# Power consumption of processor: Cubic in MIPS or linear in CPU utilisation?

I have came across two types of formulas for calculating the power consumption of a server. In the following I describe my understanding (please correct if necessary):

My questions:

• Is my understanding correct?

• According to my understanding, the higher the MIPS the higher the CPU utilization is (is this relationship linear?). How do the cubic and the linear formulas fit together then? For me it is contradicting that I have a linear as well as a cubic function for calculating the power consumption.

• My conjecture: The cubic formula applies for servers that enable DVFS whereas the linear formula applies for servers that do not enable DVFS, i.e. with a fixed frequency. Can someone confirm/disprove my conjecture with a reference?

I really need to understand this issue and I also really need references for my research.

• Without specific CPU architecture, OS and Power Management variables, this will be impossible to generalize. My i8 runs between 800MHz and 3.1MHz with many steps and reduces Vcore many times a second which Win8 Ultimate can report on right side menu, even when CPU % averages <5%. But DRAM and CPU/Core SRAM voltage and speeds are different. Start with simple Dynamic CMOS losses then look at complex parallel different CMOS . Looking at servers is pointless without these two different variables not to mention OS, Energy management BIOS tweaks .etc. DVFS is a broad topic with myopic assumptions Commented Apr 6, 2017 at 16:49
• Sry, I do not understand your comment. I do not aim to generalize anything - the two formulas are both well accepted in the literature. However, for me they seem to contradict each other. With my question I want to clarify and understand when they are used (so that there is no contradiction in my understanding any more). Commented Apr 6, 2017 at 17:00
• Is "MILP" (which appears twice in the question) a typo or some term I don't recognize? Commented Apr 6, 2017 at 17:43

One of those is an apple and the other an orange.

Systems with DVFS are capable of varying their performance in MIPS, yes. However, the frequency scaling is not exogenous but driven by the system itself in response to various factors, usually including utilization and temperature.

Utilisation varies between 0 and 1. The second formula applies whether or not you have DVFS on, but both Pmax and Pmin vary slightly according to the first formula.

So in order to really use both you need to understand how the system varies frequency according to utilisation, which may be difficult to find out.

(Other facts may complicate the picture such as ARM big/little or other systems for turning on and off whole cores in response to load.)

• "..., but both Pmax and Pmin vary slightly according to the first formula" - what exactly do you mean by this (Pmax and Pmin are constants?), could you please elaborate? Commented Apr 6, 2017 at 16:52
• Pmax is power when fully utilized. However, for a frequency scaling system, it can be fully utilized at 1GHz or fully utilized at 2GHz, which consume different amounts of power (according to 1st formula) Commented Apr 7, 2017 at 8:29

You can think of a digital chip as lots of little capacitor, each being drained and charged up to V, f times per second.

Energy needed for each charge up is 0.5 x c x V2. Or total energy in a second is c V2 x f.

Note that c (mostly gate charge,) is a function of V, typically monotonic but non linear.

That should put you into the same ballpark.