The pigtail gas tube is used on very short barrels to soften the pressure spike.
On a short barrel (10.5), the port is very large due to the barrel ending shortly after the port. This leaves very little time for the barrel gases to pressurize the gas tube before the pressure is lost out the barrel. The solution to this is to open up the barrel port to allow more gas to reach the carrier/bolt and unlock them. But, the problem that you run into on a short barrels with a large port is the ejection pressure is very violent, due to the large port size, and the case is still pressure bound to the chamber side walls causing hard extraction.
The way the pig tail works, due to the laws of physics, is once a pressure wave starts, it will continue in a single direction, which means the tube will stay pressurized after the bullet has left the barrel (some back pressure leakage, but the tube is still pressurized at over 50,000 PSI to unlock the action), while soften the gas spike (due to the added length or the pig tail tube) to allow the bolt to unlock less violently (slows carrier speed in regards to cycle rate/bolt bounce) and allowing the barrel to vent pressure out of the muzzle to un-bind the spent case from the chamber walls for easier extraction of the case.
On both a 16" and a 20" barrel, the gas port is smaller with the muzzle being several inches from the port (allows the barrel enough time to pressurize the gas tube slower than a large port), and the bolt/carrier is not subject to the excessive/violent pressure of an over sized gas port, hence the pig tail is not needed on these rifles.
Hope this clears some thing up on the tube.