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 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) 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 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 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 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 Ix 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(b) for each proper independent set b

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 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 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