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# Model and tools

Model and tools. Traffic Model. Poisson law Napoléon worries about the statistics of horse accidents of his generals Poisson confirms these are unfrequent independent events. Poisson law. A. B. Intensity λ Independence, memoriless property

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## Model and tools

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

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

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

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