  We have analyzed the implications of a phenomenological model of a built-in
cosmological constant of Rastall and Al-Rawaf -Taha type. We have produced the
presently observed cosmic acceleration with the aid of introducing a positive
cosmological constant. We have shown that the cosmological models of the type
$\Lambda\propto\frac{\ddot R}{R}, \Lambda\propto H^2$ or $\Lambda\propto 8\pi
G\rho$ are equivalent to a built-in cosmological constant model of Al-Rawaf and
Taha type. Such models are compatible with the recent observational data and do
not suffer from fine tuning problems. We predict a higher lensing probability
than in Einstein de Sitter and Abdel Rahman models. We also predict a higher
value for the luminosity and angular diameter distances than that of Abdel
Rahman model. The lens distribution peaks at relatively smaller values of the
lens red-shift and occurs at higher red-shift.
