We all know troubleshooting is an art. And it makes sense for them to charge you if there's a problem with the house power. They tested the modem by replacing it. But they haven't tested the modem power supply by replacing it. The fact that every new modem includes a power supply means that replacing a modem means replacing its power supply as well.
I'm throwing out the following steps as my five minute essay on how to show the service person that the power supply must be replaced before they can say they have replaced the modem. Refine it however you like.
Before the service person comes out again, talk to their supervisor to discuss the fact that replacing the modem means replacing its power supply, too. (You might ask what they're doing with the power supplies they have left over after supposedly replacing complete modems.)
Problem: Modem does not come on. Replaced modem, did not replace modem power supply, even though every modem comes with a power supply in the carton. This will reveal if there are problems outside the modem (a problem inside the modem should not replicate). What is not tested are the modem power supply and the house power.
Once again, modem does not come on. Replaced modem, did not replace power supply, even though every modem comes with a new power supply. We now have two modems that have failed when powered by the house power and the power supply for the first modem.
This means technician has not eliminated modem power supply as source of the problem. But power supplies go dead when they fail, right? Yes, but power supplies can add throw AC into the DC output if there are any problems with the output filter of the supply.
It is VERY unlikely that two modems in a row will fail. It is likely that a bad power supply that makes one modem fail will make every modem fail that is connected to it.
What has not been tested at this point is the modem power supply and the house power.
The power supply might be defective. It should have been replaced during the first visit, as A COMPLETE MODEM is a modem and its power supply.
The house power is not likely to be suspect because all other components are connected to house power and nothing else is failing. The one device between the house outlet and the power into the modem power supply is a _____________, which contains a switch and surge protection. This is extremely unlikely to cause failure of a product.
The problem here is VERY likely the modem power supply. Swap out the power supply, as should have been done the first time, before charging for any service visits.
A good answer is easier with a clear question giving the make and model of everything. "The biggest problem in communication is the illusion that it has taken place." -- G. “Bernie” Shaw