Fast and efficient randomized flooding on lattice sensor networks

Transcription

1 Fast and efficient randomized flooding on lattice sensor networks Ananth Kini, Vilas Veeraraghavan, Steven Weber Department of Electrical and Computer Engineering Drexel University November 19, 2004 presentation to Center for Telecommunications and Information Networking group Also submitted to the 3 rd International Symposium on Modeling and Optimizaton in Mobile, Ad Hoc, and Wireless Networks(WiOpt).

2 Abstract 1 We consider the problem of information dissemination on lattice based sensor networks. In particular, we are interested in obtaining fast, efficient, and simple mechanisms by which a source node may propagate information to all nodes in the network. Naive flooding Controlled flooding Random walk dissemination Randomized protocol We analyze and simulate this protocol, and conclude the parameter p permits valuable performance tradeoffs of efficiency and speed.

3 Outline of talk 2 What are sensor networks? Why is state promulgation important? Naive flooding Controlled flooding Random Walk Random flooding Percolation Theory

4 What are wireless sensor networks? 3 Similar to wireless Ad-Hoc Networks in that there is no centralized infrastructure and that nodes rely on each other to act as relays. They have a high failure rate and are deployed with a high spatial density. They have substantial processing capability in the aggregate, but not individually. In most settings the network must operate for long periods of time, hence energy resources limit their overall operation. Their dense deployment implies a high degree of interaction between nodes, which complicates the networking protocols.

5 Why is state promulgation important? 4 To make information available whenever, wherever To maintain control over the network For software updates For timing To check for node status

6 Topology of the Network and the protocol model Topology : the nodes are assumed to lie on the lattice Z 2. Communication : each node is capable of communicating with its four cardinal neighbors but not any other node, and all communication is assumed to be error free. Awareness : each node is aware of the identities of its four neighbors but not aware of its location or its position within the network. All nodes have sufficient memory to maintain state and identify whether or not it has received an incoming message. Timing : for simplicity and tractability we assume time is slotted and nodes are synchronized. Benefits of this model allow us to focus on the tradeoffs in protocol design. 5

7 Overview of 4 state promulgation protocols 6 Naive flooding : each node upon first receiving a packet, transmits the packet to its four cardinal neighbors. Controlled flooding : certain nodes, that are strategically placed on the lattice, are designed as transmitter nodes. These nodes, upon first receiving a packet, transmit the packet to their four cardinal neighbors. Random Walk : each node, upon first receiving a packet from a neighbor, transmits the packet to its neighbors, designating one of the three other neighbors as the next transmitter. Randomized flooding : each node, upon first receiving a packet, transmits the packet with probability p (0, 1). All subsequent receptions of the same packet are ignored by the node.

8 Naive Flooding 7

9 Efficiency 8 N t = {(x, y) Z 2 : x + y t} defines the set of possible receivers by time t. Note that N t 2t 2. R t is the set of nodes that receive the packet by time. T t is the set of nodes that transmit the packet by time t. η t = E R t E T t η = lim t η t

10 Controlled Flooding 9

11 Need for distributed protocols 10 Naive flooding where all nodes transmit the first time they receive is inefficient due to redundant transmissions. Controlled flooding demonstrates that higher efficiency can be obtained by designating certain nodes as transmitters a-priori. This protocol however requires centralized control and will not scale well for large scale networks. Hence we need a distributed protocol wherein every individual node makes a decision whom to transmit to depending on certain state information.

12 Random Walk 11

13 Speed and Spatial coverage 12 ν t = E R t N t is the instantaneous speed at time t, i.e., the total number of receivers by time t over the total number of possible receivers by time t. γ r = lim t E[ R t N r ] N r is the spatial coverage out to distance r, i.e. the fraction of receivers in N r that eventually receive the packet.

14 Characteristics of the protocols 13 The Naive flooding protocol is simple and fast. It is however inefficient in that each transmission effectively reaches only one new node The Controlled flooding protocol is fast, and can be shown to achieve twice the efficiency of naive flooding, but requires a centralized control. Effectively this protocol is not distributed and hence will not perform well for large scale networks. Random walk protocol has been analyzed previously in a paper addressing query strategies on a sensor network. This protocol is simple and efficient but extremely slow as the number of transmitters per time slot is fixed. The Randomized flooding protocol is what our paper focusses on. We have endeavored to show that the protocol is simple, fast, and efficient, and has performance tradeoffs that are parameterized by the transmission probability p.

15 Protocol Analysis 14 Naive flooding : R t = N t. Thus ν t = γ r = 1 for all t, r 0. It can be shown that T t 2t 2, hence lim t η t = 1, i.e., on average each transmission reaches only one first-time receiver. Conrolled flooding: Intelligent selection of half of the nodes as transmitters allows us to obtain the same speed and coverage as naive flooding but with twice the efficiency : i.e., T t t 2, lim t η t = 2. Random Walk : The speed in this protocol goes to zero since the number of receivers is linear in t while N t is quadratic in t.

16 15 Only partial success in analyzing the randomized protocol Observations of note are : E[R t ] = (x,y) N t P(node (x, y) receives the packet by time t). E[ T t ] = pe[ R t 1 ], i.e., the average number of transmitters by time t is p times the average number of receivers by time t 1. We see that the above two facts effectively reduces the computation of each of the performance metrics. Our analysis at this point is limited to computing the probability a node receives the packet at its earliest possible time: P(node(x, y) receives the packet by time x + y ). We compute this probability based on conditioning on all possible reception configurations of all nodes at a distance x + y 1, and then recursing back to the nodes on the axis. The problem with this technique is that it suffers from a combinatorial explosion that limits the distances from the origin, namely x + y 14.

17 Computation 16

18 Computation and Simulation Results 17 Figure 1: Two screen shots of the randomized flooding protocol. The square box contains sensors, the axis help denote the origin, and the diamonds denote the points at distance r = 10 and r = 50. The dots denote the sensors that have received the packet by time t. These screen shots are for p = 0.5. Note the rich spatial structure of the protocol.

19 18 Figure 2: Simulation results for efficiency vs. time show that the efficiency decreases in p from 2 for p = 0.5 down to 1.2 for p = 0.9.

20 19 Figure 3: Simulation results for coverage vs. distance show that coverage increases in p achieving almost perfect coverage for p 0.8.

21 20 Figure 4: Computation Results for efficiency vs. time show that efficiency decreases in p to 2 for p = 0.5 down to 1.2 for p = 0.9.

22 21 Figure 5: Computation Results for coverage vs. distance plots show that coverage increases in p. Note that the results from computation approximation runs only to t = 15 due to the combinatorial explosion and also do not consider or allow for back edges.

23 Figure 6: Computation approximation and simulation time-average efficiency versus the transmission probability p. The plot shows how higher efficiency is achieved by lower p. 22

24 Figure 7: Computation approximation and simulation spatial-average coverage versus the transmission probability p. The plot illustrates how higher coverage is achieved by higher p. 23

25 24 Figure 8: The bottom plot shows the fraction of simulations that ever reach one or more nodes at distance r versus p for r = 25, 50, 100. The plot illustrates a threshold behavior: choosing p < 0.5 means the state propagation will eventually die, while choosing p > 0.5 means the state propagation will likely continue forever.

26 Extending Computational approximations to run for higher values of t 25 As we have already mentioned we have been constrained in our ability to provide an accurate analysis of our proposed protocol. Is there a better way to analyze this protocol? The answer is Yes. Some unexpected help from another paper: Gossip-Based Ad Hoc Routing by Zygmunt J. Haas, Joseph Y. Halpern, Li Li.

27 Gossip-Based Ad Hoc Routing 26 Paper proposes a gossip-based approach, where each nodes forwards a message with some probability, to reduce overhead of routing protocols. the protocol model considers nodes placed on a two-dimensional area; with an edge placed between any pair of nodes less than a distance d apart. Implication is that gossiping exhibits bimodal behavior in sufficiently large networks Fraction of executions in which most nodes get the message depends on the gossiping probability and the topology of the network. Using gossiping probability between 0.6 and 0.8 suffices to ensure that almost every node in the network gets the message in almost every execution. Paper shows that for large networks, protocol uses up to 35 % fewer messages than (naive)flooding.

28 27 What is Percolation Theory? Percolation theory looks to answer the question : what is the probability that a large porous stone immersed in a bucket of water will be wetted at its center. Percolation Model : Z d = d-dimensional cubic lattice, with d 1. Two kinds of models in Percolation Theory : Bond model and Site Model

29 Clusters 28 C(x) = open cluster of x C(x) = Nodes in the Cluster Percolating Cluster C(x) = For the origin O, C(O) = C is defined as open cluster of origin. θ(p) = P r{ C = } If θ(p) > 0 Percolation θ(p) independent of x

30 The properties of θ(p) 29 p c (d) function of lattice dimension d. p c (1) = 1 p c (2) = 0.5 θ(p) is continuous except possibly at p c (d) θ(p) is increasing with p

31 Percolation Somewhere 30 Significance of p > p c the probability there is an I.C. somewhere in the lattice = 1. Number of I.C s if p > p c p c (d + 1) < p c (d) For p < p c size n of clusters is exponentially distributed for large n, namely P r{ C = n} exp( α(p)n) with α(p) > 0 p BOND c < p SIT E c

32 How is Percolation Theory relevant to our model 31 Given the high spatial density of Sensor Networks we see that the paths between any two nodes is negligible compared to the size of the network. The Site model of percolation theory can be used to model our network. The nodes are vertices(sites) and transmit with a probability p. We see that the theory of the existence of p c seems feasible. Our simulations(fig 8) show that there is a phase transition around the p = 0.5 point after which state promulgations will likely continue forever ( giving rise to an infinite cluster.) We are currently working on trying to use the analysis of percolation theory to carry out an analysis for our randomized protocol model.

33 Conclusion We have proposed a protocol that permits achieving a higher efficiency than naive flooding with a degradation in coverage that depends on p. 32 Randomized flooding attempts to emulate the performance of controlled flooding without incurring the associated increase in complexity. Simulations show that the optimum value of p to attain good efficiency and coverage is around the p = 0.5 mark. Simulations have shown that the choice of parameter p permits valuable performance tradeoffs of efficiency and speed. We have been unable to provide an exhaustive computational analysis of our protocol, to make a comparison with our simulation results. Our protocol model seems to be analogous to a site percolation model and it appears that we might be able to use results from percolation theory to analyze our protocol.