RE: Soliciting your opinions on Internet routing: A survey on BGP convergence

When you simply bring down an ebgp session, withdraws will propagate throughout the network.
Soon after, the alternate routes will propagate. In the interim, some routers will lose connectivity.
This problem is solved by graceful shutdown.
This only works for planned shutdown
This interim time can be many minutes because of the advertisement-interval (MRAI timer).
A possible solution to reduce this interim to seconds instead of minutes is to set the MRAI timer to 0 on all routers. A potential problem with that is that any BGP instability in the network will cause some serious flapping.
Another alternative is to use BGP add-path (rfc7911) to distribute backup routes.
This will avoid the MRAI problem, but requires more memory on routers.
This also works for accidental shutdown.