BTW, 36 boxes to handle even 40,000 requests per second seems like extreme overkill.
I strongly suspect that a decently spec'd, single box with one thread/listener per budget could handle that load. And by virtue of shared memory, greatly simplify the process along the way.
In an extremely crude analogy run on my 4-core, single socket machine, I've been able to service 60,000+ request-responses per second via UDP with average response times of the order you require.
If it were not too late to re-specify the hardware, I'd seriously consider using a single, 4-socket(32 or 40 cores with 64/80 hyperthreads) SMP shared memory server. It should easily handle the scale of loads and times you are describing.
With the rise and rise of 'Social' network sites: 'Computers are making people easier to use everyday'
Examine what is said, not who speaks -- Silence betokens consent -- Love the truth but pardon error.
"Science is about questioning the status quo. Questioning authority".
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