 Each newly created urn starts out with "k"0 balls and further balls are added to urns at a rate proportional to the number "k" that they already have plus a constant "a" > −"k"0.With these definitions, the fraction "P"("k") of urns having "k" balls in the limit of long time is given by

for "k" ≥ "k"0 (and zero otherwise), where B("x", "y") is the Euler beta function