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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

- 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

- 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

- fairness property
- maximum throughput property
- discriminators from related work

- sensor network
- notation
- congestion-free forwarding schedule
- lexicographic maxmin 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

- 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)

- 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

- 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

- 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

- 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

- 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)

- 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

- 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

- 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

- 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”

- 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”

- exploit knowledge of layer 2 to tighten 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

- 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

- 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

- 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

- 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