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

**Example text**

Instantiation u ∈ N of task τi demands execution (the job is released or the job arrives) at time moment u · πi . The period πi of any task τi is assumed to be a common multiple of all periods of its predecessor tasks (πj divides πi , where τj ∈ ◦ τi ). Let kij denote πi ◦ πj , τj ∈ τi . 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 .

Because the number of states grows rapidly even with our state reduction approach and each state has to store its probability density function, the memory space required to store the whole process can become prohibitively large. Our solution to master memory complexity is to perform the stochastic process construction and analysis simultaneously. As each arrow updates the time probability density z of the state it leads to, the process has to be constructed in topological order. The result of this procedure is that the process is never stored entirely in memory but rather that a sliding window of states is used for analysis.

Hk is a lower hyperperiod than Hk (Hk < Hk ) if k < k. Consequently, Hk is a higher hyperperiod than Hk (Hk > Hk ) if k > k . For brevity, we say that a state s belongs to a hyperperiod k (denoted s ∈ Hk ) if its PMI field is a subinterval of the hyperperiod k. In our example, three hyperperiods are considered, H0 = [0, 4), H1 = [4, 8), and H2 = [8, 12). 6, s1 , s2 , . . , s7 ∈ H0 , s8 , s9 , . . 6). In general, let us consider a state s and let Ps be the set of its predecessor states. Let k denote the order of the state s defined as the lowest hyperperiod of the states in Ps (k = min{j : s ∈ Hj , s ∈ Ps }).