Fractional approximation and queuing

That is the connection between optimally congested queues and packing a bit error space. A fractional approximation needs round off space to maintain information, the golden ratio can use it most effectively, hence its appearance. That ratio appears when tanh'' is maximum, and the hyperbolic inverse is a the closest approximation to Markov triples. Hypebolics is all about summing up a collection of overlapping queues if one looks at the derivation, it is about combinatorics. The hyperbolic line is most elastic when there are the most number of available unique combinations.

So, there was missing theory, way back when I said it; there is a spectral theory a bit more expansive for fractional approximation, the best approximations have higher rank generators, smaller bit error, make minimal use of conserved bit error space.