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

By by Sorin Manolache.

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Séminaire Bourbaki, Vol. 43, 2000-2001, Exp. 880-893

Desk of Contents

Seminaire Bourbaki quantity forty three web page 1 2000-2001 [doi UNKNOWN] Laurent Bonavero -- Factorisation faible des purposes birationnelles

Seminaire Bourbaki quantity forty three web page 39 2000-2001 [doi UNKNOWN] Marco Brunella -- Courbes entieres dans les surfaces algebriques complexes

Seminaire Bourbaki quantity forty three web page sixty three 2000-2001 [doi UNKNOWN] Michel Emery -- Espaces probabilises filtres : de los angeles theorie de Vershik au mouvement brownien, through des idees de Tsirelson

Seminaire Bourbaki quantity forty three web page eighty five 2000-2001 [doi UNKNOWN] Marc Herzlich -- Linegalite de Penrose

Seminaire Bourbaki quantity forty three web page 113 2000-2001 [doi UNKNOWN] Michele Audin -- Integrabilite et non-integrabilite de systemes hamiltoniens

Seminaire Bourbaki quantity forty three web page 137 2000-2001 [doi UNKNOWN] Pierre Cartier -- Fonctions polylogarithmes, nombres polyzetas et groupes pro-unipotents

Seminaire Bourbaki quantity forty three web page one hundred seventy five 2000-2001 [doi UNKNOWN] Antoine Chambert-Loir -- Theoremes d algebricite en geometrie diophantienne

Seminaire Bourbaki quantity forty three web page 211 2000-2001 [doi UNKNOWN] Philippe Michel -- Repartition des zeros des fonctions L et matrices aleatoires

Seminaire Bourbaki quantity forty three web page 249 2000-2001 [doi UNKNOWN] Michele Vergne -- Quantification geometrique et aid symplectique

Seminaire Bourbaki quantity forty three web page 279 2000-2001 [doi UNKNOWN] Philippe Biane -- Entropie libre et algebres d operateurs

Seminaire Bourbaki quantity forty three web page 301 2000-2001 [doi UNKNOWN] man Henniart -- Progres recents en fonctorialite de Langlands

Seminaire Bourbaki quantity forty three web page 323 2000-2001 [doi UNKNOWN] Emmanuel Peyre -- issues de hauteur bornee et geometrie des varietes

Seminaire Bourbaki quantity forty three web page 345 2000-2001 [doi UNKNOWN] Georges Skandalis -- Geometrie non commutative, operateur de signature transverse et algebres de Hopf

Seminaire Bourbaki quantity forty three web page 365 2000-2001 [doi UNKNOWN] Cedric Villani -- Limites hydrodynamiques de l equation de Boltzmann

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 }).

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