There are a couple of things this could be.
First of all, inspecting cookies in Chrome tells me that osTicket appears to be using PHP sessions and not cookies, this means each time your computer is turned off or browser is completely closed the session is destroyed and you need to restart. You could write some code for a cookie to start a new PHP session but honestly I wouldn't bother plus this creates a security threat.
How often does your desktop computer get turned off vs your laptop?
Secondly, I don't know if osTicket does this (maybe a developer or someone more familiar with osTicket sessions can say), 1 system I built requires the admin to consistently connect to the application via the same IP address for the session, this meant if a mobile data connection changed the IP address the session was destroyed and the user logged out. This was designed like this to ensure admin access to the system was never compromised with travelling executives forgetting to log out of the system from their laptops.
I hope this gives some insight and helps... well at least with your sanity... oh, try saving your password in the browser so when ever you go to the site or need to re-authenticate all you need to do is click 1 button.
Cheers,