Appendix to Temporal Analysis of Static Priority Preemptive Scheduled Cyclic Streaming Applications using CSDF Models

This is the appendix to the paper Temporal Analysis of Static Priority Preemptive Scheduled Cyclic Streaming Applications using CSDF Models [1]. The temporal analysis approach presented in [1] makes use of an iterative algorithm that computes so-called maximum busy periods over multiple task phases. The algorithm contains a stop criterion indicating after which iteration of the algorithm subsequent iterations do not need to be considered. The intuition behind that stop criterion is given in the paper and supplemented by a formal proof in this appendix. A1. VALIDITY OF THE STOP CRITERION Figure A1 recaps the algorithm presented in Figure 7 of [1]. In order to prove the validity of the stop criterion in line 14 we need to distinguish between the maximum busy periods and maximum finish times computed in different iterations. Consequently we introduce an index n that we use for w ′〈n〉 ix , w 〈n〉 ix , Z 〈n〉 ix and f̂ 〈n〉 ix′ . We define the relation between x, x′ = xn, q = qn and n as follows: n = qn ·Θi + xn − x As one can easily see this definition leads to n being initially zero and increasing by one in each iteration of the while-loop. Using this indexing and taking into account that x′ = xmust hold for exiting the while-loop we can reformulate the stop criterion more explicitly as follows (with q∗ the q for which the stop criterion is met): w 〈qΘi−1〉 ix ≤ q ∗ · Pi (A1) Note that the term −1 appears as the increase of n to qΘi (and thus x′ = x) occurs after the computation of the last maximum busy period. Moreover, the stop criterion for q∗ would not be checked if it were already true for a q′ with 0 < q′ < q∗. This implies: ∀0 q ′ · Pi (A2) In the following we prove that given these two criteria we do not have to consider any w 〈qΘi+k〉 ix with k ≥ 0 as the maximum finish times of task phases cannot become larger for any of these maximum busy periods. We conduct the proof by comparing interference characterizations, then extend these observations to maximum busy periods and finally maximum finish times. We begin with the period-andjitter interference characterization: Lemma A1. It holds: ∀k≥0 : ηjy(w 〈qΘi+k〉 ix )− ηjy(w 〈qΘi−1〉 ix ) ≤ ηjy(w ′〈k〉 ix ) w 〈qΘi+k〉 ix − w 〈qΘi−1〉 ix ≤ w ′〈k〉 ix Proof. With the subadditivity of the ceiling function da + be ≤ dae + dbe it follows with a = c − d and b = d 1 ∀0≤x<Θi : f̂ix = 0 ; 2 f o r a l l (x : ejyix ∈ E ) { 3 x′ = x ; q = 0 ; w′ ix = wix = 0 ; Zix = ∅ ; 4 do { 5 w⊕ ′ ix = Cix′ + ∑ jy∈hp(i) [ηjy(w ′ ix + w ⊕′ ix )− ηjy(w ′ ix)] · Cjy ;