How to get info if any Monitoring tool (DB performance Analyzer) stops working?
You are encountering THREADPOOL waits, which occurs when the instance has run out of available worker threads. New logins will be denied, and lots of queries will be just waiting.

To troubelshoot THREADPOOL waits, connect with the Remote DAC and check current activity. You'll likely see huge blocking chains. You'll need to resolve or reduce the blocking to resolve the THREADPOOL waits.

Some Microsoft engineers will say to increase worker threads on the instance, but that doesn't help in my experience. It just means it'll take longer before THREADPOOL is encountered.

Things that might resolve THREADPOOL waits:


Adjusting Max Degree of Parallelism and Cost Threshold for Parallelism so that less queries go parallel and thus fewer worker threads are needed
Adding indexes
Refactoring queries
RCSI
etc


For more information on THREADPOOL waits, check out these links (disclaimer: I work for Brent Ozar Unlimited):

https://www.brentozar.com/archive/2014/05/connections-slow-sql-server-threadpool/
https://www.sqlskills.com/help/waits/threadpool/
http://www.sqlpassion.at/archive/2011/10/25/troubleshooting-threadpool-waits/