When choosing compressor you have to consider if it can constantly provide desired pressure and volume of air. Little airbrush compressors little compressors boast amazing pressure figures but actually poor flow rates enough only for airbrush pistols. Also if small compressor would be able to push enough air it will likely overheat in couple of minutes.
Our laser cutter came with little membrane pump (Smallest upon the picture), I do not know what specifications it has but stream of air was extremely weak. It is adequate if all you want to do is engrave, although also then we got substantial smoke traces around engraved picture.
it was rated for volume of ~2.4cfm and had a 1 gallon tank which may be filled to 58 PSI. It managed to give us constant 1 bar (14 PSI) of pressure, but it had to run constantly and overheat badly within five minutes.
Third and current compressor we got was 6cfm rated belt driven device with 13 gallon tank which is filled to ~100psi. With this unit we are able to run at continuous ~2bar (31psi) without concern with 
overheating since compressor engine is not on all the time. Using reserve surroundings in tank we can also press up to 3.5bar (50psi) for brief jobs.
The best setup in my opinion will be compressor rated to >6cfm, huge 70+ Air Compressor For Laser Cutting gallons tank rated to 10bar (140psi) and software air control via solenoid valve.