  We introduce a new elliptic quantum toroidal algebra $U_{q,t,p}(gl_{1,tor})$.
Various representations in the quantum toroidal algebra $U_{q,t}(gl_{1,tor})$
are extended to the elliptic case including the level (0,0) representation
realized by using the elliptic Ruijsenaars difference operator. Intertwining
operators of $U_{q,t,p}(gl_{1,tor})$-modules w.r.t. the Drinfeld
comultiplication are also constructed. We show that $U_{q,t,p}(gl_{1,tor})$
gives a realization of the affine quiver $W$-algebra
$W_{q,t}(\Gamma(\widehat{A}_0))$ proposed by Kimura-Pestun. This realization
turns out to be useful to derive the Nekrasov instanton partition functions,
i.e. the $\chi_{y^-}$ and elliptic genus, of the 5d and 6d lifts of the 4d
$\mathcal{N}=2^*$ theories and provide a new Alday-Gaiotto-Tachikawa
correspondence.
