On the Complexity of Worst-Case Blocking Analysis of Nested Critical Sections

Accurately bounding the worst-case blocking for finite job sets, a special case of the classic sporadic task model of recurrent real-time systems, using either nested FIFO-or priority-ordered locks on multiprocessors is NP-hard. These intractability results are obtained with reductions from the Multiple-Choice Matching problem. The reductions are quite general and do not depend on (1) whether the locks are spin-or suspension-based, or (2) whether global or partitioned scheduling is used, or (3) which scheduling policy is employed (as long as it is work-conserving). Further, we show that, for a special case in which the blocking analysis problem is NP-hard for FIFO- and priority-ordered locks, the problem for unordered spin locks with nested critical sections can be answered in polynomial time by solving a reach ability problem on a suitably constructed graph, although (or rather, because) unordered locks do not offer any acquisition-order guarantees. Finally, we identify several challenging open problems, pertaining both to circumventing the hardness results and to classifying the inherent difficulty of the problem more precisely.