When choosing compressor you need to Air Compressor For Laser Cutting consider if it can constantly provide desired pressure and volume of air. Small airbrush compressors small compressors boast impressive pressure figures but really poor flow rates enough only for airbrush pistols. Actually if small compressor can push enough air it will likely overheat in couple of minutes.
Our laser cutter was included with little membrane pump (Smallest upon the picture), I really do not know very well what specifications it has but blast of air was extremely weak. It is adequate if all you have to to do is engrave, although actually then we got considerable smoke traces around engraved picture.
it had been rated for level of ~2.4cfm and had a 1 gallon container which may be filled to 58 PSI. It managed to give us constant 1 bar (14 PSI) of pressure, but it had to operate constantly 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 are able to run at continuous ~2bar (31psi) without fear of overheating since compressor engine isn’t on continuously. Using reserve atmosphere 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.