1. Arindam K. Das CIA Lab University of Washington Seattle, WA LIFETIME MAXIMIZATION IN ENERGY CONSTRAINED WIRELESS NETWORKS

2. with Robert J. Marks II & M.A. El-Sharkawi (UW CIA) Payman Arabshahi & Andrew Gray (JPL/NASA) LIFETIME MAXIMIZATION IN ENERGY CONSTRAINED WIRELESS NETWORKS

3. Example Static Multicast

4. Assumptions (1) We assume that there is a fixed source node which wants to communicate with some/all (multicast/broadcast) the other nodes in the wireless network. The source uses a fixed connection tree for a certain multicast group. All nodes have omni-directional antennas. Each node in the network is provided with a finite amount of battery energy which is used for signal transmission.

5. The Problem For a continuous packet transmission process at a constant bit rate and a stationary channel with no bandwidth constraints, we ask the question: “For a given multicast group, how long can the source continue to use the connection tree before any node in the system runs out of battery power”? We define the system lifetime as the time from t = 0 to the instant at which the first node in the system runs out of battery power.

6. Assumptions (2) Power is expended for signal transmission only. No power expenditure for signal reception or processing. The transmitter power is modeled as the ‘  ’ power of its distance from the receiver (2    4).

7. The Problem Define the initial node energy vector, E(0), as an N-element vector which specifies the initial battery energy at each node. The battery lifetime of node i for a particular choice of the connection tree is defined as:

8. The static case system lifetime is the minimum of all elements in the battery lifetime vector. The Problem

9. Objective of the static energy constrained multicasting (SECM) problem For a given multicast group, find the connection tree which maximizes the system lifetime.

10. The Problem maximize (system lifetime) = maximize {min (battery lifetime vector)}

11. Proposed Approach We propose a GA based approach for solving the minimax optimization problem. Key question: Encoding of chromosomes

12. Some Definitions Power matrix, P: The (i,j) th element of the power matrix is defined as where r ij is the Euclidean distance between nodes i and j. Rank matrix, R: The rank matrix is obtained by ranking each row of the power matrix from smallest to largest. P ij = r ij 

13. Examples

14. Some Definitions Cut vector,  R : The cut vector, referenced to R, is an N-element integer vector, where N is the number of nodes in the network. It indicates the location of an element on each row of the rank matrix. Threshold vector, t : An N-element vector of the elements of R specified by the cut vector  R. Elements of the threshold vector represent power settings of the individual nodes.

15. Examples  R = [3 4 5 1 2], t = [8.01 14.06 16.73 0 9.55]

16. Some Definitions Viability of a cut vector: A cut is viable if it allows all destination nodes to be reached. Otherwise, it is non-viable. A viable cut vector has an associated connection tree.

17. Outline of the Proposed Approach GA based Chromosome encoding : cut vectors Crossover : standard (e.g., random 1- point crossover), subject to a certain crossover probability. Parent selection : standard (e.g., roulette wheel)

18. Outline of the Proposed Approach Fitness function :  s = max i [t i / E i ( 0 )] Mutation :  1, on randomly chosen elements of the chromosomes, subject to a certain mutation probability. Elitism : yes

19. Viability of the Children Randomly generated cut vectors need not be viable  the children created after crossover and mutation need not correspond to viable connection trees. Use the computationally simple Viability Lemma to determine the viability of a child. - If viable, accept it. - If not, reject it, or, apply a repair operator.

20. Viability of the Children A Repair Strategy Suppose a node (say n) is not reached by a cut vector,  R. Compute the connection tree corresponding to  R. Identify the node closest to n (say m). Augment the power level of node m so that node n is reached. Recompute the cut vector corresponding to the modified connection tree.

21. Example Static Multicast Search

22. The Dynamic Energy Constrained Broadcast (DECB) Problem In the Static Energy Constrained Broadcast (SECB) problem, we assumed that the source uses only one connection tree. In the DECB problem, we assume that the source uses a set of {  : 1     } connection trees, each for a certain duration of time,   (called the duty cycle of tree  ), such that:

23. The Dynamic Energy Constrained Broadcast (DECB) Problem Clearly, the system lifetime in this case is a function of the trees used in the dictionary set and their corresponding duty cycles. The  best trees need not be the best  trees. For a given multicast group, find the  connection trees and the set of duty cycles {   : 1     } which maximize the system lifetime.

24. DECB Optimization Function Proposed optimization approach : Team optimization of co-operating systems (TOCS)

25. The TOCS Approach

26. Summary Presented optimization models for static and dynamic energy constrained broadcast / multicast problems. Outlined a GA based approach for solving the static problem. Proposed a TOCS approach for solving the dynamic problem.