Min-Plus Linear Systems Theory and Bandwidth Estimation

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

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

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

4. Linear Systems Theory • Consider an arbitrary function • Approximate by Now we let

5. Linear Systems Theory • The result of Impulse Response “convolution”

6. If input is Dirac impulse, output is the system response • Output can be calculated from input and system response: “convolution” (Classical) System Theory Linear Time Invariant (LTI) Systems

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

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

9. Min-Plus Linear System • Consider an arbitrary function • Approximate by Now we let

10. Min-Plus Linear System • The result of “min-plus convolution” Service Curve

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

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

13. 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

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

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

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

17. 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

18. Transforms • Classical LTI systems Frequency domain Time domain Fourier transform • Min-plus linear systems Rate domain Time domain Legendre transform Properties: (1) . If is convex: (2) If convex, then (3) Legendre transforms are always convex

19. 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

20. 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

21. 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

22. Min-Plus Linear Systems min-plus Linear: Time invariant: • Departures can be calculated from arrivals and service curve: • If input is burst function , output is the service curve “min-plus convolution”

23. 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 !!

24. View the network as a min-plus system that is either linear or nonlinear Bandwidth estimation scheme: 1. Timestamp probes Ap(t) - Send probes Dp(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

25. View the network as a min-plus system that is either linear or nonlinear Bandwidth estimation scheme: 1. Timestamp probes Ap(t) - Send probes Dp(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 the network calculus

26. 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)

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

28. 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 !

29. 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

30. 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

31. 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? EmuLab Measurements

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

33. 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

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

35. 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