Appendix IV : Null Word Link Distribution
The graph above shows side by side distributions, each showing the number
of links that fall within the labeled ranges for null words transmitted (i.e.
links with wasted cycles since no data was available to transmit). It stands to
reason that a cost effective graph under random traffic will use all of its
links in a near equal fashion, rather than letting certain links idle while
other links bottleneck. This is perfectly illustrated by the hypercube(11), a
graph well noted for its balance. As shown, the hypercube(11) has very few
wasted cycles, only a small number of links occur in the > 2k null words
range, and these can be attributed to random deviation. The MDSXN graphs,
however, each illustrate links which are idling away while other links remain
busy. Note that the 2nd ridge on the right (occurring at approximately 3.6k)
is growing exponentially as the deBruijn cycles of the MDSXN graphs lengthen,
and more links on the outer edge of the cycle idle.
The first ridge on the right of the MDSXN distributions
(occurring at approximately 4.5k) can be ignored because this is due to
non-existent links which are only included in the simulator data to
maintain symmetry (e.g. in a MDSXN[5,6] node 0b00000 XXXXXX would have four
symmetric links: ->0b00000 XXXXXX, ->0b00001 XXXXXX, ->0b00000 XXXXX0,
and ->0b00000 XXXXX1, and at least the first one is a non-existent link,
since it points to itself).
The deBruijn(11) topology, which would logically appear above the MDSXN[5,6]
distribution line in the above plot, has been omitted from this graph since the
deBruijn(11) was severely bottlenecked and thus required 1.65k cycles to
complete as opposed to the 1.15k cycles required by the others, and thus the
number of null words per link would have to be scaled proportionately, and
II have not yet attempted to do this.
This graph was produced using my xmesh tool which read the simulation data,
created a 3d distribution, then plotted it using the plplot utility. Here are
the commands input to xmesh which
produced this graph, which was later relabeled and colored.