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 little compressors boast amazing pressure figures but actually poor flow rates sufficient limited to airbrush pistols. Actually if small compressor would be able to push enough air it’ll likely overheat in short while.
Our laser cutter came with little membrane pump (Smallest upon the picture), I do not know what specifications it has but blast of air was extremely weak. It is enough if all you want to accomplish 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 were able to give us constant 1 bar (14 PSI) of pressure, nonetheless it had to operate on a regular basis 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 constantly. Using reserve atmosphere in tank we can also drive up to 3.5bar (50psi) for short jobs.
The best setup in my opinion will be compressor rated to >6cfm, huge 70+ gallons tank rated to 10bar (140psi) and software air control via solenoid valve.