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 adequate limited to airbrush pistols. Also if small compressor can push enough air it will likely overheat in short while.
Our laser cutter came with small membrane pump (Smallest upon the picture), I really do not know what Air Compressor For Laser Cutting specifications it has but stream of air was very weak. It is sufficient if all you want to do is engrave, although even then we got 
significant smoke traces around engraved picture.
it was rated for level of ~2.4cfm and had a 1 gallon container 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 can run at constant ~2bar (31psi) without concern with overheating since compressor engine is not on at all times. Using reserve air flow in tank we are able to also push up to 3.5bar (50psi) for short jobs.
The best setup in my opinion will be compressor rated to >6cfm, large 70+ gallons tank rated to 10bar (140psi) and software air control via solenoid valve.