1. Packet-Pair Dispersion for Bandwidth Probing: Probabilistic and Sample-Path Approaches M. J. Tunnicliffe Faculty of Computing, Information Systems and Mathematics, Kingston University, Kingston-on-Thames, Surrey, KT1 2EE. +20-85472000+62674 M.J.Tunnicliffe@king.ac.uk

2. Problem To find the bandwidth between two end-points in a network. To do this without any access to or cooperation from the intermediate routing nodes (routers, switches etc.). To do this without any synchronisation between the clocks of the end-points.

3. Networks and Network Paths Source Nodes Sink Nodes 10Mbit/s 20Mbit/s 15Mbit/s 12Mbit/s 5Mbit/s 10Mbit/s Routing Nodes Bottleneck Link Path has 6 “hops”. Bottleneck link dictates the overall bandwidth for the path. Tx Rx

4. Effect of Cross-Traffic 10Mbit/s 20Mbit/s 15Mbit/s 12Mbit/s – 8Mbit/s = 4Mbit/s Available B/W “TIGHT LINK” 5Mbit/s “NARROW LINK” 10Mbit/s The path now has two different types of bottleneck: The “Narrow Link” and the “Tight Link”. 8Mbit/s Traffic Tx Rx

5. Effect of the Tight-Link Bottleneck Latency and jitter increase as the tight-link speed is approached.

6. Assumption: Link Model Incoming packets (Differing sizes) Outgoing packets Stored packets, served in order of arrival (FIFO) Packets transferred at l bits/s Packet of size S bits requires S/l seconds for transmission. If another packet arrives less than S/l seconds behind the first, it has to wait in the queue behind the first packet. The time dispersion between the two packets is increased.

7. Packet-Pair Bandwidth Probing Packet #1 Packet #2 Departure Time Packet #1 Packet #2 Packet #1 Packet #2 Arrival Time Packet #1 Packet #2 Extra Dispersion

9. Output vs. Input Dispersion Time taken to service one packet Zero Cross Traffic

10. Output vs. Input Rate Zero Cross Traffic

11. Cross-Traffic: Fluid-Flow Analysis Probe Traffic in r bits/s l bits/s Cross Traffic in c bits/s Probe Traffic out r bits/s Cross Traffic out c bits/s Cross Traffic Probe Traffic

12. Cross-Traffic: Fluid-Flow Analysis Probe Traffic in r bits/s l bits/s Cross Traffic in c bits/s Probe Traffic out Cross Traffic out Cross Traffic Probe Traffic bits/s Bandwidth split in ratio c:r (Proportional Fair Queuing)

13. Output vs. Input Rate

14. Output vs. Input Dispersion

15. TOPP Representation Higher Order Bottleneck (Dispersion Ratio)

16. Simulation: Traffic Model Packet Size (bytes)Number Ratio ( β )Bandwidth Use Ratio ( α ) 60464.77 148112.81 500119.50 15003282.29 (Average Packet Size) (“Granularity”) Assume a Poisson arrival process. (Internet traffic is not generally Poissonian, but the Poisson model provides an adequate approximation.).

17. Single Queue Simulation Probe Pairs in 1 M bits/s Cross Traffic 500 K bits/s Probe Pairs out Cross Traffic out 100 pairs at each input spacing. Adjacent pairs 1 second apart. Individual output spacings vary. Take mean average. Available bandwidth is 500kbits/s Packet Size (bytes) Number Ratio (β) Bandwidth Use Ratio (α) 60464.77 148112.81 500119.50 15003282.29

18. TOPP Plot: Effect of Probe Size Fluid model represents asymptotic behaviour as the cross-traffic gradually loses its granular nature relative to the probe packets.

19. TOPP Plot: Effect of Granularity

20. Discrete Probe, Fluid Cross Traffic Assumption of discrete probe traffic does not alter the model’s equations. Need a model for the interaction of two discrete packet streams.

21. Need a Better Model Probabilistic approach. Represents quantities as time-evolving probability distributions. Sample Path approach. Considers possible behaviours as though they were deterministic trajectories. Analysis of Stochastic Processes:

22. Analysis of Stochastic Processes A stochastic process is a random variable that depends on time. For example X(t,ω) depends on time t and the outcome ω of a random experiment. For a particular value of ω, X ω (t) is deterministic called a sample path. For a particular value of t, X t (ω) is a random variable governed by the probability distribution behind ω.

23. Sample Paths in a Queuing System In a queuing system, V(t, ω) might represent the number of arrived packets at time t and W(t, ω) the workload (or “virtual waiting time”). In this interpretation ω represents the random processes governing packet arrivals and packet sizes. We drop the subscript and write the sample-paths V(t) and W(t). Numerous studies This analysis based mostly on Liu et al. (2005).

24. Sample Paths in a Queuing System TIME IDLE BUSY Packet Arrivals

25. Sample Paths in a Queuing System

26. Arrival of a Packet Pair 1 st Probe Packet Arrival Time Probe Packet S bits IDLE BUSY Intrusive Range Idle time is reduced by S / l seconds. Time taken to serve the probe packet S / l seconds. 2 nd Probe Packet Arrival Input Packet Separation Δ

27. Arrival of a Packet Pair 1 st Probe Packet Arrival Time Probe Packet S bits IDLE BUSY Intrusive Range. 2 nd Probe Packet Arrival Input Packet Separation Δ Packet separation is now less than the intrusive range. No idle time between packets. Waiting time of second packet is now increased by “Intrusion Residual”:.

28. Idle Time and Intrusion Residual Intrusion Residual Idle Time

29. Calculating the Output Dispersion But: Therefore: Similarly: To calculate the average output spacing, we obtain the expectation for each of the terms in the formulae. Inserting these into the equations without regard for available bandwidth variability reproduces the fluid model equations. “Nonlinearity”

30. Available Bandwidth Distribution Intrusion Residual Idle Time Frequency distribution of available bandwidth

31. Simple Model for f(x) (Departing somewhat from Liu et al.) Cross-traffic packet size = S c bits. n cross traffic packets arrive in period Δ seconds. If arrivals are Poisson, then n is governed by a Poisson distribution with a mean cΔ/S c and a standard deviation √(cΔ/S c ). Each packet reduces the available bandwidth by S c /Δ bits/s. Thus the mean available bandwidth is (l - c) with a standard deviation √(cS c / Δ). For simplicity we represent this as a Gaussian distribution:

32. Model for Output Dispersion

33. Predicted TOPP Graph

34. Probabilistic Models Park, Lim and Choi (2006) – Based on Franx’s transient state-space analysis of M/D/1 system. Haga, Diriczi, Vattay and Csabai (2007) – Based on transient solution of Takacs’ integro- differential equation for an M/G/1 system. My own approach (published 2008/9) – Discussed here. Three typical approaches:

35. Average Queue-Size Profile Finite granularity introduces: A finite average queue-size in equilibrium. A concavity in the average residual function. (This is equivalent to the “smearing” effect discussed in the sample-path analysis.) Simulation results: Mean queue-size during the impact of a probe packet.

36. Model of Probe-Packet Disturbance

37. Equilibrium Queue Behaviour

40. Transient ComponentsEquilibrium Components

42. Poisson TrafficBatch-Pareto Traffic

43. Predicted TOPP Graphs

44. Multiple-Hop Network Paths Problem with granular cross-traffic: Output dispersion of node 1 is not a determinate quantity, but a random variable governed by a probability distribution. Need a weighted integral of each possible dispersion value.

45. Multi-Hop Model

46. Dispersion Distributions

47. Present/Future Work Using intelligent algorithms to capture dispersion features from limited data. Effect of removing the “Pure FIFO” assumption (traffic shaping, wireless contentions, priority scheduling etc.) Effect of more complex traffic models (self- similarity, correlation of cross-traffic between nodes). Linking of available bandwidth concept with QoS issues. (Effective bandwidth.)