Semi-Markov PEPA Jeremy Bradley NeSC, Edinburgh 12 June 2003 What - PowerPoint PPT Presentation

Semi-Markov PEPA Jeremy Bradley NeSC, Edinburgh – 12 June 2003

What is PEPA? • a stochastic process algebra • Markovian or exponential distributions • fast, sleek, no reward cards, spreads straight from the fridge 1 Imperial College London jb@doc.ic.ac.uk 12.6.2003

PEPA: Process Algebra • Syntax: P ✄ ✁ P ::= ( a, λ ) .P P + P S P P/L A • ( a, λ ) .P : prefix operation P + P : competitive choice • P ✄ ✁ S P : component cooperation P/L : action hiding 2 Imperial College London jb@doc.ic.ac.uk 12.6.2003

PEPA: Example • A simple transmitter-receiver network: def = (transmit , λ 1 ) . (t-recover , λ 2 ) .Transmitter Transmitter def = (receive , ⊤ ) . (r-recover , µ ) .Receiver Receiver def = (transmit , ⊤ ) . (delay , ν 1 ) . (receive , ν 2 ) .Network Network def ( Transmitter ✄ ✁ ✄ ✁ = Receiver ) System { transmit , receive } Network ∅ 3 Imperial College London jb@doc.ic.ac.uk 12.6.2003

Global State Space 4 Imperial College London jb@doc.ic.ac.uk 12.6.2003

Transient and Steady State 0.05 Steady state: X_1 0.045 0.04 0.035 0.03 Probability 0.025 0.02 0.015 0.01 0.005 0 0 5 10 15 20 25 30 Time, t 5 Imperial College London jb@doc.ic.ac.uk 12.6.2003

Transient and Steady State 0.05 PEPA model: transient X_1 -> X_1 Steady state: X_1 0.045 0.04 0.035 0.03 Probability 0.025 0.02 0.015 0.01 0.005 0 0 5 10 15 20 25 30 Time, t 6 Imperial College London jb@doc.ic.ac.uk 12.6.2003

Transient and Steady State 0.05 PEPA model: transient X_1 -> X_1 Steady state: X_1 0.045 0.04 0.035 0.03 Probability 0.025 0.02 0.015 0.01 0.005 0 0 5 10 15 20 25 30 Time, t 7 Imperial College London jb@doc.ic.ac.uk 12.6.2003

Transient and Steady State 0.05 PEPA model: transient X_1 -> X_1 Steady state: X_1 0.045 0.04 0.035 0.03 Probability 0.025 0.02 0.015 0.01 0.005 0 0 5 10 15 20 25 30 Time, t 8 Imperial College London jb@doc.ic.ac.uk 12.6.2003

Transient and Steady State 0.05 PEPA model: transient X_1 -> X_1 Steady state: X_1 0.045 0.04 0.035 0.03 Probability 0.025 0.02 0.015 0.01 0.005 0 0 5 10 15 20 25 30 Time, t 9 Imperial College London jb@doc.ic.ac.uk 12.6.2003

Concurrent Interleaving 10 Imperial College London jb@doc.ic.ac.uk 12.6.2003

Concurrent Interleaving 11 Imperial College London jb@doc.ic.ac.uk 12.6.2003

Concurrent Interleaving 12 Imperial College London jb@doc.ic.ac.uk 12.6.2003

Concurrent Interleaving 13 Imperial College London jb@doc.ic.ac.uk 12.6.2003

Concurrent Interleaving 14 Imperial College London jb@doc.ic.ac.uk 12.6.2003

Concurrent Interleaving 15 Imperial College London jb@doc.ic.ac.uk 12.6.2003

Concurrent Interleaving 16 Imperial College London jb@doc.ic.ac.uk 12.6.2003

Exponential memorylessness 1.4 X~exp(1.25) 1.2 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 17 Imperial College London jb@doc.ic.ac.uk 12.6.2003

Exponential memorylessness 1.4 X~exp(1.25) 1.2 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 18 Imperial College London jb@doc.ic.ac.uk 12.6.2003

Exponential memorylessness 1.4 X~exp(1.25) 1.2 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 19 Imperial College London jb@doc.ic.ac.uk 12.6.2003

Exponential memorylessness 1.4 X~exp(1.25) 1.2 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 20 Imperial College London jb@doc.ic.ac.uk 12.6.2003

Exponential memorylessness 1.4 X~exp(1.25) 1.2 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 21 Imperial College London jb@doc.ic.ac.uk 12.6.2003

Exponential memorylessness 1.4 X~exp(1.25) 1.2 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 22 Imperial College London jb@doc.ic.ac.uk 12.6.2003

Exponential memorylessness 1.4 X~exp(1.25) 1.2 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 23 Imperial College London jb@doc.ic.ac.uk 12.6.2003

Exponential memorylessness 1.4 X~exp(1.25) 1.2 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 24 Imperial College London jb@doc.ic.ac.uk 12.6.2003

Exponential memorylessness 1.4 X~exp(1.25) 1.2 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 25 Imperial College London jb@doc.ic.ac.uk 12.6.2003

Semi-Markov Processes • Semi-Markov processes use arbitrary distributions • But no support for competitive choice or concurrent execu- tion • So what use are semi-Markov processes as an underlying for- malism for PEPA? 26 Imperial College London jb@doc.ic.ac.uk 12.6.2003

PEPA and SMPs? Two motivations: 1. Systems with Markovian concurrency that have areas of mu- tual exclusion 2. Fully generally distributed concurrent design, but run on a single threaded architecture 27 Imperial College London jb@doc.ic.ac.uk 12.6.2003

Semi-Markov PEPA • Syntax: ::= ( a, D ) .P P + P P ✄ ✁ P S P P/L A D ::= λ ω : L ( s ) • new prefix operator: ( a, D ) .P – λ : normal exponential rate parameter – ω : L ( s ): a selection weight, ω , and a general distribution description, L ( s ) 28 Imperial College London jb@doc.ic.ac.uk 12.6.2003

Semi-Markov Example I • Mutual exclusion modelling: def A = (think , λ 1 ) . (recover , λ 2 ) .A + (error , λ 3 ) . (mutex , 1 : L 1 ( s )) .A def A ✄ ✁ ∅ A ✄ ✁ · · · ✄ ✁ = S n ∅ A ∅ � �� � n • Areas of Markovian concurrency interspersed with semi-Markov sequential behaviour 29 Imperial College London jb@doc.ic.ac.uk 12.6.2003

Semi-Markov Example II • Web-server/database model: def = (get , 5 : e xp (1 . 5 , s ))) .Server 1 Server def = (static page , 1 : d et (3 , s )) .Server Server 1 + (dbase fetch , 2 : g amma (2 . 2 , 3 . 2 , s )) .Server 2 def = (dbase rtn , ⊤ ) . (dynamic page , 1 : u niform (2 , 5 , s )) .Server Server 2 def = (dbase fetch , ⊤ ) . (dbase rtn , 4 : e xp (2 . 3 , s )) .Dbase Dbase def ✄ ✁ = Sys Server rtn } Dbase { dbase fetch , dbase • Concurrent design. Single-threaded architecture with weighted process selection 30 Imperial College London jb@doc.ic.ac.uk 12.6.2003

Some Conclusions • Proposal for semi-Markov PEPA • Incorporates PEPA functionality as a subset • Has 2 genuine application areas 31 Imperial College London jb@doc.ic.ac.uk 12.6.2003

Tool Support • ipc: PEPA to DNAmaca • SM-SPN DNAmaca – Transient distributions – Passage-time distributions 32 Imperial College London jb@doc.ic.ac.uk 12.6.2003

Semi-Markov Passage-time Cumulative passage time distribution: 10.9 million state voting model 1 0.8 Cumulative probability 0.6 0.4 0.2 0 500 550 600 650 700 750 800 Time 33 Imperial College London jb@doc.ic.ac.uk 12.6.2003

Semi-Markov Passage-time Cumulative passage time distribution: 10.9 million state voting model 1 0.8 Cumulative probability 0.6 0.4 0.2 0 500 550 600 650 700 750 800 Time 34 Imperial College London jb@doc.ic.ac.uk 12.6.2003