Reboot times go up with new features with in the router and the way the HW is designed. Not much can be done about that. I usually can access the web page about 5-10 seconds before I see the UI timer run out. DIR 655 series didn't have much features and remember, the 855L is dual band, the 655 is 2.4Ghz only. So it doesn't have to load all the extra stuff that the 855L has.
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There is a security enhancement in v1.03. Not sure if anything related to disconnects.
I recommend that you have your ISP check the cabling going to the ISP modem, check signal levels going to the ISP modem. For DSL or Fiber service lines, ensure that the ISP is using good working phone or cable lines to the DSL modem and that the phone lines are filtered correctly. For cable Internet,
RG-6 coaxial cable is needed, not
RG-59. Check for t.v. line splitters and remove them as they can introduce noise on the line and lower the signal going to the ISP modem. I recommend having the ISP service physically check the lines going from the out side to the ISP modem. Connecting to the ISP modem could result in a false positive as the signal to the modem could be just enough to that point then adding on a router, could see problems. The router operation is dependent upon getting good data flow from the ISP modem and the modem is dependent upon getting good signal from the ISP Service.
I'd look into a new modem as well. This model is a few years old and has draft wireless support that probably wasnt' certified. I know you have it in bridge mode however doesn't mean other items on this modem could be a cause and if the DIR 655 is from same era with same draft N design that you never updated FW on, probably works well however the 855L is new generation with certifications and newer design and networking.