D-instanton induced S-matrix in type 0B string theory in two dimensions suffers from infrared divergences. This can be traced to the fact that these processes produce low energy rolling tachyon states that cannot be regarded as linear combination of finite number of closed string states. We compute semi-inclusive cross sections in this theory where we allow in the final state a fixed set of closed strings carrying given energies and any number of other closed string states carrying the rest of the energy. The result is infrared finite and agrees with the results in the dual matrix model, described by non-relativistic fermions moving in an inverted harmonic oscillator potential. In the matrix model the role of `any number of other closed string states' is played by a fermion hole pair on opposite sides of the potential barrier.