  In this paper we make re-analysis of a self-similarity based model of the
proton structure function at small $x$ pursued in recent years. The additional
assumption is that it should be singularity free in the entire kinematic range
$0\leq \textit{x}\leq 1$. Our analysis indicates that the singularity free
version of the model is valid in a more restrictive range of $Q^{2}$. We then
analyse the defining Transverse Momentum Dependent Parton Distributions (TMD)
occurred in the models and show that the proper generalizations and initial
conditions on them not only remove the undesired singularity but also results
in a QCD compatible structure function with logarithmic growth in $Q^2$. The
phenomenological range of validity is then found to be much larger than the
earlier versions. We also extrapolate the models to large $x$ in a parameter
free way.
