Introduce binary decision variables $y_{ij}$ to indicate that job $i$ immediately precedes job $j$, and impose indicator constraints
$$y_{ij} = 1 \implies S_j = S_i + p_i,$$
which you can alternatively linearize via big-M constraints
$$-M(1-y_{ij}) \le S_j - (S_i + p_i) \le M(1-y_{ij}).$$
To avoid the solver returning $y \equiv 0$, introduce a dummy ending job $N+1$, and impose
$$\sum_{j=1}^{N+1} y_{ij} = 1$$
for $i<N+1$ (that is, for all $i$ except the dummy).