Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-26T00:31:37.186Z Has data issue: false hasContentIssue false
  • English
  • Français

On Monte Carlo Estimates in Network Reliability

Published online by Cambridge University Press:  27 July 2009

M. Lomonosov
M. Lomonosov
Affiliation:
Department of Mathematics & Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, 84105, Israel, and ARTEMIS, IMAG, Univ. J. Fourier BP 53X, 38041 Grenoble cedex, France
Get access Rights & Permissions [Opens in a new window]

Abstract

The paper considers representations of network reliability measures as the mean value of a random variable defined on the trajectories of a certain Markov process and investigates utility of such formulae for Monte Carlo (MC) estimating. Such an MC estimator is called (ε,δ)-polynomial if its relative error is less than ε with probability >1 – δ, for any sample size equal to or greater than a polynomial of ε-1, δ-1, and the size of the network. One of the main results: The suggested MC estimator for the disconnectedness probability of a multiterminal network is (ε,δ)-polynomial, under a certain natural condition on the edge failure probabilities. The method applies also to estimating the percolation critical point and certain equilibrium characteristics of renewal networks.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 8 , Issue 2 , April 1994 , pp. 245 - 264
Copyright
Copyright © Cambridge University Press 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Bixby, R.E. (1975). The minimum number of edges and vertices in a graph with edge-connectivity n and m n-bonds. Networks 5: 253–298.CrossRefGoogle Scholar
2.Bondy, J.A. & Murty, U.R.S. (1976). Graph theory with applications. New York: American Elsevier.CrossRefGoogle Scholar
3.Colbourn, C.J. (1987). The combinatorics of network reliability. New York and Oxford: Oxford University Press.Google Scholar
4.Dinitz, E.A., Karzanov, A.V., & Lomonosov, M.V. (1976). A structure of all minimum cuts of a graph. In Issledovaniya po diskretnoy optimizatsii. Moscow: Nauka, pp. 290–306 (in Russian).Google Scholar
5.Elperin, T., Gertsbakh, I., & Lomonosov, M. (1991). Estimation of network reliability using graph evolution models. IEEE Transactions on Reliability 40(5): 572–581.CrossRefGoogle Scholar
6.Elperin, T., Gertsbakh, I., & Lomonosov, M. (1992). An evolution model for Monte Carlo estimation of equilibrium network renewal parameters. Probability in the Engineering and Informational Sciences 6: 457–469.CrossRefGoogle Scholar
7.Fishman, G.S. (1987). A Monte Carlo sampling plan for estimating reliability parameters and related functions. Networks 17: 169–186.CrossRefGoogle Scholar
8.Grimmett, G. (1989). Percolation. New York: Springer-Verlag.CrossRefGoogle Scholar
9.Karger, D.R. (1992, 11). Global min-cuts in RNC, and other ramifications of a simple min-cut algorithm. A manuscript, Department of Computer Science, Stanford University.Google Scholar
10.Keilson, J. (1979). Markov chain models—Rarity and exponentiality. New York: Springer-Verlag.CrossRefGoogle Scholar
11.Lomonosov, M.V. (1974). Bernoulli scheme with closure. Problems of Information Transmission 10: 73–81.Google Scholar
12.Lomonosov, M.V. & Polessky, V.P. (1971). An upper bound for the reliability of information networks. Problems of Information Transmission 7: 337–339.Google Scholar
13.Lomonosov, M.V. & Polessky, V.P. (1972). Lower bound of network reliability. Problems of Information Transmission 8: 118–123.Google Scholar
14.Lomonosov, M.V. & Polessky, V.P. (1972). Maximum of the connectivity probability. Problems of Information Transmission 8: 324–328.Google Scholar
15.Nash-Williams, C.St. J.A. (1961). Edge-disjoint spanning trees of finite graphs. Journal of the London Mathematical Society 36: 445–450.CrossRefGoogle Scholar
16.Polessky, V.P. (1990). Bounds for connectedness probability of a random graph. Problemy Peredatci Informatsii 26(1): 90–98.Google Scholar
17.Stepanov, V.E. (1969). Combinatorial algebra and random graphs. Theory of Probability and Its Applications 14: 373–399.CrossRefGoogle Scholar