  The most widely studied scenario in dark matter phenomenology is the thermal
WIMP scenario. Inspite of numerous efforts to detect WIMP, till now we have no
direct evidence for it. A possible explanation for this non-observation of dark
matter could be because of its very feeble interaction strength and hence,
failing to thermalise with the rest of the cosmic soup. In other words, the
dark matter might be of non-thermal origin where the relic density is obtained
by the so-called freeze-in mechanism. Furthermore, if this non-thermal dark
matter is itself produced substantially from the decay of another non-thermal
mother particle, then their distribution functions may differ in both size and
shape from the usual equilibrium distribution function. In this work, we have
studied such a non-thermal (fermionic) dark matter scenario in the light of a
new type of $U(1)_{\rm B-L}$ model. The $U(1)_{\rm B-L}$ model is interesting,
since, besides being anomaly free, it can give rise to neutrino mass by Type II
see-saw mechanism. Moreover, as we will show, it can accommodate a non-thermal
fermionic dark matter as well. Starting from the collision terms, we have
calculated the momentum distribution function for the dark matter by solving a
coupled system of Boltzmann equations. We then used it to calculate the final
relic abundance, as well as other relevant physical quantities. We have also
compared our result with that obtained from solving the usual Boltzmann (or
rate) equations directly in terms of comoving number density, $Y$. Our findings
suggest that the latter approximation is valid only in cases where the system
under study is close to equilibrium, and hence should be used with caution.
