  This paper focuses on the asymptotic behaviors of the {\it height} for a
birth-and-death process which related to a mean-field model \cite{FFS}(or the
Anick-Mitra-Sondhi model \cite{DDM}). Recently, the asymptotic mean value of
the height for the model is given in \cite{LAV}. In this paper, first, the
asymptotic variance of the height is given, and as a consequence, a weak Law of
Large Number for the height is obtained. Second, the centered and normalized
height is proved to converge in distribution to a degenerate distribution, this
indicates that the desired Central Limit Theorem fails.
