Analysis of Latency for Reliable End-to-End Batch Transmission in - - PowerPoint PPT Presentation

Analysis of Latency for Reliable End-to-End Batch Transmission in Multi-Rate Multi-Hop Wireless Networks

Teeraw at I ssariyakul ( teeraw at@trlabs.ca) Ekram Hossain ( ekram @ee.um anitoba.ca) Attahiru Sule Alfa ( alfa@ee.um anitoba.ca)

Outline

• Background and Motivations
• Modeling end-to-end transmission

(Main contribution)

• Numerical results
• Summary, conclusions, and future

studies

Why Multi-Hop?

• Use short range

communications

– Increase data rate – Reduce delay – Reduce energy consumption

• Multi-hop relay data

from the base station to the mobile

– Increase coverage of service area – Better load balance Short Range Long Range

Base st at ion

What About Ad Hoc ?

• Distributed medium access control

protocols (e.g., IEEE 802.11) lead to degradation in an end-to-end level

• More hops = > Geometric increase in

end-to-end error

• Limited energy consumption

Wireless Channel and ARQ

• Random Error in Data Transmission
• Adaptive Modulation and Coding

– Bound average hop-level packet error probability – High SNR: increase transmission rate – Low SNR: decrease transmission rate

• Automatic Repeat reQuest (ARQ) with I NFINITE

persistence SOURCE DESTI NATI ON One successf ul end-t o-end deliver y 5 t r ansmissions

Main Contribution

• Chain topology with 3 nodes
• Both hops can transmit at the

same time (e.g., ODMA)

• Objective:

– Note: Different application may need different delay requirement – Find distribution of the latency to delivery all N packets to the destination.

. . .

N packets

Source Destination

Main Contribution (cont.)

• Implication of delay distribution

– E.G., the batch becomes useless after k time slots, – Pr{ delay < = k} = Probability to deliver the batch before it becomes useless.

Absorbing Markov Process

TRANSI ENT STATES

• Start points: any state
• Finish points: any absorbing state

pij is t he t r ansit ion probabilit y f r om st at e i t o j A B X1

...

pAB pBA pAC pCA 1 Xn 1

...

ABSORBI NG STATE

Absorbing Markov Process

• Transition probability matrix (P)

I R Q

=

X … B A

To From

P=

I … X pBX … pBB pBA B pAX … pAB pAA A

. . . . . . . . .

(α,α0) = the initial probability matrix

Expectation CDF

Phase (PH) Type Distribution

• PH distribution: distribution of time to

absorption in an absorbing Markov process

• Let k be the number of transitions to

reach the absorbing state.

   > = ⋅ ⋅ − + =

−

; ; 1

1

k k F

k k

e Q α α α

1 Q I α ⋅ − ⋅ =

−1

) ( E[k]

   > = ⋅ ⋅ =

−

; ;

1

k k f

k k

R Q α α

PMF

System Model (details)

• Wireless channel

– Independent and identically distributed (i.i.d.) – Channel state m, randomly chosen from [ 1,… ,M] – State = m: transmit m packets – Each transmitted packet is in error with probability perr

• Use ARQ with infinite persistence at

each node.

Absorbing Markov Model

• Model the transmission process as an

absorbing Markov Chain

• Latency = Time to absorption

– PMF, CMF, and Expectation Absorbing state All N packets reach the destination node Finishing point Initial state N packets are supplied to the source node Starting point Markov Chain Multi-Hop Network

Queuing Model

...

Absorbing state = (0,0) All N packets reaches the destination node Finishing point Initial state = (N,0) N packets are supplied to the source node Starting point Markov Chain Multi-Hop Network

• Absorbing Markov chain (X1,X2)
• Xi = buffer size of node i

1 2 3

Mathematical Model

• Final steps
• 1. Find Relevant Matrices
• Initial probability matrix:

α= e i = [ 0 … 0 1 0 … 0]

• Transition probability matrix: P (next

page)

• 2. Use the formulae for absorbing Markov

process to find

• PMF
• CMF
• Expectation

The state where N packets are supplied to the source node

Statistics for Latency

• Latency = Time to absorption

Expectation CDF    > = ⋅ ⋅ − + =

−

; ; 1

1

k k F

k k

e Q α α α

1 Q I α ⋅ − ⋅ =

−1

) ( E[k]

   > = ⋅ ⋅ =

−

; ;

1

k k f

k k

R Q α α

PMF

Statistics for Latency

• Latency = Time to absorption

Transition Probability Matrix (P)

A33 A32 A31 A30 30 21 A22 A21 A20 20 12 11 A11 A10 10 03 02 A'00 B 01 1 00 30 21 20 12 11 10 03 02 01 00 From (X1,X2) To (X1,X2)

initial state absorbing state

X1 does not change X1 decreases

R Q R I

P= Q

Transition Probability Matrix

30 21 20 12 11 10 03 02 01 00 30 21 20 12 11 10 03 02 01 00

pi(si) = probability that node i successfully transmit s packets

X1 = 0, X2 = 2 X1 = 3, X2 = 0

Transition Probability Matrix

30 21 20 12 11 10 03 02 01 00 30 21 20 12 11 10 03 02 01 00

pi(si) = probability that node i successfully transmit s packets

Successful Transmission Probability

Complementary cumulative distribution function Probability that si packets are successfully transmitted, given that total transmitted packet is m

Si <= Xi Si > Xi

Model Validation

M (Max Tx Rate) = N (Batch Size) Equally Likely Channel

E[k]

Batch Size (N)

Effect of Number of Channel States N = 10

Deep slope: High improvement

Slope decreases: Xi< m, for some time slot

E[k]

• Max. Tx. Rate (M)

Cumulative Distribution Function

CDF (Fk)

End-to-end latency (k) 95%

Minimum Latency

Batch Size (N)

Latency for 95% Delivery

Summary and Conclusions

• End-to-end latency distribution in a multi-hop

wireless network as a function of

– link-error probability, – transmission rate, and – end-to-end latency distribution

• Validate using simulation
• Expected latency does not guarantee high

probability of batch delivery

• Increasing max. Tx rate or channel states

– Decreasing Latency – Rate under is underutilized, since packets in the buffer < current transmission rate – Decreasing rate of decrease in latency

Further Studies

• Other ARQ policies (e.g., limited

persistence) see WN27-3

• More realistic channel model (e.g.,

Rayleigh Fading or FSMC)

• Channel Access Policies
• Extension to window-based

congestion control (window= batch)

• Steady State Analysis

Thank you for Attention Question?