  The Hirsch's $h$-index is perhaps the most popular citation-based measure of
the scientific excellence. In 2013 G. Ionescu and B. Chopard proposed an
agent-based model for this index to describe a publications and citations
generation process in an abstract scientific community. With such an approach
one can simulate a single scientist's activity, and by extension investigate
the whole community of researchers. Even though this approach predicts quite
well the $h$-index from bibliometric data, only a solution based on simulations
was given. In this paper, we complete their results with exact, analytic
formulas. What is more, due to our exact solution we are able to simplify the
Ionescu-Chopard model which allows us to obtain a compact formula for
$h$-index. Moreover, a simulation study designed to compare both, approximated
and exact, solutions is included. The last part of this paper presents
evaluation of the obtained results on a real-word data set.
