Far Eastern Mathematical Journal

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Asymptotic formulas for a calculation of a lattice reliability

G. Sh. Tsitsiashvili

2010, issue 1, P. 86–90

Abstract

A calculation of a probability that there is a working way between lattice nodes has interesting physical applications. For a lattice with two columns of cells these calculations are suggested by Ch. Tanguy and are based on transform matrices. But when a number of columns increases a transform matrix dimension increases significantly also and it is difficult to use this method. So in this paper we suggest to solve the problem in cases when lattice arcs are low or high reliable. For this aim asymptotic formulas which estimate connection probabilities by the arc reliability and by integer parameters of the lattice are suggested. Algorithms to find parameters of suggested asymptotic formulas are constructed. These algorithms are based on geometric componentries.

Keywords: random lattice, ways and cross sections with minimal numbers of arcs

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References

[1] R. E. Barlow, F. Proschan, Mathematical Theory of Reliability, Wiley, London and New York, 1965. [2] I. A. Ushakov et al., Reliability of technical systems, Handbook, Radio and Communication, Moscow, 1985, 608 pp. (In Russian). [3] C. Tanguy, “What is the probability of connecting two points?”, J. Phys. A: Math. Theor., 40 (2007), 14099–14116. [4] C. Tanguy, “Asymptotic dependence of average failure rate and MTTF for a recursive meshed network architecture”, Reliability and risk analysis: theory and applications, 2(13), part 2 (2009), 45–54. [5] L. R. Ford, D. R. Fulkerson, Flows in networks, Princeton university press, Princeton, New Jersey, 1962. [6] V. V. Belov, E. M. Vorob'ev, V. E. Shatalov, Teoriya grafov, Uchebnoe posobie dlya vtuzov, Vysshaya shkola, M., 1976.

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