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