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