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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

1

Introduction and Motivations Framework Product-form solutions Motivations

2

Previous works The model of Henderson, Taylor et al. (HT)

3

The novel results Restrictions Main theorem Special cases and examples

4

Conclusion

Simonetta Balsamo, Andrea Marin Product-form solutions for models with joint-state dependent transition

Introduction and Motivations Previous works The novel results Conclusion Framework Product-form solutions Motivations

Notation

We consider Labelled Markovian Automata (LMA) defined as follows: Si =< Si, Li, Ti, qi >

Let Si be the i-th model Si = {ni, n′

i, n′′ i , . . .}: denumerable set of states of Si

Li: finite set of labels of Si Ti = {ni

ai

− → n′

i}: transition from state ni to n′ i labelled by

ai ∈ L qi : Ti → R+ is a partial function which associates a positive real number with each active transition (e.g., q(ni

ai

− → n′

i) = λ)

Transitions without rates are passive Transitions with the same label must be all active or all passive Pi, Ai: sets of passive and active labels of Si. Li = Pi ∪ Ai

Simonetta Balsamo, Andrea Marin Product-form solutions for models with joint-state dependent transition

Introduction and Motivations Previous works The novel results Conclusion Framework Product-form solutions Motivations

Closed automaton

An automaton Si is closed if Pi = ∅

Li = Ai 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 Si is an open LMA and a ∈ Pi, then Sia ← λ is the automaton Si 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 Previous works The novel results Conclusion Framework Product-form solutions Motivations

Specifying the cooperation

S1, . . . , SN is the set of cooperating models The state space is S1 × S2 × . . . × SN For each label a ∈ ∪N

i=1Li we have one of the following:

No-cooperating label: a ∈ Ai for some i = 1 . . . N and a / ∈ Lj with j = i Cooperating label: a ∈ Ai ∩ Pj and a/ ∈ Lk with k = i, j

If a ∈ Ai ∩ Pj transitions labelled by a in Si and Sj can be performed only jointly. The rate of the joint transition is given by the rate of the active transition in Si 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 Previous works The novel results Conclusion Framework Product-form solutions Motivations

An example

QUEUE 1 QUEUE 2

1 2 1 2 a λ λ λ λ (a, µ1) (a, µ1) (a, µ1) (a, ⊤) (a, ⊤) (a, ⊤) µ2 µ2 µ2 µ2 µ1

Tandem of exponential queues Arrivals according to a Poisson process Independent service times

Simonetta Balsamo, Andrea Marin Product-form solutions for models with joint-state dependent transition

Introduction and Motivations Previous works The novel results Conclusion Framework Product-form solutions Motivations

An example with joint-state dependent transition rates

QUEUE 1 QUEUE 2

1 2 1 2 a λ(n1 + n2) (??) (??) (??) (a, µ1) (a, µ1) (a, µ1) (a, ⊤) (a, ⊤) (a, ⊤) µ2 µ2 µ2 µ2 µ1

Tandem of exponential queues Arrivals according to a Poisson process whose rate depends on the total number of customers in the system Independent service times

Simonetta Balsamo, Andrea Marin Product-form solutions for models with joint-state dependent transition

Introduction and Motivations Previous works The novel results Conclusion Framework Product-form solutions Motivations

RCAT product-form

Let S1, . . . , SN a cooperation of LMAs Assume that the following conditions are satisfied: for each synchronising label a: if a ∈ Pi then ∀n ∈ Si ∃! n′ ∈ Si s.t. n

a

− → n′ ∈ Ti if a ∈ Ai then ∀n ∈ Si ∃! n′ ∈ Si s.t. n′

a

− → n ∈ Ti There exists a set of positive real value K = {K1, . . . KT} for each synchronising label a1, . . . , aT such that SC

i

= Si{at ← Kt, ∀at ∈ Pi} satisfies the following condition: ∀au ∈ Ai, ∀n ∈ Si πi(n′) πi(n) qi(n′

au

− → n) = Ku Then the steady-state distribution of π of the joint automata is in product-form: π(n) ∝

N

• i=1

πi(ni) n = (n1, . . . , nN)

Simonetta Balsamo, Andrea Marin Product-form solutions for models with joint-state dependent transition

Introduction and Motivations Previous works The novel results Conclusion Framework Product-form solutions Motivations

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(n1 + n2) µ2 µ1

π(n1, n2) =

n1+n2−1

• w=0

λ(w) 1 µn1

1

1 µn2

2

Simonetta Balsamo, Andrea Marin Product-form solutions for models with joint-state dependent transition

Introduction and Motivations Previous works The novel results Conclusion The model of Henderson, Taylor et al. (HT)

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: qi(n − 1i + 1j) = ψ(n − 1i) φ(n) χi

Simonetta Balsamo, Andrea Marin Product-form solutions for models with joint-state dependent transition

Introduction and Motivations Previous works The novel results Conclusion The model of Henderson, Taylor et al. (HT)

Why does it work?

Simonetta Balsamo, Andrea Marin Product-form solutions for models with joint-state dependent transition

Introduction and Motivations Previous works The novel results Conclusion Restrictions Main theorem Special cases and examples

Restrictions on the model class

Simonetta Balsamo, Andrea Marin Product-form solutions for models with joint-state dependent transition

Introduction and Motivations Previous works The novel results Conclusion Restrictions Main theorem Special cases and examples

Product-form theorem

Simonetta Balsamo, Andrea Marin Product-form solutions for models with joint-state dependent transition

Introduction and Motivations Previous works The novel results Conclusion Restrictions Main theorem Special cases and examples

Intuition of the conditions

Simonetta Balsamo, Andrea Marin Product-form solutions for models with joint-state dependent transition

Introduction and Motivations Previous works The novel results Conclusion Restrictions Main theorem Special cases and examples

The theorem applied to HT models

Simonetta Balsamo, Andrea Marin Product-form solutions for models with joint-state dependent transition

Introduction and Motivations Previous works The novel results Conclusion Restrictions Main theorem Special cases and examples

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