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Topology Maintenance in Asynchronous Sensor Networks Reuven Cohen Boris Kapchits

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2 Outline Neighbor Discovery in Sensor Networks Estimating the in-segment Degree of a Hidden Neighbor. An Efficient Topology Maintenance Algorithm. Simulation Study.

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3 Sensors and sensor networks Sensor is a cheap tiny device that is able to detect local events and report them to a centralized gateway using wireless comm. A sensor network consists of many sensors and a gateway –The sensors perform some common task, like smoke detection or temperature measurement, and report to the gateway. Since not all the sensors are in the transmission range of the gateway, their messages should be forwarded by other sensors Most often the network structure cannot be pre-engineered, since the sensors are placed randomly in the covered area.

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4 Energy saving Since the sensor network should operate unattended for a long time, its most scarce resource is energy. Energy is consumed when the sensor is transmitting or receiving. – If the receiving/transmitting unit is powered off, energy consumption is minimized. the sensor is said to be in “slipping mode” – In order to save energy, each sensor will switch between “sleeping mode” and “active mode”. Sensor will spend in “active mode” about 2% of its lifetime. – However, in order to communicate, two sensors should be in “active mode” simultaneously. asleepactiveasleepactiveasleep

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5 Neighbor Discovery The sensor nodes are placed randomly over the area of interest, and their first step is to detect their immediate neighbors, i.e., the nodes with which they have direct wireless communication. Although sensor nodes are static, topology changes still take place, due to the following factors: Loss of local synchronization due to accumulated clock drifts. Disruption of wireless connectivity between adjacent nodes by a temporary event, such as a passing car or animal, a dust storm, rain or fog. The ongoing addition of new nodes in some networks to compensate for nodes that do not function any more We distinguish between the neighbor discovery in the initial phase and the ongoing topology maintenance at the operation phase.

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6 Neighbor discovery (cont.) a bc Every time a node wakes up, it broadcasts HELLO. Only a node that is also awake can hear this message and establish connections. Node a receives HELLO from node c. Hence, they discover each other. However, a and b do not know about b and vise versa. a and c discover each other b sends HELLO message, which is heard by no one

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7 The differences between neighbor discovery and topology management For neighbor discovery, an aggressive protocol, one which requires the sensor to stay in active mode and expend a lot of energy until detection, is usually acceptable. Neighbor discovery is performed when the sensor has no clue about the structure of its immediate surroundings. In particular, the sensor is unable to perform any useful task. Hence, energy consumption in this state is less of an issue. When the sensor performs topology maintenance, it can perform topology maintenance together with these neighbors in order to consume less energy. Neighbor discovery Topology maintenance

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8 New approach to neighbor discovery As far as we know, this paper is the first to distinguish between neighbor discovery and topology maintenance in sensor network, and to explicitly address the topology maintenance problem. The figure below summarizes this idea: – When node u is initialized, it performs neighbor discovery. – After a certain time period in the neighbor discovery state, during which the node is expected, with high probability, to find most of its neighbors, the node moves to the topology maintenance state. The main idea behind the topology maintenance scheme is that once most of the neighbors know each other, the task of finding a new neighbor a part of the topology maintenance is divided among all the nodes in its vicinity. Connectivity lost Neighbor discovery Topology maintenance Time elapsed, or enough neighbors are detected

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9 An example for topology maintenance bca c and a discover each other c tells b to wake up at certain time, a and b discover each other During topology maintenance phase most of the nodes are organized into clusters. Thus, as soon as a new node is discovered by one of the cluster members, it can be almost immediately discovered by the rest.

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10 Outline Neighbor Discovery in Sensor Networks Estimating the in-segment Degree of a Hidden Neighbor. An Efficient Topology Maintenance Algorithm. Simulation Study.

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11 Hidden node degree estimation In order to calculate the HELLO message frequency node v should estimate the number of in-segment stations that participate in the discovery of a hidden node u. This is equal to deg S (u), namely, the number of neighbors that the hidden node u has inside the connected segment deg S (u)=4 v u deg S (v)=3

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12 The estimation methods Three methods are considered: 1. The average in-segment degree of the segment's nodes is used as an estimate of the in-segment degree of u. 2. The number of v’s in-segment neighbors, deg S (v), is used as an estimate of deg S (u). 3. A linear combination of 1 and 2. deg S (u)=4 v u deg S (v)=3

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13 Estimation methods analysis Let X be a random variable that indicates the degree deg S (v) of v, a node in the segment S. Let Y be a random variable that indicates the degree deg S (u) of u, a hidden neighbor of v, which we want to estimate. Note that u itself is also not aware of the value of Y. Let Y' be the estimated value of Y. Clearly, we want Y' to be as close as possible to Y. We use the mean square error measure, MSE, to decide how good our estimate is. – The MSE is defined as E((Y-Y') 2 ). We assume that X and Y have the same distribution. We also assume that the nodes, both in-segment and hidden, are distributed uniformly on the plane.

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14 Theorem 1 Let u, v and w be nodes in a geometric graph with the same transmission range, where nodes are distributed uniformly. If u is a neighbor of v and v is a neighbor of w, then the probability that u is also a neighbor of w is C=1-3/(4 π)√3 ~ The proof consists of some geometrical and probabilistic calculations. This Theorem is used further, in the analysis of Method 2 and 3.

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15 Method 1 analysis For the first method, where node v measures the average in-segment degree of the segment's nodes and uses this number as an estimate of the in- segment degree of u: Y’=E(X)=E(Y) The method accuracy: MSE 1 = E((Y-Y') 2 ) = E((Y-E(Y)) 2 ) = Var(Y). – Var(Y) is the variance of Y.

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16 Method 2 analysis For the second method, we have Y'=X The method accuracy: MSE 2 = E((Y-X) 2 )=… =2(E(X 2 )-E(XY)) Note that E(XY)=Σ y [y P(Y=y) E(X | Y=y)]. But from Theorem 1 : E(X | Y=y) =Cy + (1-C)E(X). Hence, substituting into the equation gives: E(XY)=…=C E(Y 2 )+(1-C)E(X)E(Y). And finally: MSE 2 =2(1-C)Var(Y)~0.84Var(Y)

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17 Method 3 analysis We estimate Y in the following way: We can divide the neighbors of u into two subsets: 1) Those that are also neighbors of v 2) Those that are not. By Theorem 1, the average size of the first subset is CX. The average size of the second subset is (1-C)E(X). Therefore, we estimate the degree of u by Y'=CX+(1-C)E(Y).

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18 Method 3 analysis (cont.) MSE 3 =E((CX+ (1-C)E(Y)-Y) 2 ) = Σ x Σ y (Cx+(1-C)E(Y)-y) 2 P(X=x,Y=y) = C 2 E(X 2 )+(1-C) 2 E(Y) 2 +E(Y 2 )+2(1-C) CE(X)E(Y)- 2CE(XY)- 2(1-C)E(Y) 2 Since E(X)=E(Y) and E(X 2 )=E(Y 2 ), we get: MSE 3 =(1+C 2 )E(Y 2 )+(2C-1-C 2 )E(Y) 2 -2CE(XY) = (1-C 2 )Var(Y)~0.67Var(Y)

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19 Which of the three methods is the best? MSE 1 = Var(Y). MSE 2 =2(1-C)Var(Y)~0.84Var(Y) MSE 3 = (1-C 2 )Var(Y)~0.67Var(Y) Hence, MSE 1 > MSE 2 > MSE 3 - The third method yields the best (smaller) MSE. - However, note that this method requires some global knowledge of the network topology, while the second method requires only local knowledge.

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20 Outline Neighbor Discovery in Sensor Networks Estimating the in-segment Degree of a Hidden Neighbor. An Efficient Topology Maintenance Algorithm. Simulation Study.

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21 A Neighbor Discovery Scheme Suppose that node u is in neighbor discovery state: It wakes up every T I seconds in average for a period of time equal to H, and broadcasts HELLO messages. The demand is that the nodes of segment S will discover u within a time period T with probability P. Each node v in the segment S is in topology maintenance state, wakes up every T N (v) seconds for a period of time equal to H, and broadcasts HELLO messages. Neighbor discovery Topology maintenance u segment S

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22 A Neighbor Discovery Scheme (cont.) In order to discover each other, nodes u and v should have an active period that overlaps by at least a portion δ, 0< δ <1, of their size H. If node u wakes up at time t for a period of H, node v should wake up between t-H(1- δ) and t+H(1- δ). The length of this valid time interval is 2H(1- δ). Since the average time interval between two wake-up periods of v is T N (v), the probability that u and v discover each other during a specific HELLO interval of u is 2H(1- δ)/T N (v). u t t+H vv t-H(1-δ)t+H(1-δ)

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23 The probability of discovery Let n be the number of in-segment neighbors of u. When u wakes up and sends HELLO messages, the probability that at least one of its n neighbors is awake during a sufficiently long time interval is 1-(1-2H(1- δ)/T N (v)) n. Consider a division of the time axis of u into time slots of length H. The probability that u is awake in a given time slot is H/T I, and the probability that u is discovered during this time slot is P 1 = H/T I (1-(1-2H(1- δ)/ T N (v)) n ). Denote by D the value of T/H. Then, the probability that u is discovered within at most D slots is P 2 =1-(1- P 1 ) D. Therefore, we seek the value of T N (v) that satisfies the following equation: P 2 =1-(1- P 1 ) D ≥ P

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24 The sufficient value of T N (v) The last expression can also be stated as P 1 ≥ 1-(1-P) 1/D Hence H/T I (1-(1-2H(1- δ)/ T N (v)) n ) ≥ 1-(1-P) 1/D and therefore T N (v)≤ 2H(1- δ)/(1-(1-T I /H(1- (1-P) 1/D )() 1/n Since v does not know the exact value of n, v can estimate it using the schemes presented earlier.

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25 T N (v) as a function of maximum tolerated delay T. The estimated number of participating nodes is n=10. We see the value of TN(v) as a function of the desired discovery time T for P=0.5, 0.8 and P is set to 0.8 and n varies between 5 and 50. TN(v) is calculated as a function of the desired discovery time.

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26 Outline Neighbor Discovery in Sensor Networks Estimating the in-segment Degree of a Hidden Neighbor. An Efficient Topology Maintenance Algorithm. Simulation Study.

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27 Simulation setup. 2,000 sensor nodes, randomly placed over a 10,000 x 10,000 grid. The transmission range is set to r=300 units. Any two nodes whose Euclidean distance is not greater than r are considered to have wireless connectivity. A half of the nodes are randomly selected to be hidden. We require that every hidden node will be detected with probability P, which ranges between 0.3 and 0.7 within a predetermined period of time T=100 time units. The hidden nodes are assumed to be in the neighbor discovery state, where they are supposed to wake up randomly, every T I =20 time units on the average. A non-hidden node v is assumed to be in the topology maintenance state, where it wakes up randomly, every T N (v) time units on the average, in order to discover hidden nodes. When a node is detected, it joins the segment. A hidden node that detects another hidden node remains in the neighbor discovery state.

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28 Ratio of hidden nodes The initial ratio is 0.5. After 100 time units, this ratio decreases to: 0.35 for P= for P= for P=0.7 After 200 time units: 93% of the nodes are detected for P=0.7 75% for P=0.3.

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29 Average frequency of HELLO intervals of the in-segment nodes We can see that for the smaller value of P (the lower curve), the frequency is almost 75% lower than the frequency for the larger value of P. We can also see that for a given value of P, the average frequency of HELLO periods decreases with the time. This is because the segment grows, and more nodes participate in the discovery process.

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30 Non-uniform distribution of nodes Another interesting case is when the hidden nodes are distributed non-uniformly in the area. To simulate this case, we randomly select some points as “dead areas," and assume that the probability of a node to be hidden increases when its distance to one of these points decreases. The rationale here is that bad weather, dust storms, or other environmental conditions may adversely affect wireless connectivity in some areas more than in others. Unlike the uniform distribution case, here we do see differences between the three estimation algorithms presented earlier.

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31 Decrease in hidden node ratio The figure shows the percent of hidden nodes as a function of time for the three estimation methods for P=0.5. Method 1 is less efficient because it assigns the same value of T N (v) to all the nodes in the segment. Method 2 underestimates the number of in-segment neighbors of the nodes that are located closer to the dead zone. The last algorithm gives the best estimate in this case, and it therefore yields the most efficient topology management scheme.

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32 A self-contained algorithm vs. our algorithm. Figure shows the ratio of hidden nodes after T time units for networks with different transmission ranges. This simulation compares our protocol with one that uses a trivial hidden neighbor detection algorithm. The simulation starts with 50% hidden nodes, and P=0.5. Our protocol guarantees that after T time units the number of hidden nodes will decrease by half to 25%. In contrast, The trivial protocol discovers half of the hidden nodes only when the transmission range is ~0.06. When the transmission range is shorter, this protocol discovers a smaller fraction of the hidden nodes. When the transmission range is longer, this protocol discovers more nodes during a time period of T.

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33 A self-contained algorithm vs. our algorithm - the figure We conclude that our algorithm can self-adjust to invest the minimum energy needed to guarantee the required discovery rate, whereas the trivial algorithm cannot.

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34 Conclusions We expose a new problem in wireless sensor networks, referred to as ongoing topology maintenance. We argued that ongoing topology maintenance is essential even if the sensor nodes are static. We showed that by having the nodes in a connected segment work together on ongoing topology maintenance, we can guarantee that (a) Hidden nodes will be detected with a certain probability P and within a certain time period T; and that (b) The energy expended by the segment nodes on the detection of hidden nodes is minimized.

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35 Conclusions cont. We showed that our scheme works well if every node connected to a segment estimates the in-segment degree of its potential hidden neighbors. We proposed three estimation algorithms and analyzed their mean square errors. We then presented an ongoing topology maintenance algorithm. Using simulations, we analyzed several aspects of our algorithms. – We showed that when the hidden nodes are uniformly distributed in the area, the simplest estimation method is good enough. – When the hidden nodes are concentrated around some dead areas, the third method was shown to be the best.

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