Expected Buildings

Chef is standing in the centre of a circle. The perimeter of the circle is divided into $h$ units (numbered $1$ through $h$) and some of those units contain buildings. There are $N$ buildings in total (numbered $1$ through $N$); let's denote the position of building $i$, i.e. the number of the unit containing this building, by $p_i$. Note that a unit may contain more than one building. Chef has a