Capture Effect in IEEE 802.11 Basic Service Area under ... - CiteSeerX

CAPTURE EFFECT IN IEEE 802.11 BASIC SERVICE AREA UNDER INFLUENCE OF RAYLEIGH FADING AND NEAR/FAR EFFECT Zoran Hadzi-Velkov 1, Boris Spasenovski 1 1

Ss. Cyril and Methodius University, Faculty of Electrical Engineering, 1000 Skopje, Macedonia, (zoranhv & boriss)@etf.ukim.edu.mk

Abstract: In this paper, we analytically analyzed the influence of capture effect over theoretical throughput and delay of a traffic-saturated IEEE 802.11b BSS in Infrastructure and Ad-hoc configurations. The capture probabilities are derived in presence of near/far effect and/or Rayleigh fading, i.e. with or without power control mechanisms. The peak aggregate throughput of the BSS operating in Basic access mode is sensitive to the capture, while its impact in RTS/CTS access mode can be disregarded. The frame delay is also insensitive to the capture effect. Keywords: IEEE 802.11b, capture effect, power control, saturation, throughput, delay I. INTRODUCTION In classical analysis of random access protocols, it is assumed that all frames involved in collision are destroyed. This is somewhat unrealistic assumption in mobile radio environment because of the differences in power levels of the transmitted signals at the receiver introduced by the deterministic path attenuation, shadowing and multipath fading. Due to the capture effect, a frame with the strongest received signal strength can be correctly decoded at the receiver even in the presence of simultaneous transmission of multiple stations [1, 2]. Capture effect is feasible in an indoor environment consisting of low-power transmitting stations, e.g. a Wireless LAN [3, 4]. In this paper, we study the influence of capture effect over the IEEE 802.11b Distributed Coordination Function (DCF) [5] in presence of near/far effect and/or Rayleigh fading. Under such conditions, we analytically estimate capture probabilities, and their impact over the channel throughput and delay in a traffic-saturated IEEE 802.11b Basic Service Set (BSS). The capture effect is considered for the two operational modes of the single circular BSS: Infrastructure Mode (stations transmitting towards the Access Point, AP), and Ad-hoc Mode (stations communicating directly among each other). II. CAPTURE MODELS Our propagation model takes the deterministic power loss and multipath fast fading of signals into account. The pathloss exponent for indoor channels in picocells is typically taken equal to 4. For the purpose of analytical tractability, we used somewhat arbitrary assumption for absence of

0-7803-7589-0/02/$17.00 ©2002 IEEE

direct path between the transmitter i and the receiver (at mutual distance ri) within the BSS, which means the envelope of transmitted signal is Rayleigh-faded. Therefore, its instantaneous power is exponentially distributed according to:

f Pi ( p) =

1 p exp(− ), p0 i p0i

p>0 ,

(1)

where p0i represents the local-mean power of the transmitted frame at the receiver. The local-mean power itself is determined by p0i = A ⋅ ri −4 ⋅ PT , where A⋅ri-4 is the deterministic path-loss law, and PT is the transmitted signal power. Constants A and PT are assumed to be identical for all transmitted frames. During simultaneous transmission of multiple stations, a receiver captures a frame if the power of detected frame Pu sufficiently exceeds the joint power (incoherent addition) of n interfering contenders Pint = ∑n Pk by a certain k =1

threshold factor for the duration of a certain time period (over which instantaneous power is assumed to remain approximately constant). Thus, the capture probability is the probability of signal-to-interference ratio γ = Pu /Pint exceeding the product z0⋅g(Sf), where z0 is known as the capture ratio, and g(Sf) is the processing gain of the correlation receiver. Actually, the processing gain introduces a reduction of interference power by factor g(Sf), which is inversely proportional to the spreading factor Sf. We assume that a receiver determines weather a possible successful capture has occurred during the transmission of the preamble/header part of the frame, which is always transmitted using the DSSS modulation of a fixed 11-chip Barker spreading sequence (i.e. Sf = 11). Given rectangularshaped chips, g(Sf) can be expressed as in [6]: 2 . (2) g (S f ) = 3⋅ S f A. BSS with Power Control The presumption of power-controlled stations in Infrastructure Mode means equal local-mean powers of all transmitted frames at the receiver, whereas the rapidly varying Rayleigh fading remains uncompensated. Given n interfering frames, the conditional capture probability can be expressed as in [4]: Pcap ( zo g (S f ) | n) = P rob(γ > zo g ( S f ) | n) = [1 + zo g (S f )]− n (3)

PIMRC 2002

B. BSS without Power Control When the stations are not power-controlled, both the localmean powers and fast fading envelopes of received frames differ and represent independent random variables. When conditioned on local-mean powers p0u, p01, …,p0n of the single captured and the n interfering frames, respectively, the capture probability can be expressed as in [2]: Pcap ( zo ⋅ g ( S f ) | r0 , r1 ,..., rn ) = Pr ob(γ > zo ⋅ g ( S f ) | r0 , r1 ,..., rn ) ∞

∞

0

0

= ∫ dp1 f P1 ( p1 ) ⋅ ⋅ ⋅ ∫ dp n f Pn ( p n )

∞

∫f

( p )dpu

Pcap ( z 0 ⋅ g ( S f ) | n) = ∫ [ I (r0 )]n ⋅ h(r0 )dr0 ,

(5)

h(ri )dri

(6)

where .

r = [r12 + r22 − 2r1r2 ⋅ cos(θ1 + θ 2 )]1/ 2 .

r 1 + z 0 ⋅ g ( S f ) ⋅ ( i ) −4 r0

C. Infrastructure Mode For the Infrastructure configuration, we assumed a single circular BSS with radius normalized to unity with the AP located in its center. Assuming uniform spatial distribution of the stations around AP, PDF of the distance between a station and the AP is given by: (7) h(r ) = 2r , 0 < r ≤ 1 ,

(9)

Given r1 and r2 distributed according to (7), and θ1 and θ2 uniformly distributed between 0 and 2π, the PDF of the distance between two arbitrary stations A and B attains the form of the beta distribution:

h( r ) =

Pu u z 0 ⋅ g ( S f )⋅( p1 +⋅⋅⋅+ pn )

∞ ∞ n  p  = ∫ dp1 f P1 ( p1 ) ⋅ ⋅ ⋅ ∫ dpn f Pn ( pn ) ⋅ exp − z 0 ⋅ g ( S f )∑ i  i =1 p0 u   0 0 n n 1 1 , (4) =∏ =∏ p r i =1 i =1 1 + z 0 ⋅ g ( S f ) ⋅ 0i 1 + z 0 ⋅ g ( S f ) ⋅ ( i ) −4 p 0u r0 where fPu, fP1,…, fPn are the power PDFs of the useful signal and of each of the n interferers, respectively, under assumption of their mutual statistical independence. r0, r1, …, rn are the random distances from the transmitting stations to the observed receiver, distributed according to identical PDF, h(r). Since all factors in the product in (4) are statistically equal, we can average over their distance distributions to obtain the averaged conditional capture probability conditioned on n interferers as the following:

I (r0 ) = ∫

station A, which is randomly positioned at location (r1,θ1), and a receiving station B, which is randomly positioned at location (r2,θ2). Their mutual distance r can be expressed as

1 1 r r (1 − ) 3 / 2 , 0 < r ≤ 2 , (10) 2 B(2,2.5) 2 2

where B(a,b) is the Beta function. A r1 r

θ1

1

r2 θ2

B

Fig. 1. Spatial distribution of two arbitrary stations in a BSS Then, (6) can be solved in closed-form as follows: 2048 I (r0 ) = 1287r04 ⋅ z0 g ( S f )  3 7 9 17 19 21 23  16  , (11) × 5 F4  (1, , ,2, ); ( , , , );− 4  2 4 4 8 8 8 8  ⋅ ( ) r z g S 0 0 f   where 5F4(.) is the generalized hypergeometric function.

Substitution of (11) into (5) yields to averaged conditional capture probability Pcap(z0 ⋅g(Sf)| n) in Ad-hoc mode, which also can only be solved numerically. Given n, probabilities Pcap(z0 ⋅g(Sf)| n) in Infrastructure and Ad-hoc mode without power control come close to each other due to their averaging over random distances within the limits of a single cell.

so that (6) can be solved as follows:   1  . (8) I ( r0 ) = 1 − r02 z0 ⋅ g ( S f ) ArcTan  2  r0 z0 ⋅ g ( S f )    By substitution of (8) into (5), the averaged conditional capture probability Pcap(z0 ⋅g(Sf)| n) can only be solved by numerical integration.

D. Ad-hoc Mode In Ad-hoc mode, we assume that the randomly distributed stations also follow uniform spatial distribution with respect to the BSS center. Fig. 1 depicts a transmitting

III. SATURATION THROUGHPUT UNDER CAPTURE In order to estimate the influence of the capture effect over channel capacity of IEEE 802.11b DCF, we used some results from [7], where the peak (saturation) throughput Smax in ideal channel conditions is expressed as

S max =

Psuc Ptr ⋅ E[ L] . (12) (1 − Ptr )σ + Ptr Psuc Ts + Ptr (1 − Psuc )Tc

E[L] is the average frame payload size, although in order to establish upper performance limit, we assumed all generated packets are fixed and maximized so that E[L] = L = 2312 octets. Ptr is the probability of at least one transmission in the observed time slot, Psuc is the probability of a successful transmission assuming at least one station is transmitting, and σ is duration of an empty slot time. Ts is the average time the channel is sensed busy by each station because of a successful transmission, and Tc is the average time the channel is sensed busy during a collision. The values of Ts and Tc differ depending on the network access mode (Basic or RTS/CTS access mode) and additional network operating parameters (Table I): Basic:  Tsbas = PHY pre / hdr + MAC hdr + L + SIFS + ACK + DIFS  bas Tc = PHY pre / hdr + MAC hdr + L + DIFS

(slots). Until the m-th retransmission, the maximal backoff timer Wi increases by factor of 2, and then it is frozen to Wm, so that  W ⋅ 2 i , i = 0,..., m , (13) Wi =  m W ⋅ 2 , i ≥ (m + 1) where W is the initial contention window, and m is the highest backoff stage. Thus, if a station reaches the backoff stage i = m, the backoff timer is reinitialized to random value between 0 and Wm – 1 after each additional collision until successful transmission of the frame. The transmission probability τ of a station in a random slot depends on W, m, and the number of contending stations N. Given W = 8 and m = 5, we estimated probability τ in function of collision size N with our C simulator, as depicted in Figure 2. 0.25

Tsrts / cts = RTS + SIFS + CTS + SIFS + PHYpre / hdr RTS/CTS:  + MAChdr + L + SIFS + ACK + DIFS   T rts / cts = RTS + DIFS  c

0.2

τ

0.15

Table I. Relevant network parameters Parameter Channel Rate PHY Preamble PHY Header MAC header ACK RTS CTS SIFS DIFS Slot_Time σ Retry limit m Initial contention window W Tsbasic Tcbasic Tsrts/cts Tcrts/cts

Default 1 Mbps 144 symbols 48 symbols 34 octets 14 octets + PHYpre/hdr 20 octets + PHYpre/hdr 14 octets + PHYpre/hdr 20 µs 50 µs 20 µs 5 8 19334 bits 19010 bits 20030 bits 402 bits

Because the transmission queue of each station is always assumed to be non-empty (saturation condition), the station always executes the Collision Avoidance procedure (i.e. binary exponential backoff mechanism). Thus, (12) is obtained assuming the finite-state model of the station is represented by the Markov chain of the backoff window size. A current state of a station is determined by the current value of the backoff timer k ∈ (0, Wi − 1) while the station is in backoff stage i ∈ (0, m), which means that it suffered i previous unsuccessful transmission attempts. Starting with the very first transmission attempt (backoff stage i = 0), the initial value of the backoff timer is uniformly chosen in the range between 0 and W0 – 1. After the i-th unsuccessful transmission attempt, the station enters backoff stage i, where its backoff timer is reinitialized to a random value between 0 and Wi –1

0.1

0.05

0 0

5

10 15 20 collision size N

25

30

Fig. 2. Transmission probability τ of each station in a random slot time estimated by simulation Given N stations contending for the channel, the probabilities Ptr and Psuc can be expressed through the probability τ of station transmitting in a randomly chosen slot time, i.e. (14) Ptr = 1 − (1 − τ ) N , and Psuc =

Nτ (1 − τ ) N −1 + Pcap ( z0 , N ) . Ptr

(15)

Eq. (15) indicates that, given at least one station is transmitting, probability of successful transmission Psuc is formed by adding the capture probability Pcap to the probability of transmission of exactly one station. It is important to emphasize that only the possibility of a single frame capture is considered. Probability τ also impacts the probability of frame capture Pcap as the following: Pcap ( z0 , N ) =

N −1

∑R ⋅ P i

cap

( zo ⋅ g ( S f ) | i ) ,

(16)

i =1

where Ri is the probability of i interfering frames being generated in the observed time slot, according to  N  i +1 τ (1 − τ ) N −i −1 . Ri =  1 i +  

(17)

Now the capture probability Pcap can be estimated according to (16) using probabilities τ from Fig. 2. 0.25

0.2

Zo = Zo = Zo = Zo = Zo = Zo =

6dB, PC 24dB, PC 6dB, uniform 24dB, uniform 6dB, beta 24dB, beta

5

10

Pcap (N )

0.15

0.1

0.05

0 0

15

Collision size N

20

25

30

Fig. 3. Probability Pcap increases with N Fig. 3 depicts Pcap vs. N contenders (the single useful plus n interfering frames, N = 1 + n), parameterized over two values of z0 (6dB and 24dB) under three separate scenarios (PC – with power control in Infrastructure mode, uniform – without power control in Infrastructure mode, and beta – without power control in ad-hoc mode). 1

The theoretical saturation throughput in Basic and RTS/CTS “handshake” access modes without power control in Infrastructure and Ad-hoc configurations vs. N is displayed in Fig. 4. The graphs refer to 1 Mbps WLAN (Table I), while corresponding system parameters must be used according to IEEE 802.11b standard for rates of 2, 5.5, and 11 Mbps. If Basic access scheme is utilized, it is obvious that presence of capture effect generates significant throughput improvement as z0 decreases. For example, given z0 = 6dB and N = 10 stations without power control, the peak theoretical throughput is estimated to 85.6% in Infrastructure mode and 84.6% in Ad-hoc mode (Fig. 4a), as opposed to 67 % in the absence of capture (zo→ ∞). Conversely, the use of the RTS/CTS “handshake” access scheme contributes significantly to the robustness of the IEEE 802.11b network under capture (Fig. 4b). IV. SATURATION DELAY UNDER CAPTURE Now let us concentrate on a single station to determine the average frame delay before its successful transmission Td under saturation. The observed station suffers an unsuccessful (re)transmission attempt with probability p due to simultaneous transmission of at least one of the N – 1 remaining stations, i.e.

p = 1 − (1 − τ ) N −1 .

0.9

While the observed station remains in the i-th stage (i.e. its backoff timer has not yet expired), the probability of transmission of at least one of the N – 1 remaining stations is p, and the probability of exactly one transmission from one of the N – 1 remaining stations (given at least one of them is transmitting) is:

S max (basic)

0.8

0.7 Zo = 6dB, PC Zo = 24dB, PC Zo = 6dB, uniform Zo = 24dB, uniform Zo = 6dB, beta Zo = 24dB, beta Zo -> Inf

0.6

0.5

p1s =

0.4 0

5

10

15

20

Collision size N

25

30

(a) 0.925

Smax (RTS/CTS)

0.92

0.915 Zo = 6dB, PC Zo = 24dB, PC Zo = Zo = Zo = Zo =

0.91

6dB, uniform 24dB, uniform 6dB, beta 24dB, beta

0

5

10

( N − 1) ⋅ τ ⋅ (1 − τ ) N − 2 + Pcap ( z0 , N − 1) . p

15

20

Collision size N

25

30

(b)

Fig. 4. Theoretical saturation throughput for both access modes: (a) Basic access; (b) RTS/CTS access

(19)

In each backoff stage i ∈ (0, m), the initial value of the backoff timer has mean of Wi / 2, so that the average time before each retransmission attempt is Wi / 2 slots. The backoff timer is decremented by 1 in each consecutive slot. W.t.r. to the observed station, these slots can be: idle slots (σ = 20µs), successful transmission slots from one of the N – 1 remaining stations, or collision slots from 2 or more simultaneous transmissions from 2 or more of the N – 1 remaining stations. The average number of consecutive idle slots nidle between two consecutive transmissions (both successful or collided) of the N – 1 remaining stations can be calculated as follows: ∞

Zo -> Inf

0.905

(18)

nidle = ∑ i(1 − p) i p = i =0

1 −1 . p

(20)

A single renewal cycle between two consecutive transmissions of the N – 1 remaining stations includes multiple consecutive idle slots and a transmission slot, i.e. nidle + 1 = 1/p slots. Since each transmission slot can be

successful transmission or collision, the average duration of a renewal cycle Trc is: (21) Trc = nidleσ + p1s Ts + (1 − p1s )Tc . Since each retransmission attempt of the observed station in backoff stage i is preceded by Wi/(2⋅nrc) = (Wi⋅p)/2 renewal cycles of the N – 1 remaining stations, the average time between two consecutive retransmissions of the observed station is (Wi ⋅p⋅Trc)/2. The average elapsed time Ttct,i before the observed station makes (i+1)-th retransmission attempt, can be calculated as: i W ⋅ p  pT i Ttct , i = ∑  k Trc + Tc  = rc ∑Wk + iTc . (22) 2 2 k =0  k =0  After substitution of (13) into (22), we have:

Ttct ,i = iTc +

 2 i +1 − 1 pTrc , i = 0,..., m . (23) ⋅ W ⋅  m +1 m 2 2 1 2 ( i m ), i ≥ (m + 1) − + − 

Now, the average delay of the first frame in the queue is:

Td =

m+ f

∑ (1 − p) ⋅ p

i

⋅ (Ttct ,i + Ts ) .

(24)

i =0

After introducing (23), (24) can be solved in closed form:

Wp 1 − p − p(2 p) m . p + Trc Td = Ts + Tc 1− p 2 (1 − p)(1 − 2 p)

(25)

In this delay analysis, the capture ratio z0 appears in (19), and its influence over the delay is neglectful. The average delay vs. N for both Basic and RTS/CTS access modes is depicted in Fig. 5. 1200 Basic access RTS/CTS access

1000

Delay (ms)

800

600 400

200

0 0

5

10

15 20 Collision size N

25

30

Fig. 5. The influence of capture over theoretical saturation delay is neglectful V. CONCLUSIONS The capture effect has significant impact over saturation throughput of an IEEE 802.11b BSS operating in Basic access mode only.

In RTS/CTS access mode, the exchange of RTS and CTS frames before actual transmission significantly reduces the likeliness of simultaneous transmissions and frame capture. Weather an ideal power control mechanism would produce higher throughput increase depends on the capture ratio: z0 < 10dB. The actual value of capture ratio depends on the receiver design. The proximity of the results for capture probabilities and throughput in Infrastructure and Ad-hoc modes without power control can be explained by: (1) averaging of the probabilities w.t.r. to distance PDFs within the limits of a single cell, and (2) the fact that we considered the possibility of capture of a single frame only. The latter assumption is certainly correct in Infrastructure mode, where all stations are trying to access a single AP. However, when stations communicate directly among each other in ad-hoc mode, simultaneous capture of multiple frames among different transmitter-receiver pairs is also feasible. This issue will be addressed elsewhere. REFERENCES [1] C.T.Lau and C.Leung, Capture Models for Mobile Packet Radio Networks, IEEE Trans. on Commun., Vol. COM-40, pp. 917-925, May 1992. [2] M. Zorzi and R.R. Rao, Capture and Retransmission Control in Mobile Radio, IEEE J. Select Areas Commun., Vol. 12, No. 8, pp.1289 – 1298, Oct. 1994. [3] A. Zahedi and K. Pahlavan, Natural Hidden Terminal and the Performance of the Wireless LANs, Proc. IEEE 6th Int. Conf. on Univ. Pers. Comm., pp. 929933, 1997. [4] Z. Hadzi-Velkov and B. Spasenovski, Capture Effect in IEEE 802.11 Wireless LANs, Proc. IEEE ICWLHN 2001, pp. 164-173, Singapore, December 5-7, 2001 [5] IEEE Standard for Wireless LAN Medium Access Control and Physical Layer Specifications, IEEE 802.11b, Nov. 1999. [6] J. Perez-Romero, L.G. Alonso and R. Agusti, Average Block Error Probability in reverse Link of a Packet DS/CDMA System under Rayleigh Fading Channel Conditions, IEEE Comm. Letters, Vol. 4, No. 4, pp. 116-118, April 2000. [7] J. Bianchi, Performance Analysis of IEEE 802.11 Distributed Coordination Function, IEEE J. Select. Areas Commun., Vol. 18, No. 3, pp. 535–547, March 2000