performance analysis of stochastic process algebra · pdf fileperformance analysis of...

Performance analysis of StochasticProcess Algebra models using

Stochastic Simulation

Jeremy Bradley Stephen Gilmore Nigel Thomas

Email: [email protected] [email protected] [email protected]

Department of Computing,

Imperial College London

LFCS,

University of Edinburgh

School of Computing Science,

Univerity of Newcastle

Produced with prosper and LATEX

JTB [18/07/2005] – p. 1/21

The story used to be...

- -��

��1

PP

PPq

PEPAipc

HYDRA

hydra Passage-time PDF

Passage-time CDF

Transient

For state spaces of less than O(109)

Very precise probabilistic results

JTB [18/07/2005] – p. 2/21

Now the story is...

- ��

��1

PP

PPq

PEPAipc/pwb

Rate equations

Dizzy Stochastic simulation

Numerical ODE solution

For very large state spaces, e.g. 101000+ states

Aggregate deterministic results

JTB [18/07/2005] – p. 3/21

Stochastic Process Algebra

PEPA syntax:

P ::= (a, λ).P P + P P ��L

P P/L A

JTB [18/07/2005] – p. 4/21


PEPA syntax:

P ::= (a, λ).P P + P P ��L

P P/L A

Action prefix: (a, λ).P

JTB [18/07/2005] – p. 4/21


PEPA syntax:

P ::= (a, λ).P P + P P ��L

P P/L A


Competitive choice: P1 + P2

JTB [18/07/2005] – p. 4/21


PEPA syntax:

P ::= (a, λ).P P + P P ��L

P P/L A



Cooperation: P1 ��L

P2

JTB [18/07/2005] – p. 4/21


PEPA syntax:

P ::= (a, λ).P P + P P ��L

P P/L A




P2

Action hiding: P/L

JTB [18/07/2005] – p. 4/21


PEPA syntax:

P ::= (a, λ).P P + P P ��L

P P/L A




P2

Action hiding: P/L

Constant label: A

JTB [18/07/2005] – p. 4/21

PEPA: Example

Sysdef= (AA ��

{run}A1) ��

{alert}(BB ��

{run}B1)

JTB [18/07/2005] – p. 5/21

PEPA: Example

Sysdef= (AA ��

{run}A1) ��

{alert}(BB ��

{run}B1)

AAdef= (run,⊤).(alert, r5 ).AA

JTB [18/07/2005] – p. 5/21

PEPA: Example

Sysdef= (AA ��

{run}A1) ��

{alert}(BB ��

{run}B1)


A1def= (start, r1 ).A2 + (pause, r2 ).A3

A2def= (run, r3 ).A1 + (fail, r4 ).A3

A3def= (recover, r1 ).A1

JTB [18/07/2005] – p. 5/21

PEPA: Example

Sysdef= (AA ��

{run}A1) ��

{alert}(BB ��

{run}B1)





BBdef= (run,⊤).(alert, r5 ).BB

JTB [18/07/2005] – p. 5/21

PEPA: Example

Sysdef= (AA ��

{run}A1) ��

{alert}(BB ��

{run}B1)





BBdef= (run,⊤).(alert, r5 ).BB

B1def= (start, r1 ).B2 + (pause, r2 ).B1

B2def= (run, r3 ).B1

JTB [18/07/2005] – p. 5/21

Types of Analysis

Steady-state and transient analysis in PEPA:





Sysdef= AA ��

{run}A1

⇒

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0 5 10 15 20 25 30

Pro

babi

lity

Time, t

Steady state: X_1

JTB [18/07/2005] – p. 6/21

Types of Analysis

Steady-state and transient analysis in PEPA:





Sysdef= AA ��

{run}A1

⇒

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0 5 10 15 20 25 30

Pro

babi

lity

Time, t

PEPA model: transient X_1 -> X_1Steady state: X_1

JTB [18/07/2005] – p. 6/21

Passage-time Quantiles

Extract a passage-time density from a PEPA model:





Sysdef= AA ��

{run}A1

⇒

JTB [18/07/2005] – p. 7/21

Example of aggregate states

Clientdef= (compute,⊤).Client1

Client1

def= (delay , µ).Client

Serverdef= (compute, λ).Server1

Server1

def= (recover , ν).Server

Sys = (Client ‖ · · · ‖ Client︸︷︷︸

N

) ��{compute}

(Server ‖ · · · ‖ Server︸︷︷︸

M

)

Cooperating clusters can be represented as tuples

JTB [18/07/2005] – p. 8/21

Rate Equation Translation

Action: delay

JTB [18/07/2005] – p. 9/21


Action: delay

Client1

n(Client1)µ

−−−−−−−→ Client

JTB [18/07/2005] – p. 9/21


Action: delay

Client1

n(Client1)µ


Action: recover

Server1

n(Server1)ν

−−−−−−−→ Server

JTB [18/07/2005] – p. 9/21


Action: delay

Client1

n(Client1)µ


Action: recover

Server1

n(Server1)ν

−−−−−−−→ Server

Action: compute

Client + Serverθ(n(Client))n(Server)λ

−−−−−−−→ Client1 + Server1

where θ(x) = 1 if x > 0, else 0.

JTB [18/07/2005] – p. 9/21

Why the θ function?

There are N client cpts enabling a compute action

There are M server cpts enabling a compute action

Overall compute rate is:

rcompute(Sys) = min(N⊤,Mλ)

JTB [18/07/2005] – p. 10/21

Why the θ function?

There are N client cpts enabling a compute action

There are M server cpts enabling a compute action

Overall compute rate is:

rcompute(Sys) = min(N⊤,Mλ)

If N = 0 then overall rate is 0, hence:

rcompute(Sys) = θ(N) Mλ

JTB [18/07/2005] – p. 10/21

Dizzy setup

JTB [18/07/2005] – p. 11/21

Dizzy simulation

JTB [18/07/2005] – p. 12/21

Voter example

Election_Preparation ��L

Electoral_Personae

Electoral_Personaedef= Voter0[N ] ��

MElectoral_App

Electoral_Appdef= Collector_0[N ] ‖ Counter_1[N ]

‖ Administrator [N ]

JTB [18/07/2005] – p. 13/21

Early voter description

Voter0def= (choose, c1).Voter0_1

Voter0_1def= (bitcommit , b1).Voter0_2

Voter0_2def= (blind1, b2).Voter0_3

Voter0_3def= (blind2, b3).Voter0_4

Voter0_4def= (voter_sign, s1).Voter0_5

Voter0_5def= (sendA, s2).Voter0_5b

Voter0_5bdef= (sendV ,⊤).Voter1

Voter1def= (unblind1, u1).Voter1_1

JTB [18/07/2005] – p. 14/21

Voter: early stage

0

2000

4000

6000

8000

10000

0 5 10 15 20 25 30

Num

ber

Time, t

Number of Voter components in derivative states

Voter0Voter0_4

Voter0_5bVoter1

JTB [18/07/2005] – p. 15/21

High-level voter description

Voter0def= (choose, c1) . . . (sendV ,⊤).Voter1

JTB [18/07/2005] – p. 16/21



Voter1def= (unblind ,u1) . . . (sendC , s6).Voter2

JTB [18/07/2005] – p. 16/21



Voter1def= (unblind ,u1) . . . (sendC , s6).Voter2

Voter2def= (check , p × c4) . . . (sendCo, s7).Voter_Finished

JTB [18/07/2005] – p. 16/21

Voter: lifecycle

0

2000

4000

6000

8000

10000

12000

0 10 20 30 40 50 60

Num

ber

Time, t

Number of Voter components in derivative states

Voter0Voter1Voter2

Voter_Finished

JTB [18/07/2005] – p. 17/21

High-level Election description

Election_Preparation

def= (sendV ,⊤).Election_Preparation + · · ·

+ (publishA, er).Election_Voting

JTB [18/07/2005] – p. 18/21





Election_Voting

def= (sendC ,⊤).Election_Voting + · · ·

+ (publishC , er).Election_Counting

JTB [18/07/2005] – p. 18/21





Election_Voting

def= (sendC ,⊤).Election_Voting + · · ·

+ (publishC , er).Election_Counting

Election_Counting

def= (sendCo,⊤).Election_Counting + · · ·

+ (final_publish, er).Election_Finished

JTB [18/07/2005] – p. 18/21

Election: population of 1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 5 10 15 20 25 30

Num

ber

Time, t

Election component in derivative states

Election_PreparationElection_Voting

Election_CountingElection_Finished

JTB [18/07/2005] – p. 19/21

Election + Voter: interaction

0

2000

4000

6000

8000

10000

0 10 20 30 40 50 60

Num

ber

Time, t

Number of Voter derivatives against Election state

Election_PreparationElection_Voting

Election_CountingElection_Finished

Voter0Voter1Voter2

Voter_Finished

JTB [18/07/2005] – p. 20/21

Conlcusion

Novel simulation techniques based on chemical rateequations

Orders of magnitude larger state spaces can beanalysed

Complexity of simulation method (Gibson-Bruck) isO(log n) where n is number of rate equations

JTB [18/07/2005] – p. 21/21

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