Cooperating stochastic automata: approximate lumping an reversed process Simonetta Balsamo Gian-Luca Dei Rossi Andrea Marin Dipartimento di Scienze Ambientali, Informatica e Statistica Universit` a Ca’ Foscari, Venezia ISCIS ’12, Paris, 3-4 October 2012
Context: Cooperating stochastic models • Models with underlying Continuous Time Markov Chain (CTMC) • Exploitation of compositionality in model definition • Each component is specified in isolation • Semantics of cooperation is defined so that the joint model can be algorithmically derived • Stochastic automata considered here synchronise on the active/passive semantics • Performance Evaluation Process Algebra (PEPA) active/passive synchronisation • Buchholz’s Communicating Markov Processes • Plateau’s stochastic automata networks (SAN) with master/slave cooperation • . . . Cooperating stochastic automata: approximate lumping an reversed process 2 of 22
Motivation • In general, the state-space’s cardinality of the joint model grows exponentially with the number of components • Steady-state analysis becomes quickly unfeasible • Space cost • Time cost • Numerical stability issues • Workarounds • Approximate analysis (e.g. fluid) • Exploitation of the geometry of the state space • Product-form decomposition • Lumping • Approximate lumping Cooperating stochastic automata: approximate lumping an reversed process 3 of 22
Previous work: Lumping on cooperating automata Definition (Lumping condition) Given active automaton M 1 , a set of labels T , and a partition of the states of M 1 into N 1 clusters C = {C 1 , C 2 , . . . , C N 1 } , we say that C is an exact lumping for M 1 if: � 1 ∈C k q 1 ( s 1 → s ′ 1 ∀C i , C j , C i � = C j , ∀ s 1 ∈ C i 1 ) = ˜ q 1 ( C i → C j ) not s ′ synchronising label � 1 ∈C k q t 1 ( s 1 → s ′ q t 2 ∀ t ∈ T , ∀C i , C j , ∀ s 1 ∈ C i 1 ) = ˜ 1 ( C i → C k ) s ′ where ϕ t s ′ 1 ) = � 1 q t 1 ( s 1 , s ′ 1 ( s 1 , ˜ 1 ) . s ′ 1 ∈ ˜ s ′ • Reduce complexity GBEs’ solution through component-wise lumping • If both automata have a spate-space of cardinality M , time cost reduces from O (( MM ) 3 ) to O (( NM ) 3 ) , where N is the number of clusters in the lumping • Intuition: for each synchronising label the original and lumped automata must behave (in steady-state) equivalently • We treat non-synchronising transitions as a special case • Conditions are stronger than the ones for regular lumpability and weaker than for PEPA strong equivalence Cooperating stochastic automata: approximate lumping an reversed process 4 of 22
Example a, λ 1 a, λ 1 µ 1 s 0 a, 3 4 λ 2 µ 1 a, λ 2 µ 2 µ 3 s 2 C 0 C 1 a, λ 2 a, 1 µ 2 4 λ 2 a, λ 3 s 1 a, λ 3 µ 1 a, λ 1 Cooperating stochastic automata: approximate lumping an reversed process 5 of 22
Marginal distribution Theorem Let M 1 and M 2 be two cooperating automata, where M 2 is passive and M 1 active. If: • M 2 never blocks M 1 ˜ • M 1 is a lumped automaton of M 1 Then the marginal steady state distribution of M 2 in the cooperations M 1 ⊗ M 2 and ˜ M 1 ⊗ M 2 are the same. Note that ergodicity is assumed and the state-space of the joint process is the Cartesian product of the single automata state-spaces. Cooperating stochastic automata: approximate lumping an reversed process 6 of 22
A trivial example a, λ a, λ a, λ P 1 0 1 2 λ µ 1 µ 1 µ 1 a, ⊤ a, ⊤ a, ⊤ P 1 P 2 0 1 2 P 2 µ 2 µ 2 µ 2 µ 1 µ 2 C 0 ˜ a, λ P 1 � � λ � � λ � n � n � � 1 − λ 1 − λ π 1 ( n ) = π 2 ( n ) = µ 1 µ 1 µ 2 µ 2 Cooperating stochastic automata: approximate lumping an reversed process 7 of 22
Another trivial one Q 1 Q 2 µ 2 µ 1 ( n ) λ λ λ λ 0 2 1 Q 1 a, µ 1 (1) a, µ 1 (2) ˜ a, µ 1 (3) Q R a, λ 1 µ 1 (2) µ 1 (1) µ 1 (3) 2 C 1 0 1 Q R 1 a, λ a, λ a, λ a, ⊤ a, ⊤ a, ⊤ 0 2 1 Q 2 µ 2 µ 2 µ 2 Cooperating stochastic automata: approximate lumping an reversed process 8 of 22
Reversed lumping and product-forms • Both previous examples allowed for a lumping into a single cluster • First is derived from the forward automaton • Second is derived from the reversed automaton • In both cases we obtain the marginal distribution, but in the latter we also have product-form! • product-form ⇒ the joint distribution is the product of the marginal ones Corollary (Product-forms) A synchronisation is in product-form if the reversed active automaton can be lumped into a single state Note that, in general the marginal steady state distribution of M 2 in ˜ 1 ⊗ M 2 ≃ the one in ˜ M R M 1 ⊗ M 2 , and is equal in product-form models. Cooperating stochastic automata: approximate lumping an reversed process 9 of 22
Approximation of marginal SSD through aggregation • With our theorem we can reduce the cost to compute marginal steady state distributions of a cooperating automaton if we’re able to find an exact lumping of the other one . • What if this is not feasible or even possible? • We could try to find an approximated lumping. • Can be applied also to the reversed process. • How we evaluate the quality of an approximation? • How we can adapt clustering algorithms to use our definition of (approximated) exact lumping? Cooperating stochastic automata: approximate lumping an reversed process 10 of 22
Evaluating the quality of an approximate lumping How close is an arbitrary state partition W to an exact lumping? • We measure the coefficient of variation of the outgoing fluxes φ t 1 ( s 1 ) of the states in ˜ s 1 . • We further refine that measurement. Cooperating stochastic automata: approximate lumping an reversed process 11 of 22
ǫ -error Definition ( ǫ -error) Given model M 1 and a partition of states W = { ˜ 1 , . . . , ˜ N 1 } , for all ˜ s 1 ∈ W and t > 2 , we define: s 1 π 1 ( s 1 ) φ t � 1 ( s 1 ) t s 1 ∈ ˜ 1 (˜ s 1 ) = φ � s 1 π 1 ( s 1 ) s 1 ∈ ˜ � t π 1 ( s 1 )( φ t � s 1 )) 2 1 ( s 1 ) − φ 1 (˜ ǫ t (˜ � � � . s 1 ) = 1 − exp − � s 1 π 1 ( s 1 ) s ∈ ˜ s 1 ∈ ˜ s 1 1 ( s 1 ) = � N 1 where φ t 1 =1 q t 1 ( s 1 , s ′ 1 ) . s ′ Cooperating stochastic automata: approximate lumping an reversed process 12 of 22
δ -error Definition ( δ -error) 1 , . . . , ˜ Given model M 1 and a partition of states W = { ˜ N 1 } , for all s ′ ˜ s 1 , ˜ 1 ∈ W , we define: s ′ 0 s 1 = ˜ ˜ 1 ∧ t = 1 ϕ t s ′ 1 (˜ s 1 , ˜ 1 ) = ( s 1 π 1 ( s 1 ) ϕ t s ′ 1 ) ) � 1 ( s 1 , ˜ s 1 ∈ ˜ otherwise � s 1 π 1 ( s 1 ) s 1 ∈ ˜ π 1 ( s 1 )( ϕ t s ′ 1 ) − ϕ t s ′ 1 )) 2 1 ( s 1 , ˜ 1 (˜ s 1 , ˜ � 2 = σ t (˜ s ′ � � s 1 , ˜ 1 ) � s 1 π 1 ( s ) s ∈ ˜ s 1 ∈ ˜ s 1 s ′ δ t (˜ s ′ 1 ) = 1 − e − σ (˜ s 1 , ˜ 1 ) s 1 , ˜ where function ϕ t 1 has been defined in Lumping conditions. Cooperating stochastic automata: approximate lumping an reversed process 13 of 22
An ideal algorithm Definition (Ideal algorithm) • Input: automata M 1 , M 2 , T , tolerances ǫ ≥ 0 , δ ≥ 0 • Output: marginal distribution π 1 of M 1 ; approximated marginal distribution of M 2 1 Find the minimum ˜ N ′ 1 such that there exists a partition W = { ˜ 1 , . . . , ˜ N ′ 1 } of the states of M 1 such that ∀ t ∈ T , t > 2 and ∀ ˜ s 1 ∈ W ǫ (˜ s 1 ) ≤ ǫ 2 Let W ′ ← W 3 Check if partition W ′ is such that ∀ t ∈ T , ∀ ˜ s 1 , ˜ s 2 ∈ W , ˜ s 1 � = ˜ s 2 , δ t (˜ s ′ s 1 , ˜ 1 ) ≤ δ . If this is true then return the marginal distribution of M 1 and the approximated of M 2 by computing the marginal distribution of ˜ M 1 ⊗ M 2 and terminate. 4 Otherwise, refine partition W to obtain W new such that the number of clusters of W new is greater than the number of clusters in W ′ . W ′ ← W new . Repeat from Step 3 Cooperating stochastic automata: approximate lumping an reversed process 14 of 22
Constructing the approximate lumped automata Definition (Approx. lumped automata) Given active automaton M 1 , a set of transition types T , and a partition of the states of M 1 into ˜ N 1 clusters W = { ˜ 1 , ˜ 2 , . . . , ˜ N 1 } , then we define the automaton M ≃ 1 as follows: � 1 )˜ λ − 1 ϕ 1 s ′ 1 (˜ s 1 , ˜ if ˜ s 1 � = ˜ s 2 ˜ s ′ 1 E 11 (˜ s 1 , ˜ 1 ) = 0 otherwise ˜ E 12 = I , ˜ ϕ t s ′ 1 ) λ − 1 E 1 t (˜ s 1 , ˜ s 1 ) = 1 (˜ s 1 , ˜ t > 2 t where ˜ N 1 ˜ � ϕ t s ′ λ t = max 1 (˜ s 1 , ˜ 1 ) s 1 =1 ,..., ˜ ˜ N 1 s ′ ˜ 1 =1 are the rates associated with the transition types in the cooperation between M ≃ 1 and M 2 . Cooperating stochastic automata: approximate lumping an reversed process 15 of 22
Initial clustering and refinement phase Initial clustering: • similarity measure can be Euclidean distance between ( φ 3 1 ( s 1 ) , . . . , φ T 1 ( s 1 )) and ( φ 3 1 ( s ′ 1 ) , . . . , φ T 1 ( s ′ )) • can be implemented using various algorithm • hierarchical clustering • K-means (but number of clusters must be decided a priori...) • . . . Refinement phase: • using the tolerance constant δ • distances between clusters depend on clusters themselves = ⇒ K-means cannot be used. • spectral analysis or iterative algorithms Cooperating stochastic automata: approximate lumping an reversed process 16 of 22
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