Maximal Lifetime Scheduling for Wireless Sensor Surveillance Networks

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Maximal Lifetime Scheduling for Wireless Sensor
Surveillance Networks

• Prof. Xiaohua Jia
• Dept. of Computer Science
• City University of Hong Kong

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Wireless Sensor Networks

• A sensor network consists of many low-cost and
  low-powered sensor devices. A wireless sensor
  node has three basic components
• A processor
• A set of radio communication devices
• Sensing devices

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Maximum Lifetime Target Surveillance Systems

• Given a set of sensors to watch a set of targets
• Each sensor has a given energy reserve. It can
  watch at most one target at a time.
• A target can be inside several sensors watching
  range. It should be watched by at least one
  sensor at any time.
• Problem find a schedule for sensors to watch the
  targets in turn, such that the lifetime is
  maximized.
• Lifetime is the duration up to when a target can
  no longer be watched by any sensor.

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Solving the Maximum Lifetime Problem

• Our solution consists of three steps
• compute the upper bound of the maximal lifetime
  and a workload matrix of sensors.
• decompose the workload matrix into a sequence of
  schedule matrices.
• obtain a target watching timetable for each
  sensor.

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Finding Maximum Lifetime

• S / T set of sensors / targets, nS, mT.
• Ei initial energy reserve of sensor i.
• S(j) set of sensors able to watch target j.
• T(i) set of targets within watching range of
  sensor i.
• xij the total time sensor i watching target j.
• Objective Max L

(1) (2)
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The Workload Matrix
targets
Xnm
sensors

• Xnm is a workload matrix, specifying the total
  time a sensor watching a target
• the sum of all elements in each column is equal
  to L (from eq. (1) in the LP formulation) .
• the sum of all elements in each row is less than
  or equal to L (from ineq. (2) in the LP
  formulation).

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Decompose Workload Matrix into a Sequence of
Scheduling Matrices

• A scheduling matrix specifies the schedule of
  sensors to watch targets during a session
• only one non-zero number in each column (i.e., a
  target is watched by only one sensor during the
  session).
• at most one non-zero number in each row (i.e., a
  sensor can watch at most one target at a time and
  there is no switching in a session).
• all non-zero elements having the same value,
  which is the duration of the session.

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A Special Case of nm

• When n m, we have
• Ri Cj L, for 1 i, j n.
• (Ri sum of row i, Cj sum of column j).
  Because
• The workload matrix Xnn can be represented as
• Xnn L Ynn
• Ynn is a Doubly Stochastic Matrix. The sum of
  each row and each column is equal to 1.

L
and
L
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A Special Case of nm (contd)

• Theorem 1. Matrix Ynn can be decomposed as
• Ynn c1P1 c2P2 ctPt,
• where t(n-1)21, each Pi, 1it, is a
  permutation matrix and c1, c2,, ct, are
  positive real numbers and c1c2ct1.

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Convert to Perfect Matching

• Represent Xnm as a bipartite graph, with xij as
  edge weight.
• Compute a perfect matching in the graph. Let ci
  be the smallest weight of the n edges in the
  matching.
• Deduct ci from the weight of the n matching-edges
  and remove the edges whose weight is zero.
• Repeat step 2) 3) until there is no edge in the
  bipartite graph.

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Complete Decomposibility

• Does there exist a perfect matching in every
  round of the decomposition process?
• Theorem 5. For any square matrix Wnn of
  nonnegative numbers, if Ri Cj for 1 i, j n,
  there exists a perfect matching in the
  corresponding bipartite graph.
• The workload matrix can be exactly decomposed
  into a sequence of schedule matrices!

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General case of ngtm

• Fill matrix Xnm with dummy columns to transform
  to the case of n m

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Fill Matrix
L
L

• Record the remaining numbers of row-sums and
  column-sums.
• Determine dummy matrix Zn(n-m) from z11.
• Assign zij to the largest possible number without
  violating the above two constraints Ri and Cj.

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An example for filling matrix
0 7 6 7
0 3 6 7

• 4
• 7
• 6
• 7

8 8 8
4 8 8
0 8 8
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DecomposeMatrix Algorithm

• Input workload matrix Xnm.
• Output a sequence of schedule matrices.
• Begin
• if ngtm then
• Fill matrix Xnm to obtain a square matrix
  Wnn
• Construct a bipartite graph G from Wnn
• while there exist edges in G do
• Find a perfect matching M (i.e., Pi) on G
• Let ci be smallest weight in M
• Deduct ci from all edges in M and remove edges
  with weight 0
• endwhile
• Output Wnn c1P1 c2P2 ctPt
• End

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A Walkthrough Example
Tab. 1. Energy reserve of sensors
6 sensors (clear color) and 3 targets (grey
color)
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Compute the LP formulation

• L 40.5643
• Workload matrix

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Fill Xnm to a square matrix

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Decompose the workload matrix

• W66 c1P1 c2P2 c5P5.
• By removing the dummy columns, we have

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Obtain scheduling timetable for sensors
Tab. 2. The schedule timetable for 6 sensors
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Simulation Results
Fig. 2(a). t versus N when M10
Fig. 2(b). t versus M when N100
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Simulation Results
Fig. 3(a). Lifetime versus surveillance range
Fig. 3(b). Lifetime versus N when M10
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Summary

• Discussed the maximal lifetime scheduling problem
  in sensor surveillance networks.
• Proposed an optimal solution to the max lifetime
  scheduling problem.
• The number of decomposition steps for finding the
  optimal schedule is linear to the network size.

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• Thank You !