When choosing compressor you need to consider if it can constantly provide desired pressure and level of air. Small airbrush compressors small compressors boast amazing pressure figures but really poor flow rates sufficient limited to airbrush pistols. Also if small compressor would be able to push enough air it will likely overheat in couple of minutes.
Our laser cutter was included with small membrane pump (Smallest upon the picture), I do not know what specifications it has but blast of air was very weak. It is sufficient if all you want to accomplish is engrave, although actually then we got Air Compressor For Laser Cutting china considerable 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 continuous 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 unit with 13 gallon container 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 on a regular basis. Using reserve air in tank we can also force 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.
