It does not immediately grow towards infinity, but it is unbounded. Over a sufficiently long period of time, any given queue size will be observed.
Assuming the input and output are just a little bit random, and are only equal on average. You will have periods during which the input and output balances out. You will have periods during which the output is faster, and the queue will shrink to 0 (but no lower). And you will have periods where the output is slower, and the queue grows, with no bound.
Assuming the input and output are just a little bit random, and are only equal on average. You will have periods during which the input and output balances out. You will have periods during which the output is faster, and the queue will shrink to 0 (but no lower). And you will have periods where the output is slower, and the queue grows, with no bound.