Analyzing Preemptive Global Scheduling

Abstract

Recently, there have been several promising techniques developed for schedulability analysis and response time analysis for multiprocessor systems based on over-approximation.

Appendix: Proof of Lemma 4.3

Proof.

We recall that by definition, Ω _k (x) consists of two sums over the sets τ ^NC and τ ^CI which are a partitioning of τ such that Ω _k (x) is maximal:

$$\displaystyle{\varOmega _{k}(x) =\sum _{\tau _{i}\in \tau ^{\mathsf{CI}}}I_{k}^{\mathsf{CI}}(\tau _{ i},x) +\sum _{\tau _{i}\in \tau ^{\mathsf{NC}}}I_{k}^{\mathsf{NC}}(\tau _{ i},x)}$$

Let $\vartheta ^{\mathsf{CI}} \subseteq \tau ^{\mathsf{CI}}$ and $\vartheta ^{\mathsf{NC}} \subseteq \tau ^{\mathsf{NC}}$ be subsets of both partitions, such that

$$\displaystyle{\forall \tau _{i} \in \vartheta ^{\mathsf{CI}}: W_{ k}^{\mathsf{CI}}(\tau _{ i},x) > x - C_{k} + 1}$$

$$\displaystyle{\forall \tau _{i} \in \vartheta ^{\mathsf{NC}}: W_{ k}^{\mathsf{NC}}(\tau _{ i},x) > x - C_{k} + 1}$$

Thus, $\vartheta:=\vartheta ^{\mathsf{CI}} \cup \vartheta ^{\mathsf{NC}}$ captures the relatively “dense” tasks of τ. Using this, I _k ^CI(τ _i , x, h) and I _k ^NC(τ _i , x, h) can be rewritten using W _k ^CI(τ _i , x) and W _k ^NC(τ _i , x) in the definition of Ω _k (x):

$$\displaystyle{ \varOmega _{k}(x) = \vert \vartheta \vert \cdot (x - C_{i} + 1) +\sum _{\tau _{i}\in \tau ^{\mathsf{CI}}\setminus \vartheta ^{\mathsf{CI}}}W_{k}^{\mathsf{CI}}(\tau _{ i},x) +\sum _{\tau _{i}\in \tau ^{\mathsf{NC}}\setminus \vartheta ^{\mathsf{NC}}}W_{k}^{\mathsf{NC}}(\tau _{ i},x) }$$

(4.11)

We consider the case of $\vert \vartheta \vert < M$. (Otherwise, the lemma obviously holds.)

Now let x < f _k − t ₀, as in the assumption of the lemma, so the Job J _k is still active at time point x. Thus, only at strictly less than C _k time points of the interval [t ₀, t ₀ + x), J _k was able to run. Now we know that all tasks from $\vartheta$ could keep at most $\vert \vartheta \vert$ processors busy at each time unit during the interval. It follows that the remaining tasks (those from $\tau \setminus \vartheta$) kept the remaining $M -\vert \vartheta \vert$ processors busy for at least x − C _k + 1 time units during the interval (otherwise, J _k would have been able to execute for C _k time units and thus finish until t ₀ + x). Consequently, the tasks from $\tau \setminus \vartheta$ must have generated a workload of at least $(M -\vert \vartheta \vert ) \cdot (x - C_{k} + 1)$ over the considered x time units. Since W _k ^CI(τ _i , x) and W _k ^NC(τ _i , x) are upper bounds of their workloads, we have

$$\displaystyle{ \sum _{\tau _{i}\in \tau ^{\mathsf{CI}}\setminus \vartheta ^{\mathsf{CI}}}W_{k}^{\mathsf{CI}}(\tau _{ i},x) +\sum _{\tau _{i}\in \tau ^{\mathsf{NC}}\setminus \vartheta ^{\mathsf{NC}}}W_{k}^{\mathsf{NC}}(\tau _{ i},x) \geq (M -\vert \vartheta \vert ) \cdot (x - C_{k} + 1) }$$

(4.12)

From (4.11) and (4.12) it follows

$$\displaystyle{\varOmega _{k}(x) \geq M \cdot (x - C_{k} + 1),}$$

which is equivalent to the lemma.