  A standard model in network synchronised distributed computing is the LOCAL
model. In this model, the processors work in rounds and, in the classic
setting, they know the number of vertices of the network, $n$. Using $n$, they
can compute the number of rounds after which they must all stop and output. It
has been shown recently that for many problems, one can basically remove the
assumption about the knowledge of $n$, without increasing the asymptotic
running time. In this case, it is assumed that different vertices can choose
their final output at different rounds, but continue to transmit messages. In
both models, the measure of the running time is the number of rounds before the
last node outputs. In this brief announcement, the vertices do not have the
knowledge of $n$, and we consider an alternative measure: the average, over the
nodes, of the number of rounds before they output. We prove that the complexity
of a problem can be exponentially smaller with the new measure, but that
Linial's lower bound for colouring still holds.
