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NUMERICAL ALGORITHMS FOR TRANSIENT ANALYSIS OF FLUID QUEUES Stéphane Mocanu Laboratoire d’Automatique de Grenoble FRANCE. Basic tandem model. Two machines separated by a finite buffer Unreliable machines Deterministic service times Infinite arrivals an machine M 1 - PowerPoint PPT Presentation
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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
AIC05 - S. Mocanu 3
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
AIC05 - S. Mocanu 4
Versions
• Non-blocking, Time Dependent Failures
Communication systems (Mitra)
• Blocking, Operation Dependent Failures
Production systems (Gershwin)
AIC05 - S. Mocanu 5
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 !
AIC05 - S. Mocanu 6
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
AIC05 - S. Mocanu 7
An example: homogeneous case
State = {M1 state, M2 state, buffer level}
U U
0 0
Machines driven by two state Markov chains
AIC05 - S. Mocanu 8
Joint evolution
(1,1,h)
(1,0,h)
(0,1,h)
(0,0,h)
2
2
1
1
1
1
2
2
U
U
AIC05 - S. Mocanu 9
Evolution equations
MxtVx
xt
t
xt,
,,
A PDE system
Markov chain generator
Drift matrix In the example
0000
000
000
0000
U
UV
AIC05 - S. Mocanu 10
Boundary conditions for ODF systems
Discontinuities of the probability distribution
CCc tPtxtPtx )(),()(),( 00 (t,x)
X
c(t,x)
P0(t)
PC(t)
AIC05 - S. Mocanu 11
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
AIC05 - S. Mocanu 12
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,
,
AIC05 - S. Mocanu 13
Initial conditions
Specify
Example : machine state (1,1) (both ON), buffer empty
)(),0(),,0( 0 hPPx C
AIC05 - S. Mocanu 14
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
AIC05 - S. Mocanu 15
The integration scheme
• Decompose the system in – Linear evolution
– Wave evolution
• Apply b.c.
V
x
xt
t
xt
,,
Mxtt
xt,
,
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Recurrent solution
• Linear transform
• Wave transform
iwavei
wavei
iW Vxkxkxk ,,1,,
Mi
lineari
lineari
iL exkxkxk ,,1,,
ini V
x
,...,1max
AIC05 - S. Mocanu 17
Recurrent form of the b.c.
Vx
xCkCkMkPtkPkP
Vx
kxkMktkk
CCCC
),(),()0,()()1(
)0,(),()0,()0,()0,1( 0
AIC05 - S. Mocanu 18
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
AIC05 - S. Mocanu 19
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
AIC05 - S. Mocanu 20
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
AIC05 - S. Mocanu 21
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