When choosing compressor you should consider if it can constantly provide desired pressure and level of air. Small airbrush compressors small compressors boast impressive pressure figures but actually poor flow rates adequate limited to airbrush Air Compressor For Laser Cutting china pistols. Actually if small compressor can push enough air it will likely overheat in couple of minutes.
Our laser cutter came with little membrane pump (Smallest on the picture), I do not know what specifications 
it has but blast of air was extremely weak. It is adequate if all you have to to do 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 container which may be filled to 58 PSI. It managed to give us continuous 1 bar (14 PSI) of pressure, nonetheless it had to operate constantly 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 can run at constant ~2bar (31psi) without concern with overheating since compressor engine is not on constantly. Using reserve air flow in tank we can also force 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.