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Corki's missiles have an ammo system which refills on a timer. When Corki uses Missile Barrage, the ammo decrements one, AND resets the cooldown timer for getting another missile.
This means, for example, that if Corki has 7 missiles ready, fires one, he has 6 missiles, and his next missile will be available in 12/10/8 seconds. If he waits 4s, then fires another missile, he has 5 missiles, and will have his next missile available in 12/10/8 seconds.
This can be tested by creating a custom game as Corki, getting to level 6, and firing a missile, while observing the cooldown timer.
Presume intended behavior is similar to other ammo systems (Heimer, Zyra, Teemo, Vi) where using the ability doesn't cause the timer to reset. In the above example, Corki should have 5 missiles and the next available in 8/6/4 seconds.
Hopefully this is just an oversight and won't be too much work to fix. It seriously impacts his missile availability, which is a huge part of his damage and harass.