The time isn't arbitrary. The time and intensities are based on scientific studies, something you seem to not understand. Time and intensity best describes exercise, and the human bodies response to it, end of.
As a rule of thumb, most studies take exercise longer than 20 minutes at steady-state (defined as; activity that achieves a balance between the energy required by working muscles and the rate of oxygen and delivery for aerobic ATP production), to cause measurable cardio-vascular changes.
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3699221/
http://www.sciencedaily.com/releases/2013/02/130213173127.htm
http://integralcrossfit.com/2015/01/understanding-the-energy-systems/