You write "Scaling an application is an arithmetic exercise. If one user consumes X amount of CPU time on the web server, how many web servers do you need to support 100,000 simultaneous users?"
And it is arithmetic (linear), until it isn't.
When you hit a saturation point, the response time stops growing linearly with load, and increases insanely. The response time heads skyward like a homesick angel, and even the sysadmin thinks it's hung. If you plot it, you get a curve that looks a lot like a hockey-stick, with a short horizontal blade, a nice gentle bend... and a long straight handle that just keeps going up.
You add linear amounts of resources to get it back under control, of course, so the cure is arithmetic. The response time and the customer response, however, are hyperbolic (;-))
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