When choosing compressor you need to consider if it will be able to constantly provide desired pressure and volume of air. Small airbrush compressors little compressors boast amazing pressure figures but really poor flow rates adequate limited to airbrush pistols. Actually if small compressor would be able to push enough air it will likely overheat in short while.
Our laser cutter came with small membrane pump (Smallest on the picture), I do not know very well what specifications it has but blast of air was very weak. It is sufficient if all you want to do is engrave, although also then we got substantial smoke traces around engraved picture.
it had been rated for level of ~2.4cfm and had a 1 gallon tank which can be filled to 58 PSI. It were able to give us continuous 1 bar (14 PSI) of pressure, nonetheless it had to operate constantly and overheat badly within five minutes.
Third and current compressor we got was 6cfm rated belt driven device with 13 gallon tank 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 atmosphere in tank we are able to also press up to 3.5bar (50psi) for short jobs.
The best setup in my Air Compressor For Laser Cutting china opinion would 
be compressor rated to >6cfm, huge 70+ gallons tank rated to 10bar (140psi) and software air control via solenoid valve.