Available Stabilizing Heaps
  This paper describes a heap construction that supports insert and delete operations in arbitrary (possibly illegitimate) states. After any sequence of at most O(m) heap operations, the heap state is guarantee to be legitimate, where m is the initial number of items in the heap. The response from each operation is consistent with its effect on the data structure, even for illegitimate states. The time complexity of each operation is O(lg K) where K is the capacity of the data structure; when the heap's state is legitimate the time complexity is O(lg n) for n equal to the number items in the heap. 
Ted Herman, Toshimitsu Masuzawa