By by Sorin Manolache.

ISBN-10: 9185457604

ISBN-13: 9789185457601

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**Extra resources for Analysis and optimisation of real-time systems with stochastic behaviour**

**Example text**

Thus, the execution time probability density ti accounts for this intra-processor communication time. Next, we discuss the modelling of the communication time between two processing tasks that are mapped on different processing elements. Let ti and tj be two processing tasks, let χ be a communication task, let P Ea and P Eb be two distinct processing elements and let B be a bus such that all of the following statements are true: • Processing tasks ti and tj are mapped on processing elements P Ea and P Eb respectively (M apP (ti ) = P Ea and M apP (tj ) = P Eb ).

Also, we determine the multiset Discarded of those tasks that are ready to execute when task τi starts, but are discarded in the mean time, as the execution of task τi spans beyond their deadlines. Once task τi completes its execution, some of its successor tasks may become ready to execute. The successor tasks, which become ready to execute as a result of task τi ’s completion, form the set Enabled. The new multiset of ready tasks, W , is the union of the old multiset of ready tasks except the ones that are discarded during the execution of task τi , Wi \Discarded, and the set Enabled and those newly arrived tasks that have no predecessor and therefore are immediately ready to run.

Instantiation u ∈ N of task τi may start executing only if instantiations u · kij , u · kij + 1, . . , u · kij + kij − 1 of tasks τj , ∀τj ∈ ◦ τi , have completed their execution. Let ΠΓ = {πΓ1 , πΓ2 , . . , πΓg } denote the set of task graph periods where πΓj denotes the period of the task graph Γj . πΓj is equal to the least common multiple of all πi , where πi is the peπΓ riod of τi and τi ∈ Vj . Task τi ∈ Vj is instantiated Ji = πij times during one instantiation of task graph Γj . The k th instantiation of task graph Γj , k ≥ 0, denoted (Γj , k), is composed of the jobs (τi , u), where τi ∈ Vj and u ∈ {k · Ji , k · Ji + 1, .

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

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