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

▶
semi markov pepa
SMART_READER_LITE
LIVE PREVIEW

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

Sep 27, 2022 382 likes •748 views

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

slide-1
SLIDE 1

Semi-Markov PEPA

Jeremy Bradley

NeSC, Edinburgh – 12 June 2003

slide-2
SLIDE 2

Imperial College London jb@doc.ic.ac.uk 12.6.2003

1

What is PEPA?

  • a stochastic process algebra
  • Markovian or exponential distributions
  • fast, sleek, no reward cards, spreads straight from the fridge
slide-3
SLIDE 3

Imperial College London jb@doc.ic.ac.uk 12.6.2003

2

PEPA: Process Algebra

  • Syntax:

P ::= (a, λ).P 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

slide-4
SLIDE 4

Imperial College London jb@doc.ic.ac.uk 12.6.2003

3

PEPA: Example

  • A simple transmitter-receiver network:

Transmitter

def

= (transmit, λ1).(t-recover, λ2).Transmitter Receiver

def

= (receive, ⊤).(r-recover, µ).Receiver Network

def

= (transmit, ⊤).(delay, ν1).(receive, ν2).Network System

def

= (Transmitter ✄

✁

∅

Receiver)

✄ ✁

{transmit,receive} Network

slide-5
SLIDE 5

Imperial College London jb@doc.ic.ac.uk 12.6.2003

4

Global State Space

slide-6
SLIDE 6

Imperial College London jb@doc.ic.ac.uk 12.6.2003

5

Transient and Steady State

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 5 10 15 20 25 30 Probability Time, t Steady state: X_1

slide-7
SLIDE 7

Imperial College London jb@doc.ic.ac.uk 12.6.2003

6

Transient and Steady State

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 5 10 15 20 25 30 Probability Time, t PEPA model: transient X_1 -> X_1 Steady state: X_1

slide-8
SLIDE 8

Imperial College London jb@doc.ic.ac.uk 12.6.2003

7

Transient and Steady State

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 5 10 15 20 25 30 Probability Time, t PEPA model: transient X_1 -> X_1 Steady state: X_1

slide-9
SLIDE 9

Imperial College London jb@doc.ic.ac.uk 12.6.2003

8

Transient and Steady State

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 5 10 15 20 25 30 Probability Time, t PEPA model: transient X_1 -> X_1 Steady state: X_1

slide-10
SLIDE 10

Imperial College London jb@doc.ic.ac.uk 12.6.2003

9

Transient and Steady State

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 5 10 15 20 25 30 Probability Time, t PEPA model: transient X_1 -> X_1 Steady state: X_1

slide-11
SLIDE 11

Imperial College London jb@doc.ic.ac.uk 12.6.2003

10

Concurrent Interleaving

slide-12
SLIDE 12

Imperial College London jb@doc.ic.ac.uk 12.6.2003

11

Concurrent Interleaving

slide-13
SLIDE 13

Imperial College London jb@doc.ic.ac.uk 12.6.2003

12

Concurrent Interleaving

slide-14
SLIDE 14

Imperial College London jb@doc.ic.ac.uk 12.6.2003

13

Concurrent Interleaving

slide-15
SLIDE 15

Imperial College London jb@doc.ic.ac.uk 12.6.2003

14

Concurrent Interleaving

slide-16
SLIDE 16

Imperial College London jb@doc.ic.ac.uk 12.6.2003

15

Concurrent Interleaving

slide-17
SLIDE 17

Imperial College London jb@doc.ic.ac.uk 12.6.2003

16

Concurrent Interleaving

slide-18
SLIDE 18

Imperial College London jb@doc.ic.ac.uk 12.6.2003

17

Exponential memorylessness

0.2 0.4 0.6 0.8 1 1.2 1.4 0.5 1 1.5 2 2.5 3 X~exp(1.25)

slide-19
SLIDE 19

Imperial College London jb@doc.ic.ac.uk 12.6.2003

18

Exponential memorylessness

0.2 0.4 0.6 0.8 1 1.2 1.4 0.5 1 1.5 2 2.5 3 X~exp(1.25)

slide-20
SLIDE 20

Imperial College London jb@doc.ic.ac.uk 12.6.2003

19

Exponential memorylessness

0.2 0.4 0.6 0.8 1 1.2 1.4 0.5 1 1.5 2 2.5 3 X~exp(1.25)

slide-21
SLIDE 21

Imperial College London jb@doc.ic.ac.uk 12.6.2003

20

Exponential memorylessness

0.2 0.4 0.6 0.8 1 1.2 1.4 0.5 1 1.5 2 2.5 3 X~exp(1.25)

slide-22
SLIDE 22

Imperial College London jb@doc.ic.ac.uk 12.6.2003

21

Exponential memorylessness

0.2 0.4 0.6 0.8 1 1.2 1.4 0.5 1 1.5 2 2.5 3 X~exp(1.25)

slide-23
SLIDE 23

Imperial College London jb@doc.ic.ac.uk 12.6.2003

22

Exponential memorylessness

0.2 0.4 0.6 0.8 1 1.2 1.4 0.5 1 1.5 2 2.5 3 X~exp(1.25)

slide-24
SLIDE 24

Imperial College London jb@doc.ic.ac.uk 12.6.2003

23

Exponential memorylessness

0.2 0.4 0.6 0.8 1 1.2 1.4 0.5 1 1.5 2 2.5 3 X~exp(1.25)

slide-25
SLIDE 25

Imperial College London jb@doc.ic.ac.uk 12.6.2003

24

Exponential memorylessness

0.2 0.4 0.6 0.8 1 1.2 1.4 0.5 1 1.5 2 2.5 3 X~exp(1.25)

slide-26
SLIDE 26

Imperial College London jb@doc.ic.ac.uk 12.6.2003

25

Exponential memorylessness

0.2 0.4 0.6 0.8 1 1.2 1.4 0.5 1 1.5 2 2.5 3 X~exp(1.25)

slide-27
SLIDE 27

Imperial College London jb@doc.ic.ac.uk 12.6.2003

26

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?

slide-28
SLIDE 28

Imperial College London jb@doc.ic.ac.uk 12.6.2003

27

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

slide-29
SLIDE 29

Imperial College London jb@doc.ic.ac.uk 12.6.2003

28

Semi-Markov PEPA

  • Syntax:

P ::= (a, D).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)

slide-30
SLIDE 30

Imperial College London jb@doc.ic.ac.uk 12.6.2003

29

Semi-Markov Example I

  • Mutual exclusion modelling:

A

def

= (think, λ1).(recover, λ2).A + (error, λ3).(mutex, 1 : L1(s)).A Sn

def

= A ✄

✁

∅ A ✄

✁

∅

· · · ✄

✁

∅ A

  • n
  • Areas of Markovian concurrency interspersed with semi-Markov

sequential behaviour

slide-31
SLIDE 31

Imperial College London jb@doc.ic.ac.uk 12.6.2003

30

Semi-Markov Example II

  • Web-server/database model:

Server

def

= (get, 5 : exp(1.5, s))).Server1 Server1

def

= (static page, 1 : det(3, s)).Server + (dbase fetch, 2 : gamma(2.2, 3.2, s)).Server2 Server2

def

= (dbase rtn, ⊤).(dynamic page, 1 : uniform(2, 5, s)).Server Dbase

def

= (dbase fetch, ⊤).(dbase rtn, 4 : exp(2.3, s)).Dbase Sys

def

= Server

✄ ✁

{dbase fetch,dbase rtn} Dbase

  • Concurrent design. Single-threaded architecture with weighted

process selection

slide-32
SLIDE 32

Imperial College London jb@doc.ic.ac.uk 12.6.2003

31

Some Conclusions

  • Proposal for semi-Markov PEPA
  • Incorporates PEPA functionality as a subset
  • Has 2 genuine application areas
slide-33
SLIDE 33

Imperial College London jb@doc.ic.ac.uk 12.6.2003

32

Tool Support

  • ipc: PEPA to DNAmaca
  • SM-SPN DNAmaca

– Transient distributions – Passage-time distributions

slide-34
SLIDE 34

Imperial College London jb@doc.ic.ac.uk 12.6.2003

33

Semi-Markov Passage-time

0.2 0.4 0.6 0.8 1 500 550 600 650 700 750 800 Cumulative probability Time Cumulative passage time distribution: 10.9 million state voting model

slide-35
SLIDE 35

Imperial College London jb@doc.ic.ac.uk 12.6.2003

34

Semi-Markov Passage-time

0.2 0.4 0.6 0.8 1 500 550 600 650 700 750 800 Cumulative probability Time Cumulative passage time distribution: 10.9 million state voting model