When choosing compressor you should consider if it will be able to constantly provide desired pressure and volume of air. Small airbrush compressors little compressors boast impressive pressure figures but really poor flow rates enough only for airbrush pistols. Actually 
if small compressor would be able to push enough air it will likely overheat in short while.
Our laser cutter was included with small membrane pump (Smallest on the picture), I really do not know what specifications it has but blast of air was very weak. It is enough if all you want 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 tank which may be filled to 58 PSI. It managed to give us constant 1 bar (14 PSI) of pressure, nonetheless it had to operate on a Air Compressor For Laser Cutting china regular basis 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 continuous ~2bar (31psi) without fear of overheating since compressor engine isn’t on at all times. Using reserve atmosphere in tank we can also drive up to 3.5bar (50psi) for brief jobs.
The best setup for me will be compressor rated to >6cfm, large 70+ gallons tank rated to 10bar (140psi) and software air control via solenoid valve.