When choosing compressor you have to consider if it can constantly provide desired pressure and level of air. Little airbrush compressors little compressors boast impressive pressure figures but actually poor flow rates adequate only for airbrush pistols. Also if small compressor can push Air Compressor For Laser Cutting china enough air it will likely overheat in short while.
Our laser cutter was included with small membrane pump (Smallest on the picture), I really 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 also 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 can be filled to 58 PSI. It were able to give us continuous 1 bar (14 PSI) of pressure, 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 device with 13 gallon tank which is filled to ~100psi. With this unit we can run at constant ~2bar (31psi) without concern with overheating since compressor engine isn’t on continuously. Using reserve air flow in tank we can also press up to 3.5bar (50psi) for brief 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.