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

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

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 the sum of all inbound forwarding rates is less than or equal to the maximum forwarding rate of that sensor

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 I

x

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() for each proper independent set

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 T

x

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