fax modem/fax machine/phone line issue

Hi - I work for a doctors office that has a fax machine, and are looking to send faxes from a computer as well, using the same fax line. So, I purchased a usb fax modem for the PC, and a 3 port phone line splitter for the incoming phone line. I then connected the line from the usb fax modem, and the fax machine to the splitter. What happens is, i can send a fax from the fax machine (I get a dial tone just fine), but then when I go to fax from the USB modem, it hangs and then says "the line is busy" (or something to that effect) - Ill then go back to the fax machine, thinking that when i press the "dial tone" button on the machine that its going to have NO dial tone, there is a dial tone! Then about 15 minutes later, i am successfully able to send a fax from the USB modem, so at least i know its working (somewhat) Its very sporadic, and I was only able to send 2 faxes from the USB modem over the course of an hour. It seems what is happening is, that when i send a fax from the USB modem, its not releasing the phone line from that transmission, but I'm really not sure. I'm wondering if anyone has seen this before, any help would be appreciated!
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Is your fax machine a multifunction?  If it is you don't need the fax modem.  You can send faxes from your computer through the fax machine if they are connected and the multifunction's software is installed.  

If that is not the case then your fax modem may be defective.

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Have you tried to connect just fax modem to phone line directly (without splitter)?
Does it behave correctly?
If it does not work also in that way I would blame modem (or fax modem sw).
Fax machines normally have two connectors - incoming phone line and connector to connect phone device. Have you tried to connect devices in line?
phone line->Line connector on fax machine and phone connector on fax machine->fax modem line connector
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