  The $d$-process generates a graph at random by starting with an empty graph
with $n$ vertices, then adding edges one at a time uniformly at random among
all pairs of vertices which have degrees at most $d-1$ and are not mutually
joined. We show that, in the evolution of a random graph with $n$ vertices
under the $d$-process with $d$ fixed, with high probability, for each $j \in
\{0,1,\dots,d-2\}$, the minimum degree jumps from $j$ to $j+1$ when the number
of steps left is on the order of $\ln(n)^{d-j-1}$. This answers a question of
Ruci\'nski and Wormald. More specifically, we show that, when the last vertex
of degree $j$ disappears, the number of steps left divided by $\ln(n)^{d-j-1}$
converges in distribution to the exponential random variable of mean
$\frac{j!}{2(d-1)!}$; furthermore, these $d-1$ distributions are independent.
