Abstract
AbstractIt is a well-known fact that, in general, the combinatorial problem of finding the reliability polynomial of a two-terminal network belongs to the class of \( \# P \)-complete problems. In particular, hammock (aka brick-wall) networks are particular two-terminal networks introduced by Moore and Shannon in 1956. Rather unexpectedly, hammock networks seem to be ubiquitous, spanning from biology (neural cytoskeleton) to quantum computing (layout of quantum gates). Because computing exactly the reliability of large hammock networks seems unlikely (even in the long term), the alternatives we are facing fall under approximation techniques using: (i) simpler ‘equivalent’ networks; (ii) lower and upper bounds; (iii) estimates of (some of) the coefficients; (iv) interpolation (e.g., Bézier, Hermite, Lagrange, splines, etc.); and (v) combinations of (some of) the approaches mentioned above. In this paper we shall advocate—for the first time ever—for an approximation based on an ‘equivalent’ statistical distribution. In particular, we shall argue that as counting (lattice paths) is at the heart of the problem of estimating reliability for such networks, the binomial distribution might be a (very) good starting point. As the number of alternatives (lattice paths) gets larger and larger, a continuous approximation like the normal distribution naturally comes to mind. Still, as the number of alternatives (lattice paths) becomes humongous very quickly, more accurate and flexible approximations might be needed. That is why we put forward the beta distribution (as it can match the binomial distribution), and we use it in conjunction with a few exact coefficients (which help fitting the tails) to approximate the reliability of hammock networks.KeywordsNetwork reliabilityHammock networksReliability polynomialApproximationsProbability distributions
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