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Evaluating Reliability of Stochastic Flow Networks

Published online by Cambridge University Press:  27 July 2009

George S. Fishman  and
Tien-Yi Danny Shaw
George S. Fishman
Affiliation:
Department of Operations ResearchUniversity of North Carolina Chapel Hill, North Carolina 27599
Tien-Yi Danny Shaw
Affiliation:
Department of Operations ResearchUniversity of North Carolina Chapel Hill, North Carolina 27599
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Abstract

This paper describes a highly efficient Monte Carlo sampling plan for evaluating the probability that the flow value in a stochastic flow network is greater than or equal to a prespecified level d. A stochastic flow network can characterize communication, transportation, and water or oil distribution systems. The paper first derives lower and upper bounds on the probability of interest and then describes how one can concentrate sampling in a specialized region of the arc capacity state space to increase the statistical efficiency of the resulting estimate. The paper also gives expressions for worst-case sample sizes needed to meet specified bounds on variances and coefficients of variation and illustrates the proposed sampling plan with an example.

Type
Articles
Information
Probability in the Engineering and Informational Sciences , Volume 3 , Issue 4 , October 1989 , pp. 493 - 509
Copyright
Copyright © Cambridge University Press 1989

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References

Bukowski, J.V. (1982). On the determination of large-scale system reliability. IEEE Transaclions on Systems, Man, and Cybernetics SMC 12: 538–548.Google Scholar
Doulliez, P. & Jamoulle, E. (1972). Transportation networks with random arc capacities. Revue Francaise d'Automatique, Informatique, Recherche Operationelle, Series V, Vol. 6, 1972.Google Scholar
Evans, J.R. (1976). Maximum flow in probabilistic graphs–the discrete case. Networks 6: 161–183.Google Scholar
Fishman, G.S. & Moore, L.R. (1984). Sampling from a discrete distribution while preserving monotonicity. The American Statistician 38: 219–223.CrossRefGoogle Scholar
Fishman, G.S. (1986). A Monte Carlo sampling plan for estimating network reliability. Operctions Research 34: 581–594.Google Scholar
Fishman, G.S. (1986). Monte Carlo estimation of the maximal flow distribution with discrete stochastic arc capacity levels. Naval Logistics Research Quarterly (to appear).Google Scholar
Garey, M.R. & Johnson, D.S. (1979). Computers and intractability: A guide to NP- completeness. New York: W.H. Freeman and Company.Google Scholar
Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. Journal American Statistical Association 58: 13–29.CrossRefGoogle Scholar
Itai, A.M & Shiloach, Y. (1979). Maximum flow in planar networks. SIAM Journal of Computers 8 (2): 135–150.CrossRefGoogle Scholar
Karzanov, A.V. (1974). Determining the maximal flow in a network by the method of preflows. Soviet Mathematics Doklady 15: 434–437.Google Scholar
Lee, S.H. (1980). Reliability in a flow network. IEEE Transactions on Reliability 29: 24–26.Google Scholar
Malhotra, V.M., Kumar, M.P. & Maheshwari, S.N. (1978). An O(V3) algorithm for finding maximum flows in networks. Information Processing Letters 7: 277–278.Google Scholar
Papadimitriou, C.H. & Steiglitz, K. (1982). Combinatorial optimization. Englewood Cliffs, New Jersey: Prentice Hall.Google Scholar
Walker, A.J. (1977). An efficient method for generating discrete random variables with general distributions. ACM Transactions on Mathematical Software 3: 253–256.CrossRefGoogle Scholar