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Motivated by the fact that any nonzero $\Lambda$ can introduce a length scaleor a time scale into Einstein's theory,$r_{\Lambda}=ct_{\Lambda}=\sqrt{3/|\Lambda|}$. Conversely, any cosmologicallength scale or time scale can introduce a $\Lambda (t)$,$\Lambda(t)=3/r^2_{\Lambda}(t)=3/(c^2t^2_{\Lambda}(t))$. In this letter, weinvestigate the time varying $\Lambda(t)$ corresponding to the length scales,including the Hubble horizon, the particle horizon and the future eventhorizon, and the time scales, including the age of the universe and theconformal time. It is found out that, in this scenario, the $\Lambda(t)$CDMmodel can be taken as the unified origin of the holographic and agegraphic darkenergy models with interaction between the matter and the dark energy, wherethe interacting term is determined by $Q=-\dot{\rho}_{\Lambda}$. We placeobservational constraints on the $\Lambda(t)$CDM models originating fromdifferent cosmological length scales and time scales with the recently compiled"Union2 compilation" which consists of 557 Type Ia supernovae (SNIa) covering aredshift range $0.015\leq z \leq 1.4$. In conclusion, an accelerating expansionuniverse can be derived in the cases taking the Hubble horizon, the futureevent horizon, the age of the universe and the conformal time as the lengthscale or the time scale.

$\Lambda(t)$CDM Model as a Unified Origin of Holographic and Agegraphic Dark Energy Models