1. Verification of Chess TDMA for a Simple Fully Connected Wireless Sensor Network Faranak Heydarian Frits Vaandrager Radboud University Nijmegen

2. Outline  Introduction  Mac Model  U PPAAL Model  A Collision-Free Network  U PPAAL Numerical Results  Theorem of having a Collision-Free Network  Conclusion & Future Work 2

3. Introduction 3 A Wireless Sensor Network (WSN) is a wireless network consisting of spatially distributed autonomous devices using sensors to cooperatively monitor physical or environmental conditions, such as temperature, sound, vibration, pressure, motion or pollutants, at different locations.

4. MAC Model 4 TXRX idle Time is considered as a sequence of Time Frames. A time frame is composed of a fixed number (C) of Time Slots. In a time slot the hardware clock of the sensor node ticks a fixed number (k 0 ) of times. A Time Frame tsn

5. Synchronization 5  The clock synchronization is requested each time a node receives a message, keeping the internal oscillator to not drift from the given T 0 moment.  Whenever a message is received, the transmitter T 0 moment is adopted by the receiver nodes.  When the nodes are powered on, they are all unsynchronized. In order to get synchronized, a node needs information about the relative time frame start, called T 0 moment.

6. Guard Time 6 RX Time Slot TX Time Slot Guard Time

7. U PPAAL Model 7 X0X0 x<=max S0S0 S1S1 x>=min tick[id]! x:=0, clk[id]:=(clk[id]+1)%k 0 csn[id]<=n start_message? tick[id]? clk[id]:=guardtime+1 csn[id]==tsn[id] urg! clk[id]==k 0 -guardtime tick[id]! csn[id]!=tsn[id] tick[id]? clk[id]==guardtime start_message! clk[id]==0 urg! csn[id]:=(csn[id]+1)%C INIT Start_TX_Slot Sending Wait_For_End_Of_Slot

8. 2 A Collision- Free Network 8 1 12 12 3 3 3 Colliison TX

9. Numerical Results Nodes234 C681068 68 n4 k0k0 g2 min496989395979294969 max507090406080305070 9 Nodes234 C6 n4 k0k0 510152051015205101520 g2 min244974991939597914294459 max2550751002040608015304560 Nodes234 C6 n4 k0k0 10 g234234234 min49241639191329149 max502517402014301510

10. Theorem   A fully-connected wireless sensor network is collision-free if and only if: 10

11. Necessity Proof by Counter Example 11 End of Frame NO COMMUNICATION g+1 g g RX TX RX S F

12. Necessity Proof 12 End of Frame NO COMMUNICATION g+1 g g RX TX RX S F

13. Sufficiency Proof using Invariants ☺ We proved if holds, then Invariant is an Invariant. 13 primed formulas ☺ We started proving the necessity part of the theorem by describing the network of timed automata by primed formulas.

14. System Invariants 1. 2. 3. 4. 5. 14

15. A Technical Lemma 15 is equivalent to

16. System Invariants (cont.) 16 ☺ ☺ If then invariant Is an invariant: Sending g+1 g RX TX t.k 0 -1 time units t.k 0 -g time units the last synchronization point before the conflict

17. Proving Sufficiency 17

18. A Fully-Connected Network vs. A non-Fully-Connected one C681012 n4444 k0k0 10 g3333 min19293949 max20304050 ratio0.950.96670.9750.98 C681012 n4444 k0k0 10 g3333 min587898118 max597999119 ratio0.98300.98730.98990.9916 BB CC AA BB CC AA 18

19. Conclusion & Future Work 19

20. Thank You ? ? ? ?