Basic tandem model

AIC05 - S. Mocanu 1

NUMERICAL ALGORITHMS FOR TRANSIENT ANALYSIS OF FLUID

QUEUESStéphane Mocanu

Laboratoire d’Automatique de GrenobleFRANCE

AIC05 - S. Mocanu 2

Basic tandem model

• Two machines separated by a finite buffer• Unreliable machines • Deterministic service times

• Infinite arrivals an machine M1

• Infinite available places at the exit of M2

SM1 M2

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Fluid (continuous) modem

Arrival Waiting Service

X1X1 X2 X2Y1 Y1 Y2 Y2C or

T1 T2

Arrival Waiting Service

X1X1 X2 X2Y1 Y1 Y2 Y2C or

U1=1/T1 U2=1/T2

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Versions

• Non-blocking, Time Dependent Failures

Communication systems (Mitra)

• Blocking, Operation Dependent Failures

Production systems (Gershwin)

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Operation depending failures

Suppose M1 slowed down by M2 (U1>U2, x=C)

Work Blocked

T1=1/U1

T2=1/U2

Work Blocked

The failure rate is reduced to: ' 21 1

1

U

U

A completely blocked (starved) machine cannot fail !A completely blocked (starved) machine cannot fail !

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Internal equations

• Not an ordinary Markov chain

• Continuous transitions on the “fluid direction”

• Infinitesimal variation of the probability mass

U

Discrete transitions

Discrete state

Continuous transition

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An example: homogeneous case

State = {M1 state, M2 state, buffer level}

U U

0 0

Machines driven by two state Markov chains

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Joint evolution

(1,1,h)

(1,0,h)

(0,1,h)

(0,0,h)

2

2

1

1

1

1

2

2

U

U

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Evolution equations

MxtVx

xt

t

xt,

,,

A PDE system

Markov chain generator

Drift matrix In the example

0000

000

000

0000

U

UV

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Boundary conditions for ODF systems

Discontinuities of the probability distribution

CCc tPtxtPtx )(),()(),( 00 (t,x)

X

c(t,x)

P0(t)

PC(t)


Difficulties

Boundary condition does NOT verify the PDE– Some boundary states are of 0 probability

– Some transitions are modified (due to ODF)

MxtVx

xt

t

xt,

,,

M0 on lower boundary

MC on upper boundary


Homogeneous case

Example: state (0,0,C)

Matrix form

)()()( )0,0(21)1,0(2 tPtP CC (0,0,C)(1,0,C)

(0,1,C)

0)(0)( )0,0()1,0( tPtP CC

CCCx

C MxtPVx

xt

t

tP,

,


Initial conditions

Specify

Example : machine state (1,1) (both ON), buffer empty

)(),0(),,0( 0 hPPx C


The problem

Find an integration algorithm for

– under boundary conditions b.c.

– with initial conditions i.c.

MxtVx

xt

t

xt,

,,

)(),0(),,0( 0 hPPx C


The integration scheme

• Decompose the system in – Linear evolution

– Wave evolution

• Apply b.c.

V

x

xt

t

xt

,,

Mxtt

xt,

,


Recurrent solution

• Linear transform

• Wave transform

iwavei

wavei

iW Vxkxkxk ,,1,,

Mi

lineari

lineari

iL exkxkxk ,,1,,

ini V

x

,...,1max


Recurrent form of the b.c.

Vx

xCkCkMkPtkPkP

Vx

kxkMktkk

CCCC

),(),()0,()()1(

)0,(),()0,()0,()0,1( 0


Numerical results

0 10 20 30 40 50 600

0.01

0.02

0.03

0.04

0.05

0.06

0.07

x

10

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

01

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x

11

0 10 20 30 40 50 600

0.005

0.01

0.015

0.02

x

00

Initial state : (0,1) buffer half full


Numerical results

Initial state : (0,1) buffer half full

0 0.2 0.4 0.6 0.8 1 1.2

0.88

0.9

0.92

0.94

0.96

0.98

1

time

Exp

ecte

d ef

ficie

ncy

First starvation


Numerical results

0 1 2 3 4 5 620

25

30

time

Exp

ecte

d bu

ffer

leve

l

0 0.1 0.2 0.3 0.4 0.5 0.6

22

24

26

28

30

time

Exp

ecte

d bu

ffer

leve

l


Some limitations

• Needs compatible i.c.– Warning : machine state (1,1), buffer empty is

NOT compatible– But : machine state (1,1), buffer = x, it IS

• Some boundaries propagates bad• For the instance we need explicit analysis of

boundary conditions• Actual numerical implementation is limited to

ON/OFF machines

Basic tandem model

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