In this paper we study small hammock networks, more precisely the 29 hammock networks presented by Moore and Shannon in their prescient paper from 1956, where this type of network was introduced. We will first of all review the concept of the reliability polynomial, and define hammock networks, emphasizing some of their properties. We will then review the state-of-the-art from three different angles: theoretical, algorithmic, and design oriented. These will show that results obtained over many years advocate very strongly for small hammock networks. Because array-based designs (e.g., FinFETs) seem to be a perfect fit for small hammocks (as are nanoelectromecanical systems, or NEMS), we have decided to start investigating them closely. The analyses we pursue here are exact, the coefficients of the reliability polynomials being determined using our own program. The reliability polynomials for nontrivial hammock networks are reported here as functions of the probability (p) that the device is closed. As far as we know, this is the first time they have appeared in this form. Our purpose in determining exact results is a practical one, as we foresee that small hammock networks could play a key role in any arraybased design. That is why their reliability enhancements are evaluated thoroughly, using both classical and non-standard cost functions. Finally, conclusions are followed by a longer list of future directions of research, some of them practical while others more theoretical.

After a comprehensive review of reliability bounds for consecutive- k-out-of- n: F systems with statistically independent components having the same failure probability q (i.i.d. components), we introduce new classes of lower and upper bounds. Our approach is different from previous ones, and relies on alternating summation as well as on the monotony of some particular sequences of real numbers. The starting point is represented by the original formula given by de Moivre; and, by considering partial sums approximating it, new lower and upper bounds on the reliability of a consecutive- k-out-of- n: F system have been established. Simulation results show that all the lower and upper bounds considered here present very similar behaviors, as all of them are exponentially closing in on the exact reliability of a consecutive- k-out-of- n: F system. Additionally, the accuracy of the different bounds depends not only on the particular values of k and n, but also on the particular range of q (some of the new bounds being the most accurate ones over certain ranges).

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.

This paper deals with reliability schemes, and in particular with hammock networks. Inspired by the regular structure of the cytoskeleton inside axons we present one way of extending the classical (2D) hammock networks to 3D. As an introductory example, we consider and analyze here the smallest 3D hammock network. This is compared against equivalent 2D hammocks as well as straightforward cylindrical (2.5D) versions of the hammocks. The simulation results rely on very fresh upper and lower bounds on the associated reliability polynomials. These reveal that 3D hammocks are significantly improving upon 2D and 2.5D hammocks, being an enabling reliability scheme.