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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
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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
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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
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Maxmin Fairness and Related Work
fairness property maximum throughput property discriminators from related work
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Maxmin Fairness Property
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Network Model and Problem Definition
sensor network notation congestion-free forwarding schedule lexicographic maxmin rate assignment
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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
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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)
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Maxmin Subset and Maxmin Subassignment
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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
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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
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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
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Finding Maxmin Assignment and Forwarding Schedule
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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
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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)
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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
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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
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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
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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”
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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”
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CDMA and Adjacent-Link Constraints
exploit knowledge of layer 2 to tighten upper and lower bounds
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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
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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
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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
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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
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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