1. © J. Liebeherr, All rights reserved, 2018

3. Switch Model

4. Components of a Packet Switch

5. Modeling a Packet Switch
Model with input and output buffers
Simplified model (only output buffers)

6. A Path of Network Switches
Each switch is modeled by the traversed output buffer
Output buffer is shared with other traffic (“cross traffic”)

7. A Path of Network Switches
Each switch is modeled by the traversed output buffer
Output buffer is shared with other traffic (“cross traffic”)

8. Arrival Model

9. Arrival Scenarios
Packet arrives to buffer only after all bits of the packet have been received
Packet arrival appears instantaneous
Multiple packets can arrive at the same time

10. Modeling Traffic Arrivals
We write arrivals as functions of time:
A(t) : Arrivals until time t, measured in bits (with A(0)=0 for t ≤ 0)
Does A(t) include the arrival at time t, or not ?
right-continuous
left-continuous
A(t) considers arrivals in (0,t]
A(t) considers arrivals in [0,t) (Note: A(0) = 0 !)
Most people take a left-continuous interpretation

11. Buffered Link

12. Buffered Link Standard model of an output buffer at a switch
Link rate of output is C
Scheduler is work-conserving (always transmitting when there is a backlog)
Transmission order is FIFO
Infinite Buffers (no buffer overflows)

13. Arrivals of packets to a buffered link
Backlog at the buffered link
slope -C

14. Definitions We write: \documentclass[10pt]{article}
\usepackage{color} %used for font color
\usepackage{amssymb} %maths
\usepackage{amsmath} %maths
\usepackage[utf8]{inputenc} %useful to type directly diacritic characters
\begin{document}
{\sf
\begin{tabbing}
xx \=xxxxxxx\= \kill
\> {\color{blue} $A(t)$ } \> Arrivals in $[0, t)$, with $ A(t) = 0$ if $t \leq 0$ \\[10pt]
\> {\color{blue}$D(t)$ } \> Departures in $[0, t)$, with $D(t) \leq A(t)$ \\[10pt]
\> {\color{blue} $B(t)$} \> Backlog at $t$. \\
\> \> {\color{blue} $B(t) = A (t) - D(t)$} \\[10pt]
\> {\color{blue} $W(t)$} \> (Virtual) delay at $t$: \\
\> \>
{\color{blue} $ W(t) = \inf \left\{ y > 0 \;|\; D(t+y) \geq A(t) \right\} $}
\end{tabbing} }
\end{document}
We write:

16. Reich's Backlog Equation (1958)
\documentclass[10pt]{article}
\usepackage{color} %used for font color
\usepackage{amssymb} %maths
\usepackage{amsmath} %maths
\usepackage[utf8]{inputenc} %useful to type directly diacritic characters
\begin{document}
{\sf
\fbox{\parbox[t]{3in}{
{\sc Reich's backlog equation:}
Given a left-continuous arrival function $A$ and a buffered link with capacity $C$.
Then for all $t \geq 0$ it holds that \[
\label{eq:reich-backlog}
B(t) = \sup_{0\leq s \leq t} \left\{ A(t) - A(s) - C(t-s) \right\} \ . \]
}}}
\end{document}

18. Introduction to Network Calculus (min, +) Algebra

21. Burst and Delay Functions

25. Service Curves

30. Traffic Envelopes and Traffic Regulation

38. Performance Bounds

45. Add per-node bounds
Add per-node bounds