I concur. Pound for pound, number on numbers, the Romans cannot possibly outspam the Persians. But they don't need to.

With such a vast empire to protect and defend, it is impossible to throw everything at the Romaioi. All the Romaioi need to do is spam enough to make the Persikoi believe that they're not worth the man-spam to crush. The persikoi will withdraw, and the Romaioi will 'win'.

As for Alexandros' army against archers, practically everything the phalangitai uses have a precautionary purpose against archers (look at the previous posts). it's no surprise that the arrow rain would have minimal effect on them. That being said, it is Alexandros' genius and his professionally integrated army that won his empire. The Makedones simply do not have even the fraction of resource and manpower to man-spam the persians in an invasion.