Product-form solutions for models with joint-state dependent transition rates Simonetta Balsamo, Andrea Marin Universit` a Ca’ Foscari - Venezia Dipartimento di Informatica Italy 2010
Introduction and Motivations Previous works The novel results Conclusion Presentation outline Introduction and Motivations 1 Framework Product-form solutions Motivations Previous works 2 The model of Henderson, Taylor et al. (HT) The novel results 3 Restrictions Main theorem Special cases and examples Conclusion 4 Simonetta Balsamo, Andrea Marin Product-form solutions for models with joint-state dependent transition
Introduction and Motivations Framework Previous works Product-form solutions The novel results Motivations Conclusion Notation We consider Labelled Markovian Automata (LMA) defined as follows: S i = < S i , L i , T i , q i > Let S i be the i -th model S i = { n i , n ′ i , n ′′ i , . . . } : denumerable set of states of S i L i : finite set of labels of S i a i T i = { n i − → n ′ i } : transition from state n i to n ′ i labelled by a i ∈ L q i : T i → R + is a partial function which associates a positive a i − → n ′ real number with each active transition (e.g., q ( n i i ) = λ ) Transitions without rates are passive Transitions with the same label must be all active or all passive P i , A i : sets of passive and active labels of S i . L i = P i ∪ A i Simonetta Balsamo, Andrea Marin Product-form solutions for models with joint-state dependent transition
Introduction and Motivations Framework Previous works Product-form solutions The novel results Motivations Conclusion Closed automaton An automaton S i is closed if P i = ∅ L i = A i All the transitions have an associated rate The transition rates are the parameters of the exponential distributed time needed to carry a transition on The process underlying a closed automaton is a Continuous Time Markov Chain (CTMC) If S i is an open LMA and a ∈ P i , then S i a ← λ is the automaton S i in which each transition labelled by a takes λ as a rate (closure) Simonetta Balsamo, Andrea Marin Product-form solutions for models with joint-state dependent transition
Introduction and Motivations Framework Previous works Product-form solutions The novel results Motivations Conclusion Specifying the cooperation S 1 , . . . , S N is the set of cooperating models The state space is S 1 × S 2 × . . . × S N For each label a ∈ ∪ N i =1 L i we have one of the following: No-cooperating label: a ∈ A i for some i = 1 . . . N and a / ∈ L j with j � = i Cooperating label: a ∈ A i ∩ P j and a / ∈ L k with k � = i , j If a ∈ A i ∩ P j transitions labelled by a in S i and S j can be performed only jointly. The rate of the joint transition is given by the rate of the active transition in S i The automaton resulting from a cooperation has still an underlying CTMC We can specify only pairwise cooperations! Simonetta Balsamo, Andrea Marin Product-form solutions for models with joint-state dependent transition
Introduction and Motivations Framework Previous works Product-form solutions The novel results Motivations Conclusion An example QUEUE 1 QUEUE 2 a λ µ 1 µ 2 Tandem of exponential queues λ λ λ Arrivals according to a 1 2 0 Poisson ( a, µ 1 ) ( a, µ 1 ) ( a, µ 1 ) process ( a, ⊤ ) ( a, ⊤ ) ( a, ⊤ ) Independent service times 1 2 0 µ 2 µ 2 µ 2 Simonetta Balsamo, Andrea Marin Product-form solutions for models with joint-state dependent transition
Introduction and Motivations Framework Previous works Product-form solutions The novel results Motivations Conclusion An example with joint-state dependent transition rates QUEUE 1 QUEUE 2 Tandem of a λ ( n 1 + n 2 ) exponential µ 1 µ 2 queues Arrivals according to a (??) (??) (??) Poisson process 1 2 whose rate 0 depends on the total number of ( a, µ 1 ) ( a, µ 1 ) ( a, µ 1 ) customers in the ( a, ⊤ ) ( a, ⊤ ) ( a, ⊤ ) system 1 2 0 Independent service times µ 2 µ 2 µ 2 Simonetta Balsamo, Andrea Marin Product-form solutions for models with joint-state dependent transition
Introduction and Motivations Framework Previous works Product-form solutions The novel results Motivations Conclusion RCAT product-form Let S 1 , . . . , S N a cooperation of LMAs Assume that the following conditions are satisfied: for each synchronising label a : if a ∈ P i then ∀ n ∈ S i ∃ ! n ′ ∈ S i s.t. n → n ′ ∈ T i a − if a ∈ A i then ∀ n ∈ S i ∃ ! n ′ ∈ S i s.t. n ′ a − → n ∈ T i There exists a set of positive real value K = { K 1 , . . . K T } for each synchronising label a 1 , . . . , a T such that S C = S i { a t ← K t , ∀ a t ∈ P i } satisfies the following condition: i π i ( n ′ ) a u ∀ a u ∈ A i , ∀ n ∈ S i π i ( n ) q i ( n ′ − → n ) = K u Then the steady-state distribution of π of the joint automata is in product-form: N � π ( n ) ∝ π i ( n i ) n = ( n 1 , . . . , n N ) i =1 Simonetta Balsamo, Andrea Marin Product-form solutions for models with joint-state dependent transition
Introduction and Motivations Framework Previous works Product-form solutions The novel results Motivations Conclusion Product-form solutions for model with joint-state dependent rates Values in K represent the reversed rates of the active transitions RCAT requires them to be constant How to check this condition with models in isolation? Is this a necessary condition for joint-state dependent transition rates? QUEUE 1 QUEUE 2 a λ ( n 1 + n 2 ) µ 1 µ 2 n 1 + n 2 − 1 λ ( w ) 1 1 � π ( n 1 , n 2 ) = µ n 1 µ n 2 1 2 w =0 Simonetta Balsamo, Andrea Marin Product-form solutions for models with joint-state dependent transition
Introduction and Motivations Previous works The model of Henderson, Taylor et al. (HT) The novel results Conclusion Solution for queueing networks and stochastic Petri nets in product-form We take inspiration from earlier works of Coleman, Henderson, Taylor, Lucic for Stochastic Petri nets, and Serfozo for queueing networks We explain their technique for the tandem of exponential queues with joint-state dependent arrival rate Define the joint-state dependent rates of station i as follows: q i ( n − 1 i + 1 j ) = ψ ( n − 1 i ) χ i φ ( n ) Simonetta Balsamo, Andrea Marin Product-form solutions for models with joint-state dependent transition
Introduction and Motivations Previous works The model of Henderson, Taylor et al. (HT) The novel results Conclusion Why does it work? Simonetta Balsamo, Andrea Marin Product-form solutions for models with joint-state dependent transition
Introduction and Motivations Restrictions Previous works Main theorem The novel results Special cases and examples Conclusion Restrictions on the model class Simonetta Balsamo, Andrea Marin Product-form solutions for models with joint-state dependent transition
Introduction and Motivations Restrictions Previous works Main theorem The novel results Special cases and examples Conclusion Product-form theorem Simonetta Balsamo, Andrea Marin Product-form solutions for models with joint-state dependent transition
Introduction and Motivations Restrictions Previous works Main theorem The novel results Special cases and examples Conclusion Intuition of the conditions Simonetta Balsamo, Andrea Marin Product-form solutions for models with joint-state dependent transition
Introduction and Motivations Restrictions Previous works Main theorem The novel results Special cases and examples Conclusion The theorem applied to HT models Simonetta Balsamo, Andrea Marin Product-form solutions for models with joint-state dependent transition
Introduction and Motivations Restrictions Previous works Main theorem The novel results Special cases and examples Conclusion An example Simonetta Balsamo, Andrea Marin Product-form solutions for models with joint-state dependent transition
Introduction and Motivations Previous works The novel results Conclusion Conclusion Simonetta Balsamo, Andrea Marin Product-form solutions for models with joint-state dependent transition
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