Lexicographic Maxmin Fairness for
Data Collection in Wireless Sensor
Networks
authored by: Shigang Chen, Yuguang Fang and
Ye Xia
presented by: Rob Mitchell
October 23, 2007
Overview

Introduction

Maxmin Fairness and Related Work

Network Model and Problem Definition

Finding Maxmin Optimal Rate Assignment

Discussions on Media Contention

Maxmin Assignment with Edge or Mixed
Capacities

Weighted Maxmin Assignment

Conclusion
Introduction



sensor networks are distinguished by their
limited energy resources
make most efficient use of energy by not
dropping sensor data
provide the best data possible by making most
efficient use of communication capacity
Maxmin Fairness and Related Work

fairness property

maximum throughput property

discriminators from related work
Maxmin Fairness Property
Network Model and Problem
Definition

sensor network

notation

congestion-free forwarding schedule

lexicographic maxmin rate assignment
Finding Maxmin Optimal Rate
Assignment

Maxmin Subset and Maxmin Subassignment

Maximum Common Rate (MCR) Problem

Maximum Single Rate (MSR) Problem

Maxmin Assignment and Forwarding Schedule

Consider Energy Expended to Receive

Eliminating Long Forwarding Paths
Maxmin Subset and Maxmin
Subassignment


given r, the maxmin subset of A with respect to r
is the set of all x such that the maxmin rate of x
is less than or equal to r
given r, the maxmin subassignment with respect
to r is the set of all maxmin rates such that x is
a member of A(r)
Maxmin Subset and Maxmin
Subassignment
Maximum Common Rate (MCR)



the actual rate at which every active sensor whose maxmin rate is not
less than or equal to r generates data equals C
the actual rate at which every active sensor whose maxmin rate is not
less than or equal to r generates data is less than or equal to W
the actual rate at which every active sensor whose maxmin rate is
less than or equal to r generates data is the maxmin rate of that
sensor

the actual rate at which every inactive sensor generates data is 0

the forwarding rate on every link is greater than or equal to 0


for every sensor, the sum of all outbound forwarding rates equals the
sum of all inbound forward rates plus the actual rate at which a
sensor generates data
for every sensor, the sum of all outbound forwarding rates is less than
or equal to the maximum forwarding rate of that sensor
Maximum Single Rate (MSR)




the actual rate at which a given sensor generates data equals S
the actual rate at which a given sensor generates data is less than or equal
to W
the actual rate at which every active sensor whose maxmin rate is not less
than or equal to r and is not considered above generates data is C(r)
the actual rate at which every active sensor whose maxmin rate is less than
or equal to r generates data is the maxmin rate of that sensor

the actual rate at which every inactive sensor generates data is 0

the forwarding rate on every link is greater than or equal to 0


for every sensor, the sum of all outbound forwarding rates equals the sum of
all inbound forward rates plus the actual rate at which that sensor generates
data
for every sensor, the sum of all outbound forwarding rates is less than or
equal to the maximum forwarding rate of that sensor
Finding Maxmin Assignment and
Forwarding Schedule

initialize r to 0

initialize A(r) to the null set

while A(r) does not contain all active sensors

compute C(r)

make X the null set

for each active sensor, x, not in A(r)

compute S(x,r)

if S(x,r) = C(r) then



C(r) is the maxmin rate of x
 add x to X
set r to C(r)

add X to A(r)
return the congestion-free forwarding schedule
Finding Maxmin Assignment and
Forwarding Schedule
Consider Energy Expended to
Receive

Tx does not consider energy requirement associated with packet
reception

leverage MCR linear program to optimize

replace:
for every sensor, the sum of all outbound forwarding rates is less than
or equal to the maximum forwarding rate of that sensor

with:
for every sensor, the sum of all outbound forwarding rates plus l the
sum of all inbound forwarding rates is less than or equal to the
maximum forwarding rate of that sensor

l represents the ratio of energy for receiving a packet to energy for
sending a packet
Eliminating Long Forwarding Paths



use only shortest path to forward packets
additional constraint which results in a less
efficient forwarding schedule
accomplish preprocessing on E to transform
into directed acyclic graph (DAG)
Discussions on Media Contention

Impact on Finding Optimal Maxmin Rate
Assignment

Contention Graph

Independent-Set Constraints

Clique Constraints

Complete-Contention Constraints

CDMA and Adjacent-Link Constraints

Using Upper and Lower Bounds
Contention Graph

forwarding rate is affected by other sensors

contending relation: (x,y) \bowtie (w,z)



a sensor cannot transmit two packets
simultaneously
a sensor cannot transmit and receive
simultaneously
when x sends a packet, any sensor that is in Ix
should not be receiving another packet
Independent-Set Constraints




an independent set is a subset of vertices
(links) with no edge (contending relation)
between any two of them
M is the media capacity (e.g. bps)
t() is the fraction of time when a proper
independent set is scheduled for transmission
add to MCR and MSR linear programs:
the forwarding rate of each link is equal to M
times the sum of t(b) for each proper
independent set b
Clique Constraints

the “opposite” of an independent-set

add to MCR and MSR linear programs:
for every clique, the sum of the forwarding rates
of every link is less than M

resulting linear programs return an “upper
bound”
Complete-Contention Constraints


every link with which a given link has a
contending relation is in its complete-contention
set
add to MCR and MSR linear programs:
for every link, the forwarding rate of that link
plus the sum of the forwarding rates of every
link in the complete-contention set of that link is
less than or equal to M

resulting linear programs return a “lower bound”
CDMA and Adjacent-Link
Constraints

exploit knowledge of layer 2 to tighten upper
and lower bounds
Using Upper and Lower Bounds

Begin with upper bound

Apply back-pressure as congestion occurs

No upstream neighbor should have to throttle
lower than the lower bound
Maxmin Assignment with Edge or
Mixed Capacities



not all links are created equal
forwarding rates are individually constrained by
c(x,y) rather than constrained as an aggregate
by Tx
replace last constraint of MCR and MSR linear
programs with:
the forwarding rate of every link is less than or
equal to the capacity of that link
Weighted Maxmin Assignment

not all sensors are created equal

replace MCR constraint:
the actual rate at which every active sensor whose maxmin rate is not
less than or equal to r generates data equals C

with:
the actual rate at which every active sensor whose maxmin rate is not
less than or equal to r generates data equals sensor weight times C

replace MSR constraint:
the actual rate at which a given sensor generates data equals S

with:
the actual rate at which a given sensor generates data equals sensor
weight times S
Conclusion




allows multipath/load balancing
polynomial-time solution for low-rate sensor
networks
initial treatment of same problem without
constraints associated with low-rate
configuration
solution appropriate for use at a base station in
stable network conditions
Recap

Introduction

Maxmin Fairness and Related Work

Network Model and Problem Definition

Finding Maxmin Optimal Rate Assignment

Discussions on Media Contention

Maxmin Assignment with Edge or Mixed
Capacities

Weighted Maxmin Assignment

Conclusion