What you are looking for is Real user monitoring (I've indeed misread your question initially) - it is certainly possible to implement your own solutions via a combination of the various Open Source monitoring frameworks available, but I'm not aware of a readily available one; however, at least two (likely several more) commercial providers are offering just this and have free tiers available one way or another:
New Relic is offering active Real User Monitoring, which is an excellent tool to Get browser performance data directly from real end users and especially correlate these with other monitoring metrics for the ultimate insight in your operational and development optimization efforts:
See exactly what real usersʼ experiences are by monitoring
- Please note that New Relic is provided as a sponsored offering by many IaaS/PaaS providers in turn, and as of recently AWS customers can use New Relic Standard for free to manage the performance of their applications in the AWS cloud as well.
Cedexis is offering Cedexis Radar to capture the experience of your actual visitors:
Cedexis brings unrivaled transparency to the global performance and
availability of the world's largest public infrastructures, from Cloud
Computing and Storage to Content and Application Delivery Networks [...]
- Please note that while Cedexis offer some kind of free community edition as well, they don't publish their prices (yet?), which I consider increasingly at odds with nowadays commodity/utility computing and resource consumption models, see Alex Payne's How Not To Sell Software in 2012 for a nice summary and rant concerning this. Accordingly, I'm unable to judge whether you'll actually be able to derive the data you are looking for, though their approach most certainly allows this and the resulting Free Country Reports are pretty impressive indeed - here's how it is supposed to work:
into various pages or applications. When implemented on a popular web
property, Radar generates a very rich data set—potentially capturing
every user to a particular site/page/application—giving you a clear
picture of your global users' site experience.
Most interestingly, both offerings can meanwhile be integrated into Cedexis' other product Openmix, see Cedexis Adds New Relic Support To Its Real-Time Cloud Routing Platform for details.
As easily figured from my phrasing, I've used New Relic with great success for the use case at hand (and lots of others) and can certainly recommend it without hesitation, whereas I've only started to look into the Cedexis platform and am not able to judge their offering yet accordingly.
I have recently provided an answer to How could I determine which AWS location is best for serving customers from a particular region?, outlining two approaches easily available for Amazon EC2 specifically:
Please see the answer itself for more details on these two options (including pointers to background and additional information) - if you can't solve your problem right away via one of these approaches, the implied information might offer a good starting point at least.