When choosing compressor you need to consider if it can constantly provide desired pressure and volume of air. Air Compressor For Laser Cutting china Little airbrush compressors small compressors boast impressive pressure figures 
but really poor flow rates adequate only for airbrush pistols. Even if small compressor would be able to push enough air it’ll likely overheat in couple of minutes.
Our laser cutter came with small membrane pump (Smallest upon the picture), I do not know very well what specifications it has but blast of air was very weak. It is enough if all you want to accomplish is engrave, although even then we got considerable smoke traces around engraved picture.
it was 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, but it had to operate on a regular basis and overheat badly within five minutes.
Third and current compressor we got was 6cfm rated belt driven unit with 13 gallon tank which is filled to ~100psi. With this unit we can run at continuous ~2bar (31psi) without concern with overheating since compressor engine isn’t on all the time. Using reserve air in tank we are able to also force up to 3.5bar (50psi) for short jobs.
The best setup for me would be compressor rated to >6cfm, huge 70+ gallons tank rated to 10bar (140psi) and software air control via solenoid valve.