Optimal Radius for Connectivity in Duty-Cycled Wireless Sensor Networks A. Bagchi 1 Cristina M. Pinotti 2 S. Galhotra 1 T. Mangla 1 1 Dept. of Comp. Sci. and Engg., IIT Delhi, India 2 Dept. of Comp. Sci. and Math.,University of Perugia, Italy 16th ACM International Conference on Modeling, Analysis and Simulation of Wireless and Mobile Systems, November 4, 2013 Cristina M. Pinotti (University of Perugia) Optimal Connectivity Radius for DC-WSN November 4th 1 / 19
Outline In a functioning network, sensors may follow a duty-cycle to save energy, but they must preserve time coverage , i.e., data generated at any time must be sensed and relayed by the network connectivity , i.e., every node should be connected to any other node and to the sink For this goal, we propose: a weak condition derived from Random Geometric Graph (RGG) a strong condition derived from Vertex-Based Random Geometric Graph (VB-RGG) We present the contiguous and the random selection duty-cycle schemes We test by simulation the connectivity of such schemes using both radii Cristina M. Pinotti (University of Perugia) Optimal Connectivity Radius for DC-WSN November 4th 2 / 19
Network Model Random Geometric Graph RGG ( n , r ) : There are n sensors distributed uniformly at random in the unit circle centred at the origin, where the sink is placed. Each sensor is modeled by a vertex. Two sensors u and v communicate if they are at distance not greater than r . Each possible transmission is modeled by an edge. Sensors are synchronized and follow a pattern of sleeping and waking period: L : length of the duty-cycle d : the number of waking slots δ = d L : the duty-cycle ratio DC-WSN A ( n , r , δ , L ) : the same vertex set as in RGG ( n , r ) each sensor u selects an awake configuration A u ⊆ { 0 , 1 ,..., L − 1 } ∈ A with | A u | = d E = { ( u , v ) : d ( u , v ) ≤ r , A u ∩ A v � = / 0 } . For two vertices u and v that are within transmission range of each other to be connected, they must share a slot where they are both awake. Cristina M. Pinotti (University of Perugia) Optimal Connectivity Radius for DC-WSN November 4th 3 / 19
Property DC-WSN A ( n , r , δ , L ) : Time Coverage Any duty-cycle scheme A that we consider guarantees: For each k ∈ { 0 , 1 ,..., L − 1 } , the probability that a node u is awake in slot k is δ k > 0, where δ k may be a function of d and L , independent of the number of sensors in the network Uniform time coverage : δ k = d / L for 0 ≤ k ≤ L − 1, i.e., the probability does not depend on the slot Cristina M. Pinotti (University of Perugia) Optimal Connectivity Radius for DC-WSN November 4th 4 / 19
Property DC-WSN A ( n , r , δ , L ) : Connectivity If δ > 1 / 2 then DC-WSN A ( n , r , δ , L ) is connected whenever RGG ( n , r ) is connected. If δ ≤ 1 / 2 to guarantee connectivity we need the Reachability property: There is a finite sequence of awake configurations that leads in the duty-cycle scheme A from any configuration to another. Example L = { 0 , 1 ,..., 5 } , A = { A 1 = { 0 , 1 , 2 } , A 2 = { 3 , 4 , 5 }} does not satisfy the reachability condition L = { 0 , 1 ,..., 9 } , A = { A 1 = { 1 , 2 , 3 } , A 2 = { 2 , 3 , 4 } , A 3 = { 3 , 4 , 5 } , A 4 = { 4 , 5 , 6 } ,..., A 10 = { 0 , 1 , 2 }} satisfies the reachability condition Cristina M. Pinotti (University of Perugia) Optimal Connectivity Radius for DC-WSN November 4th 5 / 19
The weak connectivity result Based on the connectivity condition for RGG ( n , r ) , we prove that for δ ≤ 1 / 2: Theorem The probability that DC-WSN A ( n , r ( n ) , δ , L ) is connected tends to 1 as n → ∞ if r 2 ( n ) = ( ln n + c ( n )) / n rRGG = , πδ δ with c ( n ) → ∞ as n → ∞ . Cristina M. Pinotti (University of Perugia) Optimal Connectivity Radius for DC-WSN November 4th 6 / 19
Two specific duty-cycle schemes A The contiguous model: DC-C-WSN Each sensor u independently chooses an integer i u from the set { 0 , 1 ,..., L − 1 } u is awake for the all the time slots i u to i u + d − 1 u is asleep from all the remaing L − d time slots The independent random selection model: DC-R-WSN Each sensor u chooses a set of awake slots A u of size d at random from { 0 , 1 ,..., L − 1 } Cristina M. Pinotti (University of Perugia) Optimal Connectivity Radius for DC-WSN November 4th 7 / 19
Simulations n sensors are placed uniformly at random in a unit disk with 10 5 ≤ n ≤ 10 6 δ varying between 0 . 05 and 0 . 5: to find the connected component, we make use of the Union Find algorithm each experiment is repeated at least five times Cristina M. Pinotti (University of Perugia) Optimal Connectivity Radius for DC-WSN November 4th 8 / 19
Percentage of sensors in the largest connected component In our experiments, we set c ( n ) to lnln n if not otherwise stated. DC−C−WSN, L = 100 100 99.995 99.99 Connectivity Percentage 99.985 99.98 For DC-R-WSN, the 99.975 percentage of connectivity 99.97 is almost 100% ! 99.965 99.96 δ = 0 . 25 δ = 0 . 05 99.955 δ = 0 . 10 99.95 0 2 4 6 8 10 Number of nodes (n) 5 x 10 Is the weak radius the minimum radius able to guarantee connectivity? Cristina M. Pinotti (University of Perugia) Optimal Connectivity Radius for DC-WSN November 4th 9 / 19
The vertex-based random geometric graph To improve on the radius, we introduce a generalization of the random connection model by Meester and Roy [1996]: The vertex set V is the same as RGG With each u ∈ V , we associate an independent random variable Z u which represents the awake configuration The edge ( u , v ) exists with the probability γ that Z u ∩ Z v � = / 0 if and only if d ( u , v ) ≤ r Theorem P ( VB-RGG ( n , r , γ ) is connected ) → 1 as n → ∞ if and only if r ( n ) 2 = ( ln n + c ( n )) / n rRGG = , πγ γ where lim n → ∞ c ( n ) = ∞ as n → ∞ . Cristina M. Pinotti (University of Perugia) Optimal Connectivity Radius for DC-WSN November 4th 10 / 19
The probability γ of sharing a slot DC-R-WSN DC-C-WSN γ > 1 − ( 1 − δ ) d � � γ = 2 δ − 1 / L In DC-R-WSNs, when a node u In DC-C-WSNs, a sensor v will has chosen d slots, another share at least one slot with node u node v has d possibilities to if v chooses as its starting point choose one slot in common any of the slots with u and the probability of i u − ( d − 1 ) mod L ,..., i u ,... i u + doing that is at least δ each ( d − 1 ) mod L . time. r RGG r 2 ( n ) = r RGG 2 δ − 1 / L r 2 ( n ) = ( 1 − ( 1 − δ ) d ) Cristina M. Pinotti (University of Perugia) Optimal Connectivity Radius for DC-WSN November 4th 11 / 19
Percentage of sensors in the largest connected component when L = 100, n and δ vary DC−C−WSN, L=100 DC−R−WSN, L =100 100 100 99.5 95 Connectivity Percentage Connectivity Percentage 99 90 98.5 85 98 δ = 0 . 25 80 δ = 0 . 25 δ = 0 . 10 δ = 0 . 10 δ = 0 . 05 δ = 0 . 05 97.5 75 0 2 4 6 8 10 0 2 4 6 8 10 Number of nodes (n) Number of nodes (n) 5 5 x 10 x 10 Cristina M. Pinotti (University of Perugia) Optimal Connectivity Radius for DC-WSN November 4th 12 / 19
The ratio of the weak to the strong radius in DC-C-WSN and DC-R-WSN for different δ and L L = 200 L = 100 δ DC-C-WSN DC-R-WSN DC-C-WSN DC-R-WSN 0.05 1.378 2.832 1.341 2.127 0.10 1.396 2.963 1.378 2.552 0.15 1.402 2.572 1.390 2.466 0.20 1.405 2.236 1.396 2.249 Recalling that the energy is proportional to n δ r 2 ( n ) , the strong radius halves the energy spent in DC-C-WSNs and divides at least by four the energy spent in DC-R-WSNs Cristina M. Pinotti (University of Perugia) Optimal Connectivity Radius for DC-WSN November 4th 13 / 19
Is the strong radius optimum? Percentage of connectivity with different values of c ( n ) DC−R−WSN, d = 10, L =100 DC−C−WSN, d=10, L=100 0.25 1.8 c ( n ) = − 2 sqrt ( log ( n )) c(n) = −5 � c ( n ) = − 2 . 5 ( log ( n )) 1.6 c(n) = −4.5 c ( n ) = − ( log ( log ( n ))) 2 c(n) = −4 0.2 1.4 Connectivity Percentage Connectivity Percentage c(n) = −3.5 c(n) = −3 1.2 0.15 1 0.8 0.1 0.6 0.4 0.05 0.2 0 0 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 Number of nodes (n) Number of nodes (n) 6 5 x 10 x 10 Cristina M. Pinotti (University of Perugia) Optimal Connectivity Radius for DC-WSN November 4th 14 / 19
Conclusions We have studied the optimal radius for achieving connectivity in DC-WSNs Two specific duty-cycled schemes, the DC-C-WSN and DC-R-WSN, have been proposed to stress the differences between the weak and the strong radii We have presented a general and foundational contribution because these results apply also to: the key-predistribution for secure communication L : the size of the overall set of keys d : the size of the subset of keys assigned to each node the study of connectivity in WSNs with directional antennas where the direction is fixed at random independently at each node Cristina M. Pinotti (University of Perugia) Optimal Connectivity Radius for DC-WSN November 4th 15 / 19
Questions Cristina M. Pinotti (University of Perugia) Optimal Connectivity Radius for DC-WSN November 4th 16 / 19
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