Time variable $\Lambda$ and the accelerating Universe

We perform a deductive study of accelerating Universe and focus on the importance of variable time-dependent $\Lambda$ in the Einstein's field equations under the phenomenological assumption, $\Lambda =\alpha H^2$ for the full physical range of $\alpha$. The relevance of variable $\Lambda$ with regard to various key issues like dark matter, dark energy, geometry of the field, age of the Universe, deceleration parameter and barotropic equation of state has been trivially addressed. The deceleration parameter and the barotropic equation of state parameter obey a straight line relationship for a flat Universe described by Friedmann and Raychaudhuri equations. Both the parameters are found identical for $\alpha = 1$.