When choosing compressor you need to consider if it will be able to constantly provide desired pressure and volume of air. Small Air Compressor For Laser Cutting china airbrush compressors small compressors boast impressive pressure figures but actually poor flow rates sufficient limited to airbrush pistols. Also if small compressor can push enough air it will likely overheat in couple of minutes.
Our laser cutter came with small membrane pump (Smallest upon the picture), I really do not know very well what 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 considerable 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 were able to give us constant 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 unit with 13 gallon container which is filled to ~100psi. With this unit we can run at continuous ~2bar (31psi) without fear of overheating since compressor engine is not on all the time. Using reserve atmosphere in tank we can also force up to 3.5bar (50psi) for short jobs.
The best setup for me will be compressor rated to >6cfm, huge 70+ gallons tank rated to 10bar (140psi) and software air control via solenoid valve.