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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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**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 • are independent if • Poisson law • conditionnally uniform • If then the k events are uniformly distributed over A time**Exponential inter-arrival**• Inter-arrival I are independent and exponential • Density: • Memoriless property:**Super-position of Poisson processes**• Two flows 1and 2 • The union is a Poisson flow of intensity 1+ 2**Test**• Random sequence • Discrepant sequence.**Convergence**• Small independent processes • N unfrequent events each with proba • Bernoulli law. When N→∞**Time varying intensity**• Generalization:**Probability generating functions**• Let X be a non negative integer random variable • Probability generating function for z complex**P.g.f. property**• Moment generating function**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)**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.**Multiple access protocols**• In wireless networks, medium channel is unique and must be shared • One or several of frequencies**Wireless Communication Architecture**• Access point architecture • Wifi infrastructure mode • GSM, UMTS • Wimax • Ad hoc architecture • Mesh networks • Mobile ad hoc • Sensor networks**Multiple access protocols**• Frequency Division Multiple Access • Frequency set is split between users • Time Division Multiple Access (TDMA) frequencies time time**Wireless Access Protocols**• Periodic TDMA • Time slot periodically allocated to terminal in round robin. • Examples: GSM, bluetooth. time slot**Wireless Access Protocols**• Random access protocols • More than two transmitters over one slot→ collision • Collision detection (no d’ACK) • Collision resolution algorithm. time slot**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**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.**Terminal network interface model**Packets internally generated Network interface buffer Network interface Server (one packet max)**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**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)**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**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**Random TDMA Performance**• Packet generation over all nodes • Poisson process, cumulated rate packet per slot • No packet retransmission :**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**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**Aloha and finite population**• N nodes • In this case max{B(t)}=N • System is stable when B=N and • When • And max throughput**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 }**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 }**Aloha under small load**• Infinite population with • Transmission and retransmission is a Poisson process • cumulated rate packet per slot • Equilibrium equation:**Takes exponential time**Stable point unstable point**Random Access Performance**• Maximum throughput • Average Delay in interface • BEB**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.**Random Access performance**• Workload W of interface of node i**Random Access performance**• Workload is greater than packet delay • It satisfies • With p.g.f • We know how to solve…**Random Access performance**• Interface idle probability • System is stable as long as • When • Average waiting time in buffer is as long as**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**Protocol CSMA (Wifi)**• Mini-slots**Performances of CSMA**• Poisson model: • ρ: per mini-slot load • L: packet length (in mini-slots) • Net throughput**RTS-CTS**RTS packet emitter CTS ack Vorbidden period Intended receiver**CSMA/CA performances**• Net throughput with RTS-CTS**RTS-CTS with R=10**CSMA Max throughput L

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