The directed Abelian sandpile models are defined on a square lattice by introducing a parameter, c, representing the degree of anisotropy in the avalanche processes, in which c = 1 is for the isotropic case. We calculate the expected number of the topplings per added particle, 〈T〉, which depends on the lattice size L as Lx for large L. Our exact solution gives that x = 1 when any anisotropy is included in the system, while x = 2 in the isotropic case. These results allow us to introduce a new critical exponent, θ, defined by χ ≡ 1imL→∞ 〈T〉/L with c ≠ 1 as χ approx. |c - 1|-θ for |c - 1| ≪ 1. From the explicit expression of 〈T〉, we obtain θ = 1.