When choosing compressor you have to consider if it will be able to constantly provide desired pressure and level of air. Little airbrush compressors little compressors boast amazing pressure figures but really poor flow rates adequate limited to airbrush pistols. Also if small compressor would be able to push enough air it’ll likely overheat in short while.
Our laser cutter was included with little membrane pump (Smallest on the picture), I really do not know what specifications it has but blast of air was very weak. It is sufficient if all you 
have to to accomplish is engrave, although even then we got significant smoke traces around engraved picture.
it had been rated for volume of ~2.4cfm and had a 1 gallon tank which may be filled to 58 PSI. It managed to give us continuous 1 bar (14 PSI) of pressure, Air Compressor For Laser Cutting china nonetheless it had to run all the time and overheat badly within 5 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 are able to run at constant ~2bar (31psi) without concern with overheating since compressor engine is not on on a regular basis. Using reserve air in tank we are able to also push up to 3.5bar (50psi) for brief jobs.
The best setup in my opinion would be compressor rated to >6cfm, large 70+ gallons tank rated to 10bar (140psi) and software air control via solenoid valve.