  A systematic study of charged hadron multiplicities ($n_{ch}$) at various
collision energies is very much important in understanding the basic production
mechanism of the hadrons in nucleus-nucleus collision experiments. Furthermore,
the variations of $n_{ch}$ in nucleus-nucleus collisions with respect to the
colliding energy and mass number can provide a potential probe for the
formation of quark gluon plasma (QGP) in the laboratory. In this paper, we
propose a phenomenological model based on the constituent quark-quark
interactions to calculate the average multiplicity ($n_{ch}$) and
pseudorapidity density at mid-rapidity ($(dn_{ch}/d\eta)_{\eta=0}$) of charged
hadrons at various center-of-mass energies $(\sqrt{s_{NN}})$ for
nucleus-nucleus ($A-A$) collisions. We first propose a new parametrization for
$n_{ch}^{pp}$ and $(dn_{ch}/d\eta)^{pp}_{\eta=0}$ in $p-p$ interactions based
on some initial inputs which fit the experimental data very well. We further
extend this parametrization by using simple phenomenological assumptions
regarding mean number of participating quarks and mean number of collisions to
obtain the $n_{ch}$ and $(dn_{ch}/d\eta)_{\eta=0}$ for $A-A$ collisions and
show their dependencies on the mass number of colliding nuclei as well as on
$\sqrt{s_{NN}}$. We also compare the results obtained from our model with the
results obtained from the modified Glauber model in order to demonstrate the
difference between the two formalisms. Finally, we compare the charged hadron
multiplicity and pseudorapidity density at mid-rapidity for $A-A$ collisions
obtained from our model with the available experimetal data from various
heavy-ion collision experiments and give our predictions for $A-A$ collisions
at the Large Hadron Collider (LHC) and at Compressed Baryonic Matter (CBM)
experiments.
