3. Priority-Inheritance Rule: When Ji becomes suspended, the job Jl whose priorityceiling function is equal to the system ceiling at the time inherits the current priority πi (t) of Ji
By definition, the priority-ceiling function of any job J (and hence the ceiling of the system) never falls below the priority ceiling of any data object already read and written by J until J no longer requires the object. In this way, the convex-ceiling protocol prevents other jobs from reading and writing the objects required by J between the reads and writes of J .The formal proof that the protocol ensures serializability can be found in [Naka].
In principle, the convex-ceiling protocol is the same as the basic priority-ceiling protocol. (The system ceiling used by the basic priority-ceiling protocol can also be defined as the highest priority of the priority-ceiling functions of all the jobs: The priority-ceiling function of a job is equal to the highest priority ceiling of all the resources being used by the job.) That the convex-ceiling protocol prevents deadlock and transitive blocking follows straightforwardly from this fact.
From its definition, we see that that the convex-ceiling protocol can be implemented entirely in the application level. An application-level object manager can serve as the scheduler. It maintains the priority-ceiling functions of jobs and controls the concurrent accesses to data objects. It can do so without relying on a locking mechanism. Since the basic priority-ceiling and PCP-2PL protocols can also be defined in the same manner, the same can be said for these protocols as well.
Comparison with Basic and PCP-2PL Protocols. To illustrate the differences between this protocol and the basic priority-ceiling protocol, we return to the example in Figure 8–23. Suppose that after Ji accesses Black, a job J2 with priority 2 were to request access to a data object. According to the convex-ceiling protocol, this request would be denied and J2 would be suspended until time 6 when the system ceiling is lowered to 3. In contrast, according to the basic priority-ceiling protocol, the system ceiling is given by the solid line in Figure 8–23(c). J2 would be allowed to access its required data object starting from time 4. J2 would
be blocked for a shorter amount of time at the expense of potential violation of serializability. On the other hand, according to the PCP-2PL protocol, the system ceiling would be given by the dotted line in Figure 8–23(c) because Ji does not release any data object until it no longer requires additional objects. As a consequence, J2 would be blocked for 1 more unit of time when compared with the convex-ceiling protocol. A job with priority 1 can be blocked by Ji for only 1 unit of time under the convex-ceiling protocol, but can be blocked for 5 units of time under PCP-2PL protocol.
