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 Intensity λ – Independence, memoriless property are independent if – Poisson law – conditionnally uniform If then the k events are uniformly distributed over A AB time

Exponential inter-arrival Inter-arrival I are independent and exponential – Density: – Memoriless property:

Super-position of Poisson processes Two flows 1 and 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)

p.g.f of Poisson

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) time frequencies 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 A B

Collision Resolution Algorithm Access point configuration (D=0) Aloha – Random Backoff Uniform over (0, W max ) (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 }

Stack algorithm stability condition ABCAB- ABC C CBC CC

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 }

Upper bound on colision resolution algorithms stability

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

Stable point unstable point Takes exponential time

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

L=100

L Max throughput

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

CSMA/CA performances Net throughput with RTS-CTS

L RTS-CTS with R=10 CSMA Max throughput