1. Scheduling Algorithms for Wireless Ad-Hoc Sensor Networks Department of Electrical Engineering California Institute of Technology. [Cedric Florens, Robert McEliece]

2. 2 Outline Introduction Model and Problem Statement Directional Antennas System  Line Networks  2-branch Networks  Multi-branch Networks  Tree Networks Omni directional Antennas System Conclusion and Future work

3. 3 Introduction As we known, Sensor nodes collect data, and then transmit the data to BS (Base Station). The problem here is how to route the packets efficiently? Assume there are two main phases of sensor operations.  First phase: Collect data  Second phase: Transmit data to some processing center located within the sensor network

4. 4 Model and Problem Statement(1/3) Define a sensor network as a finite collection of n identical sensor nodes. Each node is associated with an integer that represents the number of data packets stored at this node. There is one special node, the processing center (also called Base Station here).

5. 5 Model and Problem Statement(2/3) Assume that time is slotted and a one hop transmission consumes one time slot (TS). Assume sensor can only transmit/receive one data packet per TS. Multiple transmissions may occur within the network in one TS.

6. 6 Model and Problem Statement(3/3) Our goal is to route the data contained at each node to the BS as efficiently as possible. Represent our sensor network as weighted graph {V,E,p}.  V =.  E denote the set of links (the set of edges). 

7. 7 Directional Antennas System- Line Networks (Assumption) Goal: spend minimal TS to send data packet. We consider a line network. A BS is placed at one end of the network. Let’s denote node by its distance to the BS in number of hops, that is i. We denote i  i+1 a transmission from node i to node i+1. For purpose of solving this problem we look initially at the following converse problem.

8. 8 Directional Antennas System- Line Networks (Figure) Distribution Collection

9. 9 Directional Antennas System- Line Networks (Assumption) The BS is to send first data packets destined for the furthest node, then data packets for the second furthest one and so on. Nodes between the BS and its destinations are required to forward packets as soon as they arrive. Depend on above assumption, we propose the following simple greedy algorithm.

10. 10 Directional Antennas System- Line Networks (Algorithm) Force the BS to remain idle one TS When sending two consecutive packet The BS can send packet to The nearest node without Remain idle one TS.

11. 11 Directional Antennas System- Line Networks (Example)

12. 12 Directional Antennas System- Line Networks (Example) Denote the last busy TS at node i in the execution of our distribution algorithm. In the example, we have Clearly, our algorithm run in max(Ti). Next, we determine the performance of our algorithm in general.

13. 13 Directional Antennas System- Line Networks (Proof)

14. 14 Directional Antennas System- Line Networks (Proof) We define, for a given sensor network, Tu(p) the minimum length of a time schedule over all time schedules that perform the distribution job. In particular we have,

15. 15 Directional Antennas System- Line Networks (Proof)

16. 16 Directional Antennas System- Line Networks (Proof)

17. 17 Directional Antennas System- 2-branch Networks (Assumption) Consider now a line network and place the BS anywhere on that line. We look at this problem is to consider it as a two branch line network. Let p and q represent the two branches data packet. We denote the optimal performance achievable on a 2-branch network.

18. 18 Directional Antennas System- 2-branch Networks (Assumption) The BS transmits in the direction that require most work first and then alternates transmissions between the two branches. Furthest nodes are served first then second a furthest and so on. A data packet is automatically forward in the slot that followed its reception by a non destination node.

19. 19 Directional Antennas System- 2-branch Networks (Figure)

20. 20 Directional Antennas System- 2-branch Networks (Assumption)

21. 21 Directional Antennas System- Multi-branch Networks (Assumption) The direction of transmission is greedily decided, base on estimates of the completion time of the data transfer. Initial estimate for a given branch is determined by previous equation of Tu(p). The legal direction associated with the biggest estimate is chosen. After a decision has been made the estimates at each branch are updated according to the following rules:  If a legal direction was not chosen, its new estimate is its old estimate plus one  Illegal direction estimates remain unchanged.

22. 22 Directional Antennas System- Multi-branch Networks (Figure)

23. 23 Directional Antennas System- Tree-branch Networks (Assumption) We define the equivalent linear network of a network If and then, ={(i-1,i); 1<=i<=m) and where m = and The equivalent linear network’s schedule may serve as a schedule for the initial tree network. Next, we explain how transmission time slots for may be mapped onto

24. 24 Directional Antennas System- Tree-branch Networks (Mapping) Consider an element in E, say such that Define Each packet P follows a path path(P) from the BS to its detination node where path(P) denotes the finite sequence of edges(e1,…,ek). We define 

25. 25 Directional Antennas System- Tree-branch Networks (Mapping) We define a one to one correspondence g between and. We finally define is associated with time slots

26. 26 Directional Antennas System- Tree-branch Networks (Figure) We illustrate a tree network in below figure

27. 27 Directional Antennas System- Tree-branch Networks (example) In the previous figure, we have {P} = We have The edge (N1,N2) is associated with time slots

28. 28 Directional Antennas System- Tree-branch Networks (example) Thus algorithm 1 run on the equivalent linear network provides a BS transmission schedule. Next problem is to generalize Tree-branch networks  Linearize the sub-trees attached to the BS according to the procedure.  Apply multi-branch algorithm described before to the resulting multi-branch system

29. 29 Omni directional Antennas System Line-networks

30. 30 Omni directional Antennas System Line-networks Omnidirection antennas generate more channel reuse constraints and as a result longer time schedules. Ex: if the network of first figure was equipped with omnidirectional antennas, the minimum schedule length would be 14TS This is a 27% increase over the directional antenna system. In general,

31. 31 Conclusion and Future work Looking at the impact of network cycles on the optimality of our algorithms.