0
On the Validity of IEEE 802.11 MAC Modeling
Hypotheses
K. D. Huang, K. R. Duffy and D. Malone
Hamilton Institute, National University of Ireland, Maynooth, Ireland.
Abstract— We identify common hypotheses on which a large
number of distinct mathematical models of WLANs employing
IEEE 802.11 are founded. Using data from an experimental
test bed and packet-level ns-2 simulations, we investigate the
veracity of these hypotheses. We demonstrate that several of these
assumptions are inaccurate and/or inappropriate.
We consider hypotheses used in the modeling of: saturated
and unsaturated 802.11 infrastructure mode networks; saturated
802.11e networks; and saturated and unsaturated 802.11s mesh
networks. In infrastructure mode networks we find that, even
for small numbers of stations, common hypotheses hold true for
saturated stations and also for unsaturated stations with small
buffers. However, despite their widespread adoption, common
assumptions used to incorporate station buffers are erroneous.
This raises questions about the predictive power of all models
based on these hypotheses.
For saturated 802.11e models that treat differences in AIFS,
we find that one of the two fundamental hypotheses is accurate.
The other is reasonable for small differences in AIFS, but
unreasonable for large differences. For 802.11s mesh networks,
we find that assumptions are appropriate only if stations are
lightly loaded and are highly inappropriate if they are saturated.
By identifying these flawed suppositions, this work identifies
areas where mathematical models need to be revisited and revised
if they are to be used with confidence by protocol designers and
WLAN network planners.
I. INTRODUCTION 1
Since its introduction in 1997, IEEE 802.11 has become
the de facto WLAN standard. Its widespread deployment has
led to considerable research effort to gain understanding of
its Carrier Sensing Multiple Access / Collision Avoidance
(CSMA/CA) algorithm. This includes the use of simulation
tools, experiments with hardware, and the building of mathematical
models. In particular, analytical models have developed
significantly in recent years and, due to the speed with
which they can make predictions, they have been proposed as
powerful tools to aid protocol designers and WLAN network
planners.
Despite the differences in the details of published models,
most of them share common hypotheses. In this article we
identify these common assumption and investigate their validity.
This is an important task as authors do not typically check
the validity of these assumptions directly, but infer them from
the accuracy of model predictions of coarse grained quantities
1 A preliminary report on this work appeared in the Proceedings of IEEE
PIMRC 2008 [1]. The hypotheses of 802.11e, 802.11s networks, and p-
persistent protocols were not identified and investigated in that article and
simulation results were used exclusively rather than experimental measurements.
such as long run throughput or average MAC delay. If these
models are to be used with confidence for the prediction of
quantities beyond those validated within published articles, it
is necessary that their fundamental hypotheses be sound.
In the present article, we identify the following assumptions
that are adopted by numerous authors, often implicitly. For a
single station, define Ck := 1 if the kth transmission attempt
results in a collision and Ck := 0 if it results in a success. For
a station in an 802.11 network employing DCF, irrespective of
whether it is saturated (always having packets to send) or not,
many authors (e.g. [2][3][4][5][6][7][8][9][10]) assume that:
(A1) The sequence of outcomes of attempted transmissions,
{Ck}, forms a stochastically independent sequence.
(A2) The sequence {Ck} consists of identically distributed
random variables that, in particular, do not depend on
past collision history.
For models where stations have non-zero buffers, in addition
define Qk := 1 if there is at least one packet awaiting
processing after the kth successful transmission and Qk := 0
if the buffer is empty. Then it is commonly assumed (e.g.
[11][12][13][14]) that:
(A3) The sequence {Qk} consists of independent random
variables.
(A4) The sequence {Qk} consists of identically distributed
random variables that, in particular, do not depend on
backoff stage.
Many authors consider networks employing the 802.11 EDCA
in which stations have distinct AIFS parameters. For a network
with two distinct AIFS values, let {Hk} denote the sequence
of the number of slots during which stations with a lower
AIFS value can decrement their counters while stations with
the higher AIFS observe the medium as being continuously
busy for longer. The commonly adopted assumptions (e.g.
[15][16][17][18][19][20]) are:
(A5) The sequence {Hk} consists of independent random
variables.
(A6) Each element of the sequence {Hk} is identically distributed,
with a distribution that can be identified with
one derived in Section VI.
For 802.11s mesh networks, let {Dk} denote the interdeparture
times of packets from an element of the network.
That is Dk is the difference between the time at which the kth successful transmission and the k−1 th successful transmission
occurs from a tagged station. If the station’s arrival process is
Poisson, then one pair of hypotheses (e.g. [21]) used to enable
a tractable mathematical model of 802.11s mesh networks is:1
Assumption Saturated Small Buffer Big Buffer
(A1) {Ck} independent ̌ (pairwise) ̌ (pairwise) ̌ (pairwise)
(A2) {Ck} identical distributed ̌ ̌ ×
(A3) {Qk} independent - - ̌ (pairwise)
(A4) {Qk} identical distributed - - ×
(A5) {Hk} independent ̌ (pairwise) - -
(A6) {Hk} specific distribution ̌ - -
(A7) {Dk} independent ̌ (pairwise) ̌ (pairwise) ̌ (pairwise)
(A8) {Dk} exponential distributed × only if lightly loaded only if lightly loaded
TABLE I
SUMMARY OF FINDINGS: {Ck} COLLISION SEQUENCE; {Qk} QUEUE-OCCUPIED SEQUENCE; {Hk} HOLD SEQUENCE; {Dk} INTER-DEPARTURE TIME
SEQUENCE
(A7) The sequence {Dk} consists of independent random
variables.
(A8) The elements of {Dk} are exponentially distributed.
By performing statistical analysis on large volumes of
data collected from an experimental test bed, as described
in Section III, we investigate the veracity of the assumptions
(A1), (A2), (A3), (A4), (A7) and (A8). Due to instrumentation
difficulties, we use data from ns-2 packet-level simulations to
check (A5) and (A6). Our findings are summarized in Table
I.
Autocovariance, Runs Tests, Maximum Likelihood Estimators
and Goodness-of-Fit Statistics lead us to deduce that (A1)
and (A2) are reasonable hypotheses for saturated stations and
unsaturated stations with small buffers, but are not as accurate
for unsaturated stations with large buffers. Of greater concern
for unsaturated stations with large buffers, we find that (A4)
is a dubious and inaccurate assumption. In particular, the
queue-busy probability is clearly shown to be a function of
backoff stage. For 802.11e models that treat differences in
AIFS, we find that (A5) and (A6) are reasonable (albeit not
perfect). For 802.11s mesh networks, we find that (A7) and
(A8) are appropriate for lightly loaded unsaturated stations,
but that (A8) fails as stations become heavily loaded. This is
particularly true when stations are saturated as inter-departure
times coincide with MAC service times. Although these are
independent, which validates (A7), they are not distributed
exponentially, which contradicts (A8).
Clearly these findings give rise to serious concerns about
the appropriateness of many commonly adopted modeling
assumptions. As these hypotheses are inaccurate or inappropriate,
it is hard to have confidence in the predictions of models
based on them beyond their original validation. Our aim in
this article is to draw attention to these deficits and guide the
802.11 modeling community in its ongoing research effort.
This is crucial if these models are to be used by network
designers.
The rest of this paper is organized as follows. In Section II,
we introduce two of the popular 802.11 modeling approaches:
p-persistent and mean-field Markov. In Section III, the instrumented
test bed used to collect experimental data is introduced.
In Section IV we treat the fundamental decoupling hypothesis,
(A1) and (A2), for saturated stations, as well as unsaturated
stations with both small and large buffers. In Section V we
consider the additional queue decoupling assumptions, (A3)
and (A4), that is adopted when treating stations with buffers. In
Section VI, the assumptions, (A5) and (A6), that lie behind the
treatment of different AIFS values in 802.11e are considered.
In Section VII, the mesh assumptions, (A7) and (A8) that
Poisson input gives rise to Poisson output is treated. In Section
VIII we discuss our findings.
II. POPULAR ANALYTIC APPROACHES TO 802.11 DCF
AND EDCA
At its heart, the 802.11 CSMA/CA algorithm employs
Binary Exponential Backoff (BEB) to share the medium
between stations competing for access 2 . As this BEB algorithm
couples stations service processes through their shared
collisions, its performance cannot be analytically investigated
without judiciously approximating its behavior. There are two
popular modeling paradigms: the p-persistent approach and
the mean-field Markov model approach. The former has a
long history in modeling random access protocols, such as
Ethernet and Aloha [22], and the latter has its foundations in
Bianchi’s seminal papers [2][3]. While these approaches differ
in their ideology and details, we shall see that they share basic
decoupling hypotheses. Irrespective of the paradigm that is
adopted, most authors validate model predictions, but do not
directly investigate the veracity of the underlying assumptions.
All of the models we shall consider are based on the
assumption of idealized channel conditions where errors occur
only as a consequence of collisions. With this environmental
conditioning, the key decoupling approximation that enables
predictive models in all p-persistent and mean-field analytic
models of the IEEE 802.11 random access MAC is that given
a station is attempting transmission, there is a fixed probability
of collision that is independent of the past.
In p-persistent models this arises as each station is assumed
to have a fixed probability of attempted transmission, τ, per
idle slot that is independent of the history of the station and
independent of all other stations. In a network of N identical
stations, the likelihood a station does not experience a collision
given it is attempting transmission, 1 − p, is the likelihood
that no other station is attempting transmission in that slot:
1−p = (1−τ) N−1 . Thus the sequence of collision or successes
is an independent and identically distributed sequence. In the
p-persistent approach, the attempt probability, τ, is chosen in
I.
2 A brief overview of the DCF and EDCA algorithms is given in Appendix2
such a way that the average time to successful transmission
matches that in the real system, which is an input to the
model. If this average is known, this methodology has been
demonstrated to make accurate throughput and average delay
predictions [4][5].
In the mean-field approach, the fundamental idea is similar,
but the calculation of τ, and thus p, does not require
external inputs. One starts by assuming that p is given and
each station always has a packet awaiting transmission (the
saturated assumption). Then the backoff counter within the
station becomes an embedded, semi-Markov process whose
stationary distribution can be determined [2][3]. In particular,
the stationary probability that the station is attempting
transmission, τ(p), can be evaluated as an explicit function
of p and other MAC parameters (eq. (7) [3]). In a network
of N stations, as τ(p) is known, the fixed point equation
1 − p = (1 − τ(p)) N−1 can be solved to determine the ‘real’
p for the network and ‘real’ attempt probability τ. Once these
are known, network performance metrics, such as long run
network throughput, can be deduced.
Primarily through comparison with simulations, models
based on these assumptions have been shown to make accurate
throughput and delay predictions, even for small number of
stations. This is, perhaps, surprising as one would expect
that the decoupling assumptions would only be accurate in
networks with a large number of stations. The p-persistent
paradigm has been developed to encompass, for example,
saturated 802.11 networks where every station always has
a packet to send [4] and saturated 802.11e networks [8].
However, due to its intuitive appeal, its self contained ability
to make predictions, and its predictive accuracy, Bianchi’s
basic paradigm has been widely adopted for models that
expand on its original range of applicability. A selection of
these extensions include: [6][7][9][10], which consider the
impact of unsaturated stations in the absence of station buffers
and enable predictions in the presence of load asymmetries;
[11][12][13][14], which treat unsaturated stations in the presence
of stations with buffers; [15][16][17], which investigate
the impact of the variable parameters of 802.11e, including
AIFS, on saturated networks; [18][19][20] which treat unsaturated
802.11e networks; [21], which extends the paradigm
from single hop networks to multiple-radio 802.11s mesh
networks. Note that the work cited here is a small, selective
sub-collection within a vast body of literature. To appreciate
just how large this literature is, as of November 2009, the p-
persistent modeling paper [4] has been cited over 400 times
according to ISI Knowledge and over 800 times according to
Google Scholar, while the mean-field modeling paper [3] been
cited over 1300 times according to ISI Knowledge and over
3200 times according to Google Scholar.
All of the extensions that we cite are based on the idealized
channel assumption, as well as the decoupling approximation.
Some of these extensions require further additional
hypotheses. The purpose of the present article is to dissect
these fundamental assumptions to determine the range of the
applicability of models based on them.
III. EXPERIMENTAL APPARATUS
The experimental apparatus is set up in infrastructure mode.
It employs a PC acting as an Access Point (AP), another
PC and 9 PC-based Soekris Engineering net4801 embedded
Linux boxes acting as client stations. For every transmitted
packet, the client PC records the transmitting timestamp (the
time when it receives an ACK), the number of retry attempts
experienced and the absence or presence of another packet in
station’s buffer, but otherwise behaves as an ordinary client
station. All systems are equipped with an Atheros AR5215
802.11b/g PCI card with an external antenna. All stations,
including the AP, use a version of the MADWiFi wireless
driver modified to allow packet transmissions at a fixed, 11
Mb/s, rate with RTS/CTS disabled and a specified queue size.
The 11 Mb/s rate was selected as in the absence of noisebased
losses the MAC’s operation is rate independent and more
observations of transmission can be made at higher rates for an
experiment of given real-time duration. The channel on which
experiments were conducted was confirmed to be noise free
by use of a spectrum analyzer and by conducting experiments
with single transmitter-receiver pairs at 11 Mb/s.
All stations are equipped with a 100 Mbps wired Ethernet
port that is solely used for control of the test bed from a
distinct PC. In the experiments, UDP traffic is generated by
the Naval Research Laboratory’s MGEN in Poisson mode. All
UDP packets have a 1000 byte payload and are generated in
client stations before being transmitted to the AP. At the AP,
tcpdump is used to record traffic details.
Hoeffding’s concentration inequality (described in Section
IV) was used to determine how many observations were necessary
to ensure statistical confidence in estimated quantities.
Consequently, saturated and large buffer experiments were run
for 2 hours while short buffer experiments were run for 4
hours.
Care must be exercised when performing experiments with
IEEE 802.11 devices. Bianchi et al. [23] and Giustiniano et al.
[24] report on extensive validation experiments which clearly
demonstrate that cards from many vendors fail to implement
the protocol correctly. The precision of our experimental
apparatus was established using the methodology described
in [23] with additional statistical tests.
For example, to check if the back-off counters are uniformly
distributed, the sequence of transmission times of a single
saturated station with fixed packet sizes were recorded. The
back-off counter values were inferred from this sequence by
evaluating the inter-transmission times less the time taken for
a packet transmission (DIFS+Payload+SIFS+Ack) and then
dividing this quantity by the idle slot length. When the contention
window is 32 or 64, Figure 1 reports on a comparison
of the protocol’s back-off distribution with empirical distributions
based on sample sizes of 8, 706, 941 and 7, 461, 686
respectively. With a null hypothesis that the distributions are
uniform, Pearson’s χ2-test (described in Appendix II) gives p-
values of 0.7437 and 0.2036 so that the null hypothesis would
not be rejected.
There is one place where our 802.11 cards do not implement
the standard correctly, but it does not impact on our deduc-3
Probability
0.05
0.045
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
! 2 = 25.5260
P−value = 0.7437
Contention Window Size = 32
Experimental
Theoretical
0
0 5 10 15 20 25 30
Back−off Counter Value
0.025
0.02
! 2 = 72.0458
P−value = 0.2036
Contention Window Size = 64
Experimental
Theoretical
Number of stations Saturated Small buffer Big buffer
N = 2 2,549,550 2,134,187 1,846,049
N = 5 1,220,622 975,601 749,295
N = 10 711,326 502,955 380,139
TABLE II
NUMBER OF ATTEMPTED TRANSMISSIONS K (C) . EXPERIMENTAL DATA
AutoCovariance Coefficient
0.2
0.15
0.1
0.05
0
Saturated
N=2 λ=750
N=5 λ=300
N=10 λ=150
−0.05
Probability
0.015
0.01
−0.1
0 5 10 15 20
Lag
0.005
Fig. 2. Saturated collision sequence normalized auto-covariances. Experimental
data
0
0 10 20 30 40 50 60
Back−off Counter Value
Fig. 1. Comparison of protocol’s uniform back-off distribution and empirical
distribution for contention windows of size 32 and 64 based on sample sizes
of 8, 706, 941 and 7, 461, 686 respectively. Pearson’s χ2 does not reject the
hypothesis that the distributions are uniform. Experimental data
tions; both ACKTimeout and EIFS are shorter than suggested
in the standard. While this must be taken into account when,
for example, predicting throughput, it has no impact on the
aspects of the MAC’s operation that are of interest to us.
For added confirmation, all of the results that are reported
here based on experimental data were shadowed in parallel
by ns-2 based simulations that gave agreement in every case 3 .
Thus we are satisfied that none of the observations reported
in this article are a consequence of peculiarities of the cards,
drivers or experimental environment.
IV. ASSUMPTIONS (A1) AND (A2)
For a single station, define Ck := 1 if the k th transmission
attempt results in a collision and Ck := 0 if it results in a success.
The two key assumptions in [2][3][4][5] are effectively
these: (A1) the sequence {Ck} consists of independent random
variables; and (A2) the sequence {Ck} consists of identically
distributed random variables. That is, there exists a fixed
collision probability conditioned on attempted transmission,
P (Ck = 1) = p, that is assumed to be the same for all backoff
stages and independent of past collisions or successes.
3 Data from these simulations are not shown due to space constraints.
The assumptions (A1) and (A2) are common across all models
developed from the p-persistent and mean-field paradigms.
Here we investigate these for saturated stations, for unsaturated
stations with small buffers and for unsaturated stations with
big buffers. All network parameters correspond to standard
11Mbps IEEE 802.11b [25].
We begin by investigating (A1), the hypothesized independence
of the outcomes (success or collision) in the sequence of
transmission attempts. We draw deductions regarding pairwise
independence from the normalized auto-covariance of the
sequence C1, C2, . . . , CK (C) obtained experimentally, where
K (C) is the number of attempted transmissions that a single
tagged station makes during the experiment. For each experiment
K (C) is recorded in Table II. Assuming {Ck} is wide
sense stationary, the normalized auto-covariance, which is a
measure of the dependence in the sequence, is always 1 at lag
0 and if the sequence {Ck} consisted of independent random
variables, as hypothesized by (A1), then for a sufficiently large
sample it would take the value 0 at all positive lags. Non-zero
values correspond to apparent dependencies in the data.
Experiments were run for a saturated network with N =
2, 5 &10. As the number of stations is increased, the number
of attempts by the tagged station decreases due to the backingoff
effects of the MAC. Figure 2 reports the normalized
auto-covariances for these sequences at short lags. The plot
quickly converges to zero indicating little dependence in the
the success per attempt sequence, even for N = 2.
The (A1) and (A2) assumptions are also adopted in unsaturated
models with small buffers [6][7][9][10] and big buffers
[11][12][13][14]. From experimental data for unsaturated net-4
AutoCovariance Coefficient
AutoCovariance Coefficient
0.1
0.08
0.06
0.04
0.02
0
Unsaturated, Small Buffer
N=2 λ=400
N=5 λ=160
N=10 λ=80
−0.02
0 5 10 15 20
Lag
0.1
0.08
0.06
0.04
0.02
0
Unsaturated, Big Buffer
N=2 λ=250
N=5 λ=100
N=10 λ=50
−0.02
0 5 10 15 20
Lag
Fig. 3. Unsaturated, small and big buffer collision sequence normalized
auto-covariances. Note the short y-range. Experimental data
works large station buffers, Figure 3 plots the normalized autocovariances
of the attempted transmissions for N = 2, 5 &10
with both small and large buffers. As in all unsaturated models
that we are aware of, packets arrive at each station as a Poisson
process with rate λ packets per second. In the big buffer
experiments, the overall network load is kept constant at 500
packets per second, equally distributed amongst the N stations,
corresponding to a network-wide offered load of 4.25Mbps. In
the small buffer experiments network load is kept constant at
800 packets per second. Again we only show short lags as the
auto-covariance quickly drops to 0 indicating little
pairwise
dependency in the C1, . . . , C K (C) sequence and supporting the
(A1) hypothesis.
We have seen graphs similar to Figures 2 and 3 for a range
of offered loads, which are not shown due to space constraints.
These support the (A1) hypothesis that the sequence of collision
or success at each attempted transmission is close to
being a pairwise independent one.
Before investigating the (A2) hypothesis on its own, we use
the Runs Test (described in Appendix III) to jointly test (A1)
and (A2). Given a binary-valued sequence, C1, . . . , C K (C) this
test’s null hypothesis is that it was generated by a Bernoulli
sequence of random variables. The test is non-parametric and
does not depend on P (C1 = 1). The Runs Tests statistic for
each of our nine collision sequences range from 11.6617 for
the saturated, 2-station sequence to 68.5831 for the unsaturated
big-buffer, 2-station sequence. The likelihood that the data was
generated by a Bernoulli sequence is a decreasing function of
the test value and if this value is 2.58, there is less than 1%
chance that it was generated by a Bernoulli sequence. Even
the lower end of the range gives a p-value of 0, leading to
rejection of the hypothesis that the collision sequences are
i.i.d. The reason for this failure will become apparent when we
demonstrate that P (Ck = 1) depends heavily on an auxiliary
variable, αk, the back-off stage at which attempt k was made.
To investigate the (A2) hypothesis on its own, we reuse
the same collision sequence data C1, C2, . . . , CK (C) with some
additional information. For each attempted transmission k ∈
{1, . . . , K (C)}, we record the backoff stage αk at which
it was made. Assume that there is a fixed probability pi
that the tagged station experiences a collision given it is
attempting transmission at backoff stage i. A consequence
of Assumption (A2) is that pi = p for all backoff stages i.
The maximum likelihood estimator for pi is given by
∑ K
(C)
k=1
ˆpi =
Ckχ(αk = i)
∑K (C)
k=1 χ(αk
, (1)
= i)
where χ(αk = i) = 1 if αk = i and 0 otherwise. The
numerator in equation (1) records the number of collisions
at backoff stage i, while the denominator records the total
number of attempts at backoff stage i. As {Ck} is a sequence
of bounded random variables that appear to be nearly independent
(although not i.i.d.), we apply Hoeffding’s inequality
[26] to determine how many samples we need to ensure we
have confidence in the estimate ˆpi:
P (|ˆpi − pi| > x)
⎛
n∑
= P ⎝
(Ck − E(Ck))χ(αk = i)
∣
∣
k=1
⎛
⎞
≤ 2 exp ⎝−2x
K (C)
∑
k=1
χ(αk = i) ⎠ .
K
> x
(C)
∑
k=1
⎞
χ(αk = i) ⎠
Using this concentration inequality, to have at least 95%
confidence that |ˆpi − pi| ≤ 0.01 requires ∑K (C)
k=1 χ(αk = i) =
185 attempted transmissions at backoff stage i. If we have
less than 185 observations at backoff stage i, we do not have
confidence in the estimate’s accuracy so that it is not plotted.
Starting with the saturated networks, Figure 4 plots the
estimates ˆpi for the tagged station as well as the predicted
value from [2][3]. For N = 2, we only report backoff stages
0 to 3 due to lack of observations. It can be seen that the ˆpi are
similar for all i. To quantify this, with S := max ˆpi − min ˆpi,
S = 0.01 for N = 2, S = 0.038 for N = 5 and S = 0.074
for N = 10. Note that while the estimated values are not
identical to those predicted by Bianchi’s model, they are close.
These observations support the (A2) assumption for saturated
stations, even for N = 2.5
0.5
0.45
0.4
0.35
Saturated
N=2 λ=750
N=5 λ=300
N=10 λ=150
Bianchi
0.16
0.14
0.12
Unsaturated, Big Buffer
N=2 λ=250
N=5 λ=100
N=10 λ=50
Collision Probability
0.3
0.25
0.2
Collision Probability
0.1
0.08
0.06
0.15
0.1
0.04
0.05
0.02
0
0 2 4 6 8 10 12
Backoff Stage
0
0 2 4 6 8 10 12
Backoff Stage
Fig. 4.
Saturated collision probabilities. Experimental data
Fig. 6.
Unsaturated, big buffer collision probabilities. Experimental data
Collision Probability
Fig. 5.
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
Unsaturated, Small Buffer
N=2 λ=400
N=5 λ=160
N=10 λ=80
0 2 4 6 8 10 12
Backoff Stage
Unsaturated, small buffer collision probabilities. Experimental data
For unsaturated WLANs, we do not include theoretical
predictions for comparison as, unlike the saturated setting,
there are a large range of distinct models to choose from.
Plotting the predictions from any single model would not be
particularly informative and could, reasonably, be considered
unfair. The significant thing to note is that all of the models
we cite assume that pi = p for all back-off stages i, so that if
pi varies as a function of i, none can provide a perfect match.
Figure 5 is a plot of the estimates ˆpi for each backoff stage
i for the tagged station in the unsaturated 3 packet buffer
case with N = 2, 5, &10, with a network arrival rate of 800
packets per second, equally distributed amongst the N stations,
corresponding to an offered load of 6.8Mbps. In comparison
to the saturated setting, the absolute variability the estimates
is similar with for S defined above giving S = 0.045 for
N = 2, S = 0.038 for N = 5 and S = 0.074 for N = 10.
This suggests that (A2) is reasonably appropriate. There is,
however, clear structure in the graphs. For each N, the collision
probability appears to be dependent on the backoff stage.
The collision probability at the first backoff stage is higher
than at the zeroth stage. For stations that are unsaturated, we
conjecture that this occurs as many transmissions happen at
backoff stage 0 when no other station is has a packet to send
so that collisions are unlikely and ˆp0 is small. Conditioning
on the first backoff stage is closely related to conditioning that
at least one other station has a packet awaiting transmission,
giving rise to a higher conditional collision probability at stage
1, so that ˆp1 > ˆp0.
Figure 6 is analogous to Figure 5, but for stations with 100
packet station buffers. The networks are unsaturated with the
queues at each station repeatedly emptying. As with the small
buffer case, we again have that p1 > p0 and conjecture that
this occurs for the same reasons. In comparison to the values
reported in Figures 4 and 5, the absolute variability is similar
with S = 0.028 for N = 2, S = 0.042 for N = 5 and
S = 0.073 for N = 10, but with φ being the average of
{ˆpi}, relative variability, S/φ, of the estimates in Figure 6 is
consistently higher than the saturated case: S/φ = 0.65 versus
0.17 for N = 2, 0.67 versus 0.23 for N = 5 and 0.73 versus
0.22 for N = 10. This suggest that (A2) is not as good an
approximation in the presence of big station buffers.
In this Section we have investigated the veracity of the
assumptions of independence and identical distribution of the
outcomes of the collision attempt sequence. The findings are
summarized in Table I. In the next section we consider the
additional hypotheses introduced to model buffering.
V. ASSUMPTIONS (A3) AND (A4)
To model stations with buffers serving Poisson traffic, the
common idea across various authors, e.g. [11][12][13][14], is
to treat each station as a queueing system where the service
time distribution is identified with the MAC delay distribution
based on a Bianchi-like model. The assumptions (A1) and (A2)
are adopted, so that given conditional collision probability, p,
each station can be studied on its own and a standard queueing
theory model is used to determine the probability of attempted
transmission, τ(p), which is now also a function of the offered6
1
0.9
0.8
Unsaturated, Big Buffer
N=2 λ=250
N=5 λ=100
N=10 λ=50
0.9
0.8
0.7
Unsaturated, Big Buffer
N=2 λ=250
N=5 λ=100
N=10 λ=50
AutoCovariance Coefficient
0.7
0.6
0.5
0.4
0.3
P(Q>0)
0.6
0.5
0.4
0.3
0.2
0.2
0.1
0.1
0
0 5 10 15 20
Lag
0
0 2 4 6 8 10 12
Backoff Stage
Fig. 7. Unsaturated, big buffer queue-non-empty sequence normalized autocovariances.
Experimental data
Fig. 8. Unsaturated, big buffer queue-non-empty probabilities. Note the large
y-range. Experimental data
load. For symmetrically loaded stations with identical MAC
parameters, the same network coupling equation as used in the
saturated system, 1 − p = (1 − τ(p)) N−1 , identifies the ‘real’
operational conditional collision probability p.
Each time the MAC successfully transmits a packet, it
checks to see if there is another packet in the buffer awaiting
processing. Define Qk := 1 if there is at least one packet
awaiting processing after the kth successful transmission and
Qk := 0 if the buffer is empty. As it is technically challenging
to fully model these queueing dynamics while still
obtaining tractable equations that can be solved more quickly
than a simulation can be run, authors typically employ a
second queueing-based decoupling assumption that can be
distilled into the following two hypotheses: (A3) the sequence
{Qk} consists of independent random variables; and (A4)
the sequence {Qk} consists of identically distributed random
variables, with P (Qk = 1) = q. The value of q is identified
with the steady state probability that an associated M/G/1
or M/G/1/B queueing system has a non-empty buffer after a
successful transmission (e.g. [27]).
Clearly (A3) and (A4) are more speculative than (A1)
and (A2) as both disregard obvious dependencies in the
real Q1, . . . , QK (Q) sequence, where K (Q) is the number of
successful transmissions from the tagged station. These occur
as if there is two or more packets awaiting processing after
a successful transmission, there will still be another packet
awaiting transmission after the next successful transmission
and, in the presence of station buffers, the longer a packet
has been awaiting transmission, the more likely it is to have
another waiting in its buffer.
To investigate (A3) we evaluate the normalized autocovariance
of the empirical sequences Q1, . . . , QK (Q) for N =
2, 5, &10 with K (Q) = 1, 799, 250, K (Q) = 720, 044 and
K (Q)
= 359, 413 respectively. These auto-covariances are
reported in Figure 7 where it can be seen that there is a
small amount of correlation structure, but that by lag 5 this
is less than 0.2 and so we do not regard it as significant. As
one would expect, this is a function of the load. As stations
become more heavily loaded, we have seen this correlation
structure become more prevalent, until stations are saturated,
whereupon the correlation disappears as Qk = 1 for all k.
To jointly test (A3) and (A4), again we use the Runs
Test statistic described in Appendix III. In comparison to the
collision sequences, the test statistics are even more extreme
with 397.46 for N = 2, 171.39 for N = 5 and 130.23 for
N = 10. Thus this test leads to p-values of 0 in all cases
and the rejection of an i.i.d. queue-busy sequence. As with
the collision sequences, it will be clear that this happens as
P (Qk = 1) depends on the back-off stage of the k th successful
transmission.
To investigate (A4), let βk denote the backoff stage at the
k th successful transmission. With qi denoting the probability
there is another packet awaiting transmission after a successful
transmission at backoff stage i, its maximum likelihood
estimator is
ˆqi =
∑ K
(Q)
k=1 Qkχ(βk = i)
∑K (Q)
k=1 χ(βk
. (2)
= i)
Although (A3) does not appear to hold at short lags, we can
again use Hoeffding’s bound to heuristically suggest we need
at least 185 observations at a given backoff stage in order to
be confident in its accuracy.
Figure 8 shows these ˆqi estimates for all stations in each
network. They show a strong increasing trend as a function
of backoff stage. This is as one might expect, given that the
longer a packet spends while awaiting successful transmission,
the more likely it is that there will be another packet awaiting
processing when it is sent. Note that this dependency on
backoff stage raises questions over all buffered models that
adopt the assumption (A4). These findings are summarized in
Table I.7
VI. ASSUMPTIONS (A5) AND (A6)
The 802.11e standard, ratified in 2005, enables service
differentiation between traffic classes. Each station is equipped
with up to four distinct queues, with each queue effectively
being treated as a distinct station. Each queue has its
own MAC parameters including minimum contention window
(CWmin), retry limit, transmission opportunity (TXOP) and
arbitration inter-frame space (AIFS). To model the first three
of these, no additional modeling assumptions are necessary
beyond (A1)-(A2) for saturated stations or for unsaturated
stations with small buffers (e.g. [10]). For unsaturated stations
with large buffers, no additional assumptions are necessary
beyond (A1)-(A4) (e.g. [14]). However, to capture the full
power of 802.11e’s service differentiation, one must model
AIFS and this requires additional innovation and hypotheses
[15][16][17][18][19][20].
Consider N1 + N2 stations, each of which is serving one
of two traffic classes with distinct AIFS values: N1 class one
stations with AIFS1 and N2 class 2 stations with AIFS2 =
AIFS1 + Dσ, where σ is the length of an idle slot and D is
a positive integer. After every attempted transmission, class 1
stations decrement their backoff counters by a minimum of D
before class 2 stations see the medium as being idle. If the
backoff counter of a class 1 stations becomes 0 during these D
slots, it attempts transmission and, once it is complete, class
1 stations can decrement their counters by at least another D
before class 2 stations see the medium as being idle.
Consider a network of homogeneous saturated class 1 stations
and homogeneous class 2 stations. To model the impact
of different AIFS values, using the terminology in [15] we
have the notion of hold states for class 2 stations. A class 2
station is in a hold state if class 1 stations can decrement their
counters while it cannot. As all class 2 stations have the same
AIFS value, they all experience the same class 1 preemption.
Once in a hold state, they cannot begin to decrement their
backoff counters again (while class 1 stations continue to do
so) unless all class 1 stations are silent for D consecutive slots.
Given class 2 stations have just entered a hold state, let
H ∈ {D, D + 1, . . . } represent the hold time: the number of
slots that pass before class 2 stations escape the hold states.
Let {Hk} denote the sequence of observations of hold times.
Implicitly, the commonly adopted assumptions used to treat
AIFS are: (A5) the sequence {Hk} consists of independent
random variables; and (A6) each element of the sequence
{Hk} is identically distributed with the same distribution as
H defined in equation (3) below and that we now derive.
Within the analytic modeling context, this escape from hold
states can be formalized mathematically. Let τ1 denote the
stationary probability a class 1 station attempts transmission.
Define Ps1
= (1 − τ1) N1 , which is the stationary probability
that all class one stations are silent (no class 1 station is
attempting transmission). Let {Xn} denote the sequence of
hold states. After a transmission, whether successful or not,
class 2 stations enter a hold state and this process starts in
hold state X0 = 1. If the medium is idle (no class 1 stations
attempt transmission), which happens with probability Ps1, the
station moves to hold state X1 = 2, otherwise it is reset to
Fig. 9.
1−Ps 1
Ps
Ps 1 1
1 3
2 D D+1
1−Ps
1
1−Ps
1
1−Ps
1
Markov chain for modeling a difference in AIFS of D slots
X1 = 1, and so forth. The process stops the first time that the
hold state D +1 is reached, whereupon all class 2 stations see
the medium as being idle and can decrement their counters.
This system forms a Markov chain, portrayed in Figure 9, with
(D + 1) × (D + 1) the transition matrix:
⎛
⎞
1 − Ps1
Ps1
0 . . . 0 0
1 − Ps1
0 Ps1
. . . 0 0
Π =
.
.
. .
. . .
.
⎜
⎝1 − .
Ps1
. . 0 ⎟
Ps1⎠
0
.
. . 0 1
With X0 = 1, we define H := inf{i : Xi = D + 1} to be
the first time that the D consecutive idle slots are observed.
Using the form of the Markov chain, we have
P (H = i) = P (Xi = D + 1) − P (Xi−1 = D + 1)
= (Π i )1,D+1 − (Π i−1 )1,D+1. (3)
Thus P (H = i) is solely a function of Ps1
and can be readily
calculated from in equation (3), albeit not in closed form
unless D = 1 or D = 2.
Due to experimental instrumentation difficulties, the results
in this section are based exclusively on ns-2 simulations. In
order to determine how many slots the lower class stations
have spent in hold states, it is necessary to know the start
and finish times of every packet transmitted on the medium.
In an experimental setup, no one station is in possession of
this information and, due to the time-scales involved, accurate
reconciliation of the time-line from data recorded at each
station is particularly challenging. In simulation, however, this
data is readily accessible.
For a saturated network, the methodology used to measure
H is as follows. For each station in the network, a timestamp
is recorded at the start and end of every transmission. These
timestamps are collected from all stations and combined into
a single ordered list: s1, e1, s2, e2, . . . where si is the start
time of the ith packet and ei is the end time. Define Ti :=
(si+1 − ei − AIFS1)/σ for each i ≥ 1. The hold times are
determined from this sequence by first identifying the indices
at which the hold states are delineated: N0 := 0 and Ni+1 :=
inf{n ≥ Ni : Tn > D} for each i ≥ 1. The hold states are
then the sum of the hold times between transmission, with the
Ps 1
18
0.25
0.2
D=2
D=12
D=20
D=32
0.8
0.7
D=2
Sim
Theory
AutoCovariance Coefficient
0.15
0.1
0.05
P(H=i)
0.6
0.5
0.4
0.3
0.2
0
0.1
0 2 4 6 8 10 12 14 16 18 20
Lag
Fig. 10. Auto-covariance for period of hold states for class 2 stations
in a network of five class 1 and five class 2 saturated station with D =
2, 12, 20 &8. ns-2 data
final term making a contribution of D slots:
Hi =
Ni+1−1
∑
j=Ni+1
Tj + D,
where the empty sum is defined to be zero.
To consider the independence assumption (A5), we assume
that the sequence of observed hold times H1, . . . , HK (H) are
wide sense stationary and plot this data’s auto-covariance function,
thus investigating pairwise dependence. For a network
of five class 1 saturated stations and five class 2 saturated
stations that are identical apart from AIFS2 = AIFS1 + Dσ,
where D = 1, 2, 4, 8, 12, 16, 20 & 32, these plots suggest that
the sequences have little dependence at short lags. Figure 10
is a representative graph shown for D = 2, 12, 20 & 32 with
K (H) = 1, 717, 545, K (H) = 706, 032, K (H) = 533, 675 and
K (H) = 366, 298, respectively. These D values, particularly
the smaller ones, are typical of those proposed for traffic
differentiation in the 802.11e standard [28]. As the autocovariance
is less than 0.2 by lag 5 the independence
assumption (A5) is not unreasonable for differences in AIFS
values that are proposed in the standard.
To test the assumption (A6), that the probability density
of hold state idle periods has the form given in equation
(3), rather than use any specific model prediction for the
distribution, as it is a function of single parameter, Ps1, we
estimate Ps1
based on the following observation. Given D, if
the likelihood that all class 1 stations do not transmit is i.i.d.,
then P (H = D) = P D s1
. This suggests using the following
estimate of Ps1:
( ∑K (H)
)1/D
k=1 χ(Hk = D)
K (H)
Note that in using this estimate we are ensuring that P (H =
D) coincides with the empirical observation. However, unless
the model is accurate, P (H = i) for i ̸= D calculated from
equation (3) need not coincide with the empirically observed
value.
.
0
0 2 4 6 8 10 12 14 16
i
Fig. 11. Empirical and predicted probability density for the length of a hold
period for class 2 stations in a network of five class 1 and five class 2 saturated
station with D = 2. ns-2 data
P(H=i)
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
D=12
Sim
Theory
0 10 20 30 40 50 60 70 80
i
Fig. 12. Empirical and predicted probability density for the length of a hold
period for class 2 stations in a network of five class 1 and five class 2 saturated
station with D = 12. ns-2 data
Figures 11 and 12 show a sample of plots for a ten
station network, with five saturated stations in class 1 and five
saturated stations in class 2, for D = 2 (K(H) = 5, 152, 635)
and D = 12 (K(H) = 2, 118, 096). Conditioned on having a
good estimate of Ps1
, the accuracy of the distribution predicted
equation (3) appear to be remarkable. This apparent accuracy
can be explored quantitatively through a test statistic. We
use the Kolmogorov-Smirnov test (described in Appendix
II), but do not give p-values as our distribution is purely
discrete. Figure 13 plots supk |Fn(k) − F (k)| against n for
D = 2, 4, 8 &12. It is clear from the graph that the discrepancies
are small for moderate values of sample size. However,
supk |Fn(k) − F (k)| is not converging to 0 as n becomes
large. This suggests that the the predicted distribution of H is
accurate for all practical purposes, even though the distribution
is not a perfect fit. These findings are summarized in Table I.9
0.06
0.05
D=2
D=4
D=8
D=12
0.2
0.15
Saturated
N=2 λ=750
N=5 λ=300
N=10 λ=150
sup|F
n
(k)−F(k)|
0.04
0.03
0.02
AutoCovariance Coefficient
0.1
0.05
0
0.01
−0.05
0 2 4 6 8 10
x 10 4
0
n
−0.1
0 5 10 15 20
Lag
Fig. 13. Largest discrepancy between empirical and predicted distributions,
supk |Fn(k) − F (k)|, as a function of sample size n. D = 2, 4, 8 &12. ns-2
data
VII. ASSUMPTIONS (A7) AND (A8)
The 802.11s standard is a draft amendment to enable
Wireless Mesh Networks (WMNs). One approach to building
a multi-hop, multi-radio mathematical model of a WMN
that employs 802.11 is to build on the mean-field Markov
ideas, but with more involved coupling that captures medium
access dependencies across the mesh. In order to do so, it is
necessary to make hypotheses about the stochastic nature of
the departures process from mesh points, as these form the
arrivals processes to other parts of the mesh.
Let {Dk} denote the inter-departure times of packets from
an element of the network. That is Dk is the difference
between the time at which the k th
successful transmission
and the k − 1 th successful transmission occurs from a tagged
station. One hypothesis (e.g. [21]) is that if the arrivals process
to the station is Poisson, then the departure process is also
Poisson. That is: (A7) {Dk} is a stochastically independent
sequence; and (A8) the elements of {Dk} are exponentially
distributed.
Having observed K (D) inter-departure times,
D1, . . . , D K (D), we investigate these hypotheses. These
times were recorded in the same experiment as the collision
data used in Section IV, so that K (D) = K (C) where K (C)
is reported in Table II. Figures 14, 15 and 16 report the
autocovariance for saturated arrivals, unsaturated arrivals
with small buffers and unsaturated arrivals with large station
buffers for networks of N = 2, 5, &10 stations. There is
little dependency beyond short lags, suggesting that the
independence hypothesis (A7) is not inappropriate.
For hypothesis (A8) and unsaturated stations with large
buffers, due to space constraints, we only report the interdeparture
time distributions for the N = 5 network. These
are representative of our observations of other networks sizes.
Figure 17 plots the logarithm of one minus the empirical cumulative
distribution function of the inter-departure times from
Fig. 14. Saturated inter-departure time sequence normalized autocovariances.
Experimental data
AutoCovariance Coefficient
0.2
0.15
0.1
0.05
0
−0.05
Unsaturated, Small Buffer
N=2 λ=400
N=5 λ=160
N=10 λ=80
−0.1
0 5 10 15 20
Lag
Fig. 15. Unsaturated, small buffer inter-departure time sequence normalized
auto-covariances. Experimental data
AutoCovariance Coefficient
0.2
0.15
0.1
0.05
0
−0.05
Unsaturated, Big Buffer
N=2 λ=250
N=5 λ=100
N=10 λ=50
−0.1
0 5 10 15 20
Lag
Fig. 16. Unsaturated, big buffer inter-departure time sequence normalized
auto-covariances. Experimental data10
the tagged station. Also plotted is the exponential distribution
corresponding to the empirical mean. It can be seen that they
overlay each other nearly perfectly, suggesting that (A8) is a
good hypothesis in this case. Note that this implies that in the
lightly loaded, unsaturated big buffer setting, the statistics of
a Poisson arrivals process is largely unaffected when passing
through an 802.11 network element.
Figure 18 reports the equivalent quantity for the small buffer
experiment with N = 2, but here the network arrival rate of
800 packets per second is chosen to be in the regime near
saturation. For N = 2, after the successful transmission of
a packet, 56% of the time there is another packet awaiting
transmission by the time the medium is sensed idle. The
effective transition between under-loaded and saturated can
be seen as short inter-departure times possesses the features
of MAC service times (convolutions of Uniform distributions),
where as longer inter-departure times follow an exponential
distribution. With a larger number of stations and the same
offered load shared evenly across stations, this effect is less
pronounced and the inter-departure times look exponentially
distributed. This effect is independent of the buffering used
and can also be observed with in big buffer experiments when
traffic loads are closer to saturation. Thus, again the evidence
supports the assumption (A8) assumption if the network is
away from saturation.
However, Figure 19 reports the same plot for a saturated
network. Clearly the inter-departure times are not exponentially
distributed. This is unsurprising as when stations are
saturated, the inter-departure times correspond to the MAC
service times and BEB service times are not well approximated
by an exponential distribution. Qualitatively, the N = 2 and
N = 10 networks show the same features, where the (A8)
assumption is a appropriate for lightly loaded unsaturated
networks, but inaccurate one for saturated networks. This can
be statistically substantiated through the use of a Kolmogorov-
Smirnoff test. The null hypothesis that the inter-departure
times are exponentially distributed is rejected unless one trims
the data by conditioning solely large inter-departure times or
if the load is sufficiently light.
VIII. DISCUSSION
Table I summarizes our conclusions. It seems appropriate
at this stage to discuss another fundamental assumption: (A0)
all stations in the WLAN observe the same sequence of busy
and idle slots on the medium. This assumption is a cornerstone
of all CSMA/CA models that allow idle slots to be of distinct
real-time length from collisions and successful transmissions,
as is the case in 802.11 networks, and include collisions in
their considerations. Both p-persistent models and all of the
mean-field models described here are based on this premise,
which is true in the absence of hidden nodes and interfering
neighboring WLANs.
In order to model situations where (A0) is false, such as
relay topologies that do not have multiple radios and so cannot
mitigate interference at non-communicating distances, new
approximations are necessary (e.g. [29][30][31][32]). Some
of these models also use mean-field ideas, usually inspired
P(D>t)
10 0
10 −1
10 −2
10 −3
10 −4
10 −5
Unsaturated, Big Buffer, N=5 !=100
Experimental Data
Theoretical Data
0 2 4 6 8 10 12 14
x 10 4
10 −6
Inter−departure Time, t (µs)
Fig. 17. Unsaturated big buffer inter-departure time distribution (log y-scale),
N = 5. Experimental data
P(D>t)
10 0
10 −1
10 −2
10 −3
10 −4
10 −5
Unsaturated, Small Buffer, N=2 !=400
Experimental Data
Theoretical Data
0 0.5 1 1.5 2 2.5 3 3.5
x 10 4
10 −6
Inter−departure Time, t (µs)
Fig. 18. Unsaturated small buffer inter-departure time distribution (log y-
scale), N = 2. Experimental data
P(D>t)
10 0
10 −1
10 −2
10 −3
10 −4
10 −5
Saturated, N=5 !=300
Experimental Data
Theoretical Data
0 1 2 3 4 5 6
x 10 4
10 −6
Inter−departure Time, t (µs)
Fig. 19. Saturated inter-departure time distribution (log y-scale), N = 5.
Experimental data11
by percolation theory, and typically assume away the BEB
aspect of 802.11. This approach gives insight into scenarios
that are distinct from those that are amenable to analysis by
the models studied in the present paper and, as suggested
to us by an anonymous referee, the appropriateness of these
multi-hop mean-field approximations would be deserving of
an experimental investigation as they facilitate mathematical
investigation of situations that are otherwise analytically intractable.
We do not pursue this investigation in the present
article due to space constraints.
This validation of the standard decoupling assumption,
(A1) and (A2), for saturated networks helps to explain why
the predictions in [2][3][4][5] are so precise. Even though
intuitively one expects the main model assumptions to be
valid for large networks, in fact they are accurate even for
small networks. As the assumptions are reasonable, deductions
from that model should be able to make predictions regarding
detailed quality of service metrics.
The (A1) assumption continues to hold for both the unsaturated
setting with either small or big buffers, suggesting that
the attempt sequences have little dependencies. With small
buffers, the (A2) assumption that collision probabilities are
independent of backoff stage appears to be valid for stations
that are not saturated. There is, however, some structure with
p1 > p0, but this is not quantitatively significant. For larger
buffers, this discrepancy is more apparent in both relative
and absolute terms, suggesting that (A2) is an imprecise
approximation in that setting. For large buffer models, this
inaccuracy is less dramatic than the failure of the additional
queueing decoupling assumption (A3) and (A4).
Our investigations indicate that while (A3) is reasonable at
lighter loads, neither (A3) or (A4) are appropriate in general.
In particular, contradicting (A4), the probability that the queue
is non-empty after a successful transmission is strongly dependent
on backoff stage. Despite the apparent inappropriateness
of the assumptions (A3) and (A4), models based on them
continue to make accurate predictions of goodput and average
delay. One explanation is that with an infinite buffer, unless
the station is saturated, the goodput corresponds with the
offered load. Thus, to have an accurate goodput model it is
only important that the model be accurate when offered load
leads a station to be nearly saturated. Once saturated, (A4)
is true as Bianchi’s model is recovered. Thus, for goodput,
the inaccuracy of the approximation (A4) is not significant if
this phase transition is predicted by the model. However, one
would expect that for more subtle quantities the adoption of
(A4) would lead to erroneous deductions. Clearly caution must
be taken when making deductions from big buffer models that
incorporate these hypotheses. Extrapolations of that kind from
these models should be made with care by network designers.
The hold state hypotheses that were introduced to incorporate
802.11e AIFS differentiation in saturated networks are
labeled (A5) and (A6). The independence hypothesis (A5)
appears to be appropriate. The distributional assumption (A6)
appears to be accurate for any difference in AIFS once one has
a good estimate of the probability that no higher class station
attempts transmission in a typical slot. This lends confidence
to the use of these models for network design and detailed
predictions.
The 802.11s mesh network assumptions (A7) and (A8) hold
true for lightly loaded, unsaturated networks, where stations
can have either large or small buffers. In particular, the output
of an unsaturated 802.11 station with Poisson arrivals again
appears to be nearly Poisson, so long as saturation is not
being approached. However, if the station is saturated, the
inter-departure times correspond to MAC delays, which are
not similar to an exponential distribution. If stations are close
to being saturated, short inter-departure times are similar to
MAC delays, where as long inter-departure times correspond
to long inter-arrival times and are Poissonian. The impact of
this non-Poisson traffic on the accuracy of unsaturated model
predictions must be investigated before they can be used with
confidence.
We also make a comment regarding experimentation. It was
challenging to emulate the fundamental explicit hypothesis
of all the models that we investigated: that of idealized
channel conditions where errors occur only as a consequence
of collisions. As 802.11 operates in an unlicensed range of the
spectrum and other devices are free to operate in this range,
these devices lead to interference. There are extensions to the
WLAN modeling paradigms that include failed transmissions
due to noise on the medium, e.g. [33]. This approach assumes
that packet losses due to noise are i.i.d. and independent of all
other stochastic processes in the model. Whether this assumption
is appropriate is dependent on the particular environment
at hand and, clearly, cannot be subject to general validation.
Due on the failure of several of these fundamental hypotheses,
clearly there is more work to be done on analytic modeling
of 802.11. In particular, models that incorporate buffers at
stations are based on flawed hypotheses, but are important
for network designers. We suggest that it is an important
challenge for the analytic modeling community to revisit and
revise models based on these inappropriate assumptions.
Based on the observations in this article, for example, a
natural alternative to (A3) and (A4) is to use the approximation
that: (A3’) given βk = i, {Qk} is an independent sequence;
and (A4’) given βk = i, P (Qk = 1) = qi. As a first step in
this direction, in [34] negative consequences of adopting the
assumptions (A3) and (A4) are identified. A typical validation
scenario employed by modelers is to consider a symmetrically
loaded network. While this is unlikely to occur in practice,
mathematically it leads to homogeneous fixed point equations
whose solution can be quickly identified by standard numerical
techniques. For stations that are asymmetrically loaded, results
in [34] demonstrate that a model based on these assumptions
provides inaccurate throughput predictions. That this is a
consequence of (A3) and (A4) is established by considering
the setting where all stations can buffer one packet beyond the
MAC, as it is then possible to analyze a model based on (A3’)
and (A4’).
We expect that other researchers will have alternate hypotheses
that they wish to check. To facilitate this research,
all the data used in the present study is available at: http:
//www.hamilton.ie/kaidi/.12
IX. ACKNOWLEDGMENTS
We thank the anonymous reviewers for their helpful comments
and suggestions. This work was supported by Science
Foundation Ireland grant RFP-07-ENEF530.
APPENDIX I
A BRIEF OVERVIEW OF 802.11’S BEB ALGORITHM
On detecting the wireless medium to be idle for a period
DIFS, each station initializes a counter to a random number
selected uniformly in the range {0, 1, . . . , CWmin − 1}. Time
is slotted and this counter is decremented once during each
slot that the medium is observed idle. The count-down halts
when the medium becomes busy and resumes after the medium
is idle again for a period DIFS. Once the counter reaches
zero the station attempts transmission and if a collision does
not occur it can transmit for a duration up to a maximum
period TXOP (defined to be one packet except in the Quality
of Service MAC extension 802.11e). If two or more stations
attempt to transmit simultaneously, a collision occurs. Colliding
stations double their Contention Window (CW) (up to a
maximum value), select a new backoff counter uniformly and
the process repeats. If a packet experiences more collisions
than the retry limit (11 in 802.11b), the packet is discarded.
After the successful transmission of a packet or after a packet
discard, CW is reset to its minimal value CWmin and a new
count-down starts regardless of the presence of a packet at
the MAC. If a packet arrives at the MAC after the countdown
is completed, the station senses the medium. If the
medium is idle, the station attempts transmission immediately;
if it is busy, another backoff counter is chosen from the
minimum interval. This bandwidth saving feature is called
post-backoff. The revised 802.11e MAC enables the values of
DIFS (called the Arbitration Inter-Frame Spacing, AIFS, in
802.11e), CWmin and TXOP to be set on a per-class basis for
each station. That is, traffic is directed to up to four different
queues at each station, with each queue assigned different
MAC parameter values.
APPENDIX II
TESTING GOODNESS OF FIT
Given a sequence of observations of independent and identically
distributed random variables X1, . . . , Xn, we wish to
test the hypothesis that the {Xk} have common distribution F .
For discrete-valued random variables taking N distinct values,
we use Pearson’s χ 2 -test (e.g. [35]) Assume a null hypothesis
that X1 has a distribution such that P (X1 = i) = f( ⃗ θ, i),
where ⃗ θ is a collection of M parameters estimated from the
data. For each possible outcome i define
ni :=
n∑
j=1
1 {Xj=i} and X 2 :=
N∑ (ni − nf( ⃗ θ, i)) 2
nf( ⃗ .
θ, i)
i=1
For large sample sizes n, the test statistic X 2 has a χ 2
distribution with between N − 1 and N − 1 − M degrees
of freedom. We use the later, more stringent, test to determine
the p-value P (χ 2 (N − 1 − M) ≥ X 2 ).
For real-valued random variables and unbounded random
variables with a discrete distribution we evaluate the
Kolmogorov-Smirnov statistic (e.g. [36][35]). We can use the
former to determine a test for goodness-of-fit, but exact critical
levels are not possible to determine in the latter [37]. Let
Fn(k) = 1
n
n∑
χ(Xi ≤ k)
i=1
denote the empirical distribution given n observations. The
L∞ distance supk |Fn(k) − F (k)| is the greatest discrepancy
between the two distributions. It is used in the the
Kolmogorov-Smirnov test based on the observation that if the
null hypothesis that {Xk} are identically distributed were true,
then supk |Fn(k) − F (k)| is of order n−1/2 and, in particular,
supk |Fn(k) − F (k)| → 0. If F is continuous, we also have
the following weak convergence result
√
nDn ⇒ sup |B(F (t))|,
t
where B(t) is a Brownian bridge [38] from which a p-value
can be determined.
APPENDIX III
RUNS TEST FOR BINARY VALUED RANDOM VARIABLES
If {Xn} are binary valued random variables then the null
hypothesis that the sequence is independent and identically
distributed can be efficiently tested using the Runs Test
[39][35]. Given a sequence of observations X1, . . . , Xn, a run
is defined to be a maximal non-empty segment of the sequence
consisting of adjacent equal elements. Let R be the number
of runs in X1, . . . , Xn and define
n0 =
n∑
i=1
1 {Xi=0}, µ = 2n0(n − n0)
, σ
n + 1
2 =
(µ − 1)(µ − 2)
.
n − 1
Then, under the null hypothesis, Z = (R − µ)/ √ σ 2 is
asymptotically Normally distributed. Thus, given the sequence
of observations X1, . . . , Xn, one evaluates Z and the p-value
for the null hypothesis is min(P (N(0, 1) ≥ Z), P (N(0, 1) ≤
Z)), where N(0, 1) is a normally distributed random variable.
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