1. Bandwidth Estimation TexPoint fonts used in EMF.
Read the TexPoint manual before you delete this box.:

2. System Theory for Networks
Networks can be viewed as linear systems in a different algebra:
Addition (+)  Minimum (inf)
Multiplication (·)  Addition (+)
Network service is described by a service curve
Min-Plus Algebra

3. Back to (Classical) Systems
Time Shift System
eigenfunction
eigenvalue
eigenfunction
Now:
Eigenfunctions of time-shift systems are also eigenfunctions of any linear time-invariant system

4. Min-Plus Linear Systems
Departures can be calculated from arrivals and service curve:
“min-plus convolution”

5. Characterizing non-linear systems
Many networks are not min-plus linear
i.e., for some t:
… but can be described by a lower service curve
such that for all t:
Having a lower service curve is often enough, since it provides a lower bound on the service !!

6. Available Bandwidth
Available bandwidth is the unused capacity along a path
Available bandwidth of a link:
Available bandwidth of a path:
Goal: Use end-to-end probing to estimate available bandwidth
Edited slide from: V. Ribeiro, Rice. U, 2003

7. Probing a network with packet trains
A network probe consists of a sequence of packets (packet train)
The packet train is from a source to a sink
For each packet, a measurement is taken when the packet is sent by the source (arrival time), and when the packet arrives at the sink (departure time)
So: rate at which the packet trains are sent is crucial:
Rate too high  probes preempt existing traffic
Rate too low  probes only measure the input rate
source
sink
Edited slide from: V. Ribeiro, Rice. U, 2003

8. Rate Scanning Probing Method
Each packet trains is sent at a fixed rate r (in bits per second). This is done by:
All packets in the train have the same size
Packets of packet train are sent with same distance
If size of packets is L, transmission time of a packet is T, and distance between packets is D, the rate is:
r = L/(T+D)
Rate Scanning:
Source sends multiple packet trains, each with a different rate r
Packet train: D D D

9. Bandwidth estimation in the network calculus
View the network as a min-plus system that is either linear or nonlinear
Bandwidth estimation scheme:
1. Timestamp packets of packet train: Ap(t) - Send probes Dp(t) - Receive probes
2. Use probes to find a that satisfies for all (A,D).
is the estimate of the available bandwidth. The goal is to select as large as possible.

10. Bandwidth estimation in a min-plus linear network
If network is min-plus linear, we get
If we set , then
So: We get an exact solution when the probe consist of a burst (of infinite size and sent with an infinite rate)
However:
An infinite-sized instantaneous burst cannot be realized in practice (It also creates congestion in the network)

11. Rate Scanning (1): Theory
Backlog:
Max. backlog:
If , we can write this as:
Inverse transform: If S is convex we have

12. Rate Scanning (2): Algorithm
Step 1: Transmit a packet train at rate , compute
compute
Step 2: If estimate of has improved, increase and go to Step 1.
This method is very close to Pathload !

13. Non-Linear Systems When we exploit we assume a min-plus linear system
In non-linear networks, we can only find a lower service curve that satisfies
We view networks as system that are always linear when the network load is low, and that become non-linear when the network load exceeds a threshold.
Note: In rate scanning, by increasing the probing rate, we eventually exceed the threshold at which the network becomes non-linear

14. Example of a non-linear system: FIFO link
Rate of packet train is determined by gap between packets
How does this show that FIFO is not a min-plus linear system.

15. Non-Linear Systems (or: How about FIFO ?)
When we exploit we assume a min-plus linear system
In a linear system, the system response S does not change with the input. But in FIFO, the system response changes with the input rate. So, FIFO is not min-plus linear.
In a FIFO system with …
… we get (see Problem Set)

16. Non-Linear Systems (or: How about FIFO ?)
If we set
… we can describe a FIFO system as
This means: FIFO is a linear system if total traffic is below capacity, and non-linear otherwise.
So, we should not increase rate of probe traffic beyond
not linear
linear
Probing rate

17. Detecting Non-linearity
How to determine the critical rate at which network becomes non-linear?
Backlog convexity criterion
Suppose that we probe at constant rates
Legendre transform is always convex
In a linear system, the max. backlog is the Legendre transform of the service curve:
If we find that for some rate r
we know that system is not linear

18. EmuLab Measurements Emulab is a network testbed at U. Utah
can allocate PCs and build a network
controlled rates and latencies
Some Questions:
How well does theory translate to real networks?
What can we learn from the computed service curve
How robust are the methods to changes of the traffic?

19. Dumbbell Network
UDP packets with 1480 bytes (probes) and 800 bytes (cross)
Cross traffic: 25 Mbps

20. Constant Bit Rate (CBR) Cross Traffic
Cross traffic is sent at a constant rate (=CBR)
The “reference service curve” (red) shows the ideal results. The “service curve estimates” shows the results of the rate scanning method
Figure shows 100 repeated estimates of the service curve
Rate Scanning

21. Rate Scanning: Different Cross Traffic
Exponential: random interarrivals, low variance
Pareto: random interarrivals, very high variance
Exponential
Pareto