WordPress Sites goes becomes unreachable randomly but not it's subdomain

Hello,
I have a site. My https://www.mysite.com. For the last month or so I would notice that the site becomes unreachable. This has occurred at least 4 times.

You type in the address and the only feedback is “Waiting for server” . Sometimes, if I’m lucky I would get a 550 error. Otherwise, regardless of browser, the site just hangs there “Waiting for server”.

The only way to fix it is a hard reboot of the server and then after 20 minutes the site *may* be accessible again. Not all reboots are successful.

I also have a subdomain which is a copy of mysite.com which is used for testing: http://testsite.mysite.com. During an outage, my test subdomain has always been active. Furthermore, I have put a simple test file at webroot/mysite/test.txt. During an outage https://www.mysite.com/test.txt is actually always accessible so I’m guessing there is something funky going on with wordpress itself and not my server.

My server is running Windows Server 2k7 with IIS 8. The fact that a file on the main domain is accessible and that my subdomain is accessible during an outage truly confounds me. Any idea what cause this behavior?

Any and all methods of troubleshooting would be appreciated: Logs to view, tests to preform, possible causes or solutions to the problem, conspiracy theories or crystal balls are welcome.

Note: The site is currently up and I can bring it up by a hard reboot but this problem isn’t a one time thing and isn’t going away.