By by Sorin Manolache.

ISBN-10: 9185457604

ISBN-13: 9789185457601

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**Additional info for Analysis and optimisation of real-time systems with stochastic behaviour**

**Example text**

4 Construction and Analysis Algorithm The analysis is performed in two phases: 1. Divide the interval [0, LCM ) in PMIs, 2. Construct the stochastic process in topological order and analyse it. 46 CH. 4. e. A = {x|0 ≤ x ≤ LCM, ∃1 ≤ i ≤ N, ∃k ∈ N : x = kπi }. e. D = {x|0 ≤ x ≤ LCM, ∃1 ≤ i ≤ N, ∃k ∈ N : x = kπi + δi }. The set of PMIs of [0, LCM ) is {[a, b)|a, b ∈ A ∪ D∧ ∃x ∈ (A ∪ D) ∩ (a, b)}. If PMIs of a higher hyperperiod Hk , k > 0, are needed during the analysis, they are of the form [a + k · LCM, b + k · LCM ), where [a, b) is a PMI of [0, LCM ).

Thus, the behaviour of the system over the intervals [k·LCM, (k+1)·LCM ), k ∈ N, is statistically equivalent to the behaviour over the time interval [0, LCM ). Therefore, in the case when bi = 1, 1 ≤ i ≤ g, it is sufficient to analyse the system solely over the time interval [0, LCM ). One could choose the following state space definition: S = {(τ, W, t) : τ ∈ T, W ∈ set of all multisets of T, t ∈ R}, where τ represents the currently running task, W stands for the multiset1 of ready tasks at the start time of the running task, and t represents the start time of the currently running task.

In order to illustrate the construction of the stochastic process in the case bi > 1, when several instantiations of a task graph Γi may exist at the same time in the system, let us consider an application consisting of two independent tasks, τ1 and τ2 , with periods 2 and 4 respectively. LCM = 4 in this case. The tasks are scheduled according to a rate monotonic (RM) policy [LL73]. At most one active instantiation of τ1 is tolerated in the system at a certain time (b1 = 1) and at most two concurrently active instantiations of τ2 are tolerated in the system (b2 = 2).

### Analysis and optimisation of real-time systems with stochastic behaviour by by Sorin Manolache.

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