I'm not an IIS-with-QlikView specialist but this may be caused by IIS not being able to get in touch with the QlikView server. The Managelment Service can talk to QVS, but IIS most probably cannot. Wrong credentials? IIS executing without proper privileges?
My first suggestion would be to check the required steps for configuring IIS in the Server Reference Manual, and maybe execute them again.
I've been experiencing similar issues and have been worked with Qlik to resolve it, but nothing yet.
You'll probably want to check the Event logs in the QlikViewServer folder. My QVS service is restarting multiple time a day. Usually with a restart reason of internal inconsistency. Since the service restarts users only see the 'No Server' message sporadically.
You can also check the Windows Event Viewer under Windows Logs / Application. The type is 'error' and the event id = 300. I've added a trigger to email when this error occurs. This isn't fixing the problem, but at least we know when it's happening.
This started occurring after we updated our Server to QV 11.2 SR 12, and SR 13 has not improved the situation.
Your screen shot shows QVS running on 2 servers. Do you have separate QlikView licences for each server or a cluster licence?
If you run two QVS services with the same licence, I believe they can switch each other on and off at 10 minute intervals.