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Make vlan-introduction make at least some sense

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Daniel Gröber 2018-11-09 13:40:23 +01:00
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vlan-introduction.tex
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\usepackage{bbold}
\usepackage{commath}
\usepackage{parskip}
%\usepackage{fullpage}
\usepackage{booktabs}
\begin{document}
\section{VLAN Introduction}
For those unfamilliar with the concept of a VLAN (Virtual LAN) here is a shot
For those unfamilliar with the concept of a VLAN (Virtual LAN) here is a short
formal specification of what such a thing does.
A Switch is a 8-tuple \( \mathcal{S} = (\mathbb{A}, \mathbb{P}, \mathbb{V}, t, v, a, \beta, \epsilon) \)
A switch is a 8-tuple \( \mathcal{S} = (\mathbb{A}, \mathbb{P}, \mathbb{V}, t, v, a, \beta, \epsilon) \)
consisting of
\begin{itemize}
\item a finite set of (MAC) addresses \(\mathbb{A}\),
\item a finite set of physical ports \(\mathbb{P}\),
\item a finite set of VLANs \(\mathbb{V}\),
\item a mapping from physical ports and VLANs to two distinct symbols pronounced
``tagged'' and ``untagged'' repectively
\( t : \mathbb{P} \times \mathbb{V} \rightarrow \{ \tau, \upsilon \} \),
\item a mapping from physical ports and VLANs to VLANs (Port PVID)
\( v : \mathbb{P} \times \mathbb{V} \rightarrow \mathbb{V} \)
with \(v(p, q) \mapsto q\) when \(q \neq \epsilon\),
\item a mapping from addresses and VLANs to physical ports (ARP Table)
\( a : \mathbb{A} \times \mathbb{V} \rightarrow \mathbb{P} \) and
\item a mapping from VLANs and physical ports to three distinct symbols
pronounced ``tagged'', ``untagged'' and ``neither'' repectively
\( t : \mathbb{V} \times \mathbb{P} \rightarrow \{ \tau, \upsilon, \eta \} \),
\item a mapping from VLANs and physical ports to VLANs (Port PVID)\\
\( v : \mathbb{V} \times \mathbb{P} \rightarrow \mathbb{V} \)
with \(v(q, p) \mapsto q\) when \(q \neq \epsilon\)
% When PVID of a port is not member in a VLAN an error is thrown in the web
% interface
and \(v(\epsilon, p) \not\mapsto q \) when \( t(q,p) = \eta \),
\item a partial mapping from addresses and VLANs to physical ports (ARP Table)
\( a : \mathbb{A} \times \mathbb{V} \rightharpoonup \mathbb{P} \) and
\item the broadcast address \(\beta \in \mathbb{A}\)
\item the empty VLAN tag \(\epsilon \in \mathbb{V}\)
\end{itemize}
A Frame processed by a Switch \(\mathcal{S}\)
\paragraph{Definition}
A frame \( \mathcal{F}_\mathcal{S} \)
processed by a switch \(\mathcal{S}\)
is a tuple \( \mathcal{F}_\mathcal{S} = (d, q) \)
consisting of a destination address \(d \in \mathbb{A} \) and a VLAN tag \( q \in \mathbb{V} \).
consisting of a destination address \(d \in \mathbb{A} \)
and a VLAN tag \( q \in \mathbb{V} \).
When a Frame \( \mathcal{F}_\mathcal{S}' = (d, q') \)
% TODO: switch checks if the port is even in the VLAN and discards it if not
When a frame \( \mathcal{F}_\mathcal{S} = (d, q) \)
enters a port \( p \in \mathbb{P}\)
the Switch first ensures the Packet has a VLAN tag for internal processing
assigned by creating a new Frame \( \mathcal{F}_\mathcal{S} = (d, q)\) with \(q = v(p, q')\).
the switch first ensures the frame has a VLAN tag for internal processing
assigned by creating a new frame \( \mathcal{F}_\mathcal{S}^i = (d, q')\)
with \(q' = v(q, p)\).
\subsection{Broadcast processing}
Next the switch checks if the VLAN is allowed on this port. When
\( v(q, p) = \eta \)
the frame is dropped and processing of this frame is complete.
% This could also be before assigning the PVID because v(ε, p) can only be VLANs
% q that are not t(q, p) = η
When the Frame's destination address \( d = \beta \)
the Switch creates a new Frame for each port
\( p \in \{\, p \mid t(p, \_) \,\} \)
in the following manner:
\paragraph{Unicast processing}
When the frame's destination address \(d\)
is not the broadcast address the switch first determines the egress port
\(p = a(d, q)\).
If it is not defined the frame is dropped and processing of this frame is
complete. Next the final egress frame is created as in equation
\eqref{eq:egress}. \( \mathcal{F}_{\mathcal{S}, p}^{e} \)
is then transmitted out port \(p\) and processing of this frame is complete.
\begin{equation}\label{eq:egress}
\mathcal{F}_{\mathcal{S}, p}^{e} =
\left\{
\begin{array}{ll}
(d,q) & \mbox{if } t(p, q) = \tau \\
(d, \epsilon) & \mbox{if } t(p, q) = \upsilon\\
(d,q) & \mbox{if } t(q, p) = \tau \\
(d, \epsilon) & \mbox{if } t(q, p) = \upsilon\\
\end{array}
\right.
\end{equation}
\( \mathcal{F}_{\mathcal{S}, p}^{e} \)
is then transmitted out port \(p\) and processing of this Frame is complete.
\paragraph{Broadcast processing}
When the frame's destination address \( d = \beta \)
the switch creates a new frame for each port
\( p \in \{\, p \mid \forall q.\; t(q, p) \neq \eta \,\} \)
as in equation \eqref{eq:egress}. The frames \( \mathcal{F}_{\mathcal{S}, p}^{e} \)
are then transmitted out each port \(p\)
respectively and processing of this frame is complete.
\subsection{Unicast processing}
When the Frame's destination address \(d\) is not the broadcast address the Switch first determines the egress port \(p = a(d, q)\). Then a new Frame is created as in equation \ref{eq:egress}. \( \mathcal{F}_{\mathcal{S}, p}^{e} \)
is then transmitted out port \(p\) and processing of this Frame is complete.
\end{document}
% \section{Version history}
% Current Version: 2
% When frames are dropped was not considered,
% Arguments to \(v\) were swapped,
% Requirement for port PVIDs to actually be a member of the VLAN added
% Made ARP table a partial function (duh)