Saudi Arabia usually fires 2-5 U.S. Patriot missiles per ballistic missile. Each PAC-2/3 missile costs $2-3 million. I don't remember exact numbers, but I think KSA has an inventory of up to 1000-1500 missiles (a huge inventory).
Lets assume, conservatively, 2 patriots per missile, and 1500 inventory. 100 ballistic missiles fired by Yemeni Army into Saudi Arabia (to date, Yemeni Army has already fired more than this amount at Saudi Arabia) would require 200 patriot missiles - a full 13% of the Saudi inventory!
Also note that a large amount of this stockpile is much older PAC-2 missiles, and only a smaller proportion of the overall inventory is the newer PAC-3 missiles. In addition, Patriot systems do not offer 360 degree coverage - so each battery is limited in operational capability.
So we can say that in the case of full-scale conflict with Iran, for example, Saudi patriot missiles, would be of limited utility. If Saudis fire 5 interceptor missiles for each (more advanced) Iranian missile, their entire inventory will be depleted after 200-300 Iranian missiles. And that is without Iran directly targeting launchers and radars with anti-radar missiles/suicide drones and cruise missiles.
US would likely supply further missiles in case of such a crisis, but this would take time and be very expensive.
Most likely Iranian ballistic missile systems would be Fateh class (Fateh-110/330), Zolfaqar, Dezful and Qiam - these are said to cost from $0.2-0.4m and have already been produced in very large numbers (thousands). Even 2 patriot missiles for each Fateh class missile would cost $4m for a $0.2m missile - 20x the cost of the attacking missile! If Saudis fire 5 interceptor missiles, that is $10m+, more than 40x the cost for Iran. Although Saudi's wealth cannot be denied, this ratio is not sustainable for any conflict (but would still be less than accepting the economic/psychological damage of not intercepting the missiles, so still the logical path for Saudi Arabia).
Note that the above is just basic analysis of a hypothetical scenario and not something that I hope happens.