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Tel Hai Academic College Department of Computer Science Prof. Reuven Aviv Markov Models for Access Control in Computer Networks Resource: Fayez Gebali, Analysis of Computer and Communication Networks

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Contents ALOHA Protocol Slotted ALOHA CSMA/CD

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ALOHA Protocol

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Computers communicate over a broadcast channel Any computer: When you have a frame to send, send No medium sense before transmission Collisions (also called contentions) possible Sender senses medium during transmission (not before) If collision identified, retransmit after a random wait

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Collisions between frames Assume: Frame have same length & transmission time, T Any frame transmitted in period T before and during the transmission of the focus frame will collide with it To have a successful transmission in a time-step, the channel must be quiet in the previous time-step

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ALOHA Network: basic assumptions 1. Frames have same fixed length, same transmission time 2. Time-step T = frame transmission time 3. Propagation delay between any pair of stations < T 4. N stations 5. Any station can transmit any time 6. Collision occurs if frame sent at time t, and there are other transmissions during time interval [t-T, t+T] 7. All Stations identify collision during transmission

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Markov Chain model The states of the channel are modeled by a Markov chain Idle, transmitting, collided Idle: No frame is being transmitted Transmitting: One frame is being transmitted Collided: two or more frames are being transmitted At any time some stations transmit: Probability no station transmits: u 0 Probability 1 station transmits: u 1 Probability 2 or more stations transmit: 1-u 0 -u 1 How much are u k ?

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State Transition Diagram Transitions into transmitting only from idle requirement of idle channel before a successful transmission No transition from Transmitting to transmitting No transition from colliding to transmitting

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Transitions between channel states (1) idle (1) : remains idle with probability u 0 jumps from idle to transmitting, with probability u 1 jumps from idle to collided, probability 1-u 0 -u 1 Transmitting (3) : move to idle with probability u 0 Otherwise, (if one or more stations transmits) move to colliding with probability 1 –u 0 why? Collided (2) : Same idea. Move to idle with probability u 0, otherwise stay in collided, probability 1-u 0

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Transition Matrix Row 2: Transitions to colliding From idle: if more then 1 packet is transmitted: Pr = 1-u 0 -u 1 From colliding or transmitting if at least one station is transmitting: Pr = 1 – u 0 Row 3: Transitions to transmitting From idle: if just 1 packet is transmitted: Pr = u 1 From other state: impossible (current state is not idle)

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ALOHA State Transition Diagram

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Probabilities of transmissions Probability that a station has a frame to transmit: a Note: Either a new frame or retransmission Probability that k stations will transmit at same time: u k = [N!/k!(N-k)!]*a k (1-a) N-k New and retransmitted frames Probability that no frame will be transmitted u 0 = (1-a) N Probability that 1 frame will be transmitted u 1 = Na(1-a) N-1

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Steady State (Equilibrium) Probabilities State Probabilities at equilibrium (steady state): Ps = s s 1 + s 2 + s 3 = 1 Solution: s 1 = u 0 s 2 = 1 – u 0 –u 0 u 1 s 3 = u 0 u 1 Probability to be in state idle s 1 = (1-a) N Probability to be in state transmitting s 3 = Na(1-a) 2N-1 Probability to be in state collided s 2 = 1 – (1-a) N - Na(1-a) 2N-1

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Performance: Input, Throughput & Access : Mean rate of attempted (input) transmissions = k ku k = Na frames/time-step 1 frame is successfully transmitted if system is in state 3 Probability s 3 0 frames successfully transmitted in other states mean rate of transmitted (output) frames: Th = s 3 Th = u 0 u 1 = Na(1-a) 2N-1 frames/time-step Frames in Th are part of the set of input frames Probability of successful Access (transmit a frame): p A p A = output rate/Input rate = Th/ = (1-a) 2N-1 Also called efficiency

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Large N limit : Average rate of attempted (input) transmissions = k ku k = Na For large N, fixed : u 0 = (1-a) N = (1- /N) N u 0 e - u 1 = Na(1-a) N-1 = (1- /N) N-1 = (1- /N) N /(1- /N) u 1 e - Th = e -2 p e -2

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Retransmissions and delay Probability of successful Access (to transmit a frame): p A Probability of unsuccessful Access: 1- p A Probability of k unsuccessful Accesses followed by a successful Access: (1- p A ) k p A Retransmissions: mean number of unsuccessful attempts before successful transmission of a frame: n R n R k k* (1- p A ) k p A k goes from 1 to ∞ n R = (1- p A )/ p A p A – (1-a ) 2N-1 -1 n R is also average delay of transmitting a frame in Timesteps

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Max Throughput situation For fixed N, find for what value of a, Th is maximal: dTh/da = 0 a M = 1/(2N) Th(max) = (1/2)[1- 1/(2N)] 2N-1 For N ∞ Th(max) = 1/2e = 0.18394 frames/timestep Input rate: = Na M = 0.5 frames/time-step efficiency (probability of successful Access) : p A = (1- a M ) 2N-1 = (1 – 1/2N) 2N-1 p A 1/e = 0.367 when N ∞ n R = [1 -1/e]/(1/e) = e -1 = 1.718 Failed Accesses per frame

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Throughput vs load N = 10; horizontal axix is = Na Dotted line: ALOHA Solid line: Slotted ALOHA (next topic) Slotted Aloha Aloha

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Numerical Example ALOHA network N = 20 Probability that a station has a frame to transmit in a time- step a = 0.01 Results: Throughput Th = 0.1351 frames/timestep mean number of time-steps before success, n R = 0.4799 Maximal throughput Th(max) = 0.1839 frames/time-step value of a for maximal throughput, a M = 1/40) = 0.025

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Slotted ALOHA

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Time is divided to slots Stations are synchronized, allowed to transmit at start of slots only (not at any time as in ALOHA) Time-step = time-slot Vulnerable period = 1 timeslot (not 2 as in ALOHA) Collisions occur only if two frames transmitted at same timeslot. Less collisions, better throughput and successful Accesses

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Modeling As in ALOHA: States idle, collided, transmitting What’s the diff between ALOHA and Slotted ALOHA? Direct transitions from collided to transmitting and from transmitting to transmitting are now allowed occurs if just 1 station requests access in next timestep. Frame will be transmitted in the next timestep, and will not collide with anything transmitted in current timestep. (Vulnerable period is 1 timeslot, no requirement of calm time-step) Probability u 1

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State Transition Diagram

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States and Transitions (1) idle (1): Stay idle as long as all stations idle in next timeslot. Pr = u 0 jump to transmitting if just 1 station requests access in next timeslot. Pr = u 1 otherwise jump to collided. Pr = 1 - u 0 - u 1 Transmitting (3): stay transmitting if just 1 station requests access in next timeslot. Pr = u 1 jump to idle if all stations idle in next timeslot. Pr = u 0 Otherwise jump to collided. Pr = 1 - u 0 - u 1

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States and Transitions (2) Collided (2): jump to idle if all stations idle in next timeslot. Pr = u 0 jump to Transmitting if just 1 station requests access in next timeslot. Pr = u 1 Else stay in collided. Pr = 1 - u 0 - u 1 Values of u k : Assume requests for access are independent random events: Probability that a station on its own requests access: a Probability that in a system of N stations k stations requests access, u k – u k = [N!/(k!(N-k)!)]*a k (1-a) N-k binomial distribution

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Transition Matrix and State Probabilities Steady state probabilities: s = [s 1 s 2 s 3 ] t Ps = s s 1 + s 2 + s 3 = 1 Solution: idle: s 1 = u 0 collided: s 2 = 1- u 0 – u 1 transmitting: s 3 = u 1

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Slotted ALOHA vs. Pure ALOHA (1) idle (1): pure/slotted ALOHA have same probabilities (2) collided (2): Slotted ALOHA has a lower probability (3) Transmitting (3): slotted ALOHA has a higher probability Steady State Probabilities s1s1 s2s2 s3s3 Pure ALOHAu0u0 1-u 0 –u 0 u 1 u0u1u0u1 Slotted ALOHAu0u0 1- u 0 –u 1 u1u1

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Slotted ALOHA Performance : mean rate of attempted (input) transmissions = k ku k = Na Also called load One frame is transmitted whenever system at state 3 Probability s 3 0 frames are transmitted at any other state mean rate of transmitted frames: s 3 (per timestep) Th = s 3 = u 1 = Na(1-a) N-1 Compare Pure ALOHA: s 3 = u 0 u 1 = Na(1-a) 2N-1 Slotted ALOHA Th higher by factor (1-a) -N

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Retransmissions and delay Probability of successful attempt to transmit a frame: Access Probability: p A = Th/ = (1-a) N-1 also called efficiency, Probability of unsuccessful attempt: 1- p A Probability of k unsuccessful attempts followed by a successful attempt: (1- p A ) k p A Retransmissions: Average number of unsuccessful attempts before successful transmission of a frame: n R n R k k* (1- p a ) k p a k goes from 1 to ∞ n R = (1- p A )/ p A p A -1 = 1/ (1-a) N-1 - 1

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Max Throughput For fixed N, find a so that throughput is maximal: dTh/da = 0 a M = 1/N Th(max) = [1- 1/N] N-1 For N ∞ Th(max) = 1/e = 0.3679 frames/timestep Compare ALOHA: For N ∞ Th(max) = 1/2e = 0.18394 frames/timestep

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efficiency at max throughput, large N Input rate: = Na M = 1 frames/timestep Compare ALOHA: = Na M = 0.5 frames/timestep Efficiency (probability of successful access) : p A = (1- a M ) N-1 = (1 – 1/N) N-1 1/e = 0.367 Compare ALOHA: p A = (1-a M ) 2N-1 = (1 – 1/2N) 2N-1 1/e = 0.367 At max throughput, large N limit, both models have same efficiency and same number retransmissions! n R = [1 -1/e]/(1/e) = e -1 = 1.718 attempts per frame Compare ALOHA: same

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Throughput as a function of load N = 10; horizontal axis is = Na Dotted line: ALOHA; Solid line: Slotted ALOHA Both system show max throughput at very low traffic = 0.5, 1.0 frames/timestep respectively As input rate increases, both throughputs decrease rapidly Slotted Aloha Aloha

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CSMA/CD

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CSMA/CD (1) CSMA/CD (also Ethernet) used when bit propagation time p between farthest LAN stations is much smaller than transmission time t of a frame (transmission delay) Unlike ALOHA - Station sense the medium. Then either: 1-persistent CSMA: if idle, send. Else continuously monitor channel; send when idle nonpersistent CSMA: if idle, send. Else wait random time, sense again, send if idle. Else, wait again… p-persistent: if idle, send frame with probability p, or defer for next timeslot with probability 1-p if next timeslot channel is idle. Else sense again…

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CSMA/CD (2) within transmission time t a station will identify a collision if there is one, by Carrier Detection mechanism Upon collision: stop transmitting, wait random, sense etc.. If collision during retransmission, double the range of wait time (exponential backoff)

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Markov Model Assumptions and parameters Channel shared by N stations States of the channel: Idle, Transmitting, Collided Time-step (time-slot) = propagation time: T = p Transmitted frames have equal lengths Frame transmission time: t = n timesteps Usually n >> 1 There are n Transmitting states! t 1, t 2, …t n

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The backoff assumption After Transmitting state t n, system jumps to Idle To sense the medium Stations with frame to send will request access after the Idle Timestep After Collided: colliding stations have definitely a frame to send But we assume that the stations backoff to Idle To sense the medium why? 1 persistent CSMA/CD with backoff

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Basic Probabilities request probability a probability that a station created during T has a frame to send the frame will be sent in the next T Probability that k stations request to send in (next) T: u k = [N!/(k!(N-k)!)]a k (1-a) N-k u 0 = (1-a) N u 1 = Na(1-a) N-1

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States and transitions Idle: System stays Idle if no station request access. Pr = u 0 jump to Transmitting state 1 if just one stations request access. Pr = u 1 Otherwise jump to collided. Pr = 1-u 0 –u 1 Transmitting state j, t j : (j = 1, 2, …n-1) jump to Transmitting state t j+1. Pr = 1 why? Transmitting state n: jump to idle. Pr = 1 Collided: jump to idle. Pr = 1. why we did not have that in ALOHA ?

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CSMA/CD: State Transition Diagram

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Transition Matrix We organize the n+2 states: Idle, t 1, t 2, …t n, collided

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Steady State Probabilities s = [s i s t 1 s t 2 … s t n s c ] t Ps = s k s k = 1 Solution: Denote: K = 1/(2 + u 1 (n-1) – u 0 ) Probability to be in the Idle state s i = K Probabilities to be in Transmitting state t i s t 1 = s t 2 = s t 3 = … = s t n = Ku 1 Probability to be in Collided state s c = K(1-u 0 -u 1 )

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CSMA/CD Performance: Throughput Throughput Th: Mean rate of transmitted frames (1/n) of a frame is transmitted at each state t k : Pr = s t k 0 frames are transmitted at any other state Throughput = (1/n) k s t k = Ku 1 Packets/Time-step Th = nu 1 /[2 + u 1 (n-1) – u 0 ] Packets/Time-step Large n: Th 1 Very little time wasted during Collision Reminder: u 0 = (1-a) N u 1 = Na(1-a) N-1

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Throughput vs load Load : Average rate of attempted transmissions = k ku k = Na Draw Th as a function of N = 10, n = 10, 0 < <10 CSMA/CD Slotted ALOHA ALOHA

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CSMA/CD performance: Access Probability Probability of successful attempt to transmit a frame: p A p A = output rate/Input rate = Th/ also denoted p A = (nu 1 / ) / [2 + u 1 (n-1) – u 0 ] Draw p A v.s : N = 10, n = 10 CSMA/CDSlotted ALOHAALOHA

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Retransmissions Probability of successful attempt to transmit a frame: p A Probability of unsuccessful attempt: 1- p A Probability of k unsuccessful attempts followed by a successful attempt: (1- p A ) k p A Retransmissions: Average number of unsuccessful attempts before successful transmission of a frame: n R n R k k* (1- p A ) k p A k goes from 1 to ∞ n R = (1- p A )/ p A p A -1 n R is also the delay of transmitting a frame in timesteps

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CSMA/CD Performance: Delay Draw Delay v.s for: N= 10, n = 10 CSMA/CDSlotted ALOHAALOHA

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Carrier Sense Multiple Access Collision Avoidance (CSMA/CA)

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CSMA/CA CSMA: Carrier Sense, Multiple Access “Listen before Talk”, Used in Wireless LANs (Wifi) Transmitting station unable to determine if collision occurred while transmitting The transmitted signal will hide arriving signals Station knows about collision via NACK or timeout Collision Detection is not used Carrier Sense Multiple Access /Collision Avoidance

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Simple CSMA/CA Model: Basic properties Channel states: Idle, Transmitting, Collided Max propagation time between stations: p Time step T = p Frame transmission time t = nT n > 1 long frame, small LAN If channel Idle, and frame available, transmission starts 1-persistent CSMA/CA A transmitting station will keep transmitting a whole frame without attempting to identify collision n transmitting states: t i

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Simple CSMA/CA Model: Basic properties (2) A collided frame will continue to be collided – its transmission will not be stopped n collided states: c i When transmitting ends, or when collided ends, all stations go back to sense the medium (Idle) a: Probability that a station has a frame to transmit u k : Probability that k stations have frames to transmit u k = [N!/(k!(N-k)!)]*a k (1-a) N-k u 0 = (1-a) N u 1 = Na(1-a) N-1

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States and transitions Idle: System stays Idle if no station request access. Pr = u 0 jump to Transmitting state 1, t 1, if just one stations request access. Pr = u 1 Otherwise jump to collided state 1. Pr = 1-u 0 –u 1 Transmitting state j: jump to transmitting state t j+1. Pr = 1 transmitting state t n : jump to Idle. Pr = 1 Collided state j: jump to Collided state j+1. Pr = 1 Collided state n: jump to idle. Pr = 1.

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Simple CSMA/CA: State Transition Diagram

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Transition Matrix Organize states: [i, t 1, t 2, …t n. c 1, c 2, …c n ] Example Transition Matrix (n = 3)

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Simple CSMA/CA: Steady State Probabilities Ps = s k s k = 1 Solution: Denote: K = 1/(n(1-u 0 ) + 1) s i = K s t 1 = s t 2 = … =s t n = Ku 1 s c 1 = s c 2 = … =s c n = K(1-u 0 -u 1 )

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Simple CSMA/CA: Throughput (1) Throughput Th: Average rate of transmitted frames (1/n) frame transmitted at each transmitting state t k Pr = s t k 0 frames are transmitted at any other state Throughput: (1/n) k s t k (Packets per timestep) Th = nu 1 /[n(1- u 0 ) + 1] (Packets per Frame Time) Where: u 0 = (1-a) N u 1 = Na(1-a) N-1

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Simple CSMA/CA: Throughput (2) : Average rate of attempted (input) transmissions = k ku k = Na For large N, fixed : u 0 = (1-a) N = (1- /N) N e - u 1 = Na(1-a) N-1 = (1- /N) N-1 e - Th n e - n(1- e - For large N, large n (large net, long frames): Th e - (1- e - (e

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Simple CSMA/CA: Throughput (3) Draw Th as a function of Na N = 10, n = 50 Simple CSMA/CA ALOHA Slotted ALOHA

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Throughput: CSMA/CA vs. CSMA/CD CSMA /CA: lower throughput Probability of transmitting is lower, due to many Collided N = 50, n = 10 Simple CSMA/CA ALOHA Slotted ALOHA CSMA/CD Slotted ALOHA ALOHA

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Access Probability p A = Th/ Draw p A as a function of Na: N = 10, n = 50 CSMA/CA Slotted ALOHA ALOHA

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Retransmissions and Delay Retransmissions: Average number of unsuccessful attempts before successful transmission of a frame: n a n R k k* (1- p A ) k p A = (1- p A )/ p A p A -1 n R is also the delay of transmitting a frame in timesteps Draw n R as a function of Na N= 10, n = 50) CSMA/CA Slotted ALOHAALOHA

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Distributed Coordination Function (DCF) Medium Access Control for Ad Hoc Wireless LANs

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DCF Medium Access Control Part of IEEE802.11 Standard, Based on CSMA/CA Algorithm for a station, when it has a frame: Sense the medium, when idle, wait PCF/SIF time Choose a random value r (backoff, reservation number) If channel idle during a timeslot, decrement r by 1 else do not decrement r When r == 0 (and channel still idle), transmit Basic idea: Small Probability that two stations choose same value (collision) Note: No collision detection procedure

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DCF: Using the Backoff counter Assume Max backoff (reservation) value: 5 Station chose r =2 After PCF/SIF, channel was idle during timeslots 0, 1 Station started transmitting Note: timeslot assumed smaller than Time Step

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Using the Backoff Counter example Tag-User chose r=7 X-User chose r=2 Channel idle 2 slots X-User transmits Tag-User: r=5 Y-User chose r=1 Channel idle 1 slot Y-User transmits Tag-User: r= 4 Channel idle 4 slots Tag-User transmits

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Classification of stations to r-sets Each station choose a reservation value r from 0…w-1 Stations divided into “r-sets” according their r values Stations sense the medium at beginning of Time Step, then: A station from {r-set k} transmits (if it has a ready frame…) if none of {r-set j} stations, j = 0,1, 2,..k-1 have frames to transmit

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Markov Model: Assumptions and parameters (1) T: Time Step: max expected propagation delay p plus the time to sense the medium to determine if idle Also called Distributed Inter-frame Separation (DIFS) At least w time slots Channel states: Idle, Transmitting, Collided Frame transmission = n Timesteps n >1 n transmitting states, t 1, t 2, …t n Station finds it had collision via NACK/ Timeout After full transmission of its (colliding) frame n Collided states, c 1, c 2, …c n

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Markov Model: Assumptions and parameters (2) Reservation values 0…w-1; w sets of stations: {r-set 0}, {r-set 1}, …{r-set w-1} Each set has N’ = N/w stations N’ stations competing to access medium at each timeslot N’ < N Less competition relative to CSMA/CA

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Markov Model: Assumptions and parameters (2) A station can have at most 1 frame waiting for transmission. a: Probability of having a ready frame in the beginning of the Time Step (during the timeslot of its r-set) v k : Probability that k stations from a set, size N’, attempt transmission in a Time Step v k = N’!/[k!(N’-k)!]*a k (1-a) N’-k 0 ≤ k ≤ N’ Where N’ = N/w

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States and transitions (1) Channel states: Idle, Transmitting t k, Collided c k Idle: channel stays Idle if all stations, in all w r-sets have no frame to transmit Pr ≡ x = (v 0 ) w = (1-a) N Otherwise the channel will jump to other state (next slides)

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States and transitions (2) Channel jumps from idle to t 1 if: 1 station from {r-set 0} has ready frame; Probability v 1 OR 1 station from {r-set 1} has ready frame & no station from {r-set 0} has ready frame; Pr = v 0 v 1 OR 1 station from {r-set 2} has ready frame & no station from {r-set 0}, {r-set 1} has ready frame; Pr = (v 0 ) 2 v 1 …… OR 1 station from {r-set w-1} has ready frame & no station from {r-set j}, j < w-1 has ready frame; Pr = (v 0 ) w-1 v 1 Pr y = u 1 + v 0 v 1 + (v 0 ) 2 v 1 + (v 0 ) 3 v 1 + …(v 0 ) w-1 v 1 y = v 1 *(1-(v 0 ) w )/(1-v 0 )

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States and transitions (3) Channel jumps from Idle to Collided in all other cases: More then one station in any r-set requests access z = 1-x –y Channel jumps from transmitting state t j to transmitting state t j+1, Pr = 1 Channel jumps from transmitting state t n to idle, Pr = 1 Channel jumps from collided state c j to collided state c j+1, Pr = 1 Channel jumps from collided state c n to idle, Pr = 1

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DCF Markov Model: State Transition Diagram

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Transition Matrix Organize the states: idle, t 1, t 2, …t n, c 1, c 2, …c n Example Transition Matrix for n = 3

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Steady State Probabilities Ps = s k s k = 1 Solution: s = K*[ 1 y y …y z z … z] t K = 1/(n(1-x) +1) s t k = Ky Probability to be in transmitting state s c k = Kz Probability to be in colliding state where x = (v 0 ) w = (1-a) N ; y = v 1 *(1-(v 0 ) w )/(1-v 0 ); z = 1-x –y v k = N’!/[k!(N’-k)!]*a k (1-a) N’-k ; N’ = N/w

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DCF Performance: Throughput (1) Throughput Th: Average rate of transmitted frames 1/n frame is transmitted whenever system at states t k Probability s t k 0 frames are transmitted at any other state Throughput = (1/n) k s t k = Ky Frames per Time Step Th = ny/[n(1- x) + 1] Frames per FrameTime Probability that k frames are ready to be transmitted irrespective of the reservation number: u k = N!/[k!(N-k)!]*a k (1-a) N-k Load (Average input rate): = k ku k = Na

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DCF Performance: Throughput (2) Graph of Th vs load: N = 2, n = 10, w = 8 DCF Protocol CSMA/CA Slotted ALOHA ALOHA

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Throughput (3): Effect of the reservations Dependence on the “reservation window” size, w N = 32, n = 10, w = 4, 8, 16

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Throughput (4): Effect of the reservations Introducing reservation distributes access requests between slots less chance of collisions increase throughput Reducing n larger probability to listening (idle) Smaller throughput

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DCF Performance: Access Probability Probability of successful attempt to transmit a frame: p A p A = output rate/Input rate = Th/ p A = ny/[{n(1- x) + 1} Draw p A as a function of the load, N=32, n=10, w=8 DCF Protocol CSMA/CA Slotted ALOHA ALOHA

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DCF: Retransmissions and Delay Retransmissions: Average number of unsuccessful attempts before successful transmission of a frame: n R n R k k* (1- p A ) k p A = (1- p A )/ p A p A -1 n R is also the delay of transmitting a frame in timesteps Draw n R as a function of (N= 32, n = 10, w=8) DCF Protocol CSMA/CA Slotted ALOHA ALOHA

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Limitations of the Model 1. Errors were not consider. Is that serious? Access control is data link protocol One may assume that errors were dealt with by the physical layer 2. We used simple backoff procedure, not binary backoff 3. We assume that in every timestep there is a probability for a station to create a frame. If the frame is transmitted, fine, but if it is not new frames are not queued. That means that the station transmission buffer has only a single frame storage. One need to add another queue

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