A Statistical Analysis of Potential PowerNap Energy Savings

I was asked recently how much energy savings administrators might actually expect from an Ubuntu Enterprise Cloud, powered by Eucalyptus and using PowerNap.

I have blogged before about how much I enjoy mathematics and statistical analysis, and this is in fact another exciting, hard question!

The question was presented as the following hypothetical...

Given 10 quad-core servers in a cloud running 40 virtual machines. 20 virtual machines are simultaneously discarded. What power savings would you expect from PowerNap?

This is actually a rather complex combinatorics problem. I'll restate it as such:

Given 10 buckets, each of which can hold 0 - 4 items, how many ways can 20 items be distributed among those 10 buckets? Furthermore, what is the statistical distribution of empty buckets?

It's been 10 years since my university statistics classes, and my brain melted trying to derive the permutation formula. But a big thanks to my good friend Matt Dobson for solving that! If you're interested, you can view his combinatorics formula here.

It took me about an hour to hack a Python script that could empirically solve this problem by brute force, generating a comprehensive list of all of the permutations. You can now download and run /usr/bin/powernap_calculator provided by the powernap package in Karmic.

Here are the results running on the parameters above:

$ time powernap_calculator --hosts 10 --guests-per-host 4 --guests 20
Calculating...99%

In a cloud with [10] hosts, which can handle [4] guests-per-host, currently running [20] guests,
you may expect the following:

[ 5.2%] likely that [0/10] of your hosts would powernap, for a [0%] power savings
[ 27.5%] likely that [1/10] of your hosts would powernap, for a [10%] power savings
[ 42.5%] likely that [2/10] of your hosts would powernap, for a [20%] power savings
[ 21.8%] likely that [3/10] of your hosts would powernap, for a [30%] power savings
[ 2.9%] likely that [4/10] of your hosts would powernap, for a [40%] power savings
[ <1%] likely that [5/10] of your hosts would powernap, for a [50%] power savings

The overall expected value is [19.0%] power savings.

real 0m46.919s
user 0m46.227s
sys 0m0.276s

So at this snapshot in time, when your cloud suddenly dropped to 50% utilization, the expected value is a 19% power savings. Note that expected value is a very specific statistics term. Wikipedia says:

In probability theory and statistics, the expected value (or expectation value, or mathematical expectation, or mean, or first moment) of a random variable is the integral of the random variable with respect to its probability measure. For discrete random variables this is equivalent to the probability-weighted sum of the possible values, and for continuous random variables with a density function it is the probability density-weighted integral of the possible values.

Right, so we might expect about a 19% power savings for this random moment in time, when we simultaneously reduce our utilization by 50%.

However, if we observe the cloud over time, and with perhaps more realistic usage patterns, the distribution should be much better than random. VMs will come and go at more staggered intervals than simultaneous destruction of half the instances.

And you will have another tremendous factor working in your favor -- Eucalyptus' greedy scheduling algorithm. This ensures that each time a new VM is launched, it will land on a system that's already running. This is known as an annealing system -- one that's constantly correcting itself -- since under-utilized systems will automatically powernap, and new VMs will fill in the gaps on running hardware. Pretty cool, I think.

So I'm curious...

• What does (or would) your cloud look like?
• How many -h|--hosts, -p|--guests-per-host, and -g|--guests do you usually have?
• What does powernap_calculator say about your parameters?
• Post your results!
  

Note that the powernap_calculator algorithm is exponential, O((p+1)^h), so large values of (p, h) will take a very long time to compute! I'm totally open to code review of powernap_calculator and any performance enhancements ;-)

In my cloud, I have 8 dual-core hosts. I typically limit my usage to 2 guests-per-host. And I rarely run more than 4 VMs at a time.

$ powernap_calculator -h 8 -p 2 -g 4
In a cloud with [8] hosts, which can handle [2] guests-per-host, currently running [4] guests,
you may expect the following power savings:
[ 26.3%] likely that [4/8] of your hosts would powernap, for a [50%] power savings
[ 63.2%] likely that [5/8] of your hosts would powernap, for a [62%] power savings
[ 10.5%] likely that [6/8] of your hosts would powernap, for a [75%] power savings
The overall expected value is [60.5%] power savings.

My servers run at about 80 Watts fully loaded. My electricity is about $0.10 per kilowatt-hour. So a year's worth of electricity with all 8 servers running all of the time is 8 * .08 KW * 24 hr/day * 365 days/year * $0.10/KW-hr = $560/year.

I like the prospects of saving approximately 60.5% of that, or $339/year!

So how does the calculator work?

Let's use a slightly smaller example: hosts,h=4, guests-per-host,p=2, guests,g=3

Since each system can hold 0, 1, or 2 virtual machines, we're going to use a base-3 number (which is p+1). And we're going to generate all possible 4-digit base-3 numbers (which is (p+1)^h). In our case, that's 3⁴ or 81 scenarios to consider. For each of those scenarios, we convert the decimal integer to a list of each of the digits 0-2, and sum the list, throwing out any "invalid scenarios", where the sum does not add up to the target number of guests, 3, g. Our valid scenarios is actually much smaller, the following 16:

[2, 1, 0, 0]
[1, 2, 0, 0]
[2, 0, 1, 0]
[1, 1, 1, 0]
[0, 2, 1, 0]
[1, 0, 2, 0]
[0, 1, 2, 0]
[2, 0, 0, 1]
[1, 1, 0, 1]
[0, 2, 0, 1]
[1, 0, 1, 1]
[0, 1, 1, 1]
[0, 0, 2, 1]
[1, 0, 0, 2]
[0, 1, 0, 2]
[0, 0, 1, 2]

Now, of these, we are interested in how many 0's are in each row, as this indicates a host that has no guests, and can therefore be powernapped.

Our calculations yield: [0, 4, 12, 0, 0], or:

• 0 possible scenarios have no unused hosts
• 4 possible scenarios have 1 unused hosts
• 12 possible scenarios have 2 unused hosts
• 0 possible scenarios have 3 unused hosts
• 0 possible scenarios have 4 unused hosts

From this we can generate the following probabilities:

[ 25.0%] likely that [1/4] of your hosts would powernap, for a [25%] power savings
[ 75.0%] likely that [2/4] of your hosts would powernap, for a [50%] power savings

And weighting these probabilities, we can generate the expected value:

25%*.25 + 50%*.75 = 43.75%
The overall expected value is [43.8%] power savings.

Comments appreciated!

:-Dustin