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EE650R:
Lecture 3:
Date:
ClassNotes:
Review:
Reliability Physics of Nanoelectronic Devices
Physical Reliability Models: Acceleration and Projection
Sept. 15, 2006
Ehtesham Islam
Robert Wortman
3.1 Review:
The three types of reliability models include Empirical, Statistical, and Physical models.
Studying any natural phenomena requires the knowledge of the physics of the problem
and in some case the statistical model behind the problem should also analyzed. For
example, for deterministic system like solar system, empirical model like Keppler’s law
was first developed and later Newton’s law interpreted the empirical observations by
physical laws of gravitation. But for cases like modeling of Wild Fire, statistical analyses
(percolation theory) will be an additional step, which is performed in between empirical
(power-law dependence of magnitude and frequency) and physical modeling
(flammability of particular trees).
Returning to the BFRW model, we had been interested is determining the injection point
for having, say 99% fish in river after certain lifetime, say 3 years
Injection at x = 0
-xh (xh >> xm)
xm
……………
Fall
If the velocity of the river is small (p~0.5), then documenting problem empirically takes
too long, so we devised a scheme to accelerate the flow of the river so that we can collect
information quickly. Then we use the information gained through this accelerated testing
to project back to normal operating conditions (normal flow velocity). This brings
“projection” into picture (which requires consideration of proper physical model).
Without a physical model for extrapolation, one would typically connect the data points
with a straight line (or any other line) that seem reasonable and extrapolate back to
operating conditions, hence making projection erroneous.
T
TDesired
m increased
0.5
1.0
Probability
3.2 Statistical/Physical Model of BFRW:
To understand how the “empirical” extrapolation model can get us into over conservative
estimates of the injection point, let us consider the statistical model of the BFRW
problem.
i+1
i-1
i
q
p
Assume that the fish is injected at node i at time t = 0. Now, before it goes to the
waterfall, a fraction p of the fishes will go the right. One can consider as if that these
fishes have been injected at the node i+1 and therefore, starting from there should take,
on average, Ti+1 seconds to reach the waterfall. Similarly, another fraction q of the fishes
will go the left from the position i. These left-moving fish can be considered as being
injected at the node i-1 and again starting from there should take, on average, Ti-1 seconds
to reach the waterfall. Whatever direction (left or right) the fish moves, it will take some
time τ to move from i to i+1 or from i to i-1. So if the fish injection is at node i, then the
time fish takes, on average, to reach the waterfall, i.e. Ti can be expressed as,
Ti    pTi 1  qTi 1
Replacing Ti , Ti+1 , Ti-1 with T(x) , T(x+δ) , T(x-δ) respectively, we have,
T ( x)    pT ( x   )  qT ( x   )
T ( x)  pT ( x   )  qT ( x   )    0
1 
1 
q   , where >0 is used for representing velocity of river
Using, p   ;
2 2
2 2
flow in positive x direction.
or ,
Hence,
1
T ( x   )  T ( x   )   T ( x   )  T ( x   )  
2

 2 0
2
2
2

2
d T 2 dT 2


0
      (1)
dx 2  dx  2
T ( x) 
or ,
As velocity and diffusion co-efficient expressed as, v  
d 2T 2 dT 2

  0       (2)
dx 2 v dx D
General solution of Eqn. (1) is: T  C1  C2 x

 and D   , Eqn. (1)
becomes,
-------- (3)
2
Applying the boundary condition that if injection is near waterfall, i.e. xxm, then time
required to reach the waterfall, T0, we have
C1  C2 xm ;
T  C2  x  xm  ---- (4)
As Eqn. (4) must satisfy Eqn. (1),
2
2


C2  2  0
 C2 



Using C2 in Eqn. (4),

  xm  x 
T
--- (5)
 xm  x  

 2 p 1
2
Eqn. (5) gives us the expression for average arrival time, T, provided the injection of fish
is at position x and velocity of the river is such that the probability of rightward
movement for a fish from a particular position is p. The equation also indicates that if
injection is close to xm (the position of waterfall), then (xm - x), hence T will be lower.
Moreover, if velocity of the river is higher in +x direction, then p will be higher, resulting
in a lower value for T, indicating that on average the fish will require less time if river
has a flow in +x direction.


Projection & Design Penalty
As discussed earlier, in absence of a physical model for extrapolation, one would connect
the data points with whatever line seems reasonable and thus resulting extrapolation back
to operating conditions will be inaccurate. But from Eqn. (5), which estimates average
arrival time (T), we can easily observe that T is inversely proportional to (p-1/2). So, the
extrapolation to the condition of normal river flow (say, corresponding to p=1/2+1)
would be physical only when, the measurements are joined using a hyperbolic curve (red
line in the figure below). Use of some arbitrary line (blue line in the figure below) for
projection will require the injection to be far away from the waterfall (x1), compared to
the injection point (x2) estimated using proper physical model. Hence, the farmer will
unnecessarily lease some extra land (x2 -x1). Expressing in terms of reliability, if farmer,
being ignorant about the physical model developed above, chooses blue line instead of
red for projection, we will say that the farmer has over-designed his problem. Then the
incurred by the farmer design penalty for being ignorant is defined as, (x2 -x1).
T
TDesired
x1 < x2
Injection at x1
Design Penalty
0.5+1
Injection at x2
Probability
1.0
Physical Interpretation of Divergence at p=1/2:
ln f
Slope = -2
tm : First arrival time at waterfall
ln t
tm
Tav  @ p  0.5 


2 tm
2tm

tf dt ~
t
t
2

dt  ln t 2t
m
2 tm
As p tends to 0.5, T av tends to infinity. This indicates that when p  0.5, i.e. the
movement of fish in either left/right direction is equally possible, at any certain time there
is always a possibility of collecting a fraction of fish in the waterfall. As a result, the tail
of the distribution of f extends in time and we get a divergence in Tav. Indeed, comparing
with the Matlab code uploaded, we can see that it takes a lot of time to simulate when
p0.5. This is because as fish is making a random walk, the time it requires to reach the
fall increases.
Such power-law tails of distribution function occurs frequently in various reliability
phenomena, calculating trivial quantities like Tav tricky and dependent on sample
numbers. Since the sample size is often limited, the averages can be misleading and may
introduce errors in projection. We will pick up these issues of finite sample sizes in the
next class.
3.3 Exercise
Find the difference equation that describes the steady state population p(x) of fish in the
river. Solve the equations and interpret the results.
Solution:
The equation is given by
P  x   pP  x     qP  x   
Solution:
P  x  0 
n1
1  exp a  xm  x 

1  exp  axm  
P  x  0   n1 exp  axm 
where,
a


; n1  injection density
Solution for p = 1/2
n1
xm