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Min-Plus Linear Systems TheoryMin-Plus Linear Systems Theoryandand
Bandwidth EstimationBandwidth Estimation
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System
(Classical) System Theory(Classical) System Theory
Linear Time Invariant (LTI) Systems
Linear:Time invariant:
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Consider an input signal:
.. and its output at a system:
Note:
Linear Systems TheoryLinear Systems Theory
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Linear Systems TheoryLinear Systems Theory
• Consider an arbitrary function
• Approximate by
Now we let
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Linear Systems TheoryLinear Systems Theory
• The result of
“convolution”
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System
(Classical) System Theory(Classical) System Theory
Linear Time Invariant (LTI) Systems
• If input is Dirac impulse, output is the system response
• Output can be calculated from input and system response:
“convolution”
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Min-Plus Linear SystemMin-Plus Linear System
Networkmin-plus Linear:Time invariant:
data
time t
A(t)
D(t)
delay
backlog
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Consider arrival function:
.. and departure function:
Note:
Min-Plus Linear SystemMin-Plus Linear System
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Min-Plus Linear SystemMin-Plus Linear System
• Consider an arbitrary function
• Approximate by
Now we let
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Min-Plus Linear SystemMin-Plus Linear System
• The result of
“min-plus convolution
”
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Min-Plus Linear SystemsMin-Plus Linear Systems
Network
• If input is burst function , output is the service curve
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Min-Plus Linear SystemsMin-Plus Linear Systems
Network
• Departures can be calculated from arrivals and service curve:
“min-plus convolution”
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System
Back to (Classical) SystemsBack to (Classical) Systems
• Now:
Eigenfunctions of time-shift systems are also eigenfunctions of any linear time-invariant
system
Time Shift System
eigenfunction
eigenfunction eigenvalue
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System
Back to (Classical) SystemsBack to (Classical) Systems
• Solving:
• Gives:
eigenvalue
Fourier Transform
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System
Now Min-Plus Systems againNow Min-Plus Systems again
• Now:
Eigenfunctions of time-shift systems are also eigenfunctions of any linear time-invariant
system
Time Shift System
eigenfunction
eigenfunctioneigenvalue
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System
Back to (Classical) SystemsBack to (Classical) Systems
• Solving:
• Gives:
eigenvalue
Legendre Transform
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Min-Plus Algebra
System Theory for NetworksSystem 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
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TransformsTransforms
• Classical LTI systems
Fourier transform
• Min-plus linear systems
Legendre transform
Time domain
Frequency domain
Time domain
Rate domain
Properties: (1) . If is convex:
(2) If convex, then
(3) Legendre transforms are always convex
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Available BandwidthAvailable 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
i
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Probing a network with packet trainsProbing a network with packet trains
Edited slide from: V. Ribeiro, Rice. U, 2003
• 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
Networksource
sink
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Rate Scanning Probing MethodRate 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 the rate is:
r = L/(T+)
• Rate Scanning:
Source sends multiple packet trains, each with a different rate r
Packet train:
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Min-Plus Linear SystemsMin-Plus Linear Systems
Networkmin-plus Linear:Time invariant:
• If input is burst function , output is the service curve
• Departures can be calculated from arrivals and service curve:
“min-plus convolution”
data
time t
A(t)
D(t)
delay
backlog
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One more thing …One more thing …
• 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 !!
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Bandwidth estimation in the network Bandwidth estimation in the network calculuscalculus
• View the network as a min-plus system that is either linear or nonlinear
Bandwidth estimation scheme:
1. Timestamp probesAp(t) - Send probesDp(t) - Receive probes
2. Use probes to find a that satisfies for all (A,D).
3. is the estimate of the available bandwidth.
data
time t
Ap(t)
Dp(t)
delay
backlog
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Bandwidth estimation in the network Bandwidth estimation in the network calculuscalculus
• View the network as a min-plus system that is either linear or nonlinear
Bandwidth estimation scheme:
1. Timestamp probesAp(t) - Send probesDp(t) - Receive probes
2. Use probes to find a that satisfies for all (A,D).
3. the goal is to select as large as possible.
data
time t
Ap(t)
Dp(t)
delay
backlog
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Bandwidth estimation in a min-plus Bandwidth estimation in a min-plus linear networklinear 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)
data
time t
A(t) = d(t)
D(t) = S(t)
delay
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Rate Scanning (1): TheoryRate Scanning (1): Theory
• Backlog:
• Max. backlog:
• If , we can write this as:
• Inverse transform: If S is convex we have da
ta
time t
A(t)
D(t)
delay
backlog
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Rate Scanning (2): AlgorithmRate 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 !
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Non-Linear SystemsNon-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
• QUESTION: How to determine the critical rate at which network becomes non-linear
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Detecting Non-linearityDetecting Non-linearity
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
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EmuLab MeasurementsEmuLab 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 our theory translate to real networks?
• Does representing available bandwidth by a function (as opposed to a number) have advantages?
• How robust are the methods to changes of the traffic distribution?
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Dumbbell NetworkDumbbell Network
• UDP packets with 1480 bytes (probes) and 800 bytes (cross)
• Cross traffic: 25 Mbps
Cross traffic
Probe traffic
50 Mbps10 ms
100 Mbpsno delay
100 Mbpsno delay
100 Mbps10 ms
100 Mbps10 ms
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Constant Bit Rate (CBR) Cross TrafficConstant 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
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Rate Scanning: Different Cross Traffic Rate Scanning: Different Cross Traffic
• Exponential: random interarrivals, low variance• Pareto: random interarrivals, very high variance
Exponential Pareto
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Dirac impulseDirac impulse
• The Diract delta function, often referred to as the unit impulse function, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere. The integral from minus infinity to plus infinity is 1.
From: Wikipedia