  We extend the methods of Yennie, Frautschi and Suura (YFS) to compute, via
Monte Carlo methods, the effects of multiple gluon emission in the processes
$q+q'\to q''+q'''+n(G)$, where $G$ is a soft gluon. We show explicitly that the
infrared singularities in the respective simulations are canceled to all orders
in $\alpha_s$. Some discussion of this result from the standpoint of
confinement is given. More importantly, we present, for the first time ever,
sample numerical Monte Carlo data on multiple soft gluon emission in the
rigorously extended YFS framework. We find that such soft gluon effects must be
taken into account for precise SSC/LHC physics simulations.
