Longbi Lin, Ness B. Shroff and R. Srikant

1. Asymptotically Optimal Energy-Aware Routing for Multihop Wireless Networks with Renewable Energy Sources Longbi Lin, Ness B. Shroff and R. Srikant IEEE/ACM Transactions on Networking, Oct. 2007, 1021-1034

2. Outline • Introduction and Motivation • Problem Formulation • Energy-opportunistic Weighted Minimum Energy (E-WME) Algorithm and Discussion • Asymptotic Optimality of the E-WME Algorithm • Simulation Results • Conclusions

3. Introduction • Energy management is central to Wireless Ad-hoc Networks • Operational capabilities of these networks is limited by energy available at the nodes • Recent developments in renewable energy sources has brought some relief • Replenishment rate small • Energy management of these networks

4. Motivation • One possible technique for energy conservation: Energy-Aware Routing • Choose the most energy-efficient route • Existing approaches combine two basic approaches: Min Energy (ME) and Max-Min routing • Previous work incorporated measure of a node’s residual energy into the cost function • Cost metric exponential in residual energy is optimal in a competitive ratio sense

5. Contribution of this work • Development of a mathematical framework that considers energy replenishment, mobility and erroneous routing information • Associated analytical techniques to provide an understanding of the performance benefits • Provides distributed and scalable routing solutions that can be adapted for a variety of network topologies, traffic and mobility patterns

6. Problem Formulation: Notation • Wireless multi-hop network is described by a directed graph G(V,E) • V: set of vertices representing sensor nodes • E: set of edges representing the communication links between them • Path from source to destination consists of one or multiple edges • Unit energy requirement of node n on path R: • rn”n: reception energy associated with the (upstream) edge (n”,n) • tnn’: transmission energy associated with the (downstream) edge (n,n’)

7. Problem Formulation: System Model • Each sensor node begins with a fully charged battery with capacity un • At the end of each time slot , Pn() is the residual energy in node n • Each node either belongs to • Vr (set of nodes with energy replenishment), • Vp(set of nodes with no energy replenishment) • At the beginning of each time slot , nVrreceives (-1) energy due to replenishment (not allowed to exceed un)

8. Problem Formulation: System Model • Routing requests arrive sequentially (j)=(S(j),D(j),l(j),Ts(j),(j)) • S(j)and D(j) are the source and destination nodes of the jth routing request • l(j) is the packet length • Ts(j) is the arrival time of the request • (j) is the revenue gained • Routing request is accepted only if there is at least one feasible path (each node along the path has at least l(j)en(R(j)) residual energy)

9. Problem Formulation: System Model • For any node n in Vr, the energy model is given by • For any node n in Vp the energy model is given by • I(·) is the indicator function, an(j) is the event that (j) is accepted at  and nR(j) • Our goal is to maximize the total revenue over some time horizon [0,t]:

10. E-WME Algorithm for Constant Replenishment Rate • Basic idea: • Assign a cost to each node, which is an exponential function in its residual energy • Use shortest-path routing with respect to this metric • Define power depletion index (measure of impact of previously accepted requests) of node n as: • P’n(j) is the energy at node n right before considering request j

11. E-WME Algorithm for Constant Replenishment Rate • Cost metric associated with each node: • Cost associated with R when considering request (j) will be calculated as: • E-WME Algorithm • For an incoming route request j, check if the least cost route R from S(j) to D(j) satisfies • If yes, accept, otherwise reject the request

12. E-WME Discussion • E-WME algorithm presented here has provably good performance • Secure large revenue without any statistical information about routing requests • This algorithm requires only local information at each node • Can be easily incorporated in traditional distance-vector type of routing framework • With minor modifications can be integrated with DSR-type on-demand routing protocols

13. E-WME Discussion • The metric in E-WME algorithm for each node is • an exponential function of the nodal residual energy (related to load balancing), • a linear function of the transmit and receive energies (related to resource thriftiness), • an inversely linear function of the replenishment rate (quality of the replenishment) • If all nodes have the same constant replenishment rate • Cost function combines elements of Min energy and Max-Min approaches • If nodes have different rates of replenishment • Network directs traffic to nodes with faster energy renewal rate

14. E-WME Algorithm for the General Case • Allows time-varying replenishment rate at each node • Can be applied to hybrid network (nodes with and without renewable energy sources) • Let tn(j) be the amount of time it takes for the incoming energy, accumulated from time slot Ts(j-1), to equal un-Pn(Ts(j-1)), • is the earliest time the battery at node n would be fully recharged if no request were accepted after request (j-1)

15. E-WME Algorithm for the General Case • New power depletion index (for nodes with renewable energy source) : • For nodes with no renewable energy source

16. E-WME Algorithm for the General Case • Amount ot energy at node n assuming that no request is accepted after request (j-1)

17. E-WME Algorithm for the General Case • Routing metric for the general case: • T< is an upper bound on the time it takes to fully charge an empty battery at any given node • The main modification in the definition of the node cost metric (from the previous case) is to take into account the replenishment schedule in the immediate future

18. E-WME Algorithm for the General Case • Cost associated with R when considering request (j) will be calculated as: • E-WME Algorithm • For an incoming route request j, check if the least cost route R from S(j) to D(j) satisfies • If yes, accept, otherwise reject the request

19. E-WME Discussion • Take the example of a sensor network powered by solar cells. • Assume that a request arrives at the network right after sunset. • Each node knows its short-term energy replenishment schedule. Therefore, each node knows that the energy replenishment rate will be much smaller for several hours to come. • The time to recharge calculated will then be relatively large, so the cost of routing the packet will be higher than that during the daytime. • As compared to its daytime policy, the network is thus more conservative in accepting the request

20. E-WME Discussion • In a hybrid network both kinds of nodes are present: one with energy replenishment and one without. • Assuming that they both have the same residual energy and that the routing request takes the same communication costs from them • The cost metric for the node with energy replenishment is smaller • Therefore, this node is more likely to be used than the one without energy replenishment.

21. Asymptotic Optimality of the E-WME Algorithm • E-WME algorithm asymptotically achieves the best achievable performance of any online algorithm • Shown using the idea of competitive ratio • The competitive ratio is defined as • Jt,off is the performance achievable by any offline algorithm and Jt,on is the performance of the given online algorithm • Smaller competitive ratio means better performance

22. Asymptotic Optimality of the E-WME Algorithm • For the proof, we need the following assumptions • Assumption (A1) requires that the revenue from a packet scales with the amount of resource it requests • Assumption (A2) guarantees that the energy claimed by a packet is not larger than a certain fraction of the total energy available at any single node • L is the maximum hop count allowed for any path, F is a constant chosen large enough to satisfy (A1), = 2(LFT + 1).

23. Asymptotic Optimality of the E-WME Algorithm

24. Asymptotic Optimality of the E-WME Algorithm: Proof Outline • Establish a relationship between Jon and the residual energy in the network • The additional revenue gained by the offline algorithm (Joff-Jon) is upper bounded by a function of the residual energy • It follows that the ratio of Joff to Jonis upper bounded by a logarithmic function in the number of nodes in the network

25. Asymptotic Optimality of the E-WME Algorithm: Proof Outline • To show the lower bound, • we construct a sequence of routing requests, • and show that the total revenue secured by the online algorithm is upper bounded by the total revenue of the offline algorithm multiplied by 2/log n

26. Routing with Incremental Deployment of Nodes • The above theorem shows that the E-WME algorithm can make good use of the available energy at any time to prolong network lifetime, without any knowledge of future node deployments.

27. Numerical Results Plot of the end-to-end throughput against the number of node pairs that have experienced partition

28. Numerical Results Node energy distribution after 4200 successful end-to-end packet deliveries

29. Numerical Results The energy spent per packet for E-WME and ME routing. The average energy per packet starts at a relatively low level for the ME routing. Without load balancing, the residual energy runs out faster at the critical nodes

30. Conclusions • The authors have addressed the problem of energy-aware routing with distributed energy replenishment • Formulated the problem as an integrated admission control and routing framework • E-WME algorithm has an asymptotically optimal competitive ratio • Algorithm is easy to implement • A threshold-based scheme is also introduced, which reduces the overheads while incurring minimal performance degradation