And let us further consider the situation where there is a stream of ”equivalent” particles into S (number of particles per time unit) where each particle retains its identity while being in S and eventually - after a finite time - leaves the system irreversibly (i.e. for these particles the system is ”open”).
Let's imagine a constant flow of identical particles entering system S at a rate of particles per unit time.  These particles remain distinct throughout their stay in S and ultimately leave the system permanently (considered open).