Asymptotic Transport Capacity of Wireless Erasure Networks

September 28th, 2006

Asymptotic Transport Capacity of Wireless Erasure Networks

Brian Smith and Sriram VishwanathUniversity of Texas at Austin

Allerton Conference on Communication, Control, and Computing



Overview

Introduction Motivation Wireless Erasure Networks Erasure Model Upper Bounds Dense Networks Summary


Introduction

Capacity of Multiple-Source Multiple-Destination Networks (Multiple Unicast) Hard Problem Transport Capacity: Convenient Scalar Description

Distance Weighted Rate-Sum (bit-meters)

Work by Gupta & Kumar[2000] and Xie & Kumar[2004] Gaussian Interference Channel Model Information Theoretic Linear Bound on Transport Capacity

Growth Under high attenuation model


Wireless Erasure Networks: Background

Dana, Gowaikar, Hassibi & Effros [2006] Packetized Wireless Network Model

GF2 Source Alphabet Broadcast requirement on directed graph Links are independent erasure channels No receiver interference – Receiver gets vector

Result Multicasting from single source to multiple receivers can be

performed at generalized min-cut max-flow rate

Si Sjij

C

ε1

Modified Cut-Set Bound

12

13

23

24

34

342313 11

Example Bound Evaluation


Motivation Desire to investigate transport capacity from a

network layer point of view for a wireless network Thus, Erasure Network

Packet is either correctly decoded or nothing is known about the contents

Wireless Networking Broadcast Requirement Fully connected graph

Interference - physical layer phenomenon Main Result: Linear Bound on Transport Capacity

Growth


Erasure Probability as Function of Geographic Distance Minimum Node Separation Constraint: d>dmin

Threshold Model (d)=0, dd*; 1, d>d*

Exponential Model (d)=1-e-d/d*

Polynomial Decay Model (d)=1-1/(1+d), >3

In all cases (0)=0, (∞)=1

Connecting Transport Capacity with Wireless Erasure Networks

x5

x4

x3

x2x1

y6

y8

y7

y9


Dana, Gowaikar, Hassibi, Effros 2006 Includes a ‘directed acyclic graph’ requirement

Our model: Completely connected, many back cycles ΣR ≤ I(XS;YSC|XSC) Use data processing inequality

(as in XK2004) to show that onlysymbols from outside the cut matter

Back cycles do not increase sum-rate bound

Aside: Cycles

S SC


Bounding Transport Capacity

Examine one-dimensional case for intuition Every source-destination path

crosses at least one cut

mCmSi Sj

ijm

r dT 12 min

Rate Cut m

dmin

Source-Destination Rate Pairs


Threshold Model Converse

Rate Cut-set: One bit for every node “within range” Each node within range of less than

d*/dmin cuts to its right

2-D: Different Constants

*min

*min 2/2 ndddndTr

Rate Cut m

dmin

Length d* represents

range of transmission


Bounding Transport Capacity (2) Index nodes in order of increasing position Place one cut between consecutive nodes Every source-destination path

crosses at least one cut

Rate Cut m

dm

Source-Destination Rate Pairs

m

i

n

mjij

n

mmr dT

1 1

1

1

1

Node m


1-D Exponential Converse

Model: (d)=1-e-d/d*

Transport Capacity ≤κn

Constant depends only on dmin, d*

m

i

n

mj

ddn

mmr

j

ikk

edT1 1

*/1

1

1

11

m

i

n

mj

ddn

mmr

j

ikk

edT1 1

*/1

1

1

n

ii

n

ii aa

11

11


Squeeze

m

i

n

mj

ddijdn

mmr

medT1 1

*/1

1

min

dm

dm

*/1

11

ddn

mmr

medT

Bounded by Kn

Geometric Sequences


Model: (d)=1-(1+d), >3 Instead of a geometric summation, bound the

summation with integrals Per-node Transport Capacity Result: k dm

3-

As long as >3, upper-bounded dm>dmin

Polynomial Case Differences


For 1-D, bounded the transport capacity across every cut Doesn’t work for 2-D, because some cuts can diverge with n:

Attempt: Place one horizontal cut in between

each node. Allow overlapping cuts Lots of zeros-valued distances, though

2-D: Different Approach Needed

Rate cut-set for this cut increases (n)

Multiple overlapping cuts add zero to TC


Explanation of General Transport Capacity Bound

Notation

d1h d2

h

d34

Cut 1

Node 1

dij≥dmin

m

i

n

mjij

n

m

hmhr dT

1 1

1

1

1


Squish

dih di+1

h

Node i

2/)( min2min

21

dmjdddm

ik

hkij

Node m

dm-1h Move all nodes with

index greater than ‘m’ in as close as possible, within minimum distance constraint, to bound summation


Proof Sketch

Replace the product term with a summation Bound the summation with integrals to get transport

capacity in terms of horizontal components only Rearrange the summations Bound each of the terms in the new summation by a

constant


Drop the minimum distance requirement Place n nodes equally spaced on unit square

Without Interference: Upper bound and achievablity both (n)

Dense Networks


Super-linear Growth

Place n nodes in two groups, spaced d*ln n apart n ln n growth But, specifying erasure locations requires ln n extra

informationd* ln n

ne

neR

n

k

n

n

k

n

j

n

k

n

j

dnd

1

1

1 11 1

/ln

1

11111

**


Bhadra, Gupta & Shakkottai 2006 Capacity Bounds - Asymptotically tight in field size q for uniform fading

over the field

Received signal: Yj=ijXi

ij is a 0-1 Bernoulli r.v. with P[ij=0]=ij

Cut-Set Bound Write the vector of transmitted and received symbols on each side of

the cut as Y=HX H is a 0-1 random matrix

Current Work:Additive Finite-Field Interference

x5

x4

x3

x2x1

y6

y8

y7

y9

Modified Cut-Set Bound

E[rank(H)] lg(q)


Summary Transport capacity linearly bounded in number of nodes

with minimum distance requirement Threshold Model Exponential Model Polynomial Model with >3 for 1-D

Results correspond well to Xie-Kumar results for Gaussian Network case What properties of network models cause linear growth? Analog to CRIS in low-attenuation case?

Dense Networks Remove minimum distance requirement? Achievability with interference?

Asymptotic Transport Capacity of Wireless Erasure Networks

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