When choosing compressor you need to consider if it will be able to constantly provide desired pressure and level of air. Small airbrush compressors small compressors boast amazing pressure figures but really poor flow rates adequate limited to airbrush pistols. Also if small compressor would be able to push enough air it’ll likely overheat in couple of minutes.
Our laser cutter came with little membrane pump (Smallest upon the picture), I really do not know very well what specifications it has but Air Compressor For Laser Cutting china stream of 
air was very weak. It is sufficient if all you want to accomplish is engrave, although even then we got considerable smoke traces around engraved picture.
it was rated for level of ~2.4cfm and had a 1 gallon container which can be filled to 58 PSI. It were able to give us constant 1 bar (14 PSI) of pressure, nonetheless it had to operate all the time and overheat badly within 5 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 continuous ~2bar (31psi) without fear of overheating since compressor engine is not on continuously. Using reserve surroundings 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, huge 70+ gallons tank rated to 10bar (140psi) and software air control via solenoid valve.