  M. Embry and A. Lambert initiated the study of a weighted translation
semigroup $\{S_t\}$ in ${\cal B}(L^2({\mathbb R_+})),$ with a view to explore a
continuous analogue of a weighted shift operator. We continued the work,
characterized some special types of semigroups and developed an analytic model
for the left invertible weighted translation semigroup. The present paper deals
with the generalization of the weighted translation semigroup in multi-variable
set up. We develop the toral analogue of the analytic model and also describe
the spectral picture. We provide many examples of weighted translation
semigroups in multi-variable case. Further, we replace the space $L^2({\mathbb
R_+})$ by $L^2({\mathbb R_+^d})$ and explore the properties of weighted
translation semigroup $\{S_{\bar{t}}\}$ in ${\cal B}(L^2({\mathbb R_+^d})),$ in
both one and multi variable cases.
