Variant of interval scheduling (multiple machines with given availability)
Let us call $n$ the number of intervals and $m$ the number of available machines. 

There is an algorithm by Brucker and Nordmann running in time essentially $O(n^{m-1})$. Thus for any constant $m$, the problem admits a polynomial time algorithm. The authors claim that the algorithm is practical up to $m=5$ machines. 

When $m$ is not constant, the problem appears to be hard, however I do not know whether the version you stated is NP-hard. In this case, Faigle, Kern and Nawijn show that by a simple greedy algorithm you can schedule at least half of the intervals scheduled in an optimal solution. This is the overview of their algorithm, that they call $G$ (the jobs are denoted by $I_1,I_2,\ldots,I_n$,  and are considered ordered by their left endpoints): 


  $G$ places $I_t$ on a free feasible machine if such a machine is
  available. Otherwise $G$ tries to replace a job $I_v$ by $I_t$ on a
  feasible machine with the goal to reduce the maximal remaining
  processing time. If neither action is possible, $G$ discards $I_t$.