Towards a spatial stochastic process algebra Vashti Galpin - - PowerPoint PPT Presentation

Introduction Syntax Semantics Applications Example Conclusion

Towards a spatial stochastic process algebra

Vashti Galpin Laboratory for Foundations of Computer Science University of Edinburgh 31 July 2008

Vashti Galpin Towards a spatial stochastic process algebra PASTA 2008

Introduction Syntax Semantics Applications Example Conclusion

Introduction

◮ spatial concepts in a stochastic process algebra ◮ location can affect time taken ◮ aims

◮ generality and general results hence wide application ◮ CTMCs and steady state ◮ separation of concerns ◮ performance evaluation ◮ no unnecessary increase in state space ◮ single discovery of state space ◮ efficient experiments Vashti Galpin Towards a spatial stochastic process algebra PASTA 2008

Introduction Syntax Semantics Applications Example Conclusion

Introduction (cont.)

◮ examples of locations

◮ nodes in networks ◮ points in n-dimensional space

◮ applications – networks, epidemiology ◮ related research

◮ PEPA nets (Gilmore et al) ◮ StoKlaim (de Nicola et al) ◮ biological models – BioAmbients, attributed π-calculus

◮ work in progress ◮ general process algebra then applications and example

Vashti Galpin Towards a spatial stochastic process algebra PASTA 2008

Introduction Syntax Semantics Applications Example Conclusion

Syntax

◮ L set of locations ◮ PL = 2L powerset ◮ let L ∈ PL ◮ sequential components

S ::= (α@L, r).S | S + S | Cs@L

◮ model components

P ::= P ⊲

⊳

M P | P/M | C Vashti Galpin Towards a spatial stochastic process algebra PASTA 2008

Introduction Syntax Semantics Applications Example Conclusion

Location functions

◮ process location function

◮ ploc(Cs@L) = L ∪ ploc(S) where Cs@L def

= S

◮ ploc(P) = . . .

◮ action location function

◮ aloc((α@L, r).S) = L ∪ aloc(S) ◮ aloc(P) = . . .

◮ location function

◮ loc(P) = ploc(P) ∪ aloc(P)

◮ all static definitions ◮ current location function

◮ needs a dynamic definition Vashti Galpin Towards a spatial stochastic process algebra PASTA 2008

Introduction Syntax Semantics Applications Example Conclusion

Operational semantics

◮ relation for labels from A × PL × R+ ◮ Prefix

(α@L, r).S

(α@L′,r)

− − − − − → S L′ = apref ((α@L, r).S)

◮ Cooperation

P1

(α@L1,r1)

− − − − − − → P′

1

P2

(α@L2,r2)

− − − − − − → P′

2

P1 ⊲

⊳

M P2

(α@L,R)

− − − − − → P′

1 ⊲

⊳

M P′

2

α ∈ M L = async(P1, P2, L1, L2) R = rsync(P1, P2, L1, L2, r1, r2)

◮ other rules defined in the obvious manner

Constant, Choice, Hiding, other two Cooperation rules

Vashti Galpin Towards a spatial stochastic process algebra PASTA 2008

Introduction Syntax Semantics Applications Example Conclusion

Operational semantics (cont.)

◮ parameterised by three functions

◮ apref – determines location in Prefix rule ◮ async – determines location in Cooperation rule ◮ rsync – determines rate in Cooperation rule

◮ PEPA rate function

RPEPA(P1, P2, L1, L2, r1, r2) = r1 rα(P1) r2 rα(P2) min(rα(P1), rα(P2))

◮ can define apref and async for PEPA as well ◮ use of different functions to obtain different process algebras

Vashti Galpin Towards a spatial stochastic process algebra PASTA 2008

Introduction Syntax Semantics Applications Example Conclusion

Possible interpretations

| ploc(Cs)| = 1 process at one location | ploc(Cs)| > 1 process is mobile or process at one of some possible locations | aloc(Cs)| = 1 all actions occur at one location | aloc(Cs)| > 1 interaction is mobile or actions occur at one of some possible locations P

(α@L,r)

− − − − → P′, |L| = 1 synchronisation happens at one location P

(α@L,r)

− − − − → P′, |L| > 1 synchronisation involves actions or processes at different locations . . . . . .

Vashti Galpin Towards a spatial stochastic process algebra PASTA 2008

Introduction Syntax Semantics Applications Example Conclusion

And that’s not all

◮ beyond just adding locations, want to use them for

performance evaluation

◮ aims

◮ relate locations to each other ◮ want interesting theoretical results ◮ want to minimise time in analysis

◮ graph structure over locations ◮ weighted hypergraph: G = (L, E, w) with E ⊆ PL and

w : E → R or weighted directed graph

◮ weights modify rates on actions between locations ◮ weights assigned or calculated

Vashti Galpin Towards a spatial stochastic process algebra PASTA 2008

Introduction Syntax Semantics Applications Example Conclusion

Theoretical results – aim

◮ combine graph and known structures to obtain new results

quickly Analysis PEPA model Modified by G Form of result PEPA model P PG ↓ LMTS M MG M ⊚ G = MG ↓ CTMC Q QG Q G = QG ↓ steady state Π ΠG Π G = ΠG

Vashti Galpin Towards a spatial stochastic process algebra PASTA 2008

Introduction Syntax Semantics Applications Example Conclusion

Application – plant epidemiology

◮ epidemiology with spatial aspects ◮ grapevine diseases in vineyards ◮ mealy bug insect – vector for leaf roll disease ◮ two separate issues

◮ delay before vector arrives at uninfected plant – dependent on

distance between plants, hence dependent on vineyard layout

◮ delay before newly arrived vector infects plant

◮ three distinct patterns of infection ◮ build models to explain this

Vashti Galpin Towards a spatial stochastic process algebra PASTA 2008

Introduction Syntax Semantics Applications Example Conclusion

Application – networking

◮ networking performance ◮ scenario

◮ arbitrary topology ◮ single packet traversal through network ◮ processes can be colocated

◮ want to model different traffic situations ◮ choose functions to create process algebra

◮ each sequential component must have single fixed location ◮ communication must be pairwise and directional Vashti Galpin Towards a spatial stochastic process algebra PASTA 2008

Introduction Syntax Semantics Applications Example Conclusion

Application – networking (cont.)

◮ functions

apref (S) =

• l

if ploc(S) = {l} ⊥

• therwise

async(P1, P2, L1, L2) =

• (l1, l2)

if L1 = {l1}, L2 = {l2} ⊥

• therwise

rsync(P1, P2, L1, L2, r1, r2) =        r1 rα(P1) r2 rα(P2) min(rα(P1), rα(P2)) · w((l1, l2)) if L1 = {l1}, L2 = {l2}, (l1, l2) ∈ E ⊥

• therwise

Vashti Galpin Towards a spatial stochastic process algebra PASTA 2008

Introduction Syntax Semantics Applications Example Conclusion

Example network

Sender A P1 B P2 P3 C P4 D P5 E P6 Receiver F

Vashti Galpin Towards a spatial stochastic process algebra PASTA 2008

Introduction Syntax Semantics Applications Example Conclusion

PEPA model

Sender@A

def

= (prepare, ρ).Sending@A Sending@A

def

= (cSto1, r1).(ack, rack).Sender@A Receiver@F

def

= (c6toR, r6).Receiving@F Receiving@F

def

= (consume, γ).(ack, rack).Receiver@F . . . P2@C

def

= (c1to2, r).P2′@C + (c3to2, r).P2′@C + (c5to2, r).P2′@C P2′@C

def

= (c2to1, r).P2@C + (c2to3, r).P2@C + (c2to5, r).P2@C . . . Network

def

= (Sender@A ⊲

⊳

LS (P1@B ⊲

⊳

L1 (P2@C ⊲

⊳

L2 (P3@C ⊲

⊳

L3

(P4@D ⊲

⊳

L4 (P5@E ⊲

⊳

L5 (P6@F ⊲

⊳

LR Receiver@F))))))) Vashti Galpin Towards a spatial stochastic process algebra PASTA 2008

Introduction Syntax Semantics Applications Example Conclusion

Graphs

◮ rates: r = r1 = r6 = 10 ◮ the weighted graph G has no effect on the rates in the model

A B C D E F A 1 1 B 1 1 C 1 1 1 D 1 1 1 E 1 1 1 F 1 1 1 1

Vashti Galpin Towards a spatial stochastic process algebra PASTA 2008

Introduction Syntax Semantics Applications Example Conclusion

Graphs

◮ G1 represents heavy traffic between C and E

A B C D E F A 1 1 B 1 1 C 1 1 0.1 D 1 1 1 E 0.1 1 1 F 1 1 1 1

Sender A P1 B P2 P3 C P4 D P5 E P6 Receiver F

Vashti Galpin Towards a spatial stochastic process algebra PASTA 2008

Introduction Syntax Semantics Applications Example Conclusion

Graphs

◮ G2 represents no connectivity between C and E

A B C D E F A 1 1 B 1 1 C 1 1 D 1 1 1 E 1 1 F 1 1 1 1

Sender A P1 B P2 P3 C P4 D P5 E P6 Receiver F

Vashti Galpin Towards a spatial stochastic process algebra PASTA 2008

Introduction Syntax Semantics Applications Example Conclusion

Graphs

◮ G3 represents high connectivity between colocated processes

A B C D E F A 1 1 B 1 1 C 1 10 1 D 1 1 1 E 1 1 1 F 1 1 1 10

Sender A P1 B P2 P3 C P4 D P5 E P6 Receiver F

Vashti Galpin Towards a spatial stochastic process algebra PASTA 2008

Introduction Syntax Semantics Applications Example Conclusion

Analysis

◮ cumulative density function of passage time 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 3 4 5 6 7 8 9 10 Prob Time Comparison of different network models networkr networkr-fastl networkr-noCE networkr-slowCE

Vashti Galpin Towards a spatial stochastic process algebra PASTA 2008

Introduction Syntax Semantics Applications Example Conclusion

Further work and conclusions

◮ further work

◮ finalise general process algebra ◮ more specific applications ◮ theoretical results ◮ semantic equivalences

◮ conclusions

◮ first steps towards a very general stochastic process algebra

with locations

◮ most important aspect is that locations affect rates Vashti Galpin Towards a spatial stochastic process algebra PASTA 2008

Introduction Syntax Semantics Applications Example Conclusion

Thank you

This research was funded by the EPSRC SIGNAL Project

Vashti Galpin Towards a spatial stochastic process algebra PASTA 2008