Cache-Aware Scheduling

Abstract

The major obstacle to use multicores for real-time applications is that we may not predict and provide any guarantee on real-time properties of embedded software on such platforms; the way of handling the on-chip shared resources such as L2 cache may have a significant impact on the timing predictability.

Appendix: Improving the Interference Computation

The computation of $I_{k}^{i}$, an upper bound of the interference caused by τ _i over J _k , in Eq. (8.3) (Sect. 8.5.1) is grossly over-pessimistic. In the following we will present a more precise computation of $I_{k}^{i}$ by carefully identifying the worst-case scenario of τ _i ’s interference.

Recall that the problem window $[r_{k},l_{k}]$ is a time frame of a given length ($l_{k} - r_{k} = S_{k}$) for which we want to derive a bound of how much interference a task τ _i (or rather its jobs) can cause to possibly prevent J _k from running. We can compute $I_{k}^{i}$ using the following lemma:

Lemma 8.2.

An upper bound of the interference contributed by τ _i in the problem window of length S _k can be computed by

$$\displaystyle{ I_{k}^{i} = \left \{\begin{array}{ll} S_{k} &i <k \wedge S_{k} <C_{i} \\ \left \lfloor \frac{S_{k}-C_{i}} {T_{i}} \right \rfloor C_{i} + C_{i}+\omega &i <k \wedge S_{k} \geq C_{i} \\ 0 &i = k \\ \min (C_{i},S_{k}) &i> k \end{array} \right. }$$

(8.10)

where

$$\displaystyle{ \omega =\min \Big (C_{i},\max \big(0,(S_{k} - C_{i})\bmod T_{i} - (T_{i} - D_{i})\big)\Big) }$$

(8.11)

Proof.

The lemma is proved in the following cases:

• 1. i < k, i.e., τ _i ’s priority is higher than τ _k ’s.

  If $S_{k} <C_{i}$, i.e., a job of τ _i can execute even longer than J _k ’s slack, trivially $I_{k}^{i} = S_{k}$ is a safe bound.

  If $S_{k} \geq C_{i}$, the worst-case for $I_{k}^{i}$ occurs when

  • (a) one of τ _i ’s jobs is released at $l_{k} - C_{k}$,

  • (b) all jobs are released with period T _i , and

  • (c) the carry-in job executes as late as possible.

  See Fig. 8.8. To see that this is indeed the worst-case, we imagine to move the release times of τ _i ’s jobs leftwards for a distance $\epsilon ^{l} <T_{i} - C_{i}$ or rightwards for a distance $\epsilon ^{r} <C_{i}$, to see if it is possible to increase $I_{k}^{i}$ by doing so. (It’s easy to see that moving τ _i ’s jobs’ releases more in either direction creates a situation equivalent to one of these two cases. Further, $I_{k}^{i}$ cannot be increased if the number of τ _i ’s jobs in $[r_{k},l_{k}]$ is decreased, which means we only need to consider the scenario that all jobs are released periodically.) If it is moved leftwards by ε ^l, τ _i ’s interference cannot increase at neither the left nor the right end of the interval $[r_{k},l_{k}]$, so moving leftwards for a distance $\epsilon ^{l} <T_{i} - C_{i}$ will not increase the interference. On the other hand, when moving rightwards by ε ^r, the interference is increased by no more than ε ^r at the left end, but decreased by ε ^r at the right end, so moving rightwards for a distance $\epsilon ^{r} <C_{i}$ will also not increase the interference. In summary, based on the scenario in Fig. 8.8, $I_{k}^{i}$ cannot be increased no matter how we move the release time of τ _i . With this worst-case scenario, we can see that the interference contributed by the carry-out job is C _i , the number of the body jobs is $\lfloor (S_{k} - C_{i})/T_{i}\rfloor$ (each contributing C _i interference), and the interference contributed by the carry-in job is bounded by both C _i and the distance between r _k and the carry-in job’s deadline. Thus, for each task τ _i with $i <k \wedge S_{k} \geq C_{i}$, we can compute $I_{k}^{i}$ by

  $$\displaystyle{ I_{k}^{i} = \left \lfloor \frac{S_{k} - C_{i}} {T_{i}} \right \rfloor C_{i} + C_{i}+\omega }$$

  (8.12)

  where ω is defined as in Eq. (8.11).

  Fig. 8.8

  

  Computation of $I_{k}^{i}$ if i < k and $S_{k} \geq C_{i}$

• 2. i = k, i.e., τ _i is the analyzed task. Since $D_{k} \leq T_{k}$ holds for each task τ _k , the other jobs of τ _k cannot interfere with J _k , so in this case we have

  $$\displaystyle{ I_{k}^{i} = 0 }$$

  (8.13)

• 3. i > k, i.e., τ _i ’s priority is lower than τ _k .

  In FP _CA , a job $J_{i}^{h}$ with lower priority than J _k can interfere with J _k only if it is released earlier than r _k . Therefore, τ _i can only cause interference to J _k with at most one job, so its interference is bounded by C _i . The interference is also bounded by the length of the problem window S _k . Thus, for i > k, we can compute $I_{k}^{i}$ by

  $$\displaystyle{ I_{k}^{i} =\min (C_{ i},S_{k}) }$$

  (8.14)