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Introduction to the Beowulf Design





Robert G. Brown
Duke University Physics Department
rgb@phy.duke.edu



13 May 2003



Parallel Computation



Amdahl's Law

The speedup $S$ experienced running a task on $P$ processors is less than or equal to:

\begin{displaymath}
S \le \frac{(\rm T_s + T_p)}{(\rm T_s + (T_p/P))}
\end{displaymath} (1)

where $T_s$ is the time program spends doing ``serial work'' and $T_p$ is the time spend doing ``parallelizable work'' split up on $P$ processors.

Limiting result, not horribly useful quantitatively except to tell you when there is no point in parallelizing something. Can do much better.



For example, we can account for the time spent communicating between processors, the time spent setting things up, and changes in the times to perform various tasks with different algorithms. Defining things like:

${\bf T_s}$ The original single-processor serial time.
${\bf T_{is}}$ The (average) additional serial time spent doing things like IPC's, setup, and so forth, per processor, in all parallelized tasks.
${\bf T_p}$ The original single-processor parallizable time.
${\bf T_{ip}}$ The (average) additional time spent by each processor doing just the setup and work that it does in parallel. This may well include idle time, which is often important enough to be accounted for separately.
we can obtain improved estimates of the speedup:
\begin{displaymath}
T_{\rm tot}(P) = T_s + P*T_{is} + T_p/P + T_{ip}.
\end{displaymath} (2)

or
\begin{displaymath}
S \approx \frac{T_s + T_p}{T_s + P*T_{is} + T_p/P +
T_{ip}}.
\end{displaymath} (3)



All You Need to Know
About Code Granularity

Granularity typically is somewhat controllable. Network speed and latency, scaling of computation to communication as a function of problem size, CPU/memory speed, program organization all control variables.

Fine grained tasks are ``bad'' for scaling to many nodes $N$. Coarse grained tasks are ``good''.

Beware nonlinearities! CPU/cache/memory/disk bottlenecks can create ``superlinear speedup'' and violate Amdahl's Law!



Huh? Whaddideesay?

In all the figures below, $T_s = 10$ (which sets our basic scale, if you like) and $T_p = 10,100,1000,10000,100000$. In the first three figures we just vary $T_{is} = 0,1,10$ for $T_{ip} = 1$ (fixed). Finally, the last figure is $T_{is} = 1$, but this time with a quadratic dependence $P^2*T_{is}$.

\begin{figure}\centerline{\psfig{file=bscaling.Tis0.eps,height=2.7in}
\psfig{file=bscaling.Tis1.eps,height=2.7in}}\end{figure}

\begin{figure}\centerline{\psfig{file=bscaling.Tis10.eps,height=2.7in}
\psfig{file=bscaling.Tis1.Psq.eps,height=2.7in}}\end{figure}



Designs: NOW/COW/Beowulf

Goal is optimizing overall performance per dollar. The following are appropriate for increasing fineness of program granularity:

These are suitable for increasingly fine granularity, at increasing cost and decreasing general purpose utility.

Schematics for the general designs follow, first a ``true beowulf'' and then a workstation cluster.



A True Beowulf


file=beowulf.eps,height=7in



A Workstation Cluster


file=cluster.eps,height=7in



Node Design and Cost

Examples (Intrex-based estimate, YMMV):

Dells will run perhaps 10% higher plus shipping. Pricewatch min might save you 20%, BUT see later notes on ``Administrative Infrastructure''! Per node pricing may have to absorb cost of rack(s) or heavy duty steel shelving from Home Depot, screws, cable ties, and so forth, so estimate 10% more than your base minimum.



Turnkey Clusters

Turnkey clusters can make sense if you are building a very specialized cluster and need help designing and installing it. A turnkey integrator will typically resell the hardware components to you pretty much at standard retail marked up to cover their ``integration fee'' for designing the cluster, installing the clusterware on it, and so forth. This ends up being anywhere from a 20% markup of OTC prices on up.

At Duke it will only VERY RARELY make sense to get a turnkey cluster. That is because:



Cluster Networks



Parallel Program Support



Simple Example: xep (PVM Mandelbrot Set)

On a good day, this will work as a demo...



Physical Infrastructure Requirements



Physical Infrastructure Costs

Note well that recurring costs for operating a node can compete with the cost of the node! This favors getting relatively expensive nodes and dumping nodes quickly when obsolete!



Administrative Infrastructure

In summary, Min: $\sim$ 1 hour a day, on average, for a ``good'' 100+ node cluster; Max: full time job and then some for a ``bad'' cluster (depending on luck, hardware reliability, your general admin skills, your cluster admin skills, user support requirements, and the availability of cluster expertise in a distributed support environment).



Conclusions
Total Cost of Ownership (TCO) can range from: Wide range, provokes TCO fistfights in bars.

Still, beowulfish clusters often yield staggering productivity efficiency. Generally 3-10x more cost/benefit than comparable power ``big iron''. SO, literally everybody is buying or building them.



References and Resources




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Robert G. Brown 2003-05-13