Authored by shigang chen yuguang fang and ye xia presented by rob mitchell october 23 2007
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Lexicographic Maxmin Fairness for Data Collection in Wireless Sensor Networks PowerPoint PPT Presentation


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authored by: Shigang Chen, Yuguang Fang and Ye Xia presented by: Rob Mitchell October 23, 2007. Lexicographic Maxmin Fairness for Data Collection in Wireless Sensor Networks. Overview. Introduction Maxmin Fairness and Related Work Network Model and Problem Definition

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Lexicographic Maxmin Fairness for Data Collection in Wireless Sensor Networks

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Authored by shigang chen yuguang fang and ye xia presented by rob mitchell october 23 2007

authored by: Shigang Chen, Yuguang Fang and Ye Xia

presented by: Rob Mitchell

October 23, 2007

Lexicographic Maxmin Fairness for Data Collection in Wireless Sensor Networks


Overview

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

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

Maxmin Fairness and Related Work

  • fairness property

  • maximum throughput property

  • discriminators from related work


Maxmin fairness property

Maxmin Fairness Property


Network model and problem definition

Network Model and Problem Definition

  • sensor network

  • notation

  • congestion-free forwarding schedule

  • lexicographic maxmin rate assignment


Finding maxmin optimal 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

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 subassignment1

Maxmin Subset and Maxmin Subassignment


Maximum common rate mcr

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

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

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 schedule1

Finding Maxmin Assignment and Forwarding Schedule


Consider energy expended to receive

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

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

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

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

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

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

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

CDMA and Adjacent-Link Constraints

  • exploit knowledge of layer 2 to tighten upper and lower bounds


Using 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

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

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

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

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


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