Maximizing the Functional Lifetime of Sensor Networks

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Maximizing the Functional Lifetime of Sensor
Networks

• Arvind Giridhar, P.R. Kumar
• Coordinated Science Laboratory

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Data Gathering Sensor Networks

• Network consisting of multiple sensors, one sink
• Data collection
• Each sensor has data to be sent to sink (bi bits
  for sensor i)
• Periodically repeated
• Limited energy
• Sensor i has energy level Ei
• How to route data to maximize network lifetime?

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Related Work

• Tassiulas et al (2001)
• Maximum lifetime routing
• Derived equivalent linear programming formulation
• Developed distributed algorithms to converge to
  optimal solution
• Sankar and Liu (2004), Madan and Lall (2004)
  consider similar problems
• Bharadwaj, Chandrakasan (2001)
• Upper bounds on lifetime based on total energy
  consumption
• Our contribution Consider simple regular
  networks
• Analytically solve/provide sharp bounds, using
  properties of cost function

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Network Model

• Model wireless network as complete graph
• Any node can transmit to any other node.
• .but at a cost

• Cost function for communication

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Model of Information bits

• Information bits modeled as fluid
• Incompressible and infinitely divisible
• Node can transmit arbitrary non-negative quantity
  of information to any other node

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Flow Routing

• We seek flows which route all the bits to the
  sink while respecting the energy constraints
• That is,

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Functional Lifetime

• Let ? be a flow for task
• The functional lifetime of the network is the
  maximum number of repetitions L of the flow ?
  which is feasible given limited node energies

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No Direct Dependence on Time

• Functional lifetime is measured in number of
  rounds, not number of time units
• Serves as metric of cost-to-lifetime for data
  collection task
• Actual lifetime (in time units) will thus depend
  on the rate of data generation or sampling of the
  environment
• Functional lifetime does not depend on time to
  transmit
• Thus, interference does not appear in formulation
• Transmissions can be staggered in time so that no
  two transmitters ever interfere

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Linear Program Formulation

• z is the maximum fraction of node energy
  consumed in routing

• LEMMA The functional lifetime is the inverse of
  the
• optimal value z in the linear program (1). The
  corresponding
• flow ?ij1? i,j ? n scaled by 1/ z achieves
  the optimum.

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Cost Function

• Consider following cost function

• Inverse of distance-attenuation function of
  signal power over wireless medium

• For ?3, require

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Network Topology

• Linear regular topology
• Regularly spaced sensors on a line
• Sink node at one corner
• Inter-node distance d
• Define

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Near-Far Flow

• Near-far flow is a flow for which
• Each node transmits only to nearest node towards
  sink, and to the sink
• For each node i,
• Fraction of energy consumed same for all nodes

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Uniqueness of Near-far Flow

• The near-far flow is solution to following linear
  equations

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Main Result

• THEOREM Consider the linear regular network.
• The functional lifetime is upper bounded as
  follows
• If a near-far flow exists, then the above upper
  bound is achievable
• LEMMA If bifi/Ei and Ei are non-decreasing in i,
  then a near-far flow exists
• Lemma provides sufficient conditions

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Proof Outline

• Construct feasible solution to dual program with
  same objective function value
• Weak duality proves upper bound
• Proof of feasibility of dual solution depends on
  particular structure of the cost function f(.)
• Through the following inequality

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Crossing Paths in a Flow are Suboptimal
Equivalently
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Example 1

• b1b2bn-1bnb,
• E1/f1E2/f2En/fnE
• Optimal strategy consists of each node
  transmitting directly to sink
• Lifetime is E/b
• Does not depend on fR or n

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Example 2

• b1b2bn-10, bnb,
• E1E2EnE
• Under optimal strategy
• Node n transmits to sink and node n-1
• Every other node i transmits only to node i-1
• Amounts transmitted are determined by equalizing
  energy consumed

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Regular Planar Network

• Nodes located on concentric rings
• N rings
• Ring i has Mi nodes and radius iR
• Total of N(N1)M/2 nodes
• Sink node at center of circle
• Other regular networks, such as square lattice,
  can also be considered
• Similar results hold, but more analysis is more
  cumbersome

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Mapping to Linear Network

• Map each ring to a super-node
• Super-nodes located on a line at intervals of
  length R
• Energy and information bits of each super-node
  are sums of respective quantities over
  corresponding ring
• Lifetime of linear network upper bounds the
  lifetime of original network

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Upper bound

• Therefore

• THEOREM Under circular symmetry of node
  energies and bits, and existence of a near-far
  flow for the linear network of super-nodes, a
  lifetime of K times the above upper bound is
  achievable, where

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Scaling of Lifetime

• Take
• Choice of parameters gives best case lifetime
• For these parameters, we compare optimal flow to
  simple nearest neighbor scheme

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Linear Network Case

• Simple nearest neighbor scheme
• Transmit all to nearest neighbor towards sink

• Optimal flow
• Summation in denominator can be computed

• Ratio goes to 1 as
• network size grows large

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Planar Network Case

• Consider circular planar network
• Similar calculation gives
• While the nearest neighbor scheme has a lifetime
  lower bounded as follows

• Optimal is better by factor of log N

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Implications

• Thus, simple nearest neighbor communication
  scheme is nearly optimal
• Such a scheme is preferable from an operational
  point of view
• Lower interference and medium access contention
• Need not use multiple power levels
• Broader implication Optimal lifetime scales
  poorly with network size

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Future Work

• Problem needs to be solved for all values of
  per-node energies and information quantities
• Lifetime may be determined by the energy/number
  of bits of a single node - degenerate case
• Need to determine conditions when this happens,
  and what happens otherwise
• May need to consider more general cost functions
• Tradeoff between lifetime and delay?