To add information on this, after running for several days we will have issues connecting to the database, when looking at the ArcGIS Monitor Server, only (5) of the arcgis-monitor.exe services are running. I checked with the database admin and there were 236 sessions running for the database account. At this point we have to kill all of the connections to get Monitor Server to connect back to the postgres database.
Also, ArcGIS agent will go from (4) arcgis.monitor.exe services to (3) and will no longer register any metrics with the server. This normally just requires restarting the agent service, but I would like to have some better clarity on what is happening with the connections if possible.