One thing to know about, if you don't already, ArcGIS Server has a stats page you can access:
-> Login as admin
-> Logs
-> Statistics
Here, you can check for timed-out requests in addition to total requests, by service.
As far as your original question, you could roll your own, through something like a PowerShell or Python script... essentially, you'd create an http request every so often to evaluate responsiveness. You could send an e-mail if it were to time-out or become unavailable.
Something else to check-out:
System log Parser
https://www.arcgis.com/home/item.html?id=90134fb0f1c148a48c65319287dde2f7
System Monitor (contains link to new version)
http://www.arcgis.com/home/item.html?id=848f48b0f88e4de7a036377197453efe