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

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.

Networks and Network Paths

Source Nodes

Sink Nodes

10Mbit/s

20Mbit/s

15Mbit/s

12Mbit/s5Mbit/s 10Mbit/s

Routing Nodes

BottleneckLink

Path has 6 “hops”. Bottleneck link dictates the overall bandwidth for the path.

TxRx

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

TxRx

Effect of the Tight-Link Bottleneck

Latency and jitter increase as the tight-link speed is approached.

Assumption: Link ModelIncoming 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.

Packet-Pair Bandwidth Probing

Packet #1

Packet #2

Departure Time

Packet #1

Packet #2

Packet #1

Packet #2

out

Arrival Time

Packet #1

Packet #2

out

Extra Dispersion

lS lS

Output vs. Input Dispersion

l

S

lS0

out

out

Time taken to service

one packet

Zero Cross Traffic

Output vs. Input Rate

l

l0

out

Sm

Sr

Zero Cross Traffic

Cross-Traffic: Fluid-Flow Analysis

Probe Traffic inr bits/s

lbits/s

Cross Traffic inc bits/s

ProbeTraffic out

r bits/s

Cross Traffic out

c bits/s

Cross Traffic

Probe Traffic

lcr

Cross-Traffic: Fluid-Flow Analysis

Probe Traffic inr bits/s

lbits/s

Cross Traffic inc bits/s

ProbeTraffic out

Cross Traffic outCross

Traffic

Probe Traffic

lcr

cr

cl

cr

rl

bits/s

bits/s

Bandwidth split in ratio c:r(Proportional Fair Queuing)

Output vs. Input Rate

cl

cla 0

m

r

l

cr

rlm

rm

Output vs. Input Dispersion

lS

clS 0

out

clS

TOPP Representation

cl 0r

mr

out

1

lc

lSlope 1

Higher Order Bottleneck

(Dispersion Ratio)

Simulation: Traffic Model

Packet Size (bytes) Number Ratio (β) Bandwidth Use Ratio (α)

60 46 4.77

148 11 2.81

500 11 9.50

1500 32 82.29

bytes579i

iiSS

bytes1298i

iiSS

(Average Packet Size)

(“Granularity”)

Assume a Poisson arrival process. (Internet traffic is not generally Poissonian, but the Poisson model provides an adequate approximation.).

Single Queue Simulation

Probe Pairs in1 M

bits/s

Cross Traffic 500 Kbits/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)

60 46 4.77

148 11 2.81

500 11 9.50

1500 32 82.29

TOPP Plot: Effect of Probe Size

0 100000 200000 300000 400000 500000 600000 700000 8000000.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.41500 bytes

500 bytes

100 bytes

Fluid Approximation

Probing Rate (bits/s)

Dis

pers

ion

Ratio

Sr

out

Fluid model represents asymptotic behaviour as the cross-traffic gradually loses its granular nature relative to the probe packets.

TOPP Plot: Effect of Granularity

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.

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:

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, Xω(t) is deterministic called a sample path.

• For a particular value of t, Xt (ω) is a random variable governed by the probability distribution behind ω.

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).

Sample Paths in a Queuing System

TIMEIDLE

BUSY

Packet Arrivals

t t

Sample Paths in a Queuing System

Arrival of a Packet Pair

1st Probe Packet Arrival

Time

Probe PacketS bits

IDLE

BUSY

Intrusive Range

Idle time is reduced by S/l seconds.

Time taken to serve the probe packet S/l seconds.

l

StB

lStItI

1

11~

2nd Probe Packet Arrival

Input Packet Separation Δ1t 12 tt

ltBtI 1

Arrival of a Packet Pair

1st Probe Packet Arrival

Time

Probe PacketS bits

IDLE

BUSY

Intrusive Range

.

2nd Probe Packet ArrivalInput 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”:

l

tBS

tIlStR

1

11

.

1t 2t

Idle Time and Intrusion Residual

l

tBStR 11

l

StBtI 11

~

1tBS

l

S

Intrusion Residual

0

Idle Time

l

Calculating the Output Dispersion 111 tWtRtWout

l

tBStR 11 111 tWtWtD

l

tBStDout

11

But:

Therefore:

l

StB

l

S

l

tYout

11Similarly:

To calculate the average output spacing, we obtain the expectation for each of the terms in the formulae.

clBEcYEDE 0Inserting these into the equations without regard for available bandwidth variability reproduces the fluid model equations.

“Nonlinearity”

Available Bandwidth Distribution

S

out dxxfl

xSE

0

dxxfl

Sx

l

ScE

l

S

out

xS

l

S

Intrusion Residual

0

l

tBStDout

11

l

StB

l

S

l

tYout

11

S0

Idle Time

l

Frequency distribution of available bandwidth

x

Simple Model for f(x)(Departing somewhat from Liu et al.)

Cross-traffic packet size = Sc 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/Sc and a standard deviation √(cΔ/Sc).

Each packet reduces the available bandwidth by Sc/Δ bits/s. Thus the mean available bandwidth is (l - c) with a standard deviation √(cSc/ Δ). For simplicity we represent this as a Gaussian distribution:

2

2

2exp

2

1

x

xf ccS

cl

Model for Output Dispersion

SSS

out dxxfxl

dxxfl

Sdxxf

l

xSE

000

2

2

2

2

2exp

2exp

2

2erf

2erf

2

S

l

S

l

S

Predicted TOPP Graph

200000 300000 400000 500000 600000 700000 8000000.95

1

1.05

1.1

1.15

1.2

1.25

1.3

Line rate 1Mbit/sCross traffic 500Kbit/s, granularity 1298 bytes

Available Bandwidth 500Kbit/sProbe packet size: 1500 bytes

Simulation Data

Optimised Model Fit (Granularity 50 bytes)

Fluid Approximation

Probing Rate (bits/s)

Dis

pers

ion

Ratio

Sr

out

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:

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.

Model of Probe-Packet Disturbance

eqp

out nr

Sn

lS

r1

Equilibrium Queue Behaviour

wEwE 22 wEw

tw

tw~

Transient Components Equilibrium Components

t

Poisson Traffic Batch-Pareto Traffic

Predicted TOPP Graphs

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.

Multi-Hop Model

Dispersion Distributions

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.)