Exact Distribution of Access Delay in IEEE 802.11 DCF MAC

Exact Distribution of Access Delay in IEEE 802.11 DCF MAC Teerawat Issariyakul, Dusit Niyato, Ekram Hossain, and Attahiru Sule Alfa University of Manitoba and TRLabs Winnipeg, MB, Canada. Email: {teerawat, tao, ekram, alfa}@ee.umanitoba.ca.

Abstract— This paper presents an analytical framework to calculate the probability mass function (pmf) of channel access delay in IEEE 802.11 Distributed Coordination Function (DCF) Medium Access Control (MAC) mechanism. The access delay is defined as the time between a station chooses a new backoff value and the time it is able to access the channel for data packet transmission. Using a Markov process, the access delay is modeled as having phase-type distribution. Since the back-off is frozen when the channel is sensed busy, the access delay distribution is observed to be composed of noncontinuous clusters. The envelope of the pmf as well as the envelope of each cluster resemble hyper-exponential distribution. While the proposed model is flexible enough to accommodate any distribution of MAC data frame length, the numerical results presented in this paper are for the fixed-length data frames. The model would be useful in many aspects such as queueing analysis and/or designing energy-efficient MAC protocols compatible with the IEEE 802.11 DCF standard. Index Terms— IEEE 802.11 DCF, channel access delay, discrete time Markov chain, phase-type distribution.

I. I NTRODUCTION During the past few years, wireless local area networks (WLANs) have become very popular in a broad range of applications. Today, WLANs are deployed in many places like universities, corporate offices, airports, or even residential houses. Most of these WLANs are based on the IEEE 802.11 standard [1], first launched in 1997. Since then, the standard has been thoroughly investigated in two major aspects. The first aspect aims at improving the performance of IEEE 802.11, while the second is to analyze the performance of IEEE 802.11. In this paper, we limit our study to analyzing the medium access control (MAC) performance of the IEEE 802.11. IEEE 802.11 specifies Point Coordination Function (PCF) and Distributed Coordination Function (DCF) as its MAC protocols. Due to its distributed nature, DCF is more challenging to analyze and optimize. One of the earliest analytical works on modeling and analysis of DCF was presented in [2] assuming continuously backlogged data flows (i.e., under saturation condition) and errorfree wireless channel. Performance of DCF MAC under saturation condition in presence of random errors due to wireless channel was analyzed in [3] and [4] considering limited persistence automatic repeat request (ARQ)-based error recovery protocol. However, all these works derived only long-term average throughput and delay assuming that the back-off value is not frozen This work was supported in part by the Telecommunications Research Labs (TRLabs), Winnipeg, Canada, in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada, and in part by University of Manitoba Graduate Fellowship (UMGF).

when the channel is busy, which may not be the case for 802.11 DCF. By overcoming this limitation, the performance of DCF MAC under non-saturation condition was analyzed in [5] and [6] based on a M/G/1/K and a G/G/1 queueing model, respectively. Utilizing Z-transform, [5] and [6] developed a methodology to calculate the probability generating function (PGF) of average MAC-layer access delay1 for uniformly distributed data frame length. However, to obtain the probability mass function (pmf) from the PGF, it requires to perform inversion of Z-transforms which might be computationally very expensive. This paper resorts to a matrix-based approach, which avoids the use of complicated transformations and directly calculates the exact pmf of MAC-layer channel access delay. Unlike [5] and [6], we propose an efficient approach to iteratively determine the pmf of the access delay time. While [5] and [6] obtained the expected access delay by differentiating the calculated PGF, we derive a closed-form solution (which is computationally much more efficient) for this metric. Also, rather than assuming a uniform distribution for packet lengths, we generalize our model to support any packet length distribution. Moreover, the pmf for access delay can be obtained for any given back-off window size, which is significant in that this information can be used for buffer management/packet dropping at a mobile station. Classified as phase-type (P H) distribution, the derived results can later be extended for queueing analysis of IEEE 802.11 DCF, for example, by using G/PH/1 or BMAP/PH/1 models. The organization of the rest of the paper is as follows. Section II provides the preliminaries on the IEEE 802.11 DCF and the P H-distribution. In Section III, we present a Markov model for analyzing channel access delay for the 802.11 DCF MAC in which the access delay is modeled to have a PH distribution. Section IV discusses the numerical and the simulation results, and some useful implications of the obtained results. Finally, conclusions are stated in Section V. II. BACKGROUND A. IEEE 802.11 DCF MAC IEEE 802.11 DCF consists of two main mechanisms: contention window adjustment and back-off mechanisms [1]. Once a station (STA) is turned on, it sets the contention window to the minimum value (CWmin ). The contention window is doubled for every transmission failure until it reaches the maximum 1 Average MAC-layer access delay is the access delay averaged over all possible back-off values and contention windows.

value (CWmax ). If the transmission is successful, the contention window will be reset to CWmin . The back-off mechanism in IEEE 802.11 DCF MAC is illustrated in Fig. 1. After window adjustment, an STA waits for Distributed InterFrame Space (DIFS) period of time and calculates a back-off value which is a random variable uniformly distributed between 0 and CW (CW is the value of the current contention window). After this point, the back-off value is decreased by one for every idle2 time slot. When the channel becomes busy, the back-off is frozen until the channel is idle for DIFS period of time. At the end of DIFS, the back-off value is still the same as what it was before the channel had been busy [5],[7]. After that, the STA starts decreasing the back-off value by one for each subsequent time slot. When the back-off value reaches zero, the STA can transmit data in the next time slot.

of the corresponding transition probability matrix (TPM) P3 can be written as in (1) below   1 0 P= (1) ω Ω where the matrices ω and Ω are called absorbing and transient TPMs, respectively. Throughout this paper, we denote all-zero, all-one, and identity matrices by 0, e, and I, respectively. The pmf that the DTMC is absorbed at step d (fD (d)) and the expected time to absorption (E[D]) can be calculated from (2)-(3) [9]  α0 , d=0 fD (d) = (2) αΩd−1 ω, d > 1 E[D]

= α(I − Ω)−1 e

(3)

where α0 and α represent the probability that the DTMC starts at the absorbing and the transient states, respectively.

MAC-Layer Delay Channel Busy (L) DIFS Data Transmission Back-off = 6 DIFS = L=

Fig. 1.

5

4

3

2

2

8

7

6

5

4

3

2

1

0

3

2

1

0

1

0

A. System Model and Assumptions

Back-off mechanism in IEEE 802.11 DCF.

As an option, the IEEE 802.11 DCF MAC can use a handshake mechanism to reduce the impact of hidden nodes. As in Fig. 2, an STA intending to transmit data must first transmit a Ready To Send (RTS) packet. Upon receiving an RTS packet, the receiving node transmits a Clear To Send (CTS) packet back to the sender. Then, the sender can start sending a DATA packet. Finally, the receiver informs the sender of successful reception by sending back an ACKnowledgment (ACK) packet. Except for the RTS, each STA has to sense the channel idle for Short InterFrame Space (SIFS) period of time before sending any packet. Since SIFS is shorter than DIFS, only the RTS packet will be vulnerable to collision if all STAs are in the same area. DIFS

DATA

Transmitter

SIFS

SIFS CTS

ACK

Receiver

NAV(DATA) NAV(CTS) NAV(RTS) Other nodes

Fig. 2.

Handshake mechanism in IEEE 802.11 DCF.

B. Absorbing Markov Chain and P H Distribution An absorbing Markov chain is a Markov process which finally stops at an absorbing state [8]. Consider a discrete time Markov chain (DTMC) with the first state and all the subsequent states being absorbing and transient states, respectively. A general form 2 The

Consider an 802.11 WLAN with N mobile stations (STAs). The STAs are assumed to be in the range of each other and each STA implements the IEEE 802.11 DCF with the same parameter settings. We assume that unsuccessful data transmission is caused only by data collision (i.e., when more than one STA transmits simultaneously). In presence of the handshake mechanism, collisions are associated only with the RTS packets when the back-off values of more than one backlogged STAs expire in the same time slot. If an RTS is successfully transmitted, other STAs will refrain from transmission and no collision will occur. We model the system with variable-size data frames, and also simplify the results for a special case with fixed size data frames. B. Mathematical Model

SIFS RTS

III. M ODELING AND A NALYSIS OF C HANNEL ACCESS D ELAY U NDER IEEE 802.11 DCF MAC

channel is considered to be idle when there is no on-going transmission.

We define MAC-layer access delay as the time interval from the point where the contention window is adjusted to the point where the back-off value becomes zero (Fig. 1). The access delay is therefore conditioned on the current contention window. We also define unconditional MAC-layer access delay as an interval from the point where the contention window is set to CWmin to the point where the transmission is successful. In the following, we model the access delay in terms of transmission probability (ptx ) which is defined as the probability that an STA will have zero back-off value and have data to transmit. Under saturation condition, ptx can be calculated by using the method in [2]. Under non-saturation condition, ptx can be approximated by the model in [5]. We divide the analysis into two parts. The first part derives the pmf of channel access delay conditioned on a particular value of back-off timer. The second part utilizes the results from the first part and calculates the unconditional pmf of the channel access delay. 3 Throughout this paper, we use regular and boldface letters to represent scalar values and matrices, respectively.

1) Channel Access Delay Conditioned on a Back-off Value: Consider a two-dimensional absorbing DTMC (Bn , Tn ) defined at time slot n. The first variable, B ∈ {0, 1, · · · , B} is the backoff value. The second variable, Tn ∈ {1, · · · , L} is the length of a busy period due either to data transmission or collision. The DTMC starts (initial state) from the point where B = B and finishes when B = 0 (in an absorbing state). Based on this formulation, the corresponding TPM, P can be formulated as in (1), where

LHDR and is zero otherwise. The solutions for fD|B (d|b) and E[D|B] in this case can be calculated with reduced complexity (of O(L2 ) and O(1), respectively, compared to O(L3 )). Using the sparse structure of α and ω, we can expand (10) and (11) as follows: fD|B (d|b) = (1 − pd − pc ) · [Ωd−1 ]L(B−1)+1,1 E[D|B]

=

B·L X

[(I − Ω)−1 ]L(B−1)+1,j .

(12) (13)

j=1



   (ω|Ω) =   

[Q]i,j

[R]i,j p dj

Re Q 0 R .. .. . .

0 Q .. .



··· 0 ··· .. .. . . R Q R

     0  Q

 pc , i = 1, j = Lc    pdj , i = 1, j = {Ldmin , · · · , Ldmax } = 1, i = {2, · · · , L}, j = i − 1    0, otherwise  1 − pc − p0d , i = 1, j = 1 = 0, otherwise X = pd · fLdata (j − LHDR ), p0d = p dj .

(4)

(5)

(6) (7)

∀j

In the above equations, [A]c,d represents the entry in row c column d of matrix A, Ld = LDAT A + LHDR , LHDR = LRT S + LCT S + LACK + 3LSIF S + LDIF S , and Lc = LRT S + LEIF S . Here, LRT S , LCT S , LDAT A , LACK , LDIF S , LSIF S , and LEIF S (Extended InterFrame Space) are the lengths of the corresponding packets and fLdata (l), l ∈ {Ldmin − LHDR , · · · , Ldmax − LHDR } is the probability that the length of DAT A is l. Matrices R and Q with size L = max{Lc, Ldmax } represent the probability that the back-off counter will and will not decrease in the next step (or time slot). Embedded in R (the back-off decreases by one) is the probability (1−pc −p0d ) that the channel will be idle. With probabilities pc and pdl , the process moves to column Lc and l of Q (the back-off remains the same), where pd pc

= (N − 1) · ptx (1 − ptx )N −2 = 1 − (1 − ptx )

N −1

− pd .

(8) (9)

After Q and R matrices are formulated, we use (2) and (3) to calculate pmf (fD|B (d|b)) and the expectated channel access delay (E[D|B]). Mathematically,

The model complexity can be further reduced by considering the special structures of Q and R. Denoting row i of an arbitrary matrix A by Ai , we simplify (14) to  p c · AL c + p d · AL d , i = 1 (16) (QA)i = Ai+1 , i = {2, · · · , L}  (1 − pc − pd ) · A1 , i = 1 (RA)i = (17) 0, i = {2, · · · , L}. This special structure also leads to the closed-form solution for E[D|B] as in (18) E[D|B] =

B(1 + pc (Lc − 1) + pd (Ld − 1)) = B · tB 1 − p c − pd

(18)

where tB is the time required for the back off value to decrease by one unit. The proof of (18) is given in the Appendix. Note that, tB can be obatined by multiplying the average busy period (1 + pc (Lc − 1) + pd (Ld − 1)) with 1/(1 − pc − pd ), the average number of busy periods. The access delay for back-off B is just the time required for the back-off value to decrease by B units. 2) Unconditional Channel Access Delay: Using the total probability theorem, we derive the pmf for the unconditional channel access delay (fD (d)) as follows: fD (d) =

M X

fD|W (d|wi ) · πwi

(19)

i=1

fD|B (d|b) = αΩd−1 ω E[D|B] = α(I − Ω)

Since Ω is a bi-diagonal matrix, [Ωk ]i,j in (12) can be calculated by using the following heuristics:  i=j  Q · [Ωk ]i,j , k+1 [Ω ]i,j = R · [Ωk ]i−1,j + Q · [Ωk ]i,j , i < j (14)  0, i>j   Q, i = j  1 R, i = j + 1 Ω i,j = (15)  0, otherwise.

(10) −1

e

(11)

where α is an initial probability row vector, whose entry L(B − 1) + 1 is one and all other entries are zero. From (10) and (11), the complexity of calculating fD|B (d|b) and E[D|B] can be determined based on the complexity of computing Ωd−1 and the matrix inversion (I − Q)−1 . Since the size of Ω is fairly large, calculation of (10) and (11) could require substantial amount of memory and computational effort. Hereafter, we simplify the solution for a special case with fixedsize data length, where Lmax = Ld and fL (l) is 1 for l = Ld −

where fD|W (d|w) is the pmf of access delay conditioned on contention window w, πwi = [πw ]1,i is the probability that the contention window is wi , and M is the number of back-off windows. We calculate πw by using a DTMC Wn representing the contention window of an STA at the moment that it is adjusted. Based on the window adjustment mechanism discussed in section II-A, we formulate the TPM (W) with size M for this DTMC as in (20). By using the limiting property of the DTMC, namely, πw = πw · W, we obtain πw in (21). Then, we calculate fD|W (d|w) from (22), by using Bayes’s theorem.

TABLE I D EFAULT PARAMETER SETTING FOR IEEE 802.11 DCF

 1 − p c − pd ,    pc + p d , W=    0, πw i

i = {1, · · · , M }, j = 1 i = j − 1, j = {2, · · · , M } or i = M, j = M otherwise.

  (1 − pc − pd ) · (pc + pd )i−1 , i = {1, · · · , M − 1} =  (pc + pd )i−1 , i = M.

fD|W (d|w)

w−1 X

=

Parameter Channel Rate Slot Time LSIF S LDIF S Physical Header MAC Header LRT S LCT S LDAT A LACK Range of Contention Window Number of STAs

(20)

(21)

P r{B = b|W = w} · fD|B (d|b)

Value 2 M bps 20 µs 10 µs 50 µs 192 bits 272 bits 160 bits 112 bits 8000 bits 112 bits [32, 1024] 5

b=0

(22)

b=0

In (22), P r{B = b|W = w} = 1/w is the probability that an STA selects its back-off value b when the current contention window is w, and fD|B (d|b) is calculated by using (12). Finally, we calculate the pmf for the unconditional access delay by using (21) and (22) in (19). Similarly, we calculate the expected MAC-layer access delay conditioned on contention window w (E[D|w]) and unconditional expected MAC-layer access delay (E[D]) as follows: E[D|w]

=

w−1 X

=

M X wi − 1

b=0

E[D]

1 w−1 · b · tB = · tB w 2

i=1

2

· t B · πw i

W −1 = · tB . 2

(23)

AND

0.02 Model

(24)

S IMULATION R ESULTS

A. Parameter Settings and Model Validation Unless otherwise specified, we assume the parameter setting as in Table I [1]. Each STA operates under saturation condition and we calculate the value of ptx using the method in [2]. The results for the non-saturation case can be obtained by calculating ptx following [5]. We derive the pmf (fD (d)) of the MAC-layer access delay for a tagged STA in presence of other N − 1 STAs. Based on fixed DATA frame length specified in Table I, Lc = 27 time slots and Ld = 240 time slots. To validate our model, we run simulations with the same settings as those used for obtaining the numerical results from the model. All the simulation results are observed to be fairly close to those obtained from (19). However, we only show simulation results in Fig. 3 for the sake of better legibility of the figures. B. Numerical Results Fig. 3 reveals that the channel access time distribution is noncontinuous and consists of several clusters. Due to fixed length Ld , data transmission by one of the other STAs increases the access delay by Ld . The nth (n > 0) cluster in the pmf is contributed mostly by n data transmissions (by other STAs)

Simulation

0.015

0.01

0.005

0 0

Similar to E[D|B], E[D|w] is the product of average backoff decreasing time and the average back-off value. In the unconditional case, E[D] is obtained PMby replacing w with the average contention window W (= i=1 wi · πwi ). IV. N UMERICAL

before the tagged STA acquires a channel access. Therefore, these clusters are located at approximately Ld time slots apart and each cluster peak decreases at the rate of pd .

Probability mass function

w−1 1 X · fD|B (d|b). w

=

5

15

25

35

MAC−layer service time (ms)

Fig. 3. Probability mass function of channel access delay (for the parameters as shown in Table I).

Within cluster n, the first non-zero probability is due to the tagged STA selecting back-off value of 1 and the channel becomes idle in the slot after n consecutive data transmissions. Higher probability of channel access delay in cluster n results from increasing idle back-off slots. Each idle slot occurs with probability 1 − pc − pd . Therefore, the probability of increasing access delay decreases
geometrically with rate 1 − pc − pd . However, there are n+i combinations which correspond to n data n transmissions and i idle slots, resulting in the same access delay nLd +i. With respect to the number of idle slots, the probability of higher access delay increases due to the combination coefficient and decreases due to the geometric decreasing rate. Not until reaching the Lth c entry in cluster n, will the pmf be contributed by another possibility: n data transmissions and one collision. Again, the pmf after this point increases due to the combination coefficient, and decreases because of 1 − pc − pd . This phenomenon repeats every Lc time slots but the intensity is scaled down due to more number of collisions. Since Lc is comparatively small, the tail of a sub-cluster contributed by data collision spreads out to neighboring sub-clusters, and the tails within each cluster do not fluctuate considerably. Fig. 4 plots the pmf with similar settings but the number of STAs is now changed to 15. In this case, pd increases from 16.52% to 28.70%, and pc increases from 1.29% to 6.75%. Since 1 − pc − pd is smaller, each peak decays more quickly. Now, each spike does not spread to adjacent spike as much as

it does in Fig. 3. We can observe more clearly that each subcluster contributed by data collision decreases exponentially and occurs at Lc time slots apart. Also, since the height of the first spike in each cluster decreases geometrically, the cluster peak shifts to the right as the cluster number (n) increases. Now, we can observe that the envelope of each cluster resembles hyperexponential distribution, whose mean shifts to the right as n increases. Similarly, each spike in a particular cluster can also be modeled by the same distribution with right-shifted mean for increasing access delay. Probability mass function

L EMMA

A PPENDIX 1: For Ω defined in (4),  i=j  T, TRT, i > j [(I − Ω)−1 ]i,j =  0, otherwise

(25)

where T = (I − Q)−1 and R is defined in (6).

−3

6

such as the probability of a slot being idle, data length and collision length. The pmf could be very useful in the design of more efficient MAC protocols as well as for queueing analysis in an IEEE 802.11 network.

x 10

 −1

Proof: We prove (25) by induction. Suppose that T = (I−Ω) and that (25) is true when the size of Ω is M . From the relation (I − Ω)T = I,

4

(I − Q)[T]M +1,j − RTRT

2

0 0

5

15

25

35

MAC−layer service time (ms)

= 0, j = {1, · · · , M − 1}

(I − Q)[T]M +1,M − RT = 0 (I − Q)[T]M +1,M +1 = I.

(26)

th

Fig. 4.

Probability mass function of channel access delay for 15 STAs.

Probability mass function

Although each spike dies out more quickly, since pc is larger in this case, there are more number of discernible spikes in each cluster (as compared to those in Fig. 3). This is especially true for large n, where there are lesser number of idle slots. Compared to Fig. 3, there are more number of non-negligible clusters, each of which occupies more space in time. 0.01 0.008 0.006

Now let ti be the i row of T. From (I−Q)T = I, t1 −pc tLc − pd tLd = eT1 and ti − ti−1 = eTi , i = {2, · · · , L}, where eTi is a row vector whose entry i is one and all other entries are zero. Accordingly,  1   1−pc −pd , i = 1 pc +pd t1 = (27) 1−pc −pd , i = {2, · · · , Lc }  pd  , i = {L + 1, · · · , L } c d 1−pc −pd

and the ith row of RT is (1 − pc − pd )t1 for i = 1 and is 0 otherwise. Since [RT]1,1 = 1 and [RT]i,j = 0, j > 1, (RT)i = RT, i ∈ I+ and the first row of TRT is t1 . By applying this relation to (26), Lemma 1 is proven.  By applying Lemma 1 to (13),

0.004

E[D|B]

0.002 0 0

5

10

Probability mass function of access delay when data length is 400 bits.

Next, we set DATA length to 400 bits (correspondingly Ld = 50 slots) and plot the pmf in Fig. 5. In this case, the clusters are more closely located. The sub-clusters within a cluster spill over other sub-clusters, both in the same and different clusters. Due to this overlapping, each cluster looks smoother than each of those in Fig. 3. However, since pc and pd are the same as those in Fig. 3, there are approximately 10 significant clusters in Fig. 5 (as in Fig. 3). V. C ONCLUSIONS We have modeled the channel access delay for IEEE 802.11 DCF MAC as having phase-type distribution. We have used the special structures of the transition probability matrices to reduce the computational complexity and the memory requirements to an acceptable level. We have observed that the access delay distribution in the system with fixed data length is non-continuous. The envelope of the distribution resembles a hyper-exponential distribution. Therefore, it could be reconstructed by using only few parameters

L X

[(B − 1) · TRT + T]1,j = B ·

j=1

15

MAC−layer service time (ms)

Fig. 5.

= =

L X

t1

j=1

B(1 + pc (Lc − 1) + pd (Ld − 1)) 1 − p c − pd

(28)

which proves (18). R EFERENCES [1] IEEE 802.11 - Wireless LAN medium access control (MAC) and physical layer (PHY) specifications, IEEE inc. Std., 1999. [2] G. Bianchi, “Performance analysis of the IEEE 802.11 distributed coordination function,” IEEE Journal on Selected Areas in Commun., vol. 18, no. 3, pp. 535–547, Mar. 2000. [3] Z. Hadzi-Velkov and B. Spasenovski, “Saturation throughput - delay analysis of IEEE 802.11 DCF in fading channel,” in Proc. of IEEE ICC’04, June 2004. [4] P. Chatzimisios, A. C. Boucouvalas, and V. Vitsas, “Performance analysis of IEEE 802.11 DCF in presence of transmission errors,” in Proc. of IEEE ICC’04, June 2004. [5] H. Zhai, Y. Kwon, and Y. Fang, “Performance analysis of IEEE 802.11 MAC protocols in wireless LANs,” Wireless Commun. and Mobile Computing, vol. 4, no. 8, pp. 917–931, Dec. 2004. [6] O. Tickoo and B. Sikdar, “Queueing analysis and delay mitigation in IEEE 802.11 random access mac based wireless networks,” in Proc. of IEEE INFOCOM’04, Mar. 2004. [7] C. H. Foh and J. W. Tantra, “Comments on IEEE 802.11 saturation throughput analysis with freezing backoff counters,” IEEE Communications Letters, vol. 9, no. 2, pp. 130–132, Jan. 2005. [8] K. S. Trivedi, Probability and Statistics with Reliability, Queuing and Computer Science Applications. New York: Wiley & Sons, Inc., 2002. [9] M. F. Neuts, Matrix-Geometric Solutions in Stochastic Models. The John Hopkins University Press, 1981.