Our ADSL modem used to just disconnect randomly.. it happened for at least 2 years for the 4 or 5 years we had it. We replaced the router, thinking it was the problem. But it didnt solve anything. Finally my dad got fed up, called the company and we got a new modem/router unit and it works flawlessly. The old modem used to get very hot as you say, and lived waay upstairs for most of its life, under the sun. The white plastic was actually discolored yellowish due to heat exposure, I think. This new one is downstairs where it's cool.

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1985 CA 300D Turbo , 213K mi