Min-Plus Linear Systems Theory and Bandwidth Estimation Min-Plus Linear Systems Theory and Bandwidth Estimation TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.:

Linear Systems Linearity: Time-Invariance:

(Classical) System Theory Linear Time Invariant (LTI) Systems Linear:Time invariant:

Consider an input signal:.. and its output at a system: Note: Linear Systems Theory

Consider an arbitrary function Approximate by Now we let

Linear Systems Theory The result of “convolution”

(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”

Min-Plus Linear System min-plus Linear:Time invariant:

Consider arrival function:.. and departure function: Note: Min-Plus Linear System

Consider an arbitrary function Approximate by Now we let

Min-Plus Linear System The result of “min-plus convolution”

Min-Plus Linear Systems If input is burst function, output is the service curve

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

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

Back to (Classical) Systems Solving: Gives: eigenvalue Fourier Transform

Now Min-Plus Systems again Now: Eigenfunctions of time-shift systems are also eigenfunctions of any linear time-invariant system Time Shift System eigenfunction eigenvalue

Back to (Classical) Systems Solving: Gives: eigenvalue Legendre Transform

Min-Plus Algebra 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

Transforms 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

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

Probing 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 source sink

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  the rate is: r = L/(T+  ) Rate Scanning: Source sends multiple packet trains, each with a different rate r Packet train: 

Min-Plus Linear Systems min-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”

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 !!

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 probes A p (t) - Send probes D p (t) -Receive probes 2. Use probes to find a that satisfies for all (A,D). 3. is the estimate of the available bandwidth.

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 probes A p (t) - Send probes D p (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.

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)

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

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 !

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 QUESTION: How to determine the critical rate at which network becomes non-linear

Detecting 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

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 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?

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

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

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

Dirac 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