When choosing compressor you need to consider if it can constantly provide Air Compressor For Laser Cutting china desired pressure and volume 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 will likely overheat in couple of minutes.
Our laser cutter was included with small 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 want to do is engrave, although also then we got considerable smoke traces around engraved picture.
it was rated for volume of ~2.4cfm and had a 1 gallon container which can be filled to 58 PSI. It managed to give us continuous 1 bar (14 PSI) of pressure, but it had to operate 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 continuous ~2bar (31psi) without concern with overheating since compressor engine is not on on a regular basis. Using reserve atmosphere in tank we can also push up to 3.5bar (50psi) for short 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.