We analyze in detail the solid-on-solid (SOS) model for growth processes on a square substrate in 2 +1 dimensions. By using the Markovian surface properties, we introduce an alternative approach for determining the roughness exponent of a special type of SOS model-the restricted-solid-on-solid (RSOS) model-in 2 +1 dimensions. This model is the SOS model with the additional restriction that the height difference must be S=1. Our numerical results show that the behavior of the SOS model in 2 +1 dimensions for approximately S Z S Γ $ 8 belongs to the two different universality classes: during the initial time stage, t o t Γ it belongs to the random-deposition (RD) class, while for t Γ o t{t sat it belongs to the Kardar-Parisi-Zhang (KPZ) universality class. The crossover time (t Γ ) is related to S via a power law with exponent, Z ΒΌ 1:99 7 0:02 at 1s confidence level which is the same as that for 1+ 1 dimensions reported in Chein and Pang (2004) [8]. Using the structure function, we compute the roughness exponent. In contrast to the growth exponent, the roughness exponent does not show crossover for different values of S. The scaling exponents of the structure function for fixed values of separation distance versus S in one and two space dimensions are x ΒΌ 0:92 7 0:05 and x ΒΌ 0:86 7 0:05 at 1s confidence level, respectively.