1. Intro to Deterministic Analysis
© Jörg Liebeherr, 2014

2. Switch Model

3. Components of a Packet Switch

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

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

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. Arrival Model

8. 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

9. Modeling Traffic Arrivals
We write arrivals as functions of time:
A(t) : Arrivals until time t, measured in bits
We assume that A(0)=0 for t ≤ 0.
To plot the arrival function, there are a number of choices:
Continuous time or Discrete time domain
Discrete sized or fluid flow traffic
Left-continuous or right-continuous
Note: The presentation of stochastic network calculus uses discrete-time for conciseness , and otherwise a (left-) continuous time domain

10. Discrete-sized vs. Fluid Flow
Traffic arrives in multiples of bits (discrete time, discrete-sized)
Traffic arrives like a fluid (continuous time, fluid flow)
It is often most convenient to view traffic as a fluid flow, that allows discrete sized bursts

11. Fluid Flow Traffic with Bursts
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

12. Left-continuous vs. right continuous
How to describe the function value immediately before or immediately after the “jump”?

13. Buffered Link

14. 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)

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

16. 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:

18. 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}

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

21. Traffic process: Non-decreasing and and one-sided process

23. Burst and Delay Functions