Hi Vitalie,
First steps would be to read through the guide on Cron & Daemons linked above and check that your configuration is as it should be.
You should have a file in /etc/cron.d/ for your Alaveteli install, and that file should have a line that calls script/update-xapian-index. That task will log to the update-xapian-index.log file in the application's log directory, which should give some output so you can see what the script is doing.
You can also run the task manually from the application's root directory to update the index. If you do this, and the requests appear on your site, then that would almost certainly mean that you haven't been automatically indexing the content via the cron task.
cd /path/to/your/alaveteli
script/update-xapian-index verbose=true
Best,
Gareth