There are several utility programs for Tirex systems. Because they don't work out of the box and/or are insufficiently documented, they are not installed by default. You can find them in the utils directory. Be prepared to change the (Perl or Shell) code of those commands if you want to use them. See comments inside those scripts for more information. Feel free to improve those scripts and make them more flexible.
Used tirex-tiledir-check to create statistics for your tiledir and re-render the oldest tile. You might want to run this from cron. You have to change this script according to your needs.
Create statistics from job log created by the master.
Creates metatile age histograms from output of tirex-tiledir-check --list.
Creates heatmaps from output of tirex-tiledir-check --list.
View content of metatiles.
Creates statistics from tileserver apache logs.
Analyze PostGIS logs to find slowest queries.