Full Hermite interpolation of the reliability of a hammock network

Abstract

Although the hammock networks were introduced more than sixty years ago, there is no general formula of the associated reliability polynomial. Using the full Hermite interpolation polynomial, we propose an approximation for the reliability polynomial of a hammock network of arbitrary size. In the second part of the paper, we provide combinatorial formulas for the first two non-zero coefficients of the reliability polynomial. * Corresponding author. Leonard Dȃuş 2010 Mathematics Subject Classification. 05C31, 41A05, 68Rxx.

Markov Chain Method for Computing the Reliability of Hammock Networks

“…In spite of the regular form of hammock networks, no efficient method to calculate their reliability polynomial has been developed yet. Several approximation methods have been devised instead, using either interpolation (for example, the Hermite interpolation applied in [16]), or an equivalent statistical distribution (like in the recent approach of Cowell et al [14]).…”

“…With the advent of fin field-effect transistors (FinFETs) a decade ago and of various quantum chips lately, the last few years have witness a growing interest on hammocks, not only from a theoretical point of view (see [15,16,17,39]), but also because they are a perfect fit for both FinFETs (in fact any array-based transistors) [13], and 2D arrays of qubits, like Google's Bristlecone [1,5,37] . Besides, extensions of hammocks to 3D have also been suggested and analyzed [12], explaining the high reliability of transport on the axonal cytoskeleton while challenging consecutive systems [3].…”

“…Drȃgoi et al [17] have compared compositions of series and parallel networks of two devices to size-equivalent hammock networks, showing that compositions of series and parallel networks are not able to surpass hammock networks in terms of reliability. Dȃuş and Jianu [16] have calculated the first and second non-zero coefficients of the reliability polynomial and have approximated the reliability polynomial by full Hermite interpolation. Recently, Cowell et al [14] have used the beta distribution to approximate the reliability polynomial.…”

Generalized Convexity Properties and Shape-Based Approximation in Networks Reliability

“…In particular, duality is a characteristic that induces complementary properties on the coefficients, which are considered in the approximations. In [8], Hermite interpolation is used for hammock networks based on previous results on the shape [9]. Cubic splines are proposed in [6,7] and are suitable for any two-terminal networks.…”

Generalized Convexity Properties and Shape Based Approximation in Networks Reliability

“…In particular, duality is a characteristic that induces complementary properties on the coefficients, which are considered in the approximations. In [10] Hermite interpolation is used for hammock networks based on previous results on the shape [11]. Cubic splines are proposed in [4,6], and are suitable for any two-terminal networks.…”