  We consider an urn model, whose replacement matrix has all entries
nonnegative and is balanced, that is, has constant row sums. We obtain the
rates of the counts of balls corresponding to each color for the strong laws to
hold. The analysis requires a rearrangement of the colors in two steps. We
first reduce the replacement matrix to a block upper triangular one, where the
diagonal blocks are either irreducible or the scalar zero. The scalings for the
color counts are then given inductively depending on the Perron-Frobenius
eigenvalues of the irreducible diagonal blocks. In the second step of the
rearrangement, the colors are further rearranged to reduce the block upper
triangular replacement matrix to a canonical form. Under a further mild
technical condition, we obtain the scalings and also identify the limits. We
show that the limiting random variables corresponding to the counts of colors
within a block are constant multiples of each other. We provide an
easy-to-understand explicit formula for them as well. The model considered here
contains the urn models with irreducible replacement matrix, as well as, the
upper triangular one and several specific block upper triangular ones
considered earlier in the literature and gives an exhaustive picture of the
color counts in the general case with only possible restrictions that the
replacement matrix is balanced and has nonnegative entries.
