1. CPU Power Value Curves How much is the availability of a given amount of CPU power at some specified time worth, in terms of physics output?

2. The Value vs. CPU Power Curve Value is driven by physics output –Although we will assign a dollar value Value increases with CPU power –This is intuitively obvious, but where does the increase come from? Value is not flat because of: –Uncertainty in actual amount of CPU that would be needed –Diminishing returns on analyses that open up with more CPU availability Each of these drives a Gaussian marginal utility curve So we can combine them into one effect, driven by an overall Gaussian

3. The value curve has a shape like this:

4. A Bit Of Mathematics Approximating the effect of a deficiency in CPU power as that same factor as a reduction in physics, the model would say Computing power C is at least as large as actual computational luminosity L Computing power C is deficient compared to actual computational luminosity L

5. Mathematics We can make this much more tractable by replacing the concept of uncertainty in the need with uncertainty in how much of the computational luminosity need each CPU unit will satisfy: Effective computing power C is at least as large as actual computational luminosity Effective computing power C is deficient compared to actual computational luminosity

6. The value curve shown follows that equation

7. The units for expressing value On the previous graph, the value (vertical) axis was denominated in % of some maximum. We think of value more naturally in $ –What sets the scale? –A palpable fraction of the value of the lab operation! If you could not analyze data, expts would be worthless If you spent all your $ on analysis, you would have no data The curve always has positive first derivative, and negative second derivative Our acquisition levels lie near the top of this curve, generally a bit past the anticipated computational luminosity

8. Value Curve and Acquisition Level

9. Value Curve The available CPU power is, of course, previously acquired power plus this acquisition Notice that value is MUCH greater than the money spent on the given acquisition It is even greater that the total (cumulative) money spent to get the total node power available If CPU power is such a fantastic deal, why not spend even more? –Because the (roughly optimum) amount we are spending places us squarely in the region of diminishing returns

10. Marginal Value of Next CPU unit

11. The Value Curve is convex downward The value of a unit of CPU power exceeding the “optimum” fraction of computational luminosity is less that the value of the last unit of CPU power to reach the optimum. –If you overshoot this year, and undershoot next year by the same amount, all else being equal, the value curve says you will have hurt yourself

12. Value Curve Scaling What happens in a different year, when computational luminosity is greater? –Value scale remains about the same –Saturation point moves higher That is, there is a higher antiipated computational luminosity The uncertainty (sigma) driving the shape of the value curve will grow

13. How does the uncertainty grow? Uncertainty applies to the whole load –We could say that the computational burden due to work analogous to that being done on existing CPU’s is known completely, and that we apply uncertainty only to the increase in load –It is much more reasonable to say that the uncertainty is in the nature of the coming computations – then the uncertainty is proportional to the total anticipated computational luminosity. This reasonable assumption implies that – The value curve scales with anticipated computational luminosity

14. “The value curve scales with computational luminosity” A single curve applies to all cases –this is good, since we can sensibly evaluate tradeoffs in acquisitions! The curve is characterized by the mean and sigma in an underlying Gaussian curve –representing the effects of uncertainty, as in those equations –the mean is related to the anticipated computational luminosity Model makes the assumption that the acquisition levels in the past and future are near the optimum levels (the “reasonable- ness assumption”). Deciding what to use for sigma is the key point –But this is a single time-invariant number!!

15. The Scaled Value Curve

16. How the Value Curve Will Be Used Trading using today’s flexibility for using up tomorrow’s flexibility –When imperfectly fungible assets are involved In creating the bid formula for a given acquisition, we will make use of the first two derivatives of the value curves for this and subsequent years, near the acquisition (or optimum) points. –The slope expresses incremental value –The second derivative quantifies the penalty for balanced overshoot/undershoot The invariant curves for future years are multiplied by the applicable physics timeliness factors (which are < 1).