How Reliable are Compositions of Series and Parallel Networks Compared with Hammocks?



Two-terminal network, series and parallel network, composition, reliability polynomial.


A classical problem in computer/network reliability is that of identifying simple, regular and repetitive building blocks (motifs) which yield reliability enhancements at the system-level. Over time, this apparently simple problem has been addressed by various increasingly complex methods. The earliest and simplest solutions are series and parallel structures. These were followed by majority voting and related schemes. For the most recent solutions, which are also the most involved (e.g., those based on Harary and circulant graphs), optimal reliability has been proven under particular conditions. Here, we propose an alternate approach for designing reliable systems as repetitive compositions of the simplest possible structures. More precisely, our two motifs (basic building blocks) are: two devices in series, and two devices in parallel. Therefore, for a given number of devices (which is a power of two) we build all the possible compositions of series and parallel networks of two devices. For all of the resulting twoterminal networks, we compute exactly the reliability polynomials, and then compare them with those of size-equivalent hammock networks. The results show that compositions of the two simplest motifs are not able to surpass size-equivalent hammock networks in terms of reliability. Still, the algorithm for computing the reliability polynomials of such compositions is linear (extremely effcient), as opposed to the one for the size-equivalent hammock networks, which is exponential. Interestingly, a few of the compositions come extremely close to size-equivalent hammock networks with respect to reliability, while having fewer wires.a


Beiu, V.; Cowell, S.R.; Dr goi, V.; Hoar , S.; Gaspar, P. (2018); Hammocks versus hammock, 2018 7th International Conference on Computers Communications and Control (IC- CCC), Proc. of, Oradea, Romania, May 2018, Publisher: IEEE, 119-123, 2018. https://10.1109/ICCCC.2018.8390447

Beiu, V.; L. D us, L.; Rohatinovici, N.-C.; B las, V. E. (2018); Transport reliability on axonal cytoskeleton, Proc. Intl. Conf. Eng. Modern Electr. Syst. (EMES), Oradea, Romania, Jun. 2017, 160-163, 2017.

Ball, M.O.; Colbourn, C.J.; Provan, J.S. (1992); Network reliability, Tech. Rep. TR 92-74, Systems Research Center/ Institute for System Research, University of Maryland, College Park, MD, USA, June 1992.

Barlow, R. E.; Proschan, F. (1965); Mathematical Theory of Reliability, John Wiley & Sons, New York, NY, 1965.

Colbourn, C.J. (1991); Combinatorial aspects of network reliability, Annals of Operations Research, 33(1), 3 - 15, Jan. 1991.

Courtland, R. (2016); The next high-performance transistor, IEEE Spectr., 53(10), 11-12, Oct. 2016.

Cowell, S.R.; Beiu, V.; D us, L.; Poulin, P. (2018); On the exact reliability enhancements of small hammock networks, IEEE Access, 6(1), 25411-25426, Apr. 2018. [Early version as "On hammock networks - Sixty years after", Proc. Intl. Conf. Design & Technol. Integr. Syst. Nanoscale Era (DTIS), Palma de Mallorca, Spain, Apr. 2017, art. 7929871]

Cowell, S.R.; Beiu, V.; D us, L.; Poulin, P. (2017); On cylindrical hammock networks, Proc. Intl. Conf. Nanotech. (IEEE-NANO), Pittsburgh, PA, USA, Jul. 2017, 185-188, 2017.

Deng, H.; Chen, J.; Q. Li,Q.; Li,R.; Gao, Q. (2004); On the construction of most reliable networks, Discr. Appl. Maths., 140(1-3), 19-33, 2004.

Dragoi, V.; Cowell, S.R.; Hoar , S.; Gaspar, P.; Beiu, V. (2018); Can series and parallel compositions improve on hammocks?, Proc. of 2018 7th International Conference on Com- puters Communications and Control (ICCCC), Oradea, Romania, May 2018, Publisher: IEEE, 124-130, 2018. https://10.1109/ICCCC.2018.8390448

Duffin, R.J. (1965); Topology of series-parallel networks, Journal of Mathematical Analysis and Applications, 10(2), 303-318, 1965.

Geppert, L. (2002); The amazing vanishing transistor act, IEEE Spectr., 239(10), 8-33, 2002.

IEEE Rebooting Computing,

International Roadmap for Devices and Systems, (IRDS), 2017 [Online]. Available:

Klaschka, T.F. (1967); Two contributions to redundancy theory, Proc. Annual Symposium on Switching and Automata Theory (SWAT), Austin, TX, USA, Oct. 1967, 175-183, 1967.

Klaschka, T.F. (1971); A method for redundancy scheme performance assessment, IEEE Transactions on Computers, C-20(11), 1371-1376, 1971.

Kuo, W.; Zuo, M.J. (2003); Optimal Reliability Modeling: Principles and Applications, J. Wiley & Sons, Hoboken, NJ, USA, 2003.

Lee, C.Y. (1955); Analysis of switching networks, Bell System Technical Journal, 34(6), 1287-1315, 1955.

Li, Q.; Li, Q. (1998); Reliability analysis of circulant graphs, Networks, 31(2), 61-65, Mar. 1998.

Moore, E.F.; Shannon, C.E. (1956); Reliable circuits using less reliable relays - Part I, J. Frankl. Inst., 262(3), 191-208, 1956.

Moore, E.F.; Shannon, C.E. (1956); Reliable circuits using less reliable relays - Part II, J. Frankl. Inst., 262(4), 281-297, 1956.

von Neumann, J. (1952, 1956); Probabilistic logics and the synthesis of reliable organisms from unreliable components, Jan. 1952 . Also in C. E. Shannon, and J. McCarthy (Eds.): Automata Studies, Princeton Univ. Press, Princeton, NJ, USA, 43-98, Apr. 1956.

Theis, T.N.; Wong, H.-S.P. (2017); The end of Moore's law: A new beginning for information technology, Comput. Sci. & Eng., 19(2), 41-50, 2017.

Wald, J.A.; Colbourn, C.J. (1983); Steiner trees, partial 2-trees, and minimum IFI networks, Networks, vol. 13, no. 2, pp. 13(2), 159-167, 1983.

Weichenberg, G.E.; Chan, V.W.S.; Medard, M. (2004); High-reliability topological architectures for networks under stress, IEEE J. Sel. Areas Comm., 22(9), 1830-1845, 2004.

Williams, R.S. (2017); What's next?, Comput. Sci. & Eng., 19(2), 7-13, 2017.



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