When choosing compressor you should consider if it will be able to constantly provide desired pressure and volume of air. Small airbrush compressors small compressors boast amazing pressure figures but really poor flow rates enough 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 little membrane pump (Smallest on the picture), I really do not know very well what specifications it has but blast of air was extremely weak. It is sufficient if all you want to accomplish is engrave, Air Compressor For Laser Cutting china although also then we got substantial smoke traces around engraved picture.
it had been rated for volume of ~2.4cfm and had a 1 gallon tank which can be filled to 58 PSI. It managed to give us continuous 1 bar (14 PSI) of pressure, nonetheless it had to operate at all times 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 are able to run at constant ~2bar (31psi) without fear of 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 would be compressor 
rated to >6cfm, large 70+ gallons tank rated to 10bar (140psi) and software air control via solenoid valve.