You might want to configure something like blackfire (Blackfire.io: Continuous Performance Testing) on a staging or local box and simulate load with something like apache bench (ab - Apache HTTP server benchmarking tool - Apache HTTP Server Version 2.4) or siege (GitHub - JoeDog/siege: Siege is an http load tester and benchmarking utility).
Figure out what in your code is causing issues and resolve it. It is almost certainly some blocking IO process like file or db access, but you must isolate the problem somehow first before you can resolve the issue. Short of having a gut instinct or knowing of a specific release the issue started after this is where I would start.
Edit: Do you find it odd that this started right after hardware upgrades (or maybe this is a vps or something idk)?