability of a stochastic network without needing to know all of the minimal pathsets/cutsets (MPs/MCs) in advance. The analysis indicates that the proposed method is efficient when compared to the crude Monte Carlo method (CMC). Index Terms-- Minimal Pathsets/Cutsets, Method, Network, Reliability. I.
Monte
Carlo
this study to completely overcome both NP-hard obstacles. A HMC method is proposed in this article for estimating the system reliability of a stochastic network without needing to know all of the MPs/MCs in advance. The proposed method involves combined stratification, antithetic variate and control variate sampling. The statistical properties of the proposed estimator are analyzed and compared with CMC. II. NOMENCLATURE, NOTATION, AND ASSUMPTIONS
INTRODUCTION
I
A. Nomenclature Reliability: The system reliability of a stochastic network, R, is defined as the probability that there is at least one path from nodes 1 to n which operates successfully. MP/MC: It is a pathset/cutset between nodes s and t such that any proper subset of MP/MC is not a MP/MC, i.e. if any arc is removed from the set, then the remaining set is no longer a pathset/cutset [4].
This work was supported in part by the National Science Council Taiwan, R.O.C. under Grant 89-2213-E -035-056. Wei-Chang Yeh is with Department of Industrial Engineering, Feng Chia University, P.O. Box 67-100, Taichung, Taiwan 408 (e-mail:
[email protected]).
B. Notation G(V, A) a directed network with arc set E and node set V={1, 2, ¼, n}, where nodes 1 and n are the specified source node and sink node, respectively. p, c the number of nodes in the shortest path and min-cut between nodes 1 and n of G=(V, A), respectively. pi the reliability of node i, where i=1, 2,¼, n. Pi the probability that nodes 1 and n are still connected after removing i nodes, where i=1, 2,¼, n. m the total number of independent replications. mi, mi* the total number of independent replications of removing i nodes with its corresponding arcs in G=(V, A), and the number for nodes 1 and n are still connected in mi independent replications, respectively. |·| the number of elements in ·. A(·) the arc set {arc (i,j)| i, jηÍV}ÍE. R, RH, RC the exact reliability, the estimator of R obtained from HMC and CMC, respectively. Ri, R i the total and average (exact) probability of nodes 1 and n are connected in G(V-S, A-A(S)) for all node ù, set S with |S|=i, i.e. Ri= é( å ê Õ p k ) × Õ (1 - p k )ú S ÍV ë kÎS kÏS û nodes 1 to n are connected in G(V-S, A-A(S)), where |S|=i=0,1,¼, n.
Acronyms HMC: Hybrid Monte Carlo method. CMC: Crude Monte Carlo method. MC/MP:Minimal pathset/cutset. n recent years, network reliability theory has been applied extensively in many real-world systems such as oil/gas production system , computer and communication systems, power transmission and distribution systems, transportation systems, etc [1-3]. It is recommended to be measured and evaluated the performances of the systems which can be modeled as stochastic networks or into fault trees first. However, the exact computation of system reliability for a stochastic network is NP-hard [4]. Generally in system reliability evaluation, all MPs/MCs of the system must be known in advance [4]. However, both the problems in finding all MPs/MCs and computing the system reliability in terms of the known MPs/MCs are NP-hard [4]. To reduce the computational burdens, the simulation method was introduced to estimate the system reliability of a stochastic network [5-10]. However, most of the existing simulation methods assume that MPs/MCs have been found previously [5-10]. Even for a smaller sized problem, it is still impossible to evaluate the reliability of a system within a reasonable time using their algorithm. The need for an efficient and intuitive method thus arises. Hence the need to rely on simulations to improve the calculation efficiency without knowing the MP/MC in advance arises. The purpose of this study was to develop a Monde Carlo Simulation based on some simple concepts that were found in
ICITA2002 ISBN: 1-86467-114-9
RiH, Ria
H
a
RiH=Cin R i Pi and Ria=Cin R i Pi, where i=0,1, ¼, n.
RijH Rija
the estimator of R i in the jth simulation replication from HMC, where i=0, 1,¼, n and j=1, 2,¼, mi. the mirror value (antithetic variate) of RijH, i.e. the probability of nodes 1 and n are connected in G(S, A(S)) in the jth simulation replication from HMC, where S is a node set with |S|=i=0, 1,¼, n, and j=1, 2,¼, mi.
C. Assumptions The limited-flow network satisfies the following assumptions: Each arc is perfectly reliable. Each node and the system are either operative or failed. All failure events are stochastically independent. No maintenance is considered. III. THE CRUDE MONTE CARLO METHOD The Monte Carlo Simulation method is a straightforward approach especially for the large complex system. As mentioned in [10], each proposed method should be compared with the crude Monte Carlo sampling to reflect its efficiency. Thus one crude Monte Carlo method has to be introduced first as follows: 0. Let j=1 and rC=0. 1. Let i=1 and Si=f. 2. If pu
