It depends on what you mean. If it stopped working, like Mohamed said, it would be, "our computer system is down." If the system has been replaced with another system, you could possibly say it's expired, but only if it's meant to be funny because that's how it sounds. Otherwise I'd say "our old system has been replaced." Similarly, if the system has actually expired-expired, as in died and will not be functioning again, you could also say expired if you're trying to be funny. Otherwise, one more formal option is "our computer system died. We'll need another one." If the computer system you're referring to has a subscription it depends on in order for you to use it, and the time has elapsed, this would be a non-funny, completely legitimate reason to say "the computer system has expired," although to remove any confusion I'd say "the computer system subscription has expired."

If you want to say something like stopped working, you can say "our computer system is down"