Hummingbird minification – files stuck in "pending" state

When you enable minification on the site, files are getting stuck in “pending” state for very long. Intensively browsing a site does help a little but still with that speed it would take hours (if ever) to process them. The Page Caching has paused automatically while the files are in “pending” state but they cannot get out of it so, as a result, caching cannot be enabled too.