stochastic process algebras and ordinary differential...

Stochastic Process Algebras and OrdinaryDifferential Equations

Jane Hillston

Laboratory for Foundations of Computer Scienceand Centre for Systems Biology at Edinburgh

University of Edinburgh

5th September 2011

Outline

1 IntroductionStochastic Process Algebra

2 Continuous ApproximationState variables

3 Fluid-Flow SemanticsFluid Structured Operational Semantics

4 Conclusions

Outline

4 Conclusions

Process Algebra

• Models consist of agents which engage in actions.

α.P��*

action typeor name

agent/component

• The structured operational (interleaving) semantics of thelanguage is used to generate a labelled transition system.

Process algebra model Labelled transition system-SOS rules

Process Algebra

α.P��*

action typeor name

agent/component

Process Algebra

α.P��*

action typeor name

agent/component

Stochastic process algebras

Process algebras where models are decorated with quantitativeinformation used to generate a stochastic process are stochasticprocess algebras (SPA).

Stochastic Process Algebra

• Models are constructed from components which engage inactivities.

(α, r).P��* 6 HH

action typeor name

activity rate(parameter of an

exponential distribution)

component/derivative

• The language is used to generate a Continuous Time MarkovChain (CTMC) for performance modelling.

SPAMODEL

LABELLEDMULTI-

TRANSITIONSYSTEM

CTMC Q- -SOS rules state transition

diagram

(α, r).P��* 6 HH

action typeor name

SPAMODEL

LABELLEDMULTI-

TRANSITIONSYSTEM

diagram

(α, r).P��* 6 HH

action typeor name

SPAMODEL

LABELLEDMULTI-

TRANSITIONSYSTEM

diagram

Why use a process algebra?

• High level description of the system eases the task of modelconstruction.

• Formal language allows for unambiguous interpretation andautomatic translation into the underlying mathematicalstructure.

• Moreover properties of that mathematical structure may bededuced by the construction at the process algebra level.

• Furthermore formal reasoning techniques such as equivalencerelations and model checking can be used to manipulate orinterrogate models.

• Compositionality can be exploited both for model constructionand (in some cases) for model analysis.

Performance Evaluation Process Algebra (PEPA)

(α, f ).P PrefixP1 + P2 ChoiceP1 ��

LP2 Co-operation

P/L HidingX Variable

LP2 Co-operation

P1 ‖ P2 is a derived form for P1 ��∅ P2.

When working with large numbers of entities, we write P[n] todenote an array of n copies of P executing in parallel.

P[5] ≡ (P ‖ P ‖ P ‖ P ‖ P)

LP2 Co-operation

P1 ‖ P2 is a derived form for P1 ��∅ P2.

When working with large numbers of entities, we write P[n] todenote an array of n copies of P executing in parallel.

P[5] ≡ (P ‖ P ‖ P ‖ P ‖ P)

Rates of interaction: bounded capacity

Stochastic process algebras differ in how they define the rate ofsynchronised actions. In PEPA cooperation between componentsgives rise to shared actions, the rate of which are governed by theassumption of bounded capacity.

Bounded capacityNo component can be made to carry out an action in cooperationfaster than its own defined rate for the action.

Thus shared actions proceed at the minimum of the rates in theparticipating components.

In contrast independent actions do not constrain each other and ifthere are multiple copies of a action enabled in independentconcurrent components their rates are summed.

A simple example: processors and resources

Proc0def= (task1, r1).Proc1

Res0def= (task1, r3).Res1

Res1def= (reset, r4).Res0

Proc0 ��{task1}

?(task1, R)

Proc1 ��{task1}

��

(reset, r4)@@@R(task2, r2)

Proc1 ��{task1}

Res0��(task2, r2)

Proc0 ��{task1}

Res1AAAAAAAK (reset, r4)

R = min(r1, r3)

−R R 0 0

0 −(r2 + r4) r4 r2r2 0 −r2 0r4 0 0 −r4

Proc0 ��{task1}

?(task1, R)

Proc1 ��{task1}

��

Proc1 ��{task1}

Proc0 ��{task1}

R = min(r1, r3)

−R R 0 0

0 −(r2 + r4) r4 r2r2 0 −r2 0r4 0 0 −r4

Proc0 ��{task1}

?(task1, R)

Proc1 ��{task1}

��

Proc1 ��{task1}

Proc0 ��{task1}

R = min(r1, r3)

−R R 0 0

0 −(r2 + r4) r4 r2r2 0 −r2 0r4 0 0 −r4

Solving discrete state models

As we have seen the SOSsemantics of a SPA model ismapped to a CTMC withglobal states determined bythe local states of theparticipating components.

When the size of the statespace is not too large they areamenable to numerical solution(linear algebra) to determine asteady state or transientprobability distribution.

q1,1 q1,2 · · · q1,N

q2,1 q2,2 · · · q2,N

......

...qN,1 qN,2 · · · qN,N

π(t) = (π1(t), π2(t), . . . , πN(t))

q1,1 q1,2 · · · q1,N

q2,1 q2,2 · · · q2,N

......

...qN,1 qN,2 · · · qN,N

π(t) = (π1(t), π2(t), . . . , πN(t))

q1,1 q1,2 · · · q1,N

q2,1 q2,2 · · · q2,N

......

...qN,1 qN,2 · · · qN,N

π(t) = (π1(t), π2(t), . . . , πN(t))

Alternatively they may bestudied using stochasticsimulation. Each run generatesa single trajectory through thestate space. Many runs areneeded in order to obtainaverage behaviours.

Benefits of process algebra

Beyond the clear benefits for model construction to be derivedfrom using a high-level language with compositionality the formalnature of the process algebra specification has been exploited in anumber of ways.

For example,

• The correspondence between the congruence, Markovianbisimulation, in the process algebra and the lumpabilitycondition in the CTMC, allows exact model reduction to becarried out compositionally.

• Characterisation of product form structure at the processalgebra level allows decomposed model solution based on theprocess algebra structure of the model.

• Stochastic model checking based on the CSL family oftemporal logics allows evaluation of quantified properties ofthe behaviour of the system

For example,

Disadvantages of process algebra

The primary disadvantage of stochastic process algebras, shared byall discrete event modelling paradigms, is the problem of statespace explosion, also known as the curse of dimensionality.

Simple example revisited

Proc0[NP ] ��{task1}

Res0[NR ]

CTMC interpretationProcessors (NP ) Resources (NR ) States (2NP+NR )1 1 42 1 82 2 163 2 323 3 644 3 1284 4 2565 4 5125 5 10246 5 20486 6 40967 6 81927 7 163848 7 327688 8 655369 8 1310729 9 26214410 9 52428810 10 1048576

This is particularly a problem for population models — systemswhere we are interested in interacting populations of entities:

Large scale software systemsIssues of scalability are important for user satisfaction andresource efficiency but such issues are difficult to investigate usingdiscrete state models.

Biochemical signalling pathwaysUnderstanding these pathways has the potential to improve thequality of life through enhanced drug treatment and better drugdesign.

Epidemiological systemsImproved modelling of these systems could lead to improveddisease prevention and treatment in nature and better security incomputer systems.

Crowd dynamicsTechnology enhancement is creating new possibilities for directingcrowd movements in buildings and urban spaces, for example foremergency egress, which are not yet well-understood.

A conundrum

Process algebras are well-suited to constructing such models:

• Developed to represent concurrent behaviour compositionally;

• Represent the interactions between individuals explicitly;

• Stochastic extensions allow the dynamics of system behaviourto be captured;

• Incorporate formal apparatus for reasoning about thebehaviour of systems.

But solution techniques which rely on explicitly building the statespace, such as numerical solution, are hampered by spacecomplexity...

...whilst those that use the implicit state space, such as simulation,run into problems of time complexity.

A conundrum

The CODA project

In the CODA project we have been developing stochastic processalgebras and associated theory, tailored to the construction andevaluation of the collective dynamics of large systems ofinteracting entities.

One approach to this is to keep the discrete state representation inthe model and to evaluate it algorithmically rather thananalytically, i.e. carry out a discrete event simulation of the modelto explore its possible behaviours.

However, our main approach has been to use a countingabstraction in order to make a shift to population statistics and todevelop a fluid approximation.

The CODA project

Population statistics: emergent behaviour

A shift in perspective allows us to model the interactions betweenindividual components but then only consider the system as awhole as an interaction of populations.

This allows us to model much larger systems than previouslypossible but in making the shift we are no longer able to collectany information about individuals in the system.

To characterise the behaviour of a population we count thenumber of individuals within the population that are exhibitingcertain behaviours rather than tracking individuals directly.

Then we make a continuous approximation of how the counts varyover time.

Novelty

The novelty in this approach is twofold:

Linking process algebra and continuous mathematical models fordynamic evaluation represents a paradigm shift in how suchformal models are analysed.

The prospect of formally-based quantified evaluation of dynamicbehaviour could have significant impact in application domainswhich have traditionally worked directly at the level of fittingdifferential equation models.

Novelty

The novelty in this approach is twofold:

Linking process algebra and continuous mathematical models fordynamic evaluation represents a paradigm shift in how suchformal models are analysed.

The prospect of formally-based quantified evaluation of dynamicbehaviour could have significant impact in application domainswhich have traditionally worked directly at the level of fittingdifferential equation models.

Outline

4 Conclusions

Alternative Representations

ContinuousApproximation

of CTMCpopulation view

explicit stateCTMC

StochasticSimulationof CTMC

individual view

LargePEPA model

-��

��

��*

HHHHHH

of CTMC

population view

explicit stateCTMC

individual view

LargePEPA model

-��

��

��*

HHHHHH

explicit stateCTMC

individual view

LargePEPA model

��

��*

HHHHHH

A new approach to SPA semantics

1 Use a counting abstraction rather than the CTMC completestate space.

2 Assume that these state variables are subject to continuousrather than discrete change.

3 No longer aim to calculate the probability distribution overthe entire state space of the model.

4 Instead the trajectory of the ODEs estimates the expectedbehaviour of the CTMC.

Models suitable for counting abstraction

• In the PEPA language multiple instances of components arerepresented explicitly — we write P[n] to denote an array of ncopies of P executing in parallel.

P[5] ≡ (P ‖ P ‖ P ‖ P ‖ P)

• The impact of an action of a counting variable is

• decrease by 1 if the component participates in the action• increase by 1 if the component is the result of the action• zero if the component is not involved in the action.

P[5] ≡ (P ‖ P ‖ P ‖ P ‖ P)

• The impact of an action of a counting variable is

• decrease by 1 if the component participates in the action• increase by 1 if the component is the result of the action• zero if the component is not involved in the action.

P[5] ≡ (P ‖ P ‖ P ‖ P ‖ P)

• The impact of an action of a counting variable is• decrease by 1 if the component participates in the action

• increase by 1 if the component is the result of the action• zero if the component is not involved in the action.

P[5] ≡ (P ‖ P ‖ P ‖ P ‖ P)

• The impact of an action of a counting variable is• decrease by 1 if the component participates in the action• increase by 1 if the component is the result of the action

• zero if the component is not involved in the action.

P[5] ≡ (P ‖ P ‖ P ‖ P ‖ P)

• The impact of an action of a counting variable is• decrease by 1 if the component participates in the action• increase by 1 if the component is the result of the action• zero if the component is not involved in the action.

Res0[NR ]

• task1 decreases Proc0 and Res0

• task1 increases Proc1 and Res1

• task2 decreases Proc1

• task2 increases Proc0

• reset decreases Res1

• reset increases Res0

Res0[NR ]

dx1dt = −min(r1 x1, r3 x3) + r2 x2

x1 = no. of Proc1

• task1 decreases Proc0

• task1 is performed by Proc0

and Res0

• task2 increases Proc0

• task2 is performed by Proc1

Res0[NR ]

ODE interpretationdx1dt = −min(r1 x1, r3 x3) + r2 x2

x1 = no. of Proc1dx2dt = min(r1 x1, r3 x3)− r2 x2

x2 = no. of Proc2dx3dt = −min(r1 x1, r3 x3) + r4 x4

x3 = no. of Res0dx4dt = min(r1 x1, r3 x3)− r4 x4

x4 = no. of Res1

of CTMCpopulation viewset of ODEs

full generator matrix

(implicit)

explicit stateCTMC

Aggregatedreduced generatormatrix

generator functions abstract CTMC

individual view

LargePEPA model

-��

��

��*

HHHHHH

fluid approximation

counting abstraction

of CTMC

population viewset of ODEs

(implicit)

explicit stateCTMC

individual view

LargePEPA model

-��

��

��*

HHHHHH

fluid approximation

set of ODEs

(implicit)

explicit stateCTMC

individual view

LargePEPA model

��

��*

HHHHHH

fluid approximation

(implicit)

explicit stateCTMC

individual view

LargePEPA model

--��

��

��*

HHHHHH

fluid approximation

(implicit)

explicit state

explicit stateCTMC

individual view

LargePEPA model

--��

��

��*

HHHHHH

fluid approximation

of CTMC

population view

set of ODEs

(implicit)

explicit state

explicit stateCTMC

individual view

LargePEPA model

--��

��

��*

HHHHHH

fluid approximation

of CTMC

population view

set of ODEs

(implicit)

explicit state

explicit stateCTMC

Aggregated

reduced generatormatrix

individual view

LargePEPA model

--��

��

��*

HHHHHH

fluid approximation

Outline

4 Conclusions

Deriving a Fluid Approximation of a PEPA model

The aim is to represent the CTMC implicitly (avoiding state spaceexplosion), and to generate the set of ODEs which are the fluidlimit of that CTMC.

The exisiting SOS semantics is not suitable because it constructsthe state space of the CTMC explicitly.

Instead we define a structured operational semantics which definesthe possible transitions of an abitrary abstract state, giving theCTMC implicitly as a set of generator functions, which directlygive rise to the ODEs.

SPAMODEL

SYMBOLICLABELLED

TRANSITIONSYSTEM

ABSTRACTCTMC

ODEsFM(x)- - -SOS rules generator

functions

fluidapprox

SPAMODEL

SYMBOLICLABELLED

TRANSITIONSYSTEM

ABSTRACTCTMC

functions

fluidapprox

SPAMODEL

SYMBOLICLABELLED

TRANSITIONSYSTEM

ABSTRACTCTMC

functions

fluidapprox

SPAMODEL

SYMBOLICLABELLED

TRANSITIONSYSTEM

ABSTRACTCTMC

functions

fluidapprox

SPAMODEL

SYMBOLICLABELLED

TRANSITIONSYSTEM

ABSTRACTCTMC

functions

fluidapprox

Fluid Structured Operational Semantics

In order to get to the implicit representation of the CTMC we needto:

1 Make the counting abstraction (Context Reduction)

2 Collect the transitions of the reduced context (Jump Multiset)

3 Calculate the rate of the transitions in terms of an arbitrarystate of the CTMC.

Once this is done we can extract the vector field FM(x) from thejump multiset.

Context Reduction

Proc0def= (task1 , r1 ).Proc1

Res0def= (task1 , r3 ).Res1

Res1def= (reset, r4 ).Res0

Systemdef= Proc0 [NP ] ��

{task1}Res0 [NR ]

R(System) = {Proc0 ,Proc1} ��{task1}

{Res0 ,Res1}

Population Vector

ξ = (ξ1, ξ2, ξ3, ξ4)

Context Reduction

{task1}Res0 [NR ]

R(System) = {Proc0 ,Proc1} ��{task1}

{Res0 ,Res1}

Population Vector

ξ = (ξ1, ξ2, ξ3, ξ4)

Location Dependency

Systemdef= Proc0 [N ′C ] ��

{task1}Res0 [NS ] ‖ Proc0 [N ′′C ]

{Proc0 ,Proc1} ��{task1}

{Res0 ,Res1} ‖ {Proc0 ,Proc1}

Population Vector

ξ = (ξ1, ξ2, ξ3, ξ4, ξ5, ξ6)

Location Dependency

Population Vector

ξ = (ξ1, ξ2, ξ3, ξ4, ξ5, ξ6)

Location Dependency

Population Vector

ξ = (ξ1, ξ2, ξ3, ξ4, ξ5, ξ6)

Fluid Structured Operational Semantics by Example

{task1}Res0 [NR ]

ξ = (ξ1, ξ2, ξ3, ξ4)

{task1}Res0 [NR ]

ξ = (ξ1, ξ2, ξ3, ξ4)

Proc0task1 ,r1−−−−−−→ Proc1

Proc0task1 ,r1ξ1−−−−−−−→∗ Proc1

{task1}Res0 [NR ]

ξ = (ξ1, ξ2, ξ3, ξ4)

Res0task1 ,r3−−−−−−→ Res1

Res0task1 ,r3ξ3−−−−−−−→∗ Res1

{task1}Res0 [NR ]

ξ = (ξ1, ξ2, ξ3, ξ4)

Res0task1 ,r3ξ3−−−−−−−→∗ Res1

Proc0 ��{task1}

Res0task1 ,r(ξ)−−−−−−−→∗ Proc1 ��

{task1}Res1

Apparent Rate Calculation

Proc0task1 ,r1 ξ1−−−−−−−→∗ Proc1

Res0task1 ,r3 ξ3−−−−−−−→∗ Res1

Proc0 ��{task1}

Res0task1 ,r(ξ)−−−−−−−→∗ Proc1 ��

{task1}Res1

r(ξ)=min(r1ξ1, r3ξ4

Apparent Rate Calculation

Proc0task1 ,r1 ξ1−−−−−−−→∗ Proc1

Res0task1 ,r3 ξ3−−−−−−−→∗ Res1

Proc0 ��{task1}

Res0task1 ,r(ξ)−−−−−−−→∗ Proc1 ��

{task1}Res1

r(ξ)=min(r1ξ1, r3ξ4

f (ξ, l , α) as the Generator Matrix of the Lumped CTMC

(P0 ‖ P0 ‖ P0 ) ��{task1}

(R0 ‖ R0 )

(P1 ‖ P0 ‖ P0 ) ��{task1}

R1 ‖ R0 )

(P1 ‖ P0 ‖ P0 ) ��{task1}

(R0 ‖ R1 )

(P0 ‖ P1 ‖ P0 ) ��{task1}

(R1 ‖ R0 )

(P0 ‖ P1 ‖ P0 ) ��{task1}

(R0 ‖ R1 )

(P0 ‖ P0 ‖ P1 ) ��{task1}

(R1 ‖ R0 )

(P0 ‖ P0 ‖ P1 ) ��{task1}

(R0 ‖ R1 )

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��

XXXXXXzZZZZZZ~

SSSSSSSSw

(P0 ‖ P0 ‖ P0 ) ��{task1}

(R0 ‖ R0 )

(P1 ‖ P0 ‖ P0 ) ��{task1}

R1 ‖ R0 )

(P1 ‖ P0 ‖ P0 ) ��{task1}

(R0 ‖ R1 )

(P0 ‖ P1 ‖ P0 ) ��{task1}

(R1 ‖ R0 )

(P0 ‖ P1 ‖ P0 ) ��{task1}

(R0 ‖ R1 )

(P0 ‖ P0 ‖ P1 ) ��{task1}

(R1 ‖ R0 )

(P0 ‖ P0 ‖ P1 ) ��{task1}

(R0 ‖ R1 )

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SSSSSSSSw

(P0 ‖ P0 ‖ P0 ) ��{task1}

(R0 ‖ R0 )

(P1 ‖ P0 ‖ P0 ) ��{task1}

(R1 ‖ R0 )

(P1 ‖ P0 ‖ P0 ) ��{task1}

(R0 ‖ R1 )

(P0 ‖ P1 ‖ P0 ) ��{task1}

(R1 ‖ R0 )

(P0 ‖ P1 ‖ P0 ) ��{task1}

(R0 ‖ R1 )

(P0 ‖ P0 ‖ P1 ) ��{task1}

(R1 ‖ R0 )

(P0 ‖ P0 ‖ P1 ) ��{task1}

(R0 ‖ R1 )

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XXXXXXzZZZZZZ~

SSSSSSSSw

(3, 0, 2, 0) -min(3r1, 2r3)(2, 1, 1, 1)(3, 0, 2, 0) -min(3r1, 2r3)(2, 1, 1, 1)

Jump Multiset

Proc0 ��{task1}

Res0task1 ,r(ξ)−−−−−−−→∗ Proc1 ��

{task1}Res1

r(ξ) = min(r1ξ1, r3ξ3

Proc1 ��{task1}

Res0task2 ,ξ2r2−−−−−−−→∗ Proc0 ��

{task1}Res0

Proc0 ��{task1}

Res1reset,ξ4r4−−−−−−−→∗ Proc0 ��

{task1}Res0

Jump Multiset

Proc0 ��{task1}

Res0task1 ,r(ξ)−−−−−−−→∗ Proc1 ��

{task1}Res1

Proc1 ��{task1}

Res0task2 ,ξ2r2−−−−−−−→∗ Proc0 ��

{task1}Res0

Proc0 ��{task1}

{task1}Res0

Jump Multiset

Proc0 ��{task1}

Res0task1 ,r(ξ)−−−−−−−→∗ Proc1 ��

{task1}Res1

Proc1 ��{task1}

Res0task2 ,ξ2r2−−−−−−−→∗ Proc0 ��

{task1}Res0

Proc0 ��{task1}

{task1}Res0

Construction of f (ξ, l , α)

Proc0 ��{task1}

{task1}Res0

• Take l = (0, 0, 0, 0)

• Add −1 to all elements of l corresponding to the indices ofthe components in the lhs of the transition

l = (−1, 0, 0,−1)

• Add +1 to all elements of l corresponding to the indices ofthe components in the rhs of the transition

l = (−1 + 1, 0,+1,−1) = (0, 0,+1,−1)

f(ξ, (0, 0,+1,−1), reset

)= ξ4r4

Proc0 ��{task1}

{task1}Res0

• Take l = (0, 0, 0, 0)

l = (−1, 0, 0,−1)

l = (−1 + 1, 0,+1,−1) = (0, 0,+1,−1)

f(ξ, (0, 0,+1,−1), reset

)= ξ4r4

Proc0 ��{task1}

{task1}Res0

• Take l = (0, 0, 0, 0)

l = (−1, 0, 0,−1)

l = (−1 + 1, 0,+1,−1) = (0, 0,+1,−1)

f(ξ, (0, 0,+1,−1), reset

)= ξ4r4

Proc0 ��{task1}

{task1}Res0

• Take l = (0, 0, 0, 0)

l = (−1, 0, 0,−1)

l = (−1 + 1, 0,+1,−1) = (0, 0,+1,−1)

f(ξ, (0, 0,+1,−1), reset

)= ξ4r4

Proc0 ��{task1}

{task1}Res0

• Take l = (0, 0, 0, 0)

l = (−1, 0, 0,−1)

l = (−1 + 1, 0,+1,−1) = (0, 0,+1,−1)

f(ξ, (0, 0,+1,−1), reset

)= ξ4r4

Proc0 ��{task1}

Res0task1 ,r(ξ)−−−−−−−→∗ Proc1 ��

{task1}Res1

Proc1 ��{task1}

Res0task2 ,ξ2r ′2−−−−−−−→∗ Proc0 ��

{task1}Res0

Proc0 ��{task1}

{task1}Res0

f (ξ, (−1,+1,−1,+1), task1) = min(r1ξ1, r3ξ4

)f (ξ, (+1,−1, 0, 0), task2) = ξ2r2f (ξ, (0, 0,+1,−1), reset) = ξ4r4

Proc0 ��{task1}

Res0task1 ,r(ξ)−−−−−−−→∗ Proc1 ��

{task1}Res1

Proc1 ��{task1}

Res0task2 ,ξ2r ′2−−−−−−−→∗ Proc0 ��

{task1}Res0

Proc0 ��{task1}

{task1}Res0

f (ξ, (−1,+1,−1,+1), task1) = min(r1ξ1, r3ξ4

f (ξ, (+1,−1, 0, 0), task2) = ξ2r2f (ξ, (0, 0,+1,−1), reset) = ξ4r4

Proc0 ��{task1}

Res0task1 ,r(ξ)−−−−−−−→∗ Proc1 ��

{task1}Res1

Proc1 ��{task1}

Res0task2 ,ξ2r ′2−−−−−−−→∗ Proc0 ��

{task1}Res0

Proc0 ��{task1}

{task1}Res0

f (ξ, (−1,+1,−1,+1), task1) = min(r1ξ1, r3ξ4

)f (ξ, (+1,−1, 0, 0), task2) = ξ2r2

f (ξ, (0, 0,+1,−1), reset) = ξ4r4

Proc0 ��{task1}

Res0task1 ,r(ξ)−−−−−−−→∗ Proc1 ��

{task1}Res1

Proc1 ��{task1}

Res0task2 ,ξ2r ′2−−−−−−−→∗ Proc0 ��

{task1}Res0

Proc0 ��{task1}

{task1}Res0

f (ξ, (−1,+1,−1,+1), task1) = min(r1ξ1, r3ξ4

)f (ξ, (+1,−1, 0, 0), task2) = ξ2r2f (ξ, (0, 0,+1,−1), reset) = ξ4r4

Capturing behaviour in the Generator Function

{task1}Res0 [NR ]

Numerical Vector Form

ξ = (ξ1, ξ2, ξ3, ξ4) ∈ N4, ξ1 + ξ2 = NP and ξ3 + ξ4 = NR

Generator Function

f (ξ, l , α) :f (ξ, (−1, 1,−1, 1), task1) = min (r1ξ1, r3ξ3)

f (ξ, (1,−1, 0, 0), task2) = r2ξ2f (ξ, (0, 0, 1,−1), reset) = r4ξ4

{task1}Res0 [NR ]

Generator Function

{task1}Res0 [NR ]

Generator Function

Extraction of the ODE from f

Generator Function

Differential Equation

dt= FM(x) =

∑l∈Zd

l∑α∈A

f (x , l , α)

= (−1, 1,−1, 1) min (r1x1, r3x3) + (1,−1, 0, 0)r2x2

+ (0, 0, 1,−1)r4x4

Extraction of the ODE from f

Generator Function

Differential Equation

= −min (r1x1, r3x3) + r2x2

= min (r1x1, r3x3)− r2x2

= −min (r1x1, r3x3) + r4x4

= min (r1x1, r3x3)− r4x4

Consistency results

• The vector field F(x) is Lipschitz continuous i.e. all the ratefunctions governing transitions in the process algebra satisfylocal continuity conditions.

• The generated ODEs are the fluid limit of the family ofCTMCs generated by f (ξ, l , α): this family forms a sequenceas the initial populations are scaled by a variable n.

• We can prove this using Kurtz’s theorem:Solutions of Ordinary Differential Equations as Limits of PureJump Markov Processes, T.G. Kurtz, J. Appl. Prob. (1970).

• Moreover Lipschitz continuity of the vector field guaranteesexistence and uniqueness of the solution to the initial valueproblem.

Consistency results

Outline

4 Conclusions

Conclusions

• Many interesting and important systems can be studied as aresult of the link between stochastic process algebras andODEs.

• Particularly, we can now consider examples of collectivedynamics and emergent behaviour.

• Stochastic process algebras, are well-suited to modelling thebehaviour of such systems in terms of the individuals and theirinteractions.

• Continuous approximation allows a rigorous mathematicalanalysis of the average behaviour of such systems.

Conclusions

On-going work

• Time series plots counting the populations of componentsover time tell us a great deal about the dynamics of thesystem but are not necessarily the information we require.

• Recent work has established the validity of performancemeasures such as throughput, and average response timederived from the ODE solutions [TDGH 2012, IEEE TSE].

• On-going work is investigating the use of probes to query themodel by adding components to the model whose solepurpose is to gather statistics.

• Future work will also consider exploiting more of the formalstructure of the process algebras to assist in the manipulationand analysis of the ODEs.

On-going work

Thanks!

Acknowledgements: collaboratorsThanks to many co-authors and collaborators: Andrea Bracciali,Jeremy Bradley, Luca Bortolussi, Federica Ciocchetta, Allan Clark,Jie Ding, Adam Duguid, Vashti Galpin, Stephen Gilmore, DiegoLatella, Mieke Massink, Mirco Tribastone, and others.

Acknowledgements: fundingThanks to EPRSC for the Process Algebra for CollectiveDynamics grant and the CEC IST-FET programme for theSENSORIA project which have supported this work.

More information:http://www.dcs.ed.ac.uk/pepa

Thanks!

ODEs population view

TDSHA (hybridautomaton)

hybrid view

StochasticSimulation

individual view

LargePEPA model

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ODEs population view

TDSHA (hybridautomaton)

hybrid view

StochasticSimulation

individual view

LargePEPA model

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stochastic process algebras and ordinary differential...

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