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