Combining monte carlo estimates and bounds for network reliability

A simplified model of a communications network is a probabilistic graph in which each edge operates with the same probability. The all-terminal reliability, or probability that all nodes are connected, can be expressed as a polynomial in the edge operation probability. The coefficients of this polynomial are obtained from an interval partitioning of the cographic matroid, and only the first few coefficients can be computed efficiently. One of the best sets of efficiently computable reliability bounds is the Ball-Provan bounds. These bounds are obtained using the efficiently computable coefficients and can be improved substantially if additional coefficients are known. In this paper, we develop a Monte Carlo method for estimating additional coefficients by randomly sampling over spanning trees of the network. Confidence intervals for all-terminal reliability are obtained by using these estimates as additional constraints in the Ball-Provan bounds. This approach has some advantages over conventional Monte Carlo point estimate methods. In particular, the computational complexity does not depend on the reliability of the network.

[1]  Charles J. Colbourn,et al.  Matroid Steiner problems, the Tutte polynomial and network reliability , 1989, J. Comb. Theory, Ser. B.

[2]  J. S. Provan,et al.  Bounds on the reliability polynomial for shellable independence systems , 1982 .

[3]  George L. Nemhauser,et al.  Matroids and a Reliability Analysis Problem , 1979, Math. Oper. Res..

[4]  Charles J. Colbourn,et al.  Counting almost minimum cutsets with reliability applications , 1987, Math. Program..

[5]  R. Kevin Wood.  A factoring algorithm using polygon-to-chain reductions for computing K-terminal network reliability , 1985, Networks.

[6]  L. A. Goodman.  On Simultaneous Confidence Intervals for Multinomial Proportions , 1965 .

[7]  J. Scott Provan,et al.  The Complexity of Counting Cuts and of Computing the Probability that a Graph is Connected , 1983, SIAM J. Comput..

[8]  W. T. Tutte,et al.  The Dissection of Rectangles Into Squares , 1940 .

[9]  George S. Fishman.  A Comparison of Four Monte Carlo Methods for Estimating the Probability of s-t Connectedness , 1986, IEEE Transactions on Reliability.

[10]  Hiromitsu Kumamoto,et al.  Efficient Evaluation of System Reliability by Monte Carlo Method , 1977, IEEE Transactions on Reliability.

[11]  J. Scott Provan,et al.  Calculating bounds on reachability and connectedness in stochastic networks , 1983, Networks.

[12]  Charles J. Colbourn,et al.  Unranking and Ranking Spanning Trees of a Graph , 1989, J. Algorithms.

[13]  Andrei Z. Broder,et al.  Generating random spanning trees , 1989, 30th Annual Symposium on Foundations of Computer Science.

[14]  Alain Guénoche.  Random Spanning Tree , 1983, J. Algorithms.

[15]  George S. Fishman,et al.  A Monte Carlo Sampling Plan for Estimating Network Reliability , 1984, Oper. Res..

[16]  R. Stanley.  Cohen-Macaulay Complexes , 1977 .

[17]  Richard M. Karp,et al.  Monte-Carlo algorithms for enumeration and reliability problems , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[18]  J. Scott Provan,et al.  Computing Network Reliability in Time Polynomial in the Number of Cuts , 1984, Oper. Res..

[19]  Charles J. Colbourn,et al.  The Combinatorics of Network Reliability , 1987 .

[20]  G. Kirchhoff.  Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird , 1847 .

[21]  J. Scott Provan,et al.  Decompositions of Simplicial Complexes Related to Diameters of Convex Polyhedra , 1980, Math. Oper. Res..

[22]  J. I. Brown,et al.  A Combinatorial Study of the Reliability Plynomial , 1986 .

[23]  D. C. Hurst,et al.  Large Sample Simultaneous Confidence Intervals for Multinomial Proportions , 1964 .

[24]  Richard M. Van Slyke,et al.  Network reliability analysis: Part I , 1971, Networks.