  The directed preferential attachment model is revisited. A new exact
characterization of the limiting in- and out-degree distribution is given by
two \emph{independent} pure birth processes that are observed at a common
exponentially distributed time $T$ (thus creating dependence between in- and
out-degree). The characterization gives an explicit form for the joint degree
distribution, and this confirms previously derived tail probabilities for the
two marginal degree distributions. The new characterization is also used to
obtain an explicit expression for tail probabilities in which both degrees are
large. A new generalised directed prefererantial attachment model is then
defined and analysed using similar methods. The two extensions, motivated by
empirical evidence, are to allow double-directed (i.e.\ undirected) edges in
the network, and to allow the probability to connect an ingoing (outgoing) edge
to a specified node to also depend on the out-degree (in-degree) of that node.
