Car won't "randomly" start

I have a 2000 Saab 9-5 wagon. We bought it used and we seem to be having odd issues with it for about 2 years now. The latest issue has yet to be determined. We have had it in to the mechanic at least 3 times without success.

What is happening is that from time to time, I go to start the car and the power goes on (lights, radio, windows, etc) but the car makes absolutely no response. It does not even try to turn over. We just replaced the battery but I don’t believe that it’s related to that. The only thing that I have a thought on is that from time to time on the car’s computer, it shows the error “Service Theft Alarm”. I don’t know if the car not starting is related to that or not. I’ve read that there are batteries in the theft device that need replacing but I’m not sure if that would be the issue. When this first started happening, I could wait about 5 minutes and then it would start. Now, it takes about 20 to 30 minutes to wait before it will start. Any advice is GREATLY appreciated!