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Category: Spines

Paper Review: Active Queue Management with Non-Linear Packet Dropping Function

Paper Review: Active Queue Management with Non-Linear Packet Dropping Function

As promised in the previous article, I plan to review Reference 2, Active Queue Management with Non-Linear Packet Dropping Function, by D. Augustyn, A. Domanski, and J. Domanska, published in HET-NETs 2010, which discusses a change in the structure of the packet drop probability function using the average queue length in a buffer. I mentioned previously that…

Networking Mathematics: Random Early Detection and TCP synchronization

Networking Mathematics: Random Early Detection and TCP synchronization

Computer networks are something most of us take for granted--speed, reliability, availability are expectations. In fact, network problems tend to make us very angry, whether it's dropped packets (yielding jittery Skype calls), congestion (that huge game download eating all the bandwidth), or simply a network outage. There's an awful lot going on underneath the hood…

Little’s Law: For Estimation Only

Little’s Law: For Estimation Only

I had been intending on writing some posts on queuing theory for a while now, as this branch is the closest to my research interests and was the spark that sent me down the road that eventually led to my PhD dissertation. Most are quite familiar with the concepts of queuing theory, at least intuitively,…

Building a Ground Floor: Topology Basics Pt. 1

Building a Ground Floor: Topology Basics Pt. 1

Like some other terms in mathematics ("algebra" comes to mind), topology is both a discipline and a mathematical object. Moreover like algebra, topology as a subject of study is at heart an artful mathematical branch devoted to generalizing existing structures like the field of real numbers for their most convenient properties. It is also a…

Cayley’s Theorem: Powerful Permutations

Cayley’s Theorem: Powerful Permutations

We've discussed before how powerful isomorphisms can be, when we find them. Finding isomorphisms "from scratch" can be quite a challenge. Thankfully, Arthur Cayley proved one of the classic theorems of modern algebra that can help make our lives a bit easier. We'll explore this theorem and note the power of groups of permutations.  (more…)

Isomorphisms: Making Mathematics More Convenient

Isomorphisms: Making Mathematics More Convenient

Much of pure mathematics exists to simplify our world, even if it means entering an abstract realm (or creating one) to do it. The isomorphism is one of the most powerful tools for discovering structural similarities (or that two groups are identical structurally) between two groups that on the surface look completely unrelated. In this post,…

All the Same Opposites

All the Same Opposites

Editor's note: see this appendix for supporting proofs. Fields are among the most convenient algebraic structures, preserving much of the arithmetic we know and love from familiar fields like the rationals and the real numbers . Now, it is unnecessary that a set possess infinitely many elements to possibly constitute a field (under the right…

The Red-Headed Step-Distributions

The Red-Headed Step-Distributions

Almost every textbook in probability or statistics will speak of classifying distributions into two different camps: discrete (singular in some older textbooks) and continuous. Discrete distributions have either a finite or a countable sample space (also known as a set of Lebesgue measure 0), such as the Poisson or binomial distribution, or simply rolling a…

Mailbox Answers: Calculating New Parity After an Overwrite

Mailbox Answers: Calculating New Parity After an Overwrite

I recently did some work for Mr. Howard Marks, an independent analyst and founder of Deep Storage on the subject of data protection and data loss. He e-mailed me with a question regarding calculating the new parity for a stripe of data on a storage system.  Let us consider the case of a RAID 5…

A Note on Russell’s Paradox

A Note on Russell’s Paradox

Topics in mathematics very frequently rely on set theory (I am hard-pressed to quickly think of one that doesn't). Set theory itself is a very abstract area of study. Even so, mathematicians built it on axioms (self-evident truths taken as fundamental starting points), and, of course, debate continues to this day as to which axioms…