Lexicographic Maxmin Fairness for Data Collection in Wireless Sensor Networks

1. 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

2. 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

3. 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

4. Maxmin Fairness and Related Work • fairness property • maximum throughput property • discriminators from related work

5. Maxmin Fairness Property

6. Network Model and Problem Definition • sensor network • notation • congestion-free forwarding schedule • lexicographic maxmin rate assignment

7. 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

8. 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)

9. Maxmin Subset and Maxmin Subassignment

10. 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

11. 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

12. 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

13. Finding Maxmin Assignment and Forwarding Schedule

14. 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

15. 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)

16. 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

17. 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

18. 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

19. 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”

20. 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”

21. CDMA and Adjacent-Link Constraints • exploit knowledge of layer 2 to tighten upper and lower bounds

22. 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

23. 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

24. 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

25. 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

26. 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