We now consider another schedulability condition, OCBP-schedulability, that offers a performance guarantee (as measured by the processor speedup factor) that is superior to the performance guarantee offered by the WCR-approach. OCBP-schedulability is a constructive test: we determine off-line, before knowing the actual execution times, a total ordering of the jobs in a priority list and for each scenario execute at each moment in time the available job with the highest priority.
The priority list is constructed recursively using the approach commonly referred to in the real-time scheduling literature as the “Audsley approach” (Audsley, 1991, 1993); it is also related to a technique introduced by Lawler (Lawler, 1973). First determine the lowest priority job: Job Ji may be assigned the lowest priority if there is at least Ci(χi) time between its release time and its deadline available when every other jobJj is executed beforeJi forCj(χi) time units (the WCET of jobJj according to the criticality level of jobi). This can be determined by simulating the behavior of the schedule under the assumption that every job other thanJi has priority overJi (and ignoring whether these other jobs meet their deadlines or not — i.e., they may execute under any relative priority ordering, and will continue executing even beyond their deadlines). The procedure is repeatedly applied to the set of jobs excluding the lowest priority job, until all jobs are ordered, or at some iteration a lowest priority job does not exist. If job Ji has higher priority than jobJj we writeJiBJj.
Because the priority of a job is based only on its own criticality level, the instance I is calledOwn Criticality Based Priority OCBP)-schedulable if we find a complete ordering of the jobs.
If at some recursion in the algorithm no lowest priority job exists, we say the instance is not OCBP-schedulable. We can simply argue that this does not mean that the instance is not MC-schedulable: Suppose that scheduling according to the fixed priority listJ1, J2, J3 with χ2 = 1 and χ1 = χ3 = 2, proves the instance to be schedulable. It may not be OCBP-schedulable since this does not take into account thatJ2does not need to be executed at all if J1 receives execution time c1 > C1(1).
It is evident that the OCBP priority list for an instance of njobs can be determined in time polynomial inn: at mostn jobs need be tested to determine whether they can be the lowest-priority job; at most (n−1) jobs whether they can be the 2nd-lowest priority jobs; etc. Therefore, at most n+ (n−1) +· · ·+ 3 + 2 + 1 =O(n2) simulations need be run, and each simulation takes polynomial time.
We illustrate the operation of the OCBP priority assignment algorithm by an example: Example 3.1. Consider the instance comprised of the following three jobs. J1 is not subject to certification, whereas J2 and J3 must be certified correct.
Ji ai di χi Ci(1) Ci(2)
J1 0 4 1 2 2
J2 0 5 2 2 4
J3 0 10 2 2 4
Let us determine which, if any, of these jobs could be assigned lowest priority according to the OCBP priority assignment algorithm:
IfJ1 were assigned lowest priority,J2 andJ3 could consumeC2(1) +C3(1) = 2 + 2 = 4 units of processor capacity over [0,4), thus leaving no execution for J1 prior to its deadline.
IfJ2 were assigned lowest priority,J1 andJ3 could consume C1(2) +C3(2) = 2 + 4 = 6 units of processor capacity over [0,6), thus leaving no execution for J2 prior to its deadline at time-instant 5.
IfJ3 were assigned lowest priority,J1 andJ2 could consume C1(2) +C2(2) = 2 + 4 = 6 units of processor capacity over [0,6). This leaves 4 units of execution forJ3 prior to its deadline at time-instant 10, which is sufficient forJ3 to execute forC3(2) = 4 time units. Job J3 may therefore be assigned the lowest priority.
Next, the OCBP priority assignment algorithm would consider the instance{J1, J2}, and seek to assign one of these jobs the lower priority:
If J1 were assigned lower priority, J2 could consume C2(1) = 2 units of processor capacity over [0,2). This leaves 2 units of execution for J1 prior to its deadline at
time-instant 4, which is sufficient forJ1 to execute forC1(1) = 2 time units. Job J1
may therefore be assigned the lowest priority from among {J1, J2}.
It may be verified thatJ2 cannot be assigned the lowest priority from among {J1, J2}. If we were to do so, then J1 could consume C1(2) = 2 units of processor capacity over [0,2). This leaves 3 units of execution for J1 prior to its deadline at time-instant 5, which is not sufficient for J2 to execute for the C2(2) = 4 time units it needs to complete on time.2
The final OCBP priority ordering is therefore as follows. JobJ2 has the greatest priority, job J1 has the next-highest priority, and J3 has the lowest priority. It may be verified that scheduling according to these priorities is a correct MC scheduling strategy for the instance {J1, J2, J3}.