Model and tools

1. Model and tools

2. Traffic Model • Poisson law • Napoléon worries about the statistics of horse accidents of his generals • Poisson confirms these are unfrequent independent events

3. Poisson law A B • Intensity λ • Independence, memoriless property • are independent if • Poisson law • conditionnally uniform • If then the k events are uniformly distributed over A time

4. Exponential inter-arrival • Inter-arrival I are independent and exponential • Density: • Memoriless property:

5. Super-position of Poisson processes • Two flows 1and 2 • The union is a Poisson flow of intensity 1+ 2

6. Test • Random sequence • Discrepant sequence.

7. Convergence • Small independent processes • N unfrequent events each with proba • Bernoulli law. When N→∞

8. Time varying intensity • Generalization:

9. Probability generating functions • Let X be a non negative integer random variable • Probability generating function for z complex

10. P.g.f. property • Moment generating function

11. P.g.f. property • P.g.f. knowledge gives the distribution • X with pgf f(z), Y with pgf g(z) • X and Y independent • X+Y pgf is f(z).g(z)

12. p.g.f of Poisson

13. Composition of random variables • X and Y integer random variables of p.g.f f(z) and g(z) • Sum of X independent copies of Y: p.g.f.

14. Multiple access protocols • In wireless networks, medium channel is unique and must be shared • One or several of frequencies

15. Wireless Communication Architecture • Access point architecture • Wifi infrastructure mode • GSM, UMTS • Wimax • Ad hoc architecture • Mesh networks • Mobile ad hoc • Sensor networks

16. Multiple access protocols • Frequency Division Multiple Access • Frequency set is split between users • Time Division Multiple Access (TDMA) frequencies time time

17. Wireless Access Protocols • Periodic TDMA • Time slot periodically allocated to terminal in round robin. • Examples: GSM, bluetooth. time slot

18. Wireless Access Protocols • Random access protocols • More than two transmitters over one slot→ collision • Collision detection (no d’ACK) • Collision resolution algorithm. time slot

19. Collision resolution algorithm • Wireless Network standards • Minimal SNR (Wifi ≈10 db or more) • Collision: none goes through • Capture due to near far effect • A goes through • unfrequent B A

20. Collision Resolution Algorithm • Access point configuration (D=0) • Aloha • Random Backoff • Uniform over (0, Wmax) (retransmission window) • Repeat after each collision • Binary Exponential Backoff (BEB) • retransmission window doubles after each collision (for the same packet) • Limited number of retransmissions.

21. Terminal network interface model Packets internally generated Network interface buffer Network interface Server (one packet max)

22. Average delay analysis for periodic TDMA • Poisson model traffic per slot for node i • Average delay in network interface • Must add delay in buffer • Maximum throughput: packet per slot • Non uniform traffic : packet per slot

23. Average delay analysis for periodic TDMA • Between two periodic slots • N slots • Poisson rate per slot • Buffer queue size X • P.g.f q(z)

24. Average delay analysis for periodic TDMA • Resolution of p.g.f • From q(1)=1 • Quantity q(0) is average idle slot • Average queue size

25. Average delay analysis for periodic TDMA • The average number customers queued at time of a random arrival • Is also the average number of full periods in buffer • Average time in buffer • Average packet waiting delay

26. Random TDMA Performance • Packet generation over all nodes • Poisson process, cumulated rate  packet per slot • No packet retransmission :

27. Random TDMA Performance • Packet generation over all nodes • Poisson process, cumulated rate  packet per slot • ALOHA Packet transmission attempt process: Two model cases: • infinite population: nodes transmits only one packet and die; • Finite population nodes are permanent and manage a queue of packets • Poisson process, cumulated rate  packet per slot

28. Aloha and infinite population • Is unstable for all λ>0 • Take B large number of waiting packets: • System diverges: B(t) at time t • Also true for binary exponential backoff

29. Aloha and finite population • N nodes • In this case max{B(t)}=N • System is stable when B=N and • When • And max throughput

30. Stack collision resolution in infinite population • Stack algorithm local procedure C←0; While packet to transmit{ if (C=0) then { transmit; if collision then C←rand(0,1)} else { if listen=collision then C←C+1; else C←C-1 }

31. Stack algorithm stability condition

32. Ternary Stack collision resolution • Ternary Stack algorithm local procedure C←0; While packet to transmit{ if (C=0) then { transmit; if collision then C←rand(0,1,2)} else { if listen=collision then C←C+1; else C←C-1 }

33. Upper bound on colision resolution algorithms stability

34. Aloha under small load • Infinite population with • Transmission and retransmission is a Poisson process • cumulated rate  packet per slot • Equilibrium equation:

35. Takes exponential time Stable point unstable point

36. Random Access Performance • Maximum throughput • Average Delay in interface • BEB

37. Random Access performance • Geometric ALOHA • Packet (re)transmitted on current slot with proba • Average backoff • For a packet • Delay B in interface has p.g.f.

38. Random Access performance • Workload W of interface of node i

39. Random Access performance • Workload is greater than packet delay • It satisfies • With p.g.f • We know how to solve…

40. Random Access performance • Interface idle probability • System is stable as long as • When • Average waiting time in buffer is as long as

41. Summary periodic TDMA random TDMA • Periodic TDMA • Throughput up to • Interface delays in • Queueing delays in • Random TDMA • Throughput up to • Interface delays in • Queueing delays in

42. Protocol CSMA (Wifi) • Mini-slots

43. Performances of CSMA • Poisson model: • ρ: per mini-slot load • L: packet length (in mini-slots) • Net throughput

44. L=100

45. Max throughput L

46. RTS-CTS RTS packet emitter CTS ack Vorbidden period Intended receiver

47. CSMA/CA performances • Net throughput with RTS-CTS

48. RTS-CTS with R=10 CSMA Max throughput L