When choosing compressor you have to consider if it will be able to constantly provide desired pressure and level of air. Small airbrush compressors small compressors boast amazing pressure figures but actually poor flow rates adequate only for airbrush pistols. Also if small compressor can push enough air it’ll likely overheat in couple of minutes.
Our laser cutter was included with small membrane pump (Smallest on the picture), I really do not know very well what specifications it has but stream of air was extremely weak. It is adequate if all you want to accomplish is engrave, although also then we got considerable smoke traces around engraved picture.
it had been rated for volume of ~2.4cfm and had a 1 gallon tank which may be Air Compressor For Laser Cutting china filled to 58 PSI. It managed to give us constant 1 bar (14 PSI) of pressure, nonetheless it had to run on a regular basis and overheat badly within 5 minutes.
Third and current compressor we got was 6cfm rated belt driven device with 13 gallon container which is filled to ~100psi. With this unit we are able to run at continuous ~2bar (31psi) without fear of overheating since compressor engine isn’t on on a regular basis. Using reserve air in tank we are able to also push up to 3.5bar (50psi) for short jobs.
The best setup in my opinion would be compressor rated to >6cfm, huge 70+ gallons tank rated to 10bar (140psi) and software air control via 
solenoid valve.