kipp,
First of all you need to identify exactly what is happening and try to successfully recreate it; intermittent problems can take time to resolve. Here is how I would go about trying to identify your problem.
1. What type of number does this happen on local, long distance, or international numbers?
2. Try dialing the number slowly and exactly as it was dialed from the same extension; then try on a different extension.
3. If you know how then monitor the PRI via the command console and verify that the call is trying to leave the system; I doubt it is.
From what you already explained it sounds as though you are not getting out of the system and the "fast busy" is really the error tone from the phone system not the carrier; therefore not a "fast busy". I would think the person didn't dial the outgoing digit [Default: 8] or doesn't have access to that number type or needs an account code to accomplish it. Then after a period of time while remaining off-hook they get the incomplete extension as the system is now providing service order tone after the beeping; typically the beeping is associated with a message on the screen, but not always.
Thanks,
TE
