If you're running CF 10 or above, there was a very interesting post on the Adobe CF blog, from July 19 2015, entitled, Configuring Status Worker in Connectors. The Adobe blog post title may not have caught your attention, but it's about setting up a lightweight and built-in Tomcat monitoring feature for observing the status of the Tomcat web server connector.
You may want to consider enabling it, but I would add some caveats and observations that I share below. Note that it's really quite easy to enable, and DOES NOT require a restart of CF (only of your web server, or technically in IIS, the application pool/s) to take effect.
If you've not yet read their blog entry, go check it out and then come back here for several observations I have to share, some of which I think you'll agree could be very important. (BTW, if you don't follow that Adobe CF blog regularly, you really should. Often great content, and very little "noise".)