Cosmological consequences of a built-in cosmological constant model

We have analysed 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 proportional to R/R, Lambda proportional to H-2 or Lambda proportional to 8piGrho 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 higher values for the luminosity and angular diameter distances than those of Abdel Rahman model. The lens distribution peaks at relatively smaller values of the lens red-shift and occurs at higher red-shift.