Transcript Document 7463591

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
Contents
• ALOHA Protocol
• Slotted ALOHA
• CSMA/CD
ALOHA Protocol
ALOHA Protocol
• 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
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
ALOHA Network: basic assumptions
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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
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: u0
 Probability 1 station transmits: u1
 Probability 2 or more stations transmit: 1-u0-u1
• How much are uk?
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
Transitions between channel states (1)
• idle (1) : remains idle with probability u0
 jumps from idle to transmitting, with probability u1
 jumps from idle to collided, probability 1-u0-u1
• Transmitting (3) : move to idle with probability u0
 Otherwise, (if one or more stations transmits) move to
colliding with probability 1 –u0 why?
• Collided (2) : Same idea. Move to idle with probability
u0, otherwise stay in collided, probability 1-u0
Transition Matrix
• Row 2: Transitions to colliding
 From idle: if more then 1 packet is transmitted:
 Pr = 1-u0-u1
 From colliding or transmitting if at least one station is
transmitting: Pr = 1 – u0
• Row 3: Transitions to transmitting
 From idle: if just 1 packet is transmitted: Pr = u1
 From other state: impossible (current state is not idle)
ALOHA State Transition Diagram
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:
 uk = [N!/k!(N-k)!]*ak(1-a)N-k
 New and retransmitted frames
• Probability that no frame will be transmitted
• u0 = (1-a)N
• Probability that 1 frame will be transmitted
• u1 = Na(1-a)N-1
Steady State (Equilibrium) Probabilities
• State Probabilities at equilibrium (steady state):
 Ps = s
s1 + s2 + s3 = 1
• Solution:
• s1 = u 0
s2 = 1 – u0 –u0u1
s3= u0u1
• Probability to be in state idle
 s1 = (1-a) N
• Probability to be in state transmitting
• s3 = Na(1-a)2N-1
• Probability to be in state collided
• s2 = 1 – (1-a)N - Na(1-a)2N-1
Performance: Input, Throughput & Access
• l : Mean rate of attempted (input) transmissions
 l = Skkuk = Na frames/time-step
• 1 frame is successfully transmitted if system is in state 3
• Probability s3
 0 frames successfully transmitted in other states
 mean rate of transmitted (output) frames: Th = s3
 Th = u0u1 = 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): pA
• pA = output rate/Input rate = Th/ l = (1-a)2N-1
• Also called efficiency h
Large N limit
• l : Average rate of attempted (input) transmissions
 l = Skkuk = Na
• For large N, fixed l:
 u0 = (1-a)N = (1- l/N)N
 u0  e-l
 u1 = Na(1-a)N-1 = l(1-l/N)N-1 = l(1-l/N)N/(1-l/N)
 u1  le-l
 Th =  le-2l
pA = h  e-2l
Retransmissions and delay
• Probability of successful Access (to transmit a frame): pA
• Probability of unsuccessful Access: 1- pA
• Probability of k unsuccessful Accesses followed by a
successful Access: (1- pA)k pA
• Retransmissions: mean number of unsuccessful attempts
before successful transmission of a frame: nR
 nR = Skk* (1- pA)k pA
k goes from 1 to ∞
• nR = (1- pA)/ pA = 1/ pA – 1 = 1/ (1-a )2N-1 -1
• nR is also average delay of transmitting a frame in
Timesteps
Max Throughput situation
• For fixed N, find for what value of a, Th is maximal:
• dTh/da = 0  aM = 1/(2N)
• Th(max) = (1/2)[1- 1/(2N)]2N-1
• For N  ∞
Th(max) = 1/2e = 0.18394 frames/timestep
• Input rate: l = NaM = 0.5 frames/time-step
• efficiency (probability of successful Access) :
 pA = (1- aM)2N-1 = (1 – 1/2N) 2N-1
 pA  1/e = 0.367 when N ∞
• nR = [1 -1/e]/(1/e) = e -1 = 1.718 Failed Accesses per frame
Throughput vs load l
• N = 10;
horizontal axix is l = Na
• Dotted line: ALOHA
• Solid line: Slotted ALOHA (next topic)
Slotted Aloha
Aloha
Numerical Example
• ALOHA network N = 20
• Probability that a station has a frame to transmit in a timestep a = 0.01
• Results:
• Throughput Th = 0.1351 frames/timestep
• mean number of time-steps before success, nR= 0.4799
• Maximal throughput Th(max) = 0.1839 frames/time-step
• value of a for maximal throughput, aM = 1/40) = 0.025
Slotted ALOHA
Slotted ALOHA
• 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
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 u1
State Transition Diagram
States and Transitions (1)
• idle (1): Stay idle as long as all stations idle in next
timeslot. Pr = u0
 jump to transmitting if just 1 station requests access in
next timeslot. Pr = u1
 otherwise jump to collided. Pr = 1 - u0 - u1
• Transmitting (3): stay transmitting if just 1 station
requests access in next timeslot. Pr = u1
 jump to idle if all stations idle in next timeslot. Pr = u0
 Otherwise jump to collided. Pr = 1 - u0 - u1
States and Transitions (2)
• Collided (2): jump to idle if all stations idle in next
timeslot. Pr = u0
• jump to Transmitting if just 1 station requests access in
next timeslot. Pr = u1
• Else stay in collided. Pr = 1 - u0 - u1
 Values of uk : 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, uk
– uk = [N!/(k!(N-k)!)]*ak(1-a)N-k binomial distribution
Transition Matrix and State Probabilities
•
•
•
•
•
•
Steady state probabilities: s = [s1 s2 s3]t
Ps = s
s1 + s 2 + s 3 = 1
Solution:
idle: s1 = u0
collided: s2 = 1- u0 – u1
transmitting: s3 = u1
Slotted ALOHA vs. Pure ALOHA
Steady State Probabilities
s1
s2
s3
Pure ALOHA
u0
1-u0 –u0u1
u0u1
Slotted ALOHA
u0
1- u0 –u1
u1
• (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
Slotted ALOHA Performance
• l : mean rate of attempted (input) transmissions
 l = Skkuk = Na
 Also called load
• One frame is transmitted whenever system at state 3
 Probability s3
• 0 frames are transmitted at any other state
 mean rate of transmitted frames: s3 (per timestep)
 Th = s3 = u1 = Na(1-a)N-1
 Compare Pure ALOHA: s3 = u0u1 = Na(1-a)2N-1
• Slotted ALOHA Th higher by factor (1-a)-N
Retransmissions and delay
• Probability of successful attempt to transmit a frame:
 Access Probability: pA = Th/ l = (1-a)N-1
 also called efficiency, h
• Probability of unsuccessful attempt: 1- pA
• Probability of k unsuccessful attempts followed by a
successful attempt: (1- pA)k pA
• Retransmissions: Average number of unsuccessful
attempts before successful transmission of a frame: nR
 nR = Skk* (1- pa)k pa
k goes from 1 to ∞
• nR = (1- pA)/ pA = 1/ pA -1 = 1/ (1-a)N-1 - 1
Max Throughput
• For fixed N, find a so that throughput is maximal:
• dTh/da = 0  aM = 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
efficiency at max throughput, large N
• Input rate:
l = NaM = 1 frames/timestep
• Compare ALOHA: l = NaM = 0.5 frames/timestep
• Efficiency (probability of successful access) :
 pA = (1- aM)N-1 = (1 – 1/N) N-1  1/e = 0.367
• Compare ALOHA:
 pA = (1-aM)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!
• nR = [1 -1/e]/(1/e) = e -1 = 1.718 attempts per frame
 Compare ALOHA: same
Throughput as a function of load l
• N = 10;
horizontal axis is l = Na
• Dotted line: ALOHA; Solid line: Slotted ALOHA
• Both system show max throughput at very low traffic
 l = 0.5, 1.0 frames/timestep respectively
• As input rate increases, both throughputs decrease rapidly
Slotted Aloha
Aloha
CSMA/CD
CSMA/CD (1)
• CSMA/CD (also Ethernet) used when bit propagation
time tp between farthest LAN stations is much smaller
than transmission time tt 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…
CSMA/CD (2)
• within transmission time tt 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)
Markov Model Assumptions and parameters
• Channel shared by N stations
• States of the channel: Idle, Transmitting, Collided
• Time-step (time-slot) = propagation time: T = tp
• Transmitted frames have equal lengths
• Frame transmission time: tt = n timesteps
• Usually n >> 1
• There are n Transmitting states!
• t1, t2, …tn
The backoff assumption
• After Transmitting state tn, 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
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:
 uk = [N!/(k!(N-k)!)]ak(1-a)N-k
 u0 = (1-a)N
u1 = Na(1-a)N-1
States and transitions
• Idle:
 System stays Idle if no station request access. Pr = u0
 jump to Transmitting state 1 if just one stations request
access. Pr = u1
 Otherwise jump to collided. Pr = 1-u0 –u1
• Transmitting state j, tj: (j = 1, 2, …n-1)
 jump to Transmitting state tj+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 ?
CSMA/CD: State Transition Diagram
Transition Matrix
• We organize the n+2 states: Idle, t1, t2, …tn, collided
Steady State Probabilities
• s = [si st1 st2 … stn sc]t
• Ps = s
Sksk = 1
• Solution:
• Denote: K = 1/(2 + u1(n-1) – u0)
• Probability to be in the Idle state
 si = K
 Probabilities to be in Transmitting state ti
• st1 = st2 = st3 = … = stn = Ku1
• Probability to be in Collided state
 sc = K(1-u0-u1)
CSMA/CD Performance: Throughput
• Throughput Th: Mean rate of transmitted frames
• (1/n) of a frame is transmitted at each state tk: Pr = stk
• 0 frames are transmitted at any other state
 Throughput = (1/n)Skstk = Ku1 Packets/Time-step
 Th = nu1/[2 + u1(n-1) – u0] Packets/Time-step
 Large n: Th  1
 Very little time wasted during Collision
• Reminder:
 u0 = (1-a)N
u1 = Na(1-a)N-1
Throughput vs load l
• Load l : Average rate of attempted transmissions
 l = Skkuk = Na
• Draw Th as a function of l
• N = 10, n = 10,
0 < l <10
CSMA/CD
Slotted ALOHA
ALOHA
CSMA/CD performance: Access Probability
• Probability of successful attempt to transmit a frame: pA
• pA = output rate/Input rate = Th/ l (also denoted h)
• pA = (nu1/ l) / [2 + u1(n-1) – u0]
• Draw pA v.s l : N = 10, n = 10
ALOHA
Slotted ALOHA
CSMA/CD
Retransmissions
• Probability of successful attempt to transmit a frame: pA
• Probability of unsuccessful attempt: 1- pA
• Probability of k unsuccessful attempts followed by a
successful attempt: (1- pA)k pA
• Retransmissions: Average number of unsuccessful
attempts before successful transmission of a frame: nR
 nR = Skk* (1- pA)k pA
k goes from 1 to ∞
• nR = (1- pA)/ pA = 1/ pA -1
• nR is also the delay of transmitting a frame in timesteps
CSMA/CD Performance: Delay
• Draw Delay v.s l for: N= 10, n = 10
ALOHA
Slotted ALOHA
CSMA/CD
Carrier Sense Multiple Access
Collision Avoidance
(CSMA/CA)
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
Simple CSMA/CA Model: Basic properties
• Channel states: Idle, Transmitting, Collided
• Max propagation time between stations: tp
• Time step T = tp
• Frame transmission time tt = 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: ti
Simple CSMA/CA Model: Basic properties (2)
 A collided frame will continue to be collided – its
transmission will not be stopped
 n collided states: ci
• 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
• uk: Probability that k stations have frames to transmit
• uk = [N!/(k!(N-k)!)]*ak(1-a)N-k
 u0 = (1-a)N
u1 = Na(1-a)N-1
States and transitions
• Idle:
 System stays Idle if no station request access. Pr = u0
 jump to Transmitting state 1, t1, if just one stations
request access. Pr = u1
 Otherwise jump to collided state 1. Pr = 1-u0 –u1
• Transmitting state j: jump to transmitting state tj+1. Pr = 1
 transmitting state tn: jump to Idle. Pr = 1
• Collided state j: jump to Collided state j+1. Pr = 1
 Collided state n: jump to idle. Pr = 1.
Simple CSMA/CA: State Transition Diagram
Transition Matrix
• Organize states: [i, t1, t2, …tn. c1, c2, …cn]
• Example Transition Matrix (n = 3)
Simple CSMA/CA: Steady State Probabilities
• Ps = s
 Sksk = 1
• Solution:
• Denote: K = 1/(n(1-u0) + 1)
• si = K
• st1 = st2 = … =stn = Ku1
• sc1 = sc2 = … =scn = K(1-u0-u1)
Simple CSMA/CA: Throughput (1)
• Throughput Th: Average rate of transmitted frames
• (1/n) frame transmitted at each transmitting state tk
 Pr = stk
• 0 frames are transmitted at any other state
 Throughput: (1/n)Skstk (Packets per timestep)
 Th = nu1/[n(1- u0) + 1] (Packets per Frame Time)
• Where:
 u0 = (1-a)N
 u1 = Na(1-a)N-1
Simple CSMA/CA: Throughput (2)
• l : Average rate of attempted (input) transmissions
 l = Skkuk = Na
• For large N, fixed l:
 u0 = (1-a)N = (1- l/N)N  e-l
 u1 = Na(1-a)N-1 = l(1-l/N)N-1  le-l
 Th  n le-l / [n(1- e-l) + 1]
• For large N, large n (large net, long frames):
• Th  le-l / (1- e-l) = l / (el - 1)
Simple CSMA/CA: Throughput (3)
• Draw Th as a function of l = Na
N = 10, n = 50
Simple CSMA/CA
Slotted ALOHA
ALOHA
Throughput: CSMA/CA vs. CSMA/CD
• CSMA /CA: lower throughput
• Probability of transmitting is lower, due to many Collided
• N = 50, n = 10
CSMA/CD
Simple CSMA/CA
Slotted ALOHA
ALOHA
ALOHA
Slotted ALOHA
Access Probability
• pA = Th/l
• Draw pA as a function of l = Na: N = 10, n = 50
CSMA/CA
ALOHA
Slotted ALOHA
Retransmissions and Delay
• Retransmissions: Average number of unsuccessful
attempts before successful transmission of a frame: na
 nR = Skk* (1- pA)k pA = (1- pA)/ pA = 1/ pA -1
• nR is also the delay of transmitting a frame in timesteps
• Draw nR as a function of l = Na: N= 10, n = 50)
ALOHA
Slotted ALOHA
CSMA/CA
Distributed Coordination Function (DCF)
Medium Access Control for
Ad Hoc Wireless LANs
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
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
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
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
Markov Model: Assumptions and parameters (1)
• T: Time Step: max expected propagation delay tp 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, t1, t2, …tn
• Station finds it had collision via NACK/ Timeout
 After full transmission of its (colliding) frame
 n Collided states, c1, c2, …cn
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
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)
• vk : Probability that k stations from a set, size N’, attempt
transmission in a Time Step
• vk = N’!/[k!(N’-k)!]*ak(1-a)N’-k 0 ≤ k ≤ N’
 Where N’ = N/w
States and transitions (1)
• Channel states: Idle, Transmitting tk, Collided ck
• Idle:
 channel stays Idle if all stations, in all w r-sets have no
frame to transmit
 Pr ≡ x = (v0)w = (1-a)N
 Otherwise the channel will jump to other state (next
slides)
States and transitions (2)
• Channel jumps from idle to t1 if:
• 1 station from {r-set 0} has ready frame; Probability v1
• OR 1 station from {r-set 1} has ready frame & no station
from {r-set 0} has ready frame; Pr = v0v1
• OR 1 station from {r-set 2} has ready frame & no station
from {r-set 0}, {r-set 1} has ready frame; Pr = (v0)2v1
• ……
• OR 1 station from {r-set w-1} has ready frame & no station
from {r-set j}, j < w-1 has ready frame; Pr = (v0)w-1v1
 Pr y = u1 + v0v1 + (v0)2v1 + (v0)3v1 + …(v0)w-1v1
 y = v1*(1-(v0)w)/(1-v0)
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 tj to transmitting
state tj+1, Pr = 1
• Channel jumps from transmitting state tn to idle, Pr = 1
• Channel jumps from collided state cj to collided state cj+1,
Pr = 1
• Channel jumps from collided state cn to idle, Pr = 1
DCF Markov Model: State Transition Diagram
Transition Matrix
• Organize the states: idle, t1, t2, …tn, c1, c2, …cn
• Example Transition Matrix for n = 3
Steady State Probabilities
• Ps = s
 Sksk = 1
• Solution:
• s = K*[ 1 y y …y z z … z]t
 K = 1/(n(1-x) +1)
 stk = Ky
Probability to be in transmitting state
 sck = Kz
Probability to be in colliding state
• where
 x = (v0)w = (1-a)N ; y = v1*(1-(v0)w)/(1-v0); z = 1-x –y
 vk = N’!/[k!(N’-k)!]*ak(1-a)N’-k ; N’ = N/w
DCF Performance: Throughput (1)
• Throughput Th: Average rate of transmitted frames
• 1/n frame is transmitted whenever system at states tk
 Probability stk
• 0 frames are transmitted at any other state
 Throughput = (1/n)Skstk = 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:
• uk = N!/[k!(N-k)!]*ak(1-a)N-k
• Load (Average input rate): l = Skkuk = Na
DCF Performance: Throughput (2)
 Graph of Th vs load: N = 2, n = 10, w = 8
DCF Protocol
Slotted ALOHA
ALOHA
CSMA/CA
Throughput (3): Effect of the reservations
• Dependence on the “reservation window” size, w
• N = 32, n = 10, w = 4, 8, 16
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
DCF Performance: Access Probability
• Probability of successful attempt to transmit a frame: pA
• pA = output rate/Input rate = Th/ l
• pA = ny/[{n(1- x) + 1}l]
• Draw pA as a function of the load, l: N=32, n=10, w=8
DCF Protocol
CSMA/CA
Slotted ALOHA
ALOHA
DCF: Retransmissions and Delay
• Retransmissions: Average number of unsuccessful
attempts before successful transmission of a frame: nR
 nR = Skk* (1- pA)k pA = (1- pA)/ pA = 1/ pA -1
• nR is also the delay of transmitting a frame in timesteps
• Draw nR as a function of l (N= 32, n = 10, w=8)
ALOHA
Slotted ALOHA
CSMA/CA
DCF Protocol
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