Years ago I worked in ISP support for a range of national and international providers. Answering this question is as much about the call centre environment (and usually outsourcing) than the technical reasons.
The router your ISP gives you (often for free) is selected based on cost not reliability...
- The memory isn't error checking/correcting so glitches will happen occasionally.
- The cheapest manufacturer's firmware is probably written by the cheapest staff, it probably doesn't clean up after itself and runs itself out of resources.
- The DSLAM (or equivalent) is often crappy for the same reason as the free router and a little time of inactivity gives it plenty of time to know that the connection is actually gone rather than just a dropout. That makes sure the ATM layer, PPPoE/PPPoA, DHCP and everything else that may be needed depending on the provider's configuration is renegotiated.
All those are quickly fixed by a power cycle with a short delay. Technically that delay would be in the order of a few seconds but there's more to it than that...
Not everyone who calls up is stupid...
- Some call to ask if it is them or the ISP at fault as much as anything else and to register their displeasure.
- Some lack the confidence to know if it's ok to power-cycle the router - that is not stupidity. I lack the confidence to go even tandem skydiving and that's just walking/jumping through a door.
- Some forget the router is there. It sits in a corner out of sight silently flashing away and usually 'just works'.
- Some can't judge the 30 seconds that is more than ample to discharge the capacitors, so it's best to over estimate how long it needs than underestimate. 30 Seconds of silence in a phone conversation feels like forever.
Outsourced call centres for this kind of work are typically paid per call while the staff are paid a pittance and treated like scum. Suggesting a reboot with a wait is profitable because it is:-
- Most likely to fix the problem (it really is!)
- Easy for "techs" to explain to customers.
- Gets a caller off the line quickly (allowing "techs" to take more calls per hour since they don't have to wait.)
- If they have to call back into a queue the call centre may get termination rates.
- Many contracts have a stipulation that if the caller calls back within a certain time that it counts as a continuation the same call. This is also a good reason why you might find 5 minutes as a common value.
The power-cycle is as much about giving the network equipment a chance to reset as the router itself (customer/premises equipment) but five minutes is a business driven value.
"techs" is a term that needs inverted commas as you could train pretty much anyone who is polite and speaks the language to do the job from never having turned a computer on in 5 days easily. They are rarely technical in any way and politeness (like ability to speak the language) is frequently considered optional to save costs.