EE650R: Reliability Physics of Nanoelectronic Devices
Lecture 3:Physical Reliability Models: Acceleration and Projection
Date: Sept. 15, 2006
ClassNotes: Ehtesham Islam
Review: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 laterNewton’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
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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). Thisbrings “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.
3.2 Statistical/Physical Model of BFRW:
To understand how the “empirical” extrapolation model can get us into overconservative estimates of the injection point, let us consider the statistical model of the BFRW problem.
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,
ReplacingTi, Ti+1 , Ti-1 with T(x),T(x+δ) , T(x-δ) respectively, we have,
Using, , where >0 is used for representing velocity of river flow in positive x direction.
Hence,
As velocity and diffusion co-efficient expressed as, and , Eqn. (1) becomes,
General solution of Eqn. (1) is: ------(3)
Applying the boundary condition that if injection is near waterfall, i.e. xxm, then time required to reach the waterfall, T0, we have
---- (4)
As Eqn. (4) must satisfy Eqn. (1),
Using C2 in Eqn. (4),
--- (5)
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 beinaccurate. 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).
Physical Interpretation of Divergence at p=1/2:
As p tends to 0.5, Tav 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 p0.5. 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.
3.3Exercise
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