From the numerics point of view, there are just minimal changes from what happens in the case of a single rain record, with the addition of an iteration over the rain impulse and a different evaluation of \(i\left(k\right)\)s, at least when the IUH is not changing shape over time. 
In this case it is clear that the domain where the hydrograph due to a specific rain record \(k\) is different from zero in a domain of length \(t_c+\Delta t\ =\ \left(m+n\right)\ \delta t\). The total hydrograph, at any clock time \(t\), is easily seen to be of length \(\left(l\cdot n+m\right)\ \delta t\). Please notice that at any new record incoming the knowledge of future discharges due only to past rain record is available for all the m future time steps, being no modified by future at all. 
In a real time perspective, when we receive a new rain record (of duration \(n\ \cdot\delta t\)) we have to expand the whole hydrograph domain by \(m\) steps forward in time.