Airtime fairness in a rate separation IE

Airtime Fairness in a Rate Separation IEEE 802.11b MAC David Tung Chong Wong, Anh Tuan Hoang and Chen Khong Tham Institute for Infocomm Research, Agency for Science, Technology and Resea

Airtime Fairness in a Rate Separation IEEE 802.11b

MAC

David Tung Chong Wong, Anh Tuan Hoang and Chen Khong Tham Institute for Infocomm Research, Agency for Science, Technology and Research (A*STAR)

1 Fusionopolis Way, #21-01 Connexis, Singapore 138632 {wongtc, athoang, cktham}@i2r.a-star.edu.sg

Abstract – IEEE 802.11 distributed coordination function (DCF)

medium access control (MAC) does not provide airtime fairness

for all stations in a multi-rate scenario as it only provides max-min

throughput fairness This gives rise to the rate anomaly problem

where the maximum throughput is limited by the slowest

transmitting station In our paper, we propose airtime fairness in a

rate separation IEEE 802.11b MAC Stations are grouped

according to their transmission rates for transmitting their packets

in different data transmission periods (DTPs) for the different

groups of stations The analytical framework is formulated for N

stations, including an access point (AP) The state transition

diagram is modeled by a two-dimensional discrete-time Markov

chain One dimension of the Markov chain is for the backoff stage

and the second dimension is for the value of the backoff counter

The saturated throughput is approximated by the sum of the

product of a weighted ratio of the throughput of the DTP under

consideration and the throughput of the DTP minus the period

necessary to transmit a packet before the end of the current DTP,

the probability of the number of devices in the DTP and the

number of DTPs The DTPs to achieve airtime fairness are

formulated as non-linear equations, which are solved using

Newton-Raphson method with Jacobian functions Numerical

results of the saturated throughput corresponding to typical

parameter values are presented These results show the advantage

of the proposed rate separation IEEE 802.11b MAC with airtime

fairness in achieving airtime fairness and high saturated

throughput

Keywords – analytical formulation, airtime fairness, saturated

throughput, data transmission periods, beacon period, rate

separation IEEE 802.11b MAC

I.INTRODUCTION IEEE 802.11 distributed coordination function (DCF) medium

access control (MAC) does not provide airtime fairness for all

stations in a multi-rate scenario This is because it is designed to

provide max-min throughput fairness This design philosophy

of IEEE 802.11 MAC causes the rate anomaly problem [1],

where the maximum throughput is limited by the slowest

transmitting station Every station is given equal channel access

probability in IEEE 802.11 MAC Thus, in a multi-rate

scenario, faster stations are penalized as they need to wait

longer for slower stations to transmit their packets If the

packets are all equal, slower stations will take up a larger

portion of the airtime for transmitting their packets as compared

to that of faster stations

Reference [2] proposed a scheme for airtime fairness in

IEEE 802.11b MAC by controlling the minimum contention

window of each station All stations compete for channel access

with airtime fairness A rate separation IEEE 802.11b MAC is

proposed to achieve higher throughput in [3] Stations are

grouped according to their transmission rates for transmitting

their packets in different data transmission periods (DTPs) for

the different groups of stations Only stations belonging to the DTP can transmit in it

In our paper, airtime fairness in a rate separation IEEE 802.11b MAC is our focus The goal is to find the different DTPs such that airtime fairness per station for different classes

is achieved, given a fixed superframe time The contributions of this paper are as follows Firstly, an analytical framework for the saturated throughput is formulated for the rate separation IEEE 802.11b MAC The state transition diagram is modeled

by a two-dimensional discrete-time Markov chain One dimension of the Markov chain is for the backoff stage and the second dimension is for the value of the backoff counter The saturated throughput is approximated by the sum of the product

of a weighted ratio of the throughput of the DTP under consideration and the throughput of the DTP minus the period necessary to transmit a packet before the end of the current DTP, the probability of the number of devices in the DTP and the number of DTPs Secondly, the DTPs to achieve airtime fairness are formulated as non-linear equations, which are solved using Newton-Raphson method with Jacobian functions Thirdly, the advantage of the rate separation IEEE 802.11b MAC with airtime fairness is shown to achieve airtime fairness and high saturated throughput in the numerical results section The rest of the paper is organized as follows Section II describes the superframe of a rate separation IEEE 802.11b MAC, while Section III describes our CSMA/CA MAC scheme used in the DTPs In Section IV, we present an analytical model for transmissions in the DTPs Numerical results are presented

in Section V Both analytical and simulation results are presented Finally, concluding remarks are made in Section VI

II SUPERFRAME FORMAT OF RATE SEPARATION IEEE 802.11b

MAC The superframe format is shown in Fig 1 Each superframe consists of eight parts: four beacon periods (BPs) and four data transmission periods (DTPs) All beacon periods are assumed to

be of equal size In each beacon, the AP coordinates and

informs which stations, using the same data rate, can transmit in

the following DTP after the beacon In the DTP, the stations that can transmit in the DTP using the same data rate will

cooperate by attempting to transmit only in the DTP that is

designated to them in the beacon There is a period before the

end of each DTP that does not allow a packet transmission to be successful as it is insufficiently long to transmit the packet, short inter-frame space (SIFS) time and ACK frame This is for basic access

Figure 1 Superframe format of rate separation IEEE 802.11b MAC protocol

For request-to-send/clear-to-send (RTS/CTS) access, additional

RTS frame, CTS frame and two SIFS times have to be added

DTP i is for stations with data rate i, R i , i = 1, 2, 3, 4, where

R1=11 Mbps, R2=5.5 Mbps, R3=2 Mbps and R4=1 Mbps We

assume that station associations have been completed

III MAC PROTOCOL Our CSMA/CA MAC protocol works in the DTPs as

follows:

• If the channel is idle for more than a DCF inter-frame space

time (DIFS), a station can transmit immediately

• If the channel is busy, the station will generate a random

backoff period This random backoff period is uniformly

selected from zero to the current contention window size

The backoff counter will decrement by one if the channel is

idle for each time slot and will freeze if the channel is sensed

busy or if the its DTP is not active The backoff counter is

re-activated to count down when the channel is sensed idle

for more than a DIFS time or when its DTP is active At the

initial backoff stage, the current contention window size is

set at the minimum contention window size

• If the backoff counter reaches zero, the station will attempt

to transmit its frame if the remaining time to the end of its

active DTP is greater than or equal to the period mentioned

earlier If it is successful, the destination device will send an

acknowledgement after a SIFS and the current contention

window size is reset to the minimum contention window

size If it is not successful, it will increase the current

contention window size by doubling it and adding one in the

next backoff stage and a new random backoff period is

selected as before

• If the backoff counter reaches zero, the device will not

attempt to transmit its frame if the remaining time to the end

of its current active DTP is less than the period mentioned

earlier Instead, it will increase the current contention

window size by doubling it and adding one in the next

backoff stage and a new random backoff period is selected

as before and the count down of the backoff counter will

start after a DIFS after the next BP If the maximum retry

limit is reached, the packet will be dropped and the next

packet will start its backoff process with a minimum

contention window size after a DIFS from the beginning of

its next active DTP

• If a busy period ends within the period mentioned earlier, all

the backoff counters will freeze until after a DIFS from the

beginning of its next active DTP

• This process repeats itself until the frame is successfully

transmitted or until the maximum retry limit is reached If

the frame is still not successfully transmitted, then it is

dropped

• If a station does not receive an acknowledgement within an acknowledgement timeout period after a frame is transmitted, it will continue to attempt to re-transmit the frame according to the backoff algorithm

For each of the next backoff stage, the maximum contention window size is doubled that of the previous maximum contention window size and plus one Then, the backoff counter value is uniformly chosen from zero to the maximum contention window size This is equivalent to doubling the previous maximum contention window size and choosing the backoff counter value uniformly from zero to the maximum contention window size minus one

IV ANALYTICAL MODEL

We assume there are a generic I number of data rates, denoted

by R i , where i=1,…,I, and R 1 >R 2 >…>R I Their corresponding

coverage distances are denoted by D i , D 1 <D 2 <…<D I We also assume that all stations are uniformly distributed in a coverage

area of radius R, where R=D I, the access point is at the centre of

the circle Stations within circle of radius D 1 use data rate R 1 to communicate with the AP, while stations within concentric

circles of radii D i-1 and D i use data rate R i , i=2,…,I Let p i,

i=1,…,I, denote the probability of being in each of these regions

using direct transmission of data rate R i and it is given by

⎪

⎪

⎩

⎪⎪

⎨

⎧

=

−

=

=

R

D D

i R

D p

i i

i i

, , 2 , ) (

1 ,

2

2 1 2 2 2

π

π

Let N and n denote the total number of stations, including

the AP, and the number of stations in each region, excluding the

AP, respectively Let us first consider a region Assuming

uniform distribution of nodes around the AP, the probability of

n devices in the region using data rate i, excluding the AP,

denoted by Pr(n,i), is given by

, ) 1 (

1 )

,

n

N i

⎠

⎞

⎜⎜

⎝

⎛ −

where

, , 1

p

Let a(t) be a random process representing the backoff stage

j, j=0,1,…,L retry , at time t, where L retry is the retry limit Let b(t)

be a random process representing the value of the backoff

counter at time t The value of the backoff counter b(t) is uniformly chosen in the range of (0,1,…,W j), where

1

2 1+

= j−

W for 0≤j≤L retry

retry j

W =( +1)2 −1,0≤ ≤ (4)

where W=W 0

Let p denote the probability that a transmitted frame collides

in a DTP Let T SF be the superframe period, T BP be the Beacon

Period (BP), and T DTP,i be the ith Data Transmission Period (DTP) with data rate R i , respectively Let T Q be the time required to successfully transmit a contention period packet including header, payload, SIFS, ACK, and SIFS, at the point

of decision This is for basic access method For RTS/CTS access method, additional RTS frame, SIFS, CTS frame and SIFS are necessary If the remaining time before the end of the

current active ith DTP is greater than T Q when the backoff counter is zero, then the packet can be attempted to be

BP – beacon period

DTP – data transmission period

R i, i=1,2,3,4 – data rate i

Superframe of a rate separation IEEE 802.11b MAC

{R1 = 11 Mbps}

BP

1 DTP 1

BP

2 DTP 2

BP

3 DTP 3

BP

4 DTP 4

{R2 = 5.5 Mbps} {R3 = 2 Mbps} {R4 = 1 Mbps}

transmitted in the remaining active ith DTP, otherwise it will

not be transmitted Let δ denote a time slot in the backoff

counter

The two-dimensional random process {a(t)=j, b(t)=k} is a

discrete-time Markov chain Therefore, the state of each station

is described by {j,k}, j stands for the backoff stage taking

values from {0,1,…, L retry } and k stands for the backoff delay

taking values from {0,1,…, W j} in time slots

Let , lim Pr[a(t) j,b(t) k]

t

b j k = =

∞

→

distribution of the Markov chain The non-null transition

probabilities are listed as follow:

retry

L j W k W

p j

k)|( ,0)]=(1− ) ( +1),0≤ ≤ ,0≤ <

,

0

retry

L

k)|( ,0)]=1( +1),0≤ ≤ , =

,

0

retry

W k k

j

k

j, )|( , +1)]=1,0≤ ≤ −1 ,0≤ ≤

Pr[(

retry j

j

L j W k W

p j

k

+

=

1 )]

0

,

1

(

|

)

,

Pr[(

The first equation above represents the transition probability to

each state in backoff stage 0 from backoff stage j, 0≤j<L retry,

while the second equation represents the transition probability

to each state in backoff stage 0 from backoff stage L retry The

third equation represents the backoff counter decrementing by

one, while the fourth equation represents the transition

probability to each state in backoff stage j

From [4], we can similarly solve for b0,0 which is given by

) 1

)(

2 1 ( ) ) 2 ( 1 )(

1

)(

1

(

) 2 1 )(

1 ( 2

1 1

0

,

−

− +

−

− +

−

−

=

retry L retry

L

p p p

p W

p p

A station transmits when its backoff counter reaches zero, that

is, the device is at any of the states {j,0}, where 0≤j≤L retry The

probability that a station transmits during a generic time slot in

the network, denoted by τ, is given by

(1 )

0 0,0

0 b,0 L retry p b p L retry b p

j j retry

L

j∑ j = ∑ =⎢⎣⎡⎜⎛ − ⎟ ⎥⎦⎤ −

=

=

A transmitted frame collides when one or more station in the ith

DTP transmit during a time slot The probability that a station

in the backoff stage senses the channel busy, denoted by p, is

given by

1 1 ) 1 (

1− − −+

The additional one in the power term is to cater for the AP τ

and p can be solved numerically

The probability that the channel is busy in the ith DTP

happens when at least one device transmit during a time slot,

denoted by p b, and is given by

1 ) 1 (

b

The probability that a successful transmission occurs in a time

slot, denoted by p s, is given by

1 1 ) 1 ( ) 1

The probability that the channel is idle for a time slot is (1-p b)

The probability that the channel is neither idle nor successful

for a time slot is [1-(1-p b )- p s ]= p b - p s

Let T E(L), T s,i , and T c,i denote the time to transmit a payload

with length E(L), the average time required to successfully

transmit a packet having data rate R i and the average time that

the channel has a collision using data rate R i, respectively The

fraction of the ith DTP in a superframe, denoted by F DTP,i, is given by

SF

i DTP i

T

while, the fraction of the ith DTP minus T Q in a superframe,

denoted by F DTP,i-TQ, is given by

SF

Q i DTP TQ i

T T

=

The saturated throughput, denoted by S, is given by

1

∑

=

=

I

i S i

where

2

2

) (

) 1 ( ) , Pr(

2

2

) slot

a time of length ( E

) slot

a time in time ion transmiss payload

( E ) , Pr(

, ,

,

, ,

,

, ,

) ( 1

1

, ,

,

, ,

,

1 1

⎟

⎟

⎟

⎟

⎟

⎠

⎞

⎜

⎜

⎜

⎜

⎜

⎝

⎛

−

− +

−

×

− + +

−

∑

=

⎟

⎟

⎟

⎟

⎟

⎠

⎞

⎜

⎜

⎜

⎜

⎜

⎝

⎛

−

− +

−

×

∑

=

−

−

=

−

−

=

TQ i DTP Q i DTP

Q i DTP

i DTP Q i DTP

i DTP

i c s b i s s b

i i L E s N

n

i TQ i DTP Q i DTP

Q i DTP

i DTP Q i DTP

i DTP

N n i

F T T

T T

F T T

T

T p p T p p

R T p i

n

R F

T T

T T

F T T

T

i n S

δ

(13)

Let T H,i , L i , L i * , T E(L*) , T SIFS , T ACK , and T DIFS denote respectively time to transmit the header (including preamble and frame header consisting of physical layer header, header check sequence (HCS), Reed-Solomon parity bits, and MAC header

with data rate R i ), the length of the payload with data rate R i, including frame check sum (FCS) and pad bits, the length of the

longest frame in a collision with data rate R i, the time to

transmit a payload with length E(L *), the SIFS time, the time to transmit an acknowledgement and a DIFS time for a station

For the basic access, T s,i and T c,i are respectively given by

DIFS ACK SIFS i

L E i H i

DIFS i

L E i H i

T, = , + ( *)+ (15)

For the RTS/CTS access, T s,i and T c,i are respectively given by

DIFS ACK

SIFS i

L E i H CTS RTS i

(16)

DIFS RTS i

The normalized airtime per station using rate i, R i, is given by

4 , 3 , 2 , 1 , =

N p R

S A

i i

i

For airtime fairness by equating A1= A2, A1=A3 and 4

1 A

A = , we can form three nonlinear equations as follows

, 3 , 2 , 1 , 0 2

2

2 2

4 4

2 2

4 4

)

,

,

,

(

2 1 1 1

2 1 1 1 1 1 1

2

1

1

1 1 1 1 1

2 1 1 1 1 1

1

1

1

2 1 1 2 1 1 1 2

1 1 1 2 1 1 4

3

2

1

=

= +

−

−

+

+

− +

−

+

−

−

=

+ + + + + +

+

+ + +

+ + +

+

+ +

+ + +

+

i c c c c a x c c x

c

a

x c c a x c x x c x

x

c

a

x c x c a x x x x a x

x

x

x

F

i i i i i i

i

i i i

i i i

i

i i

i i i

i i

(19) where

, 4 , 3 , 2 , 1

1 =

p B

B p a i

i

, ) (

) 1 ( ) , Pr(

, ,

) ( 1

i L E s N

n

T p i

n B

− + +

−

∑

= −

, 4 , 3 , 2 , 1 , , =

=T i

4 , 3 , 2 , 1 , , =

Assuming that the beacon periods are of equal size, we have the

fourth equation to solve for the x i’s as follows

, 0 )

, ,

,

4 x x x x =x +x +x +x −b=

where

4 BP

T

To use Newton-Raphson method for solving the nonlinear

equations in (19) and (24), we need to find the Jacobian

functions of (19) and (24) as follows

4 , 3 , 2 , 1 , ) , , ,

∂

∂

x

x x x x F J

j

i

Using (26), we have the Jacobian functions of the nonlinear

equations in (19) and (20) as

, 3 , 2 , 1 , 2

2

4 4

4 4 8

1 1 1 2

1

1 1 1 1 1 1 1 1

2 1 1 1

1

1

= +

−

+

−

−

−

=

+ + +

+ + + + +

+ + +

+

i c c a

c

x c x c a x c a x x

x

a

J

i i

i

i i i i i

i i i

a

i

(27)

{3,4}, {2,3}, {1,2},

}

{

, 2 2 4 4

4 8

=

− +

+

− +

−

=

i,j

c c c a x c x c a x c x x

x

a

J ij j j j j j j j

(28) },

4 , 3 { }, 2 , 3 { }, 4 , 2 { }, 2 , 2 { }, 4 , 1 { }, 3 , 1 { }

,

{

,

= i j

and

4 , 3 , 2 , 1 , 1

4 = j=

Using the Newton-Raphson method [5], we have

,

F x

where

, 4 , 3 , 2 , 1 , ],

= J ij i j

[ ], =1,2,3,4,

= δx i i

[F i(x1,x2,x3,x4)],

=

and solving iteratively until all thresholds are met for (33), we

have

,

x x

xnew = old +δ (35) where

, 4 , 3 , 2 , 1 ],

= x i new i

new

4 , 3 , 2 , 1 ],

= x i old i

old

The initial values of x i,’s can be arbitrarily chosen such that (24)

is met

V NUMERICAL RESULTS

In this section, we present result for the saturated throughput, data transmission periods and normalized airtime for the rate separation IEEE 802.11b MAC protocol described

in section III The parameter values used in the numerical examples for the IEEE 802.11b MAC are tabulated as in Table

1 The RTS/CTS access method is assumed in this section The simulation model is implemented using the simulation kernel

smpl [6] and a C program The C program uses the scheme mentioned in Section III to handle the case when backoff counter is zero and the remaining time before the end of the current active DTP is greater or less than the period needed to successfully transmit a RTS frame, CTS frame, packet, an acknowledgement and three SIFSs before the end of the current active DTP

The saturated throughputs of stations using IEEE 802.11b MAC protocol and a rate separation (RS) IEEE 802.11b MAC protocol with airtime fairness is shown in Fig 2 From this figure, it can be seen that the saturated throughput of a rate separation IEEE 802.11b MAC with airtime fairness can be

much higher than that of the standard IEEE 802.11b MAC at high number of stations

The saturated throughput of class i in the rate separation IEEE 802.11b MAC, S i, is shown in Fig 3 Simulation results are represented by lines, while analytical results are represented

by symbols The saturated throughput of class 1, S1, is the highest, follow by the saturated throughput of class 2, S2, follow

by the saturated throughput of class 4, S4, and the saturated of class 3, S3, is the lowest Class 1 corresponds to a data rate of 11 Mbps, while class 2 corresponds to a data rate of 5.5 Mbps Class 3 corresponds to a data rate of 2 Mbps, while class 4 corresponds to a data rate of 1 Mbps From (1) and Table I,

p1=0.232, p2=0.218, p3=0.108 and p4=0.442 Besides the reason

of airtime fairness, one of the reasons that the saturated throughput of class 4 is higher than that of class 3 is that p4 is several times higher than p3

Fig 4 shows the DTPs for all classes as the number of stations, N, varies DTPs 1 and 2 remain relatively constant as the number of stations, N, varies DTPs 3 and 4 are more sensitive to changes in the number of stations, N, at the lower values, but are relatively constant at higher number of stations,

N

T ABLE I P ARAMETER V ALUES U SED F OR I EEE 8 02.11b M AC Symbol Value

j

Data rate, Rk ,k=1,2,3,4 {11,5.5,2,1} Mbps

Distance, Dk ,k=1,2,3,4 {48.2, 67.1, 74.7, 100 } m Physical header (including preamble) 192 bits

CTS frame=HTS frame=ACK frame Physical header + 112 bits

0.5

1

1.5

2

2.5

3

0 5 10 15 20 25 30 35 40 45 50

Number of stations, N

anal - S_RS 802.11b MAC with Airtime Fairness sim - S_RS 802.11b MAC with Airtime Fairness anal -S_802.11b MAC

sim -S_802.11b MAC

Figure 2 Saturated throughput Fig 5 shows the normalized airtime per station for all

classes in the rate separation IEEE 802.11b MAC protocol As

can be seen from this figure, the curves for all classes overlap

with one another This clearly demonstrates that airtime fairness

is achieved with each station having exactly the same

normalized airtime

VI CONCLUDING REMARKS

An analytical formulation of the saturated throughput of a

rate separation IEEE 802.11b MAC is presented A

two-dimensional discrete-time Markov chain is used for analytical

modeling of the DTPs One dimension of the Markov chain is

for the backoff stage and the second dimension is for the value

of the backoff counter The saturated throughput is

approximated by the sum of the product of a weighted ratio of

the throughput of the DTP under consideration and the

throughput of the DTP minus the period necessary to transmit a

packet before the end of the current DTP, the probability of the

number of devices in the DTP and the number of DTPs The

DTPs to achieve airtime fairness are formulated as non-linear

equations, which are solved using the Newton-Raphson method

with Jacobian functions Numerical examples with typical

parameters are used to show the performance of the saturated

throughput, DTPs and airtime of the rate separation IEEE

802.11b MAC The results also clearly demonstrated the

advantage of having airtime fairness and higher saturated

throughput for the rate separation IEEE 802.11b MAC as

compared to the standard IEEE 802.11b MAC

REFERENCES [1] M Heusse, F Rousseau, G Berger-Sabbatel and A Duda, “Performance

Anomaly of 802.11b,” IEEE INFOCOM 2003

[2] T Joshi, A Mukherjee, Y Yoo and D.P Agrawal, ”Airtime Fairness for

IEEE 802.11 Multirate Networks,” IEEE Transactions on Mobile

Computing, vol 7, no 4, April 2008

[3] C.T Im and D.H Kwon, “A Rate Separation Mechanism for Performance

Improvements of Multi-Rate WLANs,” International Conference on

Computational Science and its Applications (ICCSA), May 2005

[4] G Bianchi, “Performance Analysis of the IEEE 802.11 Distributed

Coordination Function,” IEEE Journal on Selected Areas in

Communications, vol 18, no 3, March 2000

[5] W.H Press, S.A Teukolsky, W.T Vetterling and B.P Flannery,

Numerical Recipes in C, Cambridge University Press, 1988

[6] M.H MacDougall, “Simulating Computer Systems: Techniques and

Tools,” The MIT Press, 1987

0 0.5 1 1.5

0 5 10 15 20 25 30 35 40 45 50

Number of stations, N

anal - S_1 sim - S_1 anal - S_2 sim - S_2 anal - S_3 sim - S_3 anal - S_4 sim - S_4

Figure 3 Class i saturated throughput for rate separation IEEE 802.11b MAC

with airtime fairness

0 0.5 1 1.5 2

0 5 10 15 20 25 30 35 40 45 50

Number of stations, N

anal -T_DTP_1 anal -T_DTP_2 anal -T_DTP_3 anal -T_DTP_4

Figure 4 Data transmission period i for rate separation IEEE 802.11b MAC

with airtime fairness

0 0.02 0.04 0.06 0.08 0.1

0 5 10 15 20 25 30 35 40 45 50

Number of stations, N

S i

p i

anal -Normalized airtime per station of class 1 anal -Normalized airtime per station of class 2 anal -Normalized airtime per station of class 3 anal -Normalized airtime per station of class 4

Figure 5 Normalized airtime per station of class i for rate separation IEEE

802.11b MAC with airtime fairness