Bounding the Lifetime of Sensor Networks Via Optimal Role Assignments Manish Bhardwaj, Anantha Chandrakasan Massachusetts Institute of Technology June 2002

Data Gathering Wireless Networks: A Primer Sensor Relay Aggregator Asleep r B R

Network Characteristics Sensor Types: Low Rate (e.g., acoustic and seismic) Bandwidth: bits/sec to kbits/sec Transmission Distance: 5-10m (< 100m) Spatial Density 0.1 nodes/m2 to 20 nodes/m2 Node Requirements Small Form Factor Required Lifetime: > year Maximizing network lifetime is a key challenge

Functional Abstraction of DGWN Node “Raw” Sensor Data Processed Sensor Data Radio+ Protocol Processor Analog Sensor Signal DSP+RISC +FPGA etc. Analog Pre-Conditioning Sensor+ A/D Sensor Core Computational Core Communication & Collaboration Core

Energy Models Etx = a11+ a2dn d n = Path loss index Transmit Energy Per Bit Transceiver Electronics Startup Energy Power-Amp Erx = a12 Receive Energy Per Bit Erelay = a11+a2dn+a12 = a1+a2dn Prelay = (a1+a2dn)r d Relay Energy Per Bit Esense = a3 Sensing Energy Per Bit Eagg = a4 Aggregation Energy Per Bit

The Role Assignment Problem: Jargon d A Node Roles: Sense, Relay, Aggregate, Sleep Role Attributes: Sense: Destination Relay: Source and Destination Aggregate: Source1, Source2, Destination Sleep: None Feasible Role Assignment: An assignment of roles to nodes such that valid and non-redundant sensing is performed

Feasible Role Assignment 11 6 1 5 2 15 13 12 8 7 B 14 4 3 9 10 FRA: 1  5  11  14  B

Infeasible Role Assignment (Redundant)

Infeasible Role Assignment (Invalid)

Infeasible Role Assignment (Invalid)

Infeasible Role Assignment (Invalid)

Infeasible Role Assignment (Redundant)

Feasible Role Assignment 11 6 1 5 2 15 13 12 8 7 B 14 4 3 9 10 FRA: 1  5  11  14  B; 2  3  9  14  B

Infeasible Role Assignment

Enumerating FRAs (Collinear Networks) B 5 4 3 2 1 Collinear networks: All nodes lie on a line Flavor being considered: Sensor given, no aggregation (Max Lifetime Multi-hop Routing) Property: Self crossing roles need not be considered B 1 2 3 4 5 B 1 2 3 4 5

Enumerating Candidate FRAs B 5 4 3 2 1 Property allows reduction of candidate FRAs from (N-1)! to 2N-1 R0: 1  B R1: 1  2  B R2: 1  3  B R3: 1  4  B R4: 1  5  B R5: 1  2  3  B R6: 1  2  4  B R7: 1  2  5  B R8: 1  3  4  B R9: 1  3  5  B R10: 1  4  5  B R11: 1  2  3  4  B R12: 1  2  3  5  B R13: 1  2  4  5  B R14: 1  3  4  5  B R15: 1  2  3  4  5  B

Collaborative Strategy Collaborative strategy is a formalism that precisely captures the mechanism of gathering data Is characterized by specifying the order of FRAs and the time for which they are sustained A collaborative strategy is feasible iff it ends with non-negative energies in the nodes R2, t0 R13, t1 R15, t2 R2, t4 R6, t5 R11, t8 R2, t9 R11, t10 R0, t3 R8, t6 Note that FRAs might be repeated (e.g. R2 here) R5, t7 B 5 4 3 2 1

Canonical Form of a Strategy Canonical form: FRAs are sequenced in order. Some FRAs might be sustained for zero time It is always possible to express any feasible collaborative strategy in an equivalent canonical form Ra0, t0 Ra1, t1 Ra2, t2 Ra4, t4 Ra5, t5 Ra8, t8 Ra9, t9 Ra10, t10 Ra6, t6 Ra3, t3 Ra7, t7 R0, t’0 R2, t’2 R6, t’6 R7, t’7 R9, t’9 R10, t’10 R12, t’12 R14, t’14 R1, t’1 R3, t’3 R5, t’5 R8, t’8 R11, t’11 R13, t’13 R4, t’4 R15, t’15 Canonical Form

The Role Assignment Problem How to assign roles to nodes to maximize lifetime? Same as: Which collaborative strategy maximizes lifetime? Same as: How long should each of the FRAs be sustained for maximizing lifetime (i.e. determine the t’ks)? Solved via Linear Programming: Objective: subject to: [Non-negativity of role time] [Non-negativity of residual energy]

Example dchar dchar/2 dchar/2 B 3 2 1 Min-hop Min-Energy Persistent 0.25 Min-Energy R0: 0 R1: 0 R2: 1.0 R3: 0 1.0 Persistent R0: 0.09 R1: 0.23 R2: 0 R3: 1.0 1.32 Optimal R0: 0 R1: 0.375 R2: 0.375 R3: 0.625 1.38 R0: 1  B R1: 1  2  B R2: 1  3  B R3: 1  2  3  B Total Lifetime

Strategy Polynomial time separation oracle + Interior point method Transformation to network flows Key observation (motivated by Tassiulas et al.) Broad class of RA problems can be transformed to network flow problems Network flow problems solved in polynomial time Flow solution  RA solution in polynomial time

Equivalence to Flow Problems Role Assignment View 3/11 3/11 3/11 R0: 0 (0) R1: 0.375 (3/11) R2: 0.375 (3/11) R3: 0.625 (5/11) B 3 2 1 1.375 (11/11) 5/11 5/11 3/11 3/11 Network Flow View 3/11 + 3/11 3/11 3/11 + 5/11 f12: 8/11 f13: 3/11 f1B: 0 f23: 3/11 f2B: 5/11 f3B: 6/11 B 3 2 1 5/11 3/11

Equivalent Flow Program

Extensions to k-of-m Sensors B Set of potential sensors (S), |S| = m Contract: k of m sensors must sense Flow framework easily extended Total net volume emerging from nodes in S is now k Constraints to prevent monopolies Constraints to prevent consumption

k of m sensors Program (additional constraints)

2-Sensor Example Sensing time divided equally between 1a and 1b Single Sensor Lifetime 1.375 s 3/11 R0: 0 (0) R1: 0.375 (3/11) R2: 0.375 (3/11) R3: 0.625 (5/11) B 3 2 1 1.375 (11/11) 5/11 3/11 2/15 2 Sensor Lifetime 1.816 s R0: 0.246 (2/15) R1: 0.615 (5/15) R2: 1.0 (8/15) R3: 0 (0) 1a B 3 2 1b 1.816 (15/15) 5/15 8/15 Sensing time divided equally between 1a and 1b Note the complete change in optimal routing strategy

Extensions to Aggregation B 3 2 1 Flavor: 1 and 2 must sense, aggregation permitted Roles increase from 2N-1 to 3.(2N-2)2 (for N-node collinear network with two assigned sensors) R0: 1  B; 2  B R1: 1  2  B; 2  B R2: 1  3  B; 2  B R3: 1  2  3  B; 2  B R4: 1  B; 2  3  B R5: 1  2  B; 2  3  B R6: 1  3  B; 2  3  B R7: 1  2  3  B; 2  3  B R8: 1  2  B; 2  B R9: 1  2  3  B; 2  3  B R10: 1  3  B; 2  3  B R11: 1  2  3  B; 2  3  B Non-Aggregating FRAs Aggregating FRAs

Aggregation Example Aggregation energy per bit taken as 180 nJ 3 2 1 R8: 1  2  B; 2  B (56%) R10: 1  3  B; 2  3  B (20%) R6: 1  3  B; 2  3  B (20%) Aggregation energy per bit taken as 180 nJ Total lifetime is 1.195 (1.596 for 0 nJ/bit, 0.8101 for  nJ/bit) It is NOT optimal for network to aggregate ALL the time The aggregator roles shifts from node to node

Aggregation Flavors 2-Level Flat General 9 8 B 10 3 8 9 1 2 3 4 5 6 7 11 3 4 1 2 5 6 7 1 2 5 6 7 2-Level Flat General

Flat and 2-Level are Poly-Time Key Idea: Multicommodity Flows Two classes of bits: Bits destined for aggregation Bits not destined for aggregation Already aggregated Never aggregated Total of P+1 commodities P-2 P-1 P

Multiple Sources B Constraints non-trivial due to possible overlaps …

Key: Virtual Nodes B Constraints as before (but using virtual nodes when there are overlaps) Virtual nodes connected via an overall energy constraint

Probabilistic Extension Single source, but lives at A, B and C probabilistically Discrete source location pmf What is the lifetime bound now? Previous program except weigh the flow by the probability

Extensions to Arbitrary PDFs Given topology and the source location pdf how can we derive a lifetime bound? No more difficult than the discrete problem …

Key: Partitioning R Partition into sub-regions (a through k) 1 c B 3 g e f d 2 l j i h k 4 5 a R Partition into sub-regions (a through k) Every point in a sub-region has the same S Calculate the probabilities of all the sub-regions Same as the discrete problem!

Reduction to discrete probabilistic source Growth of number of regions For fixed density and r, grows linearly with the number of nodes

“Future Work” PDFs of lifetime using PDFs of input graphs Lifetime loss in the absence of an oracle Multiple access issues Translating optimal role assignment into feasible data gathering protocols