CS Dept, City Univ.1 Maximal Lifetime Scheduling for Wireless Sensor Surveillance Networks Prof....
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CS Dept, City Univ. 1 Maximal Lifetime Scheduling for Wireless Sensor Surveillance Networks Prof. Xiaohua Jia Dept. of Computer Science City University of Hong Kong
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CS Dept, City Univ. 1
Maximal Lifetime Scheduling for Wireless Sensor
Surveillance Networks Prof. Xiaohua Jia
Dept. of Computer Science
City University of Hong Kong
CS Dept, City Univ. 2
Wireless Sensor Networks
A A sensor network consists of many low-cost and low-powered sensor devices. A wireless sensor sensor node has three basic components:node has three basic components:
• A processorA processor• A set of radio communication devicesA set of radio communication devices• Sensing devicesSensing devices
CS Dept, City Univ. 3
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 find a schedule for sensors to watch the targets in turn, such that the lifetime is the targets in turn, such that the lifetime is maximized. maximized.
LifetimeLifetime is the duration up to when a target can is the duration up to when a target can no longer be watched by any sensor.no longer be watched by any sensor.
CS Dept, City Univ. 4
Solving the Maximum Lifetime Problem
Our solution consists of three steps:
1) compute the upper bound of the maximal lifetime and a workload matrix of sensors.
2) decompose the workload matrix into a sequence of schedule matrices.
3) obtain a target watching timetable for each sensor.
CS Dept, City Univ. 5
Finding Maximum LifetimeS / T = set of sensors / targets, n=|S|, m=|T|.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)
,Tj
.Si },min{)(
i
iTj
ij ELx
LxjSi
ij )(
CS Dept, City Univ. 6
The Workload Matrix
Xn×m 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).
Xn×m=
mnnmnn
m
m
xxx
xxx
xxx
...
......
...
...
21
22221
11211
targetstargets
sensorsensor
ss
CS Dept, City Univ. 7
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.
0...00
......
0...000
...000
...
0...00
......
...000
0...00
0...00
......
...000
0...00
...
......
...
...
2
2
2
1
1
1
21
22221
11211
t
t
mnnmnn
m
m
z
z
z
z
z
z
z
z
xxx
xxx
xxx
CS Dept, City Univ. 8
A Special Case of n=m
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 Xn×n can be represented as: Xn×n = L ×Yn×n
Yn×n is a Doubly Stochastic Matrix. The sum of each row and each column is equal to 1.
nnnnnn
n
n
xxx
xxx
xxx
...
......
...
...
21
22221
11211
i j
ji LnCR ,
.LRi and
L
L
CS Dept, City Univ. 9
A Special Case of n=m (cont’d)
Theorem 1. Matrix Yn×n can be decomposed as:
Yn×n = c1P1 + c2P2 +…+ ctPt,
where t≤(n-1)2+1, each Pi, 1≤i≤t, is a permutation
matrix; and c1, c2,…, ct, are positive real numbers and
c1+c2+…+ct=1.
CS Dept, City Univ. 10
Convert to Perfect Matching
1) Represent Xn×m as a bipartite graph, with
xij as edge weight.
2) Compute a perfect matching in the graph. Let ci be the smallest weight of
the n edges in the matching.
3) Deduct ci from the weight of the n
matching-edges and remove the edges whose weight is zero.
4) Repeat step 2) & 3) until there is no edge in the bipartite graph.
CS Dept, City Univ. 11
Complete Decomposibility
Does there exist a perfect matching in every round of the decomposition process?
Theorem 5. For any square matrix Wn×n 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!
CS Dept, City Univ. 12
General case of n>m
Fill matrix Xn×m with dummy columns to transform to the
case of n = m:
nnmnnnnnmnn
mnm
mnm
nn
zzzxxx
zzxxxzzxxx
W
×
×
... ......... ......
...z ......z ...
2121
2222122221
1121111211
CS Dept, City Univ. 13
Fill Matrix
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 R’i and C’j.
nnmnnnnnmnn
mnm
mnm
nn
zzzxxx
zzxxxzzxxx
W
×
×
... ......... ......
...z ......z ...
2121
2222122221
1121111211
;)11
'i
mn
jiji RLzR
LzCn
iijj
1
')2
L
L
CS Dept, City Univ. 14
An example for filling matrix
4
767
434241
333231
232221
131211
zzzzzzzzzzzz
8 8 8
434241
333231
232221
0 0 4
zzzzzzzzz
4 8 8
0
767
4342
3332
2322
0 0 4
0 0 4
zzzzzz
0
367
0 8 8
CS Dept, City Univ. 15
DecomposeMatrix Algorithm
Input: workload matrix Xn×m.Output: a sequence of schedule matrices.Begin
if n>m thenFill matrix Xn×m to obtain a square matrix Wn×n;
Construct a bipartite graph G from Wn×n;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;
endwhileOutput Wn×n = c1P1 + c2P2 +…+ ctPt;
End
CS Dept, City Univ. 16
A Walkthrough Example
1
23
4
1
2
6
5
3
Sensors 1 2 3
Ei 15.6926 34.2627 24.8717
Sensors 4 5 6
Ei 21.7847 46.6865 34.5310
6 sensors (clear color) and 3 targets (grey color)
Tab. 1. Energy reserve of sensors
CS Dept, City Univ. 17
Compute the LP formulation
L = 40.5643
Workload matrix:
0 0 0 21.8444 12.4064 0
0 17.9125 0 0 0 24.8717
18.7199 10.2454 0 0 0 6926.15
36X
CS Dept, City Univ. 18
Fill Xn×m to a square matrix
40.5643 0 0 0 0 0 0 6.3135 0 21.8444 12.4064 0 0 22.6518 0 0 17.9125 0 0 11.5990 4.0936 0 0 24.87170 0 11.5990 18.7199 10.2454 0 0 0 24.8717 0 0 6926.15
66W
CS Dept, City Univ. 19
Decompose the workload matrix
W6×6 = c1P1 + c2P2 +…+ c5P5.
By removing the dummy columns, we have:
0 0 0 0 0 0 0 6.3135 0 0 0 6.3135
6.3135 0 0 0 0 0
0 0 0 4.0936 0 0
0 0 0 0 0 0 0 4.0936 0 0 0 0936.4
36X
0 0 0 6.1518 0 0
0 0 0 0 0 6.15180 6.1518 0 0 0 0
0 0 0 0 12.4064 0 0 0 0 0 0 12.4064
12.4064 0 0 0 0 0
0 0 0 11.5990 0 0
0 11.5990 0 0 0 0 0 0 0 0 0 11.5990
CS Dept, City Univ. 20
Obtain scheduling timetable for sensors
Sensors Watching Duty (time duration and watching targets)
10~4.0936Target 1
4.0936~28.9653Turn off
28.8953~40.5643Target 1
20~10.2454Target 2
10.2454~28.9653Target 3
28.8953~40.5643Turn off
30~4.0936Turn off
4.0936~28.9653Target 1
28.8953~40.5643Turn off
40~10.2454Turn off
10.2454~16.5589Target 2
16.5589~28.8953Turn off
28.8953~40.5643Target 2
50~10.2454Target 3
10.2454~16.5589Turn off
16.5589~28.8953Target 2
28.8953~40.5643Target 3
60~40.5643Turn off
Tab. 2. The schedule timetable for 6 sensors
CS Dept, City Univ. 21
Simulation Results
150
200
250
300
350
50 60 70 80 90 100
The number of sensors (N)
Th
en n
um
ber
of
dec
om
po
sin
g
step
s (t
)
100
200
300
400
500
600
700
800
900
5 10 15 20 25
The number of targets (M)
Th
e n
um
ber
of
deco
mp
osin
g
ste
ps
(t)
Fig. 2(a). t versus N when M=10 Fig. 2(b). t versus M when N=100
CS Dept, City Univ. 22
Simulation Results
Fig. 3(a). Lifetime versus surveillance range Fig. 3(b). Lifetime versus N when M=10
0
50
100
150
200
250
300
5 10 15 20 25
The surveillance range
Th
e lif
etim
e o
f su
rvei
llan
cesy
stem
(h
r)
Our optimal algorithm
Greedy algorithm
0
50
100
150
200
250
10 20 30 40 50 60 70 80 90 100
The number of sensors (N)
Th
e lif
etim
e o
f su
rvei
llan
ce
syst
em (
hr)
Our optimal algorithm
Greedy algorithm
CS Dept, City Univ. 23
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.
CS Dept, City Univ. 24
Thank You !
