Communication Model

The linear model - varying nr and cpr

First, a number of benchmark tests was dry-ran, with the following parameters:

• name: "test"
• num-cells-per-rank: cpr
• num-ranks: nr
• num-synapses: 1000
• duration: 100
• min-delay: 10
• spike-frequency: 100
• realtime-ratio: 0.1

The parameters nr and cpr take values from the sets (1, 4, 10, 40, 100, 400, 1000, 4000, 10000) and (100, 250, 1000, 2500, 10000), respectively, which yields a total of 9*5=45 tests. The times t all the programs spent in the walkspikes process were measured and a graph t(nr*cpr) was plotted. For visibility and distinguishability, logarithms were taken for both the input and output data. From this graph, the data sets corresponding to test runs with the same cpr were identified and the process was repeated for those with same nr, with the following results:

The preliminary goal was to fit all those lines linearly in order to create an appropriate function which would return the expected execution time, given cpr and nr as a final product. After fitting the mentioned data sets and analyzing the linear functions, two things have been observed:

• The slope of the lines has no visible correlation with cpr on nr (on appropriate graphs) - thus, it was concluded that the slope is constant (c1 for the same-cpr graph and r1 for the same-nr graph, both taken as the mean of the slopes), albeit subject to fluctuations.
• The intersections of the functions with the Y-axis seem to be somewhat equidistant - it was assumed that they are linear with log(cpr) and log(nr), appropriately.

In order to tackle Y-axis intersections, their coordinates were plotted against log(nr) (likewise for log(cpr)):

These plots provided another two pairs of constants, describing the linear functions used on the graphs:

• c21 for the new function's slope, same-cpr graph
• c22 for the new function's intersection with Y-axis, same-cpr graph
• r21 for the new function's slope, same-nr graph
• r22 for the new function's intersection with Y-axis, same-nr graph

Now the fits on the initial two graphs are parametrized - for given nr, cpr, and p corresponding to the X-axis value:

• The lines connecting measurements with same nr are fitted as: r1/p+r21log(nr)+r22
• The lines connecting measurements with same cpr are fitted as: c1/p+c21log(cr)+c22 The intersections of these parametrized functions are illustrated in the image below:

They provide us with the values of p corresponding to the given nr and cpr, which are then plugged in either of the parametrized functions in order to get the appropriate Y-axis value. This is the logarithm of the expected execution time, and thus the final expression (in Wolfram Mathematica) is:

These results were compared to the measured ones in the following graph, where blue dots represent the measured execution times, whereas the red ones represent the estimates:

Despite the two sets of points being clearly correlated, the error of the model increases when we go back to the original data values, instead of the logarithms. This is neatly summarized in a table:

The nonlinear model

In this model, our simulation is considered to be ran on only one rank (nr = 1) and time is modelled as a function of ns and cpr. At first, an approach similar to the previous model was followed, where data points were grouped by one of their arguments (slicing). Afterwards, a two-variable fit was computed based on the expected behaviour of the program, as well as conclusions from slicing.

[insert theoretical expectations]

The slicing plots are shown in the images below, taking only a fraction of all the data points for distinguishability.