First of, as it was pointed out at the workshop, we don't have a clear definition of what reliable mean. Are we talking about reliable services or reliable infrastructure and to what extent does it need to be reliable?
From the technical point of view, the core infrastructure is reliable enough. The main Internet exchange points report consistent 99.9999% uptime for la last 5 years. The Domain Name System is distributed and redundant by design so it scores quite high on the “reliable” scale. The technical problems start to be apparent on the national level and even more so on the last mile so even if the core of the Internet is quite technically reliable the end user will perceive it at times as unreliable. Even if this is the case, any solution to this problem will have to come from the national arena, most probably via some kind of co-regulation between the ISP and local users groups.
Apart from the technical point of view there is always a service reliability issue but at this time I don't believe that this is something IG should be concerned with. Time and competition between service providers will certainly deal with this problem.
The IG should be concerned with issues of a global nature and the core infrastructure ones. One useful thing might be a global monitoring body that takes measurements from key points, compile statistics and publish them. Based on this public data it is easier to make predictions, anticipate problems but also take meaningful reactive measures.
One special case of core infrastructure, global issue that IG needs to take into consideration is the Domain Name System. The DNS is quite a controversial subject and that just proves there is a need to reform it. The main problem I see with it is the fact that it is not following the Internet design. What happens if one day the ICANN falls into the wrong hands. Political regimes are not historically trustworthy and we have no reason to think that ICANN is immune that's why I believe the Internet would greatly benefit from an improved version of DNS, a version that must follow the design principles of the Internet.