1

Performance Analysis of Dynamic Channel
Bonding in Spatially Distributed
High Density WLANs

arXiv:1801.00594v1 [cs.NI] 2 Jan 2018

Sergio Barrachina-Muñoz, Francesc Wilhelmi, Boris Bellalta
Abstract—In this paper we discuss the effects on throughput and fairness of dynamic channel bonding (DCB) in spatially distributed
high density (HD) wireless local area networks (WLANs). First, we present an analytical framework based on continuous time Markov
networks (CTMNs) for depicting the phenomena given when applying different DCB policies in spatially distributed scenarios, where
nodes are not required to be within the carrier sense of each other. Then, we assess the performance of DCB in HD IEEE 802.11ax
WLANs by means of simulations. Regarding spatial distribution, we show that there may be critical interrelations among nodes – even
if they are located outside the carrier sense range of each other – in a chain reaction manner. Results also show that, while always
selecting the widest available channel normally maximizes the individual long-term throughput, it often generates unfair scenarios
where other WLANs starve. Moreover, we show that there are scenarios where DCB with stochastic channel width selection improves
the latter approach both in terms of individual throughput and fairness. It follows that there is not a unique DCB policy that is optimal for
every case. Instead, smarter bandwidth adaptation is required in the challenging scenarios of next-generation WLANs.
Index Terms—Dynamic channel bonding, spatial distribution, policy, CTMN, WLAN, throughput, IEEE 802.11ax

F

1

I NTRODUCTION

W

IRELESS local area networks (WLANs), with IEEE
802.11 as the most widely used standard, are a costefficient solution for wireless Internet access that can satisfy
most of the current communication requirements in domestic, public, and business scenarios. However, the scarcity
of the frequency spectrum, the increasing throughput demands given by new hungry-bandwidth applications, and
the heterogeneity of current wireless network deployments
give rise to substantial complexity. Such issues gain importance in dense WLAN deployments, leading to multiple
partially overlapping scenarios and coexistence problems.
In this regard, two main approaches for optimizing the
scarce resources of the frequency spectrum are being deeply
studied in the context of WLANs: channel allocation (CA)
and channel bonding (CB). While CA refers to the action of
allocating the potential transmission channels for a WLAN
or a group of WLANs (i.e., allocating both the primary
and secondary basic channels), CB is a technique whereby
nodes are allowed to use a contiguous set of idle channels
for transmitting in larger bandwidths, and thus potentially
achieving higher throughputs.
This paper focuses on CB, which was firstly introduced
in the IEEE 802.11n amendment [1], where two 20 MHz
basic channels can be aggregated to transmit in a 40 MHz
channel. Newer amendments like IEEE 802.11ac (11ac) [2]
extend the number of basic channels that can be aggregated
up to 160 MHz channel widths. It is expected that IEEE
802.11ax (11ax) will boost the use of wider channels [3].
Nonetheless, due to the fact that using wider channels
increases the contention and interference among nodes,
undesirable lower performances may be experienced when

•

All the authors are with Universitat Pompeu Fabra (UPF).
E-mails: {sergio.barrachina, boris.bellalta, francesc.wilhelmi}@upf.edu

applying static channel bonding (SCB), specially in high
density (HD) WLAN scenarios. To mitigate such a negative
effect, dynamic channel bonding (DCB) policies are used to
select the bandwidth in a more flexible way based on the
instantaneous spectrum occupancy. A well-known example
of DCB policy is always-max (AM)1 [4], [5], where transmitters select the widest channel found idle when the backoff
counter terminates.
To the best of our knowledge, the works in the literature
assessing the performance of DCB study only the SCB
and AM policies, while they also assume fully overlapping
scenarios where all the WLANs are within the carrier sense
range of the others [4], [6], [7], [8]. Therefore, there is an
important lack of insights on the performance of CB in more
realistic WLAN scenarios, where such a condition usually
does not hold.
With this work we aim to extend the state of the art
by providing new insights on the performance of DCB in
WLAN scenarios that are not required to be fully overlapping; where the effect of hidden and exposed nodes plays
a crucial role due to the spatial distribution interdependencies. Namely, a node’s operation has a direct impact on the
nodes inside its carrier sense range, which in turn may affect
nodes located outside such range in a complex and hard
to prevent way. Besides, we assess different DCB policies,
including an stochastic approach that selects the channel
width randomly.
To do so, we first introduce the Spatial-Flexible Con-

1. Some papers in the literature use the terms DCB and AM indistinctly. In this paper we notate AM as an special case of DCB.

2

tinuous Time Markov Network (SFCTMN),2 an analytical
framework based on Continuous Time Markov Networks
(CTMNs). This framework is useful for describing the different phenomena that occur in WLAN networks when considering DCB in spatially distributed scenarios, i.e., where
nodes are not required to be within the carrier sense range of
each other. In this regard, we depict such complex phenomena by means of illustration through several toy scenarios.
We then evaluate the performance of the proposed policies in HD 11ax WLAN scenarios by means of simulations
using the 11axHDWLANsSim [9] wireless network simulator.3 We find that, while AM is normally the best DCB policy
for maximizing the individual long-term throughput of a
WLAN, it may generate unfair scenarios where some other
WLANs starve. In fact, there are cases where less aggressive
policies like stochastic channel width selection improves
AM both in terms of individual throughput and fairness.
The contributions of this paper are as follows:
1) Novel insights on the effects of DCB in HD WLANs.
We depict the complex interactions that are given in
spatially distributed scenarios – i.e., considering path
loss, signal-to-interference-plus-noise ratio (SINR) and
clear channel assessment (CCA) thresholds, co-channel
and adjacent channel interference, etc. – and discuss
the influence that nodes operation may have among
them. In this regard, we show that AM is not always the
optimal policy and that different networks may require
different DCB policies. This leads to the need of local
policy adaptation.
2) Generalization of DCB policies including only-primary
(i.e., selecting just the primary channel for transmitting), SCB, AM, and probabilistic uniform (PU) (i.e.,
selecting the channel width stochastically). We show
that WLAN scenarios implementing any combination
of DCB policies can be characterized by a CTMN with
specific transition probabilities. This allows us to analytically compare the behavior of these policies.
3) Algorithm for modeling WLAN scenarios with CTMNs
that extends the one presented in [8]. This extension
allows us capturing non-fully overlapping scenarios,
taking into consideration spatial distribution implications.
4) Performance evaluation of the presented DCB policies
in HD WLAN scenarios by means of simulations. The
physical (PHY) and medium access control (MAC)
layers are set in a representative way for single user
(SU) 11ax WLANs. We show that AM does not always
maximize the individual throughput, but more flexible
policies like PU are required. Besides, we show that
policy learning and/or adaptation is required on a perWLAN basis in order to maximize both the individual
throughput and system fairness.
The remainder of this article is organized as follows. In
Section 2, we provide related work on DCB. Then, in Section
2. All of the source code of SFCTMN is open, encouraging sharing of
algorithms between contributors and providing the ability for people to
improve on the work of others under the GNU General Public License
v3.0. The code used in this work can be found at https://github.com/
sergiobarra/SFCTMN.
3. An overview of the simulator, as well as the parameters used in
the simulations are shown in the Appendix

3 we describe the system model considered throughout
the article. The SFCTMN framework for modeling WLAN
scenarios with CTMNs is detailed in Section 4. We depict the
interactions in frequency and space in Section 5 by means of
several toy scenarios. The performance of the different DCB
policies in HD WLANs is assessed in Section 6. We conclude
with some final remarks and future work at Section 7.

2

R ELATED WORK

Several works in the literature assess the performance of
CB by means of analytical models, simulations or real
testbeds. Authors in [10], [11] experimentally analyze SCB
in IEEE 802.11n WLANs and show that the reduction of
Watt/Hertz when transmitting in larger channel widths
causes lower SINR at the receivers. This lessens the coverage
area consequently and increases the probability of packets
losses due to the accentuated vulnerability to interference.
Nonetheless, they also show that DCB can provide significant throughput gains when such issues are palliated by
properly adjusting the transmission power and data rates.
With the 11ac and 11ax amendments larger channel
widths are allowed (up to 160 MHz) and the pros and
cons of CB are also accentuated. Nevertheless, it is important to emphasize that in the dense and short-range
WLAN scenarios expected in the coming years [3], the issues
concerning low SINR values may be palliated due to the
shorter distances between transceiver and receiver, and to
techniques like spatial diversity multiple-input multipleoutput (MIMO) [12]. In this regard, authors in [5], [13]
assess the performance of DCB in 11ac WLANs by means
of simulations and show that it can provide significant
throughput gains. Still, they also corroborate the fact that
these gains are severely compromised by the activity of
overlapping wireless networks.
There are also works in the literature that use an analytical approach for assessing the performance of CB. For
instance, authors in [6] analytically model and evaluate the
performance of CB in short-range 11ac WLANs, proving
significant performance boost when the presence of external
interference is low to moderate. In [4], authors show that
the use of CB can provide significant performance gains
even in scenarios with a high density of WLANs, though
it may also cause unfair situations. Likewise, authors in
[8] use a CTMN-based model to explain key properties
of DCB such as the sensitivity to the backoff and transmission time distributions, or the high switching times
between different dominant states. Non-saturation regimes
are considered in [14], where authors propose an analytical
model for the throughput performance of channel bonding in 11ac WLANs in presence of collisions under both
saturated and non-saturated traffic loads. Authors in [15]
develop an analytical framework to study the performance
of opportunistic channel bonding in WLANs where 11ac
users coexist with legacy users. Similarly, authors in [16]
assess the performance of CB in the context of opportunistic
spectrum access via an analytical model and conclude that
CB is generally beneficial where there is low primary users
activity.
Some solutions involving CB are also available in the
literature. For instance, an intelligent scheme for jointly

3

Fig. 1: Simplified set of allowed channels for transmitting in the 11ac and 11ax amendments. This particular channelization scheme is defined by C =
{{1}, {2}, ..., {1, 2}, {3, 4}, ..., {1, 2, 3, 4}, ..., {1, 2, ..., 8}}.
adapting the rate and bandwidth in MIMO IEEE 802.11n
WLANs is presented in [12]. Real experiments show that
such scheme (ARAMIS) accurately adapts to a wide variety of channel conditions with negligible overhead and
achieving important performance gains. In [17], authors
propose an spectrum-distribution algorithm where access
points (APs) adjust both the primary channel and width
to match their traffic loads. They show that the spectrum
utilization can be substantially improved when allocating
more bandwidth to APs with high traffic load. An stochastic
spectrum distribution framework accounting for WLANs
demand uncertainty is presented in [18], showing better
performance compared to the naive allocation approach.
Authors in [19] show that the maximal throughput performance can be achieved with DCB under the CA scheme with
the least overlapped channels among WLANs.
We believe that this is the first work that provides
insights on the performance of DCB in spatially distributed
scenarios, where the effect of hidden and exposed nodes
plays a crucial role due to the spatial interdependencies
among nodes. In this regard, we also provide an algorithm
for generating the CTMNs to model such kind of scenarios.
Besides, we assess the performance of different DCB policies, including a novel stochastic approach. and show that
always selecting the widest available channel is sub-optimal
in some of the studied scenarios.

3

•

•

•

•

must share the same primary channel pw . Essentially,
the primary channel is used to i) sense the medium for
decrementing the backoff when the primary channel’s
frequency band is found free, and ii) listening to decodable packets for performing the request to send (RTS)
and clear to send (CTS) handshake – or entering in
network allocation vector (NAV) state – and receiving
data or ACK packets. Hence, the primary channel is
always used when transmitting and receiving packets.
Channel C : a channel C = {c1 , c2 , ..., cN } consists of
a contiguous set of N basic channels. The width (or
bandwidth) of a channel is N |c|.
Channelization scheme C : the set of channels that can
be used for transmitting is determined by the channel access specification and the system channel (Csys ),
whose bandwidth is given by Nsys |c|. Namely, all the
nodes in the system must transmit in some channel included in C . A simplified version of the channelization
considered in the 11ac and 11ax standards is shown in
Figure 1.
WLAN’s allocated channel Cw : nodes in a WLAN w
must transmit in a channel contained in Cw ∈ C . Different WLANs may be allocated with different primary
channels and different available channel widths. We use
Nw to denote the number of basic channels allocated for
WLAN w.
tx
Transmission channel Cn
: a node n belonging to a
WLAN w has to transmit in a channel Cntx ⊆ Cw ∈ C ,
which will be given by i) the set of basic channels in Cw
found idle by node n at the end of the backoff (Cnfree ),4
and ii) on the implemented DCB policy.

In the example shown in Figure 2, the system channel
Csys counts with Nsys = 8 basic channels and WLAN X is
allocated with the channel CX = {1, 2, 3, 4} and primary
channel pX = 3. Note that in this example the AP of WLAN
X does not select the entire channel allocated, but a smaller
one, i.e., CXtx = {3, 4} ⊆ CX . Two reasons may be the cause:
i) basic channels 1 and 2 are detected busy at the end of the
backoff, or ii) the DCB policy determines not to pick them.

S YSTEM MODEL UNDER CONSIDERATION

In this Section we first depict the notation regarding channelization that is used throughout this article. We also define
the DCB policies that are studied and provide a general
description of the carrier sense multiple access with collision
avoidance (CSMA/CA) operation in IEEE 802.11 WLANs.
Finally, we expose the main assumptions considered in the
presented scenarios.
3.1

Channelization

Let us define some terms regarding channel access for the
sake of facilitating further explanation:
• Basic channel c: the frequency spectrum is split into
basic channels of width |c| = 20 MHz.
• Primary channel pw : basic channel with different roles
depending on the node state. All the nodes belonging
to the same WLAN w – i.e., AP and stations (STAs) –

Fig. 2: Channel access notation. WLAN X is allocated with 4
basic channels but only 2 are used for the transmission.

4. Note that, in order to include secondary basic channels for transmitting, a WLAN must listen them free during at least a PIFS period
before the backoff counter terminates as shown in Figure 3. While such
PIFS condition is not considered in the SFCTMN framework for the
sake of analysis simplicity, the 11axHDWLANSim simulator does.

4

3.2

CSMA/CA operation in IEEE 802.11 WLANs

According to the CSMA/CA operation, when a node n
belonging to a WLAN w has a packet ready for transmission, it measures the power sensed in the frequency band
of pw . Once the primary channel has been detected free,
i.e., the power sensed by n at pw is smaller than its CCA
threshold, the node starts the backoff procedure by selecting
a random initial value of BO ∈ [0, CW − 1] time slots. The
contention window is defined as CW = 2b CWmin , where
b ∈ {0, 1, 2, ..., m} is the backoff stage with maximum value
m, and CWmin is the minimum contention window. When
a packet transmission fails, b is increased by one unit, and
reset to 0 when the packet is acknowledged.
After computing BO, the node starts decreasing the backoff counter while sensing the primary channel. Whenever
the power sensed by n at pw is higher than its CCA, the
backoff is paused and set to the nearest higher time slot until
pw is detected free again, at which point the countdown is
resumed. When the backoff timer reaches zero, the node
selects the transmission channel Cntx based on the set of idle
basic channels Cnfree and on the DCB policy.
The selected transmission channel is then used throughout the whole packet exchanges involved in a data packet
transmission between the transceiver and receiver. Namely,
RTS (used for notifying the selected transmission channel),
CTS, and ACK packets are also transmitted in Cntx . Likewise,
any other node that receives an RTS in its primary channel
with enough power to be decoded will enter in NAV state,
which is used for deferring channel access and avoiding
packet collisions (especially those caused by hidden node
situations).
In Figure 3, the temporal evolution of a node operating under AM is shown. In this example, the WLAN
to which the node belongs is allocated with the channel
Cw = {1, 2, ..., 8} and primary channel pw = 2 . While at
the end of the first backoff the node is able to select the full
allocated channel to transmit in, i.e., Cntx = Cw , at the end of
the second one, because of the interference sensed at channel
3 during the PIFS period, a narrower channel Cntx = {1, 2}
is selected.
3.3

DCB policies

The DCB policy determines which transmission channel
a node must pick from the set of available ones. When
the backoff terminates, any node belonging to a WLAN w
operates according to the DCB policy as follows:
• Only-primary (OP): picks just the primary channel for
transmitting if it is found idle. This policy is also known
as single-channel.
• Static channel bonding (SCB): exclusively picks the
full channel allocated in its WLAN when found free.
Namely, nodes operating under SCB cannot transmit in
channels different than Cw .
• Always-max (AM): picks the widest possible channel
found free in Cw for transmitting.
• Probabilistic uniform (PU): picks with same probability any of the possible channels found free inside the
allocated channel Cw .
For the sake of illustration, let us consider the example
shown in Figure 3, where the evolution of a node imple-

menting AM is presented. Regarding the rest of DCB policies, OP would just pick the basic channel 2 after both backoff terminations. Instead, while SCB would transmit in the
entire channel after the first backoff, it would not transmit
at the end of second one because part of Cw is busy. Finally,
PU would transmit on channels {2}, {1, 2}, {1, 2, 3, 4} or
{1, 2, ..., 8} with same probability 1/4 at the first backoff
termination, and on channels {2} or {1, 2} with probability
1/2 at the end of the second one. An schematic flowchart of
the DCB policy operation is shown in Figure 4.
3.4

Main assumptions

In this paper we present results gathered via the SFCTMN
framework based on CTMNs, and also via simulations
through the 11axHDWLANsSim wireless network simulator.
While in the latter case we are able to introduce more
realistic implementations of the 11ax amendment, in the
analytical model we use relaxed assumptions for facilitating posterior analysis. This subsection depicts the general
assumptions considered in both cases.
1) Channel model: signal propagation is isotropic regardless of the selected path loss model L. Also, the propagation delay between any pair of nodes is considered
negligible due to the small carrier sense areas given
in the 5 GHz band. Besides, the transmission power
is divided uniformly among the basic channels in the
selected transmission channel. Specifically, we apply a
loss of 3 dB per basic channel each time the bandwidth
is doubled. We also consider an adjacent channel interference model that replicates the half of the power
transmitted per Hertz into the two basic channels that
are contiguous to the actual transmission channel Cntx .
2) Packet errors: we consider that a packet is lost if i)
the power of interest received at the receiver is less
than its CCA, ii) the SINR (γ ) perceived at the receiver
does not accomplish the capture effect (CE), i.e., γ <
CE, or iii) the receiver was already receiving a packet.
In the latter case, the decoding of the first packet is
ruptured only if the CE is no longer accomplished
because of a the interfering transmission. We assume
an infinite maximum number of retransmissions per
packet, whose effect is almost negligible in most of the
cases because of the small probability of retransmitting
a data packet more than few times [20].
3) Modulation coding scheme (MCS): the MCS index
used by each WLAN is the highest possible, and it is
kept constant throughout all the simulation. We assume
that the MCS selection is designed to keep the packet
error rate constant and equal to η , given that static
deployments are considered.
4) Traffic: downlink traffic is considered. In addition,
we assume a full-buffer model, i.e., APs always have
packets pending for transmission as they operate in
saturated regime.

4

T HE CTMN MODEL FOR HD WLAN S

The analysis of CSMA/CA networks through CTMN models was firstly introduced in [21]. Such models were later
applied to IEEE 802.11 networks in [4], [6], [8], [14], [19],

5

Fig. 3: CSMA/CA temporal evolution of a node operating under AM and the 11ax channelization scheme.

Fig. 4: Flowchart of the transmission channel selection. In
this example channel 5 is the primary channel and a DCB
policy D = AM is applied.

[22], [23], among others. Experimental results in [24], [25]
demonstrate that CTMN models, while idealized, provide
remarkably accurate throughput estimates for actual IEEE
802.11 systems. A comprehensible example-based tutorial
of CTMN models applied to different wireless networking
scenarios can be found in [26].
Nevertheless, to the best of our knowledge, works that
model CSMA/CA WLANs using CB study just the SCB and
AM policies, while assuming fully overlapping scenarios
where all the WLANs are inside the carrier sense range of
each other. Therefore, there is an important lack of insights
on more general WLAN scenarios, where such condition
usually does not hold and interdependencies among nodes
may have a critical impact on their performance.
In this section we depict our extended version of the
algorithm introduced in [8] for generating the CTMNs corresponding to spatially distributed WLAN scenarios, which
is implemented in the SFCTMN framework. With this extension, as the condition of having fully overlapping networks
is no longer required for constructing the corresponding
CTMNs, more factual observations can be made.
In this paper, since we consider that there is only one
transmitter-receiver pair per WLAN (i.e., the AP and one
STA), we will simply refer to the WLAN activity as a unit.
4.1

Implications

Modeling WLAN scenarios with CTMNs requires the backoff and transmission times to be exponentially distributed.
This has a main implication: because of the negligible propagation delay, the probability of packet collisions between
two or more nodes within the carrier sense range of the
other nodes is zero. The reason is that two WLANs will
never end their backoff at the same time, and therefore they
will never start a transmission at the same time either.

In overlapping single-channel CSMA/CA networks, it
is shown that the state probabilities are insensitive to the
backoff and transmission time distributions [25], [27]. However, even though authors in [8] prove that the insensitivity
property does not hold for DCB networks, the sensitivity
to the backoff and transmission time distributions is very
small. Therefore, the analytical results obtained using the
exponential assumption offer a good approximation for deterministic distributions of the backoff, data rate and packet
length as well.

4.2

Constructing the SFCTMN

In order to depict how CTMNs are generated, let us
consider the toy scenario (Scenario I) shown in Figure 5,
which is composed of two fully overlapping WLANs implementing AM.5 The channel allocation of such scenario
can be defined as CA = {1, 2, 3, 4} with pA = 2, and
CB = {3, 4} with pB = 3. That is, there are four basic
channels in the system, and the set of valid transmission
channels according to the 11ax channel access scheme is
CI = {{1}, {2}, {3}, {4}, {1, 2}, {3, 4}, {1, 2, 3, 4}} (see Figure 1).

Fig. 5: Scenario I. WLANs A and B are inside the carrier
sense range of each other with potentially overlapping basic
channels 3 and 4.
Due to the fact that both WLANs are inside the carrier
sense range of each other, their APs could be transmitting
simultaneously at any time t only if their transmission
channels do not overlap, i.e., CAtx (t) ∩ CBtx (t) = ∅. Notice
that slotted backoff collisions cannot occur because their
counters decrease continuously in time, and therefore two
transmissions cannot be neither started nor finished at the
very same time.
5. We have selected Scenario I for depicting the algorithm in a simple
way. CTMNs corresponding to non-fully overlapping scenarios (e.g.,
Scenario III in Section 5) can be also generated with the very same
algorithm.

6

4.2.1

States

4.2.2

CTMN algorithm: finding states and transitions

The first step for constructing the CTMN is to identify the
global state space Ψ, which is simply composed by all the
possible combinations given by the system channelization
scheme and the channel allocations of the WLANs. The

A22 B33

−

∅
s1

)
(s 5
A
µ
,
8
−,

λA , µA (s2 )
1, 3

A22

ψ8

ψ6

A22 B34

A21

ψ9

λ

s5

B,

9,
7

A41

B(

s4

B34
s3

)

ψ10
A21 B34

)
(s 4

B33
ψ7

A21 B33

µ

s2

)
(s 3
B
,µ
λ B 2, 5

A state in the CTMN is defined by the set of WLANs active
and the basic channels on which they are transmitting.
Essentially, we say that a WLAN is active if it is transmitting
in some channel, and inactive otherwise. We define two
types of state spaces: the global state space (Ψ) and the
feasible state space (S ).
• Global state space: a global state ψ ∈ Ψ is a state
that accomplishes two conditions: i) the channels in
which the active WLANs are transmitting comply with
the channelization scheme C , and ii) all active WLANs
transmit inside their allocated channels. That is, Ψ only
depends on the particular channelization scheme C
in use and on the channel allocation of the WLANs
in the system. In this paper, we assume that every
transmission should be made in channels inside Csys
that are composed of a = 2k contiguous basic channels,
for some integer k ≤ log2 Nsys , and that their rightmost
basic channels fall on multiples of a, as stated in the
11ac and 11ax amendments.
• Feasible state space: a feasible state s ∈ S ⊆ Ψ exists
only if each of the active WLANs in such state started
their transmissions by accomplishing the CCA requirement derived from the assigned DCB policy. Namely,
given a global state space, S depends only on the spatial
distribution and on the DCB policies assigned to each
WLAN.
The CTMN corresponding to the toy scenario being
discussed is shown in Figure 6. Regarding the notation,
we represent the states by the most left and most right
basic channels used in the transmission channels of each of
their active WLANs. For instance, state s4 = A21 B43 refers
to the state where A and B are transmitting in channels
CAtx = {1, 2} and CBtx = {3, 4}, respectively.
Concerning the state spaces, states ψ6 , ψ7 , ψ8 , ψ9 , ψ10 ,
ψ11 , ψ12 ∈
/ S are not reachable (i.e., they are global but
not feasible) for two different reasons. First, states ψ11 and
ψ12 are not feasible because of the overlapping channels
involved. Secondly, the rest of unfeasible states are so due
to the fact that AM is applied, thus at any time t that
WLAN A(B) finishes its backoff and B(A) is not active, A(B)
picks the widest available channel, i.e., CAtx (t) = {1, 2, 3, 4}
or CBtx (t) = {3, 4}, respectively. Likewise, any time A(B)
finishes its backoff and B(A) is active, A(B) picks again
the widest available channel, which in this case would be
CAtx (t) = {1, 2} for A and CBtx (t) = {3, 4} for B if A is not
transmitting in its full allocated channel, respectively.
Some states such as s5 = A21 are reachable only via backward transitions. In this case, when A finishes its backoff
and B is transmitting in CBtx (t) = {3, 4} (i.e., s3 ), A picks just
CAtx (t) = {1, 2} because the power sensed in channels 3 and
4 exceeds the CCA as a consequence of B’s transmission.
That is, s5 is only reachable through a backward transition
from s4 , given when B finishes its transmission in state s4 .

λ

A
,µ ,6
A

4

s4
A41 B33
ψ11
A41 B34
ψ12

Fig. 6: CTMN of Scenario I when applying AM. Circles represent states. All the states are global. Specifically, feasible
states are displayed in white, while non-feasible states are
gray colored. Two-way transitions are noted with forward
and backward rates λ, µ, respectively, to avoid cluttering
in the figure. The only backward transition is colored in
red. The blue pair of numbers beside the transition edges
represent the algorithm’s discovery order of the forward
and backward transitions, respectively.
feasible states in S are later identified by exploring the states
in Ψ. Algorithm 1 shows the pseudocode for identifying
both S and the transitions among such states, which are
represented by the transition rate matrix Q.
Essentially, while there are discovered states in S that
have not been explored yet, for any state sk ∈ S not explored, and for each WLAN X in the system, we determine if
X is active or not. If X is active, we then set possible backward
transitions to already known and not known states. To do
so, it is required to fully explore Ψ looking for states where:
i) other active WLANs in the state remain transmitting in the
same transmission channel,6 and ii) WLAN X is not active.
On the other hand, if WLAN X is inactive in state sk , we
try to find forward transitions to other states. To that aim,
the algorithm fully explores Ψ looking for states where i)
other active WLANs in the state remain transmitting in the
6. Notice that we use Xab ∈ s to say that a WLAN X transmits in a
range of contiguous basic channels [a, b] when the CTMN is in state
s. With slight abuse of notation, s − Xab represents the state where
all WLANs that were active in s remain active except for X, which
becomes inactive after finishing its packet transmission. Similarly, s +
Xab , represents the state were all the active WLANs in s remain active
and X is transmitting in the range of basic channels [a, b].

7

Algorithm 1: CTMN generation of spatially distributed
DCB WLAN scenarios.
1 i = 1; .........# Index of the last state found
2 k = 1; ....... # Index of the state currently being explored
3 sk = ∅; ......# State currently being explored
4 S = {sk }; # Set of feasible states
Q = [ ]; ....... # Transition rate matrix
# Generate the global state space Ψ
7 Ψ = generate_psi_space() ;
5

6

8
9
10

11
12

while sk ∈ S do
foreach WLAN X do
# If WLAN is active in sk

if ∃a, b s.t. Xba ∈ sk then
foreach ψ ∈ Ψ do
# If there exists a backward transition

13

if sk − Xba == ψ then
if ψ 6∈ S then
i = i + 1;
k 0 = i;
S = S ∪ ψ;

14
15
16
17
18
19

else

20

22

# Get index of state ψ in S
k 0 = get_index(ϕ∗ );
# New backward transition sk → sk0

23

Qk,k0 = µX (sk )

21

24

# If WLAN is NOT active in sk

25

else
Φ = ∅; . # Set of global states reachable from sk
Φ∗ = ∅; . # Set of feasible states reachable from sk

26
27
28

29
30
31
32

33
34
35
36
37
38
39

# Find possible forward states

foreach ψ ∈ Ψ do
if ∃a, b s.t. sk + Xba == ψ then
Φ = Φ ∪ ψ;
# Function f finds feasible states and corresponding
transition probabilities according to the DCB
policy D and the basic channels found free

CXfree (sk ) = {PXrx (sk , CX ) < CCAX : 0, 1};
~ } = f (D, CXfree , Φ);
{Φ∗ , α
foreach ϕ∗ ∈ Φ∗ do
if ϕ 6∈ S then
i = i + 1;
k 0 = i;
S = S ∪ ϕ∗ ;
else
# Get index of state ϕ∗ in S
k 0 = get_index(ϕ∗ );
# New forward transition sk → sk0

40

41
42

Qk,k0 = α
~ (ϕ∗ )λX ;

43
44

k = k + 1;

same transmission channel, and ii) X is active in the new
state as a result of applying the implemented DCB policy
(D) as shown in line 33. It is important to remark that in
order to apply such policy, the set of idle basic channels
in state sk , i.e., CXfree (sk ), must be identified according to
the power sensed in each of the basic channels allocated
to X, i.e., PXrx (sk , CX ), and on its CCA level. Afterwards,
the transmission channel is selected through the f function,
which applies D. In Figure 4, a simple flowchart of the
transmission channel selection is shown.
Each transition between two states si and sj has a
corresponding transition rate Qi,j . For forward transitions,
the packet transmission attempt rate (or simply backoff rate)
has an average duration λ = 1/(E[B] · Tslot ), where E[B]
is the expected backoff duration in time slots, determined
−1
by the minimum contention window, i.e., E[B] = CWmin
.
2
Furthermore, for backward transitions, the departure rate (µ)
depends on the duration of a successful transmission (i.e.,
µ = 1/tsuc ), which in turn depends on both the data rate
(r) given by the selected MCS and transmission channel
width, and on the packet length E[Ldata ]. In the algorithm
we simply say that the data rate of a WLAN X depends on
the state of the system, which collects such information, i.e.,
µX (s).
Depending on the DCB policy, different feasible forward
transitions may exist from the very same state, which are
represented by the set Φ∗ . As shown in line 43, every feasible
forward transition rate is weighted by a transition probability
vector (α
~ ) whose elements determine the probability of transiting to each of the possible global states in Φ. Namely, with
an slight abuse of mathematical notation, the probability
to transit to any given feasible state ϕ∗ ∈ Φ is α
~ (ϕ∗ ). As
aPconsequence, α
~ must follow the normalization condition
α
~ (ϕ∗ ) = 1.
For the sake of illustration, in the CTMNs of Figures 6,
8 and 10, states are numbered according to the order in
which they are discovered. Transitions between states are
also shown below the edges. Note that with SFCTMN, as
non-fully overlapping networks are allowed, transitions to
states where one or more WLANs may suffer from packet
loses due to interference are also reachable as shown in
Section 5.
4.3

Performance metrics

Since there are a limited number of possible channels to
transmit in, the constructed CTMN will always be finite.
Furthermore, it will be irreducible due to the fact that
backward transitions between neighboring states are always
feasible. Therefore, a steady-state solution to the CTMN
always exists. However, due to the possible existence of
one-way transitions between states, the CTMN is not always
time-reversible and the local balance may not hold [28], and
hence, it prevents to find simple product-from solutions to
compute the equilibrium distribution of the CTMNs.
The equilibrium distribution vector ~π represents the
fraction of time the system spends in each feasible state.
Hence, we define πs as the probability of finding the system
at state s. In order to obtain ~π we can use the transition rate
matrix Q given the system of equations ~π Q = 0.
As an example, for scenario I, considering that its elements are sorted by the discovery order of the states,

8

~π = (π∅ , πA41 , πB43 , πA21 B43 , πA21 ). Besides, the corresponding
transition rate matrix is


∗
λA
λB
0
0
 µA (s2 ) ∗

0
0
0



,
∗
λA
0
Q =  µB (s3 ) 0


0
0 µA (s4 ) ∗ µB (s4 ) 
µA (s5 ) 0
0
λB
∗
where λA , λB and µA (s), µB (s) are the packet generation and
departure rates in state s of WLANs A and B, respectively.
The diagonal elements represented by ‘*’ in the matrix
should be replaced by the negative sum of the rest of items
of their row, e.g., Q4,4 = − µA (s4 ) + µB (s4 ) , but for the
sake of format issues we do not include them in the matrix.
Once ~π is computed, estimating the average throughput
experienced by each WLAN is straightforward. Specifically,
the average throughput of an WLAN w is
X


Γw := E[L]
{γw (s) > CE : 0, 1}µw (s)πs 1 − η ,
s∈S

where E[L] is the expected data packet length, γw (s) is the
SINR perceived by the STA in WLAN w in state s, CE is the
capture effect, and η is the constant packet error probability.
The system aggregate throughput is then the sum of the
PM
throughputs of all the WLANs, i.e., Γ := w=1 Γw . Besides,
in order to evaluate the fairness of a given scenario, we can
use the proportional fairness metric,

P :=

M
X

log10 Γw ,

w=1

and the Jain’s Fairness Index (F ) [29],
2
PM
w Γw
F :=
PM 2  .
M
w Γw

5

I NTERACTIONS IN FREQUENCY AND SPACE

In this section we draw some relevant conclusions about
applying different DCB policies in CSMA/CA WLANs by
analyzing four representative toy scenarios with different
channel allocations and spatial distributions. To that aim we
use the SFCTMN analytical framework. We also validate the
gathered results by means of the 11axHDWLANsSim wireless network simulator (refer to the Appendix for details on
the parameters considered).
5.1

Feasible states dependence on the DCB policy

In Table 1 we show the effect of applying different DCB
policies on the average throughput experienced by WLANs
A and B (ΓA and ΓB , respectively), and by the whole
network (Γ), in scenarios I and II (presented in Figures 5
and 7, respectively).
Let us first consider Scenario I. As explained in Section 4 and shown in Figure 6, the CTMN reaches 5 feasible states when WLANs implement AM. Instead, due to
the fact that both WLANs overlap in channels 3 and 4
when transmitting in their whole allocated channels – i.e.,
CAtx = CA = {1, 2, 3, 4} and CBtx = CB = {3, 4}, respectively
– the SCB policy reaches just three feasible states. Such states
correspond to those with a single WLAN transmitting, i.e.,

TABLE 1: DCB policy effect on the average throughput
[Mbps] in Scenario I and Scenario II. The values obtained
through 11axHDWLANsSim are displayed in red, while the
other correspond to the SFCTMN framework.
Policy
D

|S|

OP

4

SCB

3

AM

5

PU

10

Scenario I
ΓA
ΓB

Γ

113.23 113.23 226.47
113.23 113.23 226.46

143.46 143.46 286.92
131.98 148.85 280.83

220.12 212.21 432.34
217.60 214.81 432.41

149.14 148.38 297.52
149.20 148.42 297.63

|S|

4
3
3
6

Scenario II
ΓA
ΓB

Γ

113.23 113.23 226.47
113.23 113.23 226.46

109.19 109.19 218.38
108.72 108.84 217.56

109.19 109.19 218.38
108.72 108.84 217.56

113.19 113.19 226.38
113.20 113.18 226.38

S = {∅, A41 , B43 }. In the case of OP, both WLANs are forced
to pick just their primary channel for transmitting and,
therefore, S = {∅, A22 , B33 , A22 B33 }. Notice that state A22 B33 is
feasible because A and B have different primary channels
and do not overlap when transmitting in them.
The last policy studied is PU, which is characterized by
providing further exploration on the global state space Ψ. It
usually allows expanding the feasible state space S accordingly because more transitions are permitted. In scenario I,
whenever the CTMN is in state ∅ and the backoff of A or
B expires, the WLANs pick each of the possible available
channels with same probability. Namely, the CTMN will
transit to A22 , A21 or A41 with probability 1/3 when A’s
backoff counter terminates, and to B33 or B43 with probability
1/2 whenever B’s backoff counter terminates.
Likewise, if the system is in state B33 , and A terminates its
backoff counter, the CTMN will transit to the feasible states
A22 B33 or A21 B33 with probability 1/2. Similarly, whenever the
system is in state A22 or A21 , and B finishes its backoff, B
will pick the transmission channels {3} or {3, 4} with same
probability 1/2 making the CTMN to transit to the corresponding state where both WLANs transmit concurrently.
These probabilities are called transition probabilities and are
represented by the vector α
~ X,s (s0 ). For instance, in the latter
case, the probability to transit from s = A22 to s0 = A22 B33
when B terminates its backoff is α
~ B,A22 (A22 B33 ) = 1/2.
5.2

The short vs. long-term throughput dilemma

Intuitively, one could think that, as it occurs in Scenario I,
always picking the widest channel found free by means
of AM, i.e., maximizing the throughput of the immediate
packet transmission (or short-term throughput), may be
the best strategy for maximizing the long-term throughput
as well. However, the Scenario II depicted in Figure 7, is
a counterexample that illustrates such lack of applicable
intuition. It consists of two overlapping WLANs as Scenario I, but with different channel allocation. Specifically,
CA = CB = {1, 2} with pA = 1 and pB = 2, respectively.
The CTMNs that are generated according to the different
DCB policies – generalized to any value the corresponding
transition probabilities α
~ A,∅ , α
~ B,∅ may have – are shown in
Figure 8.
Regarding the transition probabilities, Table 2 shows the
vectors α
~ A,∅ , α
~ B,∅ that are given for each of the studied
DCB policies in Scenario II. Firstly, with OP, due to the fact

9

A11 B22 ,

Fig. 7: Scenario II. WLANs A and B are inside the carrier
sense range of each other with potentially overlapping basic
channels 1 and 2.

,µ

)
s2
(
A

A11
s2

)λ A
s 2 1, 5
(
∅

,
α~ A

)λ A,

∅
s1

3)
µ A(s

(s 3
~ A,∅
α
2, 7
α
~ B,
∅ (s
4 )λ
B, µ
B,
∅(

s5

4,
1

B,

A21

λ A,
B22

)λ
B

0

6,

µ

B(

12

s5

)

s3

B (s
4)

3, 9

α~

λ

s 5)
µ A(
1
8, 1

A11 B22
s6

s4

,µ

B(

s5

)
B12
s5

Fig. 8: CTMN corresponding to Scenario II. Transitions edges
are dashed for referring to those that may be given or not
depending on the DCB policy. For instance, state s6 is only
reachable for the OP and PU policies. The discovery order
of the states and transitions (displayed in blue) corresponds
to the PU policy.

that WLANs are only allowed to transmit in their primary
channel, the CTMN can only transit from state ∅ to states
A11 or B22 , i.e., α
~ A,∅ (s2 ) = α
~ B,∅ (s4 ) = 1. Similarly, with
SCB, WLANs can only transmit in their complete allocated
channel, thus, when being in state ∅ the CTMN transits
only to A21 or B21 , i.e., α
~ A,∅ (s3 ) = α
~ B,∅ (s5 ) = 1. Notice
that AM generates the same transition probabilities (and
respective average throughput) than SCB because whenever
the WLANs have the possibility to transmit – which only
happens when the CTMN is in state ∅ – both A and B
pick the widest channel available, i.e., CAtx = CBtx = {1, 2}.
Finally, PU picks uniformly at random any of the possible
transitions that A and B provoke when terminating their
backoff in state ∅, i.e., α
~ A,∅ (s2 ) = α
~ A,∅ (s3 ) = 1/2 and
α
~ B,∅ (s4 ) = α
~ B,∅ (s5 ) = 1/2, respectively.
TABLE 2: Transition probabilities from state ∅ of WLANs A
and B in Scenario II for different DCB policies.
D

|S|

OP
SCB
AM
PU

4
3
3
6

α
~ A,∅ (s2 ) α
~ A,∅ (s3 ) α
~ B,∅ (s4 ) α
~ B,∅ (s5 )

1.0
0.0
0.0
0.5

0.0
1.0
1.0
0.5

1.0
0.0
0.0
0.5

0.0
1.0
1.0
0.5

Notice that state
when both WLANs are transmitting at the same time, is reachable from states A11 and
B22 for both OP and PU. In such states, when either A or B
terminates its backoff and the other is still transmitting in its
primary channel, only a transition to state A11 B22 is possible,
i.e., α
~ A,s2 (s6 ) = α
~ B,s4 (s6 ) = 1.
Interestingly, as shown in Table 1, applying OP in Scenario II, i.e., being conservative and unselfish, is the best
policy to increase both the individual average throughput of
A and B (ΓA , ΓB , respectively) and the system’s aggregated
one (Γ). Instead, being aggressive and selfish, i.e., applying
SCB or AM, provides the worst results both in terms of
individual and system’s aggregate throughput.
In addition, PU provides similar results than OP in
average because most of the times that A and B terminate
their backoff counter, they can only transmit in their primary
channel, as the secondary channel is most likely occupied by
the other WLAN. In fact, state A11 B22 is the dominant state
for both OP and PU. Specifically, the probability of finding
the CTMN in state A11 B22 , i.e., ~πs6 , is 0.9802 for OP and 0.9702
for PU, respectively. Therefore, the slight differences on
throughput experienced with OP and PU are given because
of the possible transition from ∅ to the states A21 and B12
in PU, where WLANs occupy entirely the allocated channel
preventing the other for decreasing its backoff.
Despite being a very simple scenario, we have shown
that it is not straightforward to determine the optimal DCB
policy that the AP in each WLAN must follow. Evidently,
in a non-overlapping scenario, AM would be the optimal
policy for both WLANs because of the non-existence of
inter-WLAN contention. However, this is not a typical case
in dense scenarios and, consequently, AM should not be
adopted as de facto DCB policy, even though it provides
more flexibility than SCB. This simple toy scenario also
serves to prove that some intelligence should be implemented in the APs in order to harness the information
gathered from the environment.
Concerning the throughput differences in the values
obtained by SFCTMN and 11axHDWLANsSim,7 we note
that the main disparities correspond to the AM and SCB
policies. It is important to remark that while SFCTMN does
not consider neither backoff collisions nor NAV periods,
11axHDWLANsSim actually does so in a more realistic way.
Therefore, in 11axHDWLANsSim, whenever there is an slotted backoff collision, the RTS packets can be decoded by the
STAs in both WLANs if the CE is accomplished. That is why
the average throughputs are increased consequently.
Regarding the NAV periods, an interesting phenomena
occurs in Scenario I when implementing SCB, AM or PU.
While the RTS packets sent by B cannot be decoded by A
because its primary channel is always outside the possible
transmission channels of B (i.e., pA = 2 ∈
/ CBtx = {3, 3} or
{3, 4}), the opposite occurs when A transmits them. Due
to the fact that the RTS is duplicated in each of the basic
channels used for transmitting, whenever A transmits in its
whole allocated channel, B is able to decode the RTS (i.e.,
pB = 3 ∈ CAtx = {1, 2, 3, 4}) and enters in NAV consequently.
7. In 11axHDWLANsSim the throughput is simply computed as the
number of useful bits (corresponding to data packets) that are successfully transmitted divided by the observation time of the simulation.

10

5.3

Cumulative interference and hidden nodes

When considering non-fully overlapping scenarios, i.e.,
where some of the WLANs are not inside the carrier sense
range of the others, complex and hard to prevent phenomena may occur. As an illustrative example, let us consider
the case shown in Figure 9a, where 3 WLANs sharing a
single channel (i.e., CA = CB = CC = {1}) are deployed
composing a line network.
As the carrier sense range is fixed and is the same for
each AP, by locating the APs at different distances we obtain
different topologies that are worth to be analyzed. We name
these topologies from T1 to T4 depending on the distance
between consecutive APs, which increases according to the
topology index. Notice that all the DCB policies discussed
in this work behave exactly the same way in single-channel
scenarios. Therefore, in this subsection we do not make
distinctions among them.
The average throughput experienced by each WLAN
in each of the regions is shown in Figure 9b. Regarding
topology T1, when APs are close enough to be inside the
carrier sense range of each other in a fully overlapping
manner, the medium access is shared fairly because of the
CSMA/CA mechanism. For that reason, the throughput is
decreased to approximately 1/3 with respect to topology T4.
Specifically, the system spends almost the same amount of
time in the states where just one WLAN is transmitting, i.e.,
π(A11 ) = π( B11 ) = π(C11 ) ≈ 1/3.
The neighbor overlapping case in topology T2, where
A and C can transmit at the same time whenever B is not
active, but B can only do so when neither A nor C are
active, is a clear case of exposed-node starvation. Namely,
B has very few transmission opportunities as A and C are
transmitting almost permanently and B must continuously
pause its backoff consequently.
An interesting and hard to prevent phenomena occurs
in the potential central node overlapping case at topology
T3. Figure 10 shows the corresponding CTMN. In this case,
the cumulated interference that B perceives when A and C
transmit at the same time prevents it to decrease the backoff.
However, B is able to decrement the backoff any time A or
C are not transmitting.
This leads to two possible outcomes regarding packet
collisions. On the one hand, if the capture effect condition is
accomplished by B (i.e., γB > CE) no matter whether A and
C are transmitting, B will be able to successfully exchange
packets and the throughput will increase accordingly. On
the other hand, if the capture effect condition is not accomplished, B will suffer a huge packet error rate because
most of the initiated transmissions will be lost due to the
concurrent transmissions of A and C (i.e., γB < CE when
A and C transmit). This phenomena may be recurrent and
have considerable impact in high density networks where
multiples WLANs interact with each other. Therefore, it
should be foreseen in order to design efficient DCB policies.
Finally, as expected, in topology T4, due to the fact that
WLANs are isolated, i.e., outside the carrier sense range of
each other, they achieve the maximum throughput because
all their transmission are successful and they are never
required to pause their backoff.
Concerning the differences on the average throughput
values estimated by SFCTMN and 11axHDWLANsSim, we

observe two phenomena regarding backoff collisions in
topologies T1 and T3. In T1, due to the fact that simultaneous transmissions (or backoff collisions) are permitted and
captured in 11axHDWLANsSim – and such transmissions do
not cause any packet losses as the capture effect is accomplished – the throughput is slightly higher accordingly.
The most notable difference is given in T3. In this topology SFCTMN estimates that B is transmitting just the 50.1%
of the time because, as A and C operate like in isolation,
most of the time they transmit concurrently, causing backoff
freezing at B. However, 11axHDWLANsSim estimates that
B transmits about the 75% of the time, capturing a more
realistic behavior. Such difference is caused by the insensitivity property of the CTMN. For instance, whenever the
system is in state s6 = A11 C11 and A finishes its transmission
(transiting to s4 = C11 ), B decreases its backoff accordingly
while C is still active. In this case it is more probable to
transit from s4 to s7 = B11 C11 than to s6 = A11 C11 again
because, in average, the remaining backoff counter of B
will be smaller than the generated by A when finishing its
transmission. This is in fact not considered by the CTMN,
which assumes the same probability to transit from s4 to s6
than to s7 because of the exponential distribution and the
memoryless property.
5.4

Variability of optimal policies

Most often, the best DCB policy for increasing the own
throughput, no matter what policies the rest of WLANs may
implement, is AM. Nonetheless, there are exceptions like the
one presented in Scenario II. Besides, if achieving throughput
fairness between all WLANs is the objective, other policies
may be required. Therefore, there is not always an optimal
common policy to be implemented by all the WLANs.
In fact, there are cases where different policies must be
assigned to different WLANs in order to increase both the
fairness and individual WLAN throughputs.
For instance, let us consider another toy scenario (Scenario IV) using the topology T2 of Scenario III, where three
WLANs are located in a line in such a way that they are in
the carrier sense range of the immediate neighbor. In this
case, however, let us assume a different channel allocation:
CA = CB = CC = {1, 2} and pA = pC = 1, pB = 2.
Table 3 shows the individual and aggregated throughputs, the proportional throughput fairness, and the Jain’s
fairness index for different combinations of DCB policies.
We note that, while implementing AM in all the WLANs
the system’s aggregated throughput is the highest (i.e.,
Γ = 428.73 Mbps), the throughput experienced by B is
the lowest (i.e., ΓB = 4.04 Mbps), leading to a very unfair
situation as indicated by J = 0.67928.
We also find another case where implementing AM does
not maximize the individual throughput. Namely, when A
and C implement AM (i.e., DA = DC = AM), it is preferable
for B to implement PU and force states in which A and
C transmit only in their primary channels. This increases
considerably both the throughput of B (i.e., ΓB = 66.46
Mbps) and the fairness (i.e., P = 6.206 and J = 0.89988).
Looking at the most fair combinations, we notice that A,
C or both must implement PU in order to let B transmit with
similar amount of opportunities. Specifically, when both A

11

(b) Average throughput.

(a) Topologies.

Fig. 9: Scenario III. Yellow and blue arrows indicate the carrier sense range of WLANs A and C, respectively. The carrier sense
range of WLAN B is not displayed. T3-noCE refers to topology T3 when B does not accomplish the capture effect condition
whenever A, B and C are active. MC and Sim refer to the values obtained through SFCTMN and 11axHDWLANsSim.

A

s2 em

4
1,

∅

2

C

−, 22

s6

6

1
ty 10,
9t,y
p 2
em 0
11, 19

15
,2

3

AC

s3 em
p
3,
1

s5

13

B

2, 8

s1

5, 14

pt
y y 7,
p6,t 1
em 7

6

AB

1
,2
18

ABC

s8

BC

s4

s7

Fig. 10: CTMN corresponding to Scenario III-T3. For the sake
of visualization, neither the transition rates nor the transmission channels are included in the figure. The discovery
order of the transitions is represented by the pairs in blue.

TABLE 3: Policy combinations effect on throughput and
fairness in the WLANs of Scenario IV.
Policy

Throughput [Mbps]

DA DB DC

AM
AM
PU
PU
AM
AM

AM
PU
AM
PU
AM
PU

AM
AM
PU
PU
PU
PU

Fairness

ΓA

ΓB

ΓC

Γ

P

212.35
155.48
113.76
113.77
115.36
115.33

04.04
66.46
112.27
112.27
110.70
110.73

113.77
113.77
113.76
113.77
114.28
114.28

428.73
377.42
339.79
339.81
340.34
340.34

5.260
6.206
6.162
6.162
6.164
6.164

J

0.67928
0.89988
0.99996
0.99996
0.99969
0.99970

and C implement PU, B experiences the highest throughput
and the system achieves the highest fairness (i.e., J ≈ 1)
accordingly. Nonetheless, the price to pay is to heavily
decrease the throughput of A and C. Therefore, as most of
the times it does not exist a global policy that satisfies all the
WLANs in the system, different policies should be adopted
depending on the parameter to be optimized.

E VALUATION OF DCB IN HD WLAN S

In this Section we study the effects of the presented DCB
policies on dense WLAN scenarios. We first draw some general conclusions from analyzing the throughput and fairness
obtained through the different policies when increasing the
number of WLANs per area unit. Then, we discuss what is
the optimal policy that a particular WLAN should locally
pick in order to maximize its own throughput.
The results gathered in this section have been obtained
using 11axHDWLANSim, an event-based wireless network
simulator that aims to capture the effects of the newest
technologies included in the IEEE 802.11 amendments. The
values of the considered parameters are shown in the Appendix.
6.1

Network density vs. throughput

Figure 11 shows the general scenario considered for conducting the experiments presented in this Section. For the
sake of speeding up 11axHDWLANSim simulations, we assume that each WLAN is composed just of 1 AP and 1 STA.
Due to the fact that APs and STAs are located randomly
in the map, the number of STAs should not have a significant impact on the results because only downlink traffic is
assumed.
Essentially, we consider a rectangular area Amap =
100 × 100 m2 , where M WLANs are placed uniformly at
random with the single condition that any pair of APs must
be separated at least dmin
AP-AP = 10 m. The STA of each
WLAN is located also uniformly at random at a distance
max
dAP-STA ∈ [dmin
AP-STA , dAP-STA ] = [1, 5] m from the AP. The
channelization C counts with Nsys = 8 basic channels
and follows the 11ax proposal (see Figure 1). The channel
allocation is also set uniformly at random, i.e., every WLAN
w is assigned a primary channel pw ∼ U [1, 8] and allocated
channel CX containing Nw ∼ U {1, 2, 4, 8} basic channels.
For each number of WLANs studied (i.e., M =
2, 5, 10, 20, 30, 40, 50), we generate ND = 50 deployments
with different random node locations and channel allocations. Then, for each of the deployments, we assign to all
the WLANs the same DCB policy. Namely, we simulate

12

Fig. 11: Example scenario where M = 11 WLANs are spread
uniformly at random in a map of size mw × mh .

NM × ND × NP = 7 × 50 × 4 = 1400 scenarios, where NM
is the number of different M values studied and NP is the
number of DCB policies considered in this paper. Besides,
all the simulations have a duration of Tobs = 20 seconds.
In Figure 12, it is shown by means of boxplots8 the
average throughput per WLAN for each of the presented
DCB policies. As expected, when there are few WLANs in
the area, the most aggressive policies (i.e., SCB and AM)
provide the higher throughputs. Instead, PU, and especially
OP, perform the worst, as they do not extensively exploit the
free bandwidth.
However, when M increases and the scenario gets
denser, the average throughput obtained by all the policies
except SCB tends to be similar. This occurs due to the fact
that WLANs implementing AM and PU will likely perform
single-channel transmissions (as OP does) because the PIFS
condition for multiple channels will most likely not be
accomplished. In the case of SCB, some of the basic channels
inside the WLAN’s allocated channel will most likely be
occupied by other WLANs, and therefore its backoff counter
will get repeatedly paused. Thus, its average throughput in
dense scenarios is considerably low with respect to the other
policies.
In order to asses the use of the spectrum, we first define
the average bandwidth usage of a WLAN w as
BWw =

Nsys
1 X tx
t (c) · |c|,
Tobs c=1 w

where ttx
w (c) is the time that WLAN w is transmitting in a
channel containing at least the basic channel c, and |c| is the
bandwidth of a basic channel (i.e., 20 MHz). We can then
define the spectrum utilization of a given scenario as the
ratio
P
ACS M
w=1 BWw
ρ=
,
Amap · Nsys · |c|
where ACS is the circular carrier sense area of each WLAN
when transmitting in a single-channel. The average of such
ratio in the considered scenarios is shown in Figure 13.
Similarly to the throughput, while OP and PU do not
leverage the free spectrum in low density scenarios, SCB
8. Matlab’s bloxplot definition: https://www.mathworks.com/help/
stats/boxplot.html

Fig. 12: Node density effect on the average WLAN throughput. On each box, the central mark indicates the median,
and the bottom and top edges of the box indicate the 25th
and 75th percentiles, respectively. The whiskers extend to
the most extreme data points not considered outliers, and
the outliers are plotted individually using the ‘+’ symbol.

Fig. 13: Node density effect on spectrum utilization.
and AM do so by exploiting the most bandwidth. Instead,
when the number of nodes per area increases, SCB suffers
from heavy contention periods, which reiterates the need
of flexibility to adapt to the channel state. In this regard,
we note that AM is clearly the policy exploiting the most
bandwidth in average for any number of WLANs.
Nonetheless, neither the average throughput per WLAN
nor the spectrum utilization may be a proper metric when
assessing the performance of the whole system. Namely,
having some WLANs experiencing high throughputs when
some others starve is often a situation preferable to be
avoided. In that sense, we focus on the fairness, which is
both indicated by the boxes and outliers in Figure 12, and

13

more clearly by the expected Jain’s fairness index shown in
Figure 14.

WLANs operating with the same primary channel due to
heavy interference, specially when OP is being used.
6.2

Fig. 14: Node density effect on Jain’s Fairness Index.
As expected, the policy providing the highest fairness
is OP. In fact, no matter the channel allocation, WLANs
only pick their primary channel for transmitting when implementing OP; hence the fairness is always maximized at
the cost of probably wasting part of the frequency spectrum,
specially when the node density is low. In this regard, PU
also provides high fairness while exploiting the spectrum
more, which increases the average throughput per WLAN
accordingly.
Regarding the aggressive policies, as expected, SCB is
clearly the most unfair policy due to its ‘all or nothing’
strategy. Therefore, it seems preferable to prevent WLANs
from applying SCB in dense scenarios because of the number of WLANs that may starve or experience really low
throughput. However, even though being aggressive, AM
is able to adapt its transmission channel to the state of
the medium, thus providing both higher throughput and
fairness.
Still, as indicated by the boxes and outliers of Figure
12, AM is not per se the optimal policy. In fact, there are
scenarios where PU performs better in terms of both fairness
and throughput. Consequently, there is room to improve
the presented policies with some smarter adaptation or
learning approaches (e.g., tunning properly the transition
probabilities α
~ when implementing stochastic DCB).
There are also some phenomena that we have observed
during the simulations that are worth to be mentioned.
Regarding backoff decreasing slowness, in some scenarios,
it can be the case that a WLAN w is forced to decrease its
backoff counter very slowly due to the fact that neighboring
WLANs operate in a channel including the primary channel
of w. That is why more fairness is achieved with PU in dense
networks as such neighboring WLANs do not always pick
the whole allocated channel. Thus, they let w to decrease
their backoff more often, and to proceed to transmit accordingly.
Finally, concerning the transmission power and channel
width, we have observed that transmitting just in the primary channel can also be harmful to other WLANs because
of the higher transmission power used per 20 MHz channel.
While this may allow to use higher MCS and respective
data rates, it may also cause packet losses in neighboring

Local optimal policy

With the following experiment we aim to identify what
would be the optimal policy that a particular WLAN should
adopt in order to increase its own throughput. In this case,
we consider an area Amap = 50 × 50 m2 with one WLAN
(A) located at the center, and M − 1 = 19 WLANs spread
uniformly at random in the area. Regarding the channel allocation, all the WLANs are again set with primary channel
and allocated channel selected uniformly at random, with
the exception of A, which is allocated the widest channel
(i.e., CA = {1, ..., 8}) in order to provide more flexibility
and capture complex effects. The primary channel of A is
also selected uniformly at random (i.e., pA ∼ U [1, 8]).
While the DCB policies of the M − 1 WLANs are also
set uniformly at random (i.e., they will implement OP, SCB,
AM or PU with same probability 1/4), A’s policy is set in
a deterministic way. Specifically, we generate ND = 400
deployments following the aforementioned conditions for
each of the DCB policies that A can implement. That is, we
simulate ND × NP = 1600 scenarios. The simulation time of
each scenario is Tobs = 25 seconds.
The first noticeable result is that, in dense scenarios,
SCB is non-viable for WLANs with wide allocated channels
because they are most likely prevented to initiate transmissions. In fact, A is not able to successfully transmit data
packets in any of the 400 scenarios simulated for SCB (i.e.,
E[ΓSCB
A ] = 0).
Regarding the rest of policies, on average, A’s throughput is 3.2 % higher when implementing AM (E[ΓAM
A ] =
41.07 Mbps) with respect to PU (E[ΓPU
A ] = 39.74 Mbps).
Besides, for dense scenarios like this one, there is a clear
trend to pick just one channel when implementing AM
and PU. That is why OP provides an average throughput
(E[ΓOP
A ] = 39.04 Mbps) close to the ones achieved by AM
and PU.
Nonetheless, as the high standard deviation of the
throughput indicates (e.g., σAM (ΓA ) = 38.25 Mbps when
implementing AM), there are important differences regarding ΓA depending on the scenario. Table 4 compares the
share of scenarios where AM or PU provide the highest
individual throughput for A. Three types of outcomes are
categorized accordingly a defined margin δΓ :
AM
(a) PU performs better than AM: E[ΓPU
A ] − E[ΓA ] > δΓ
AM
(b) PU performs similarly to AM: |E[ΓPU
]
−
E
[Γ
A
A ]| < δΓ
PU
(c) PU performs worse than AM: E[ΓAM
]
−
E
[Γ
A
A ] > δΓ
We use the margin δΓ = 0.5 Mbps for capturing the cases
where AM and PU perform similarly.
TABLE 4: Share of scenarios where AM or PU provide the
highest individual throughput for WLAN A.
AM
E[ΓPU
A ] > E[ΓA ]

AM
E[ΓPU
A ] ≈ E[ΓA ]

PU
E[ΓAM
A ] > E[ΓA ]

97/400 (24.25 %)

135/400 (33.75 %)

168/400 (42.00 %)

We see that in most of the cases AM performs better
than PU. However, in about 1/4 of the scenarios PU improves AM and this share is even higher when considering

14

smaller values of δΓ . Also, there are a few scenarios where
the throughput experienced by PU with respect to AM is
considerable (more than 30 %). This mainly occurs when the
neighboring nodes frequently occupy A’s primary channel
through complex interactions that keep frozen its backoff
accordingly for long periods of time.
Therefore, as a rule of thumb for dense networks, we
can state that, while AM reaches higher throughputs on
average, stochastic DCB is less risky and performs similarly
well. Nonetheless, even though PU is more fair than AM
on average, it does not guarantee the absence of starving
WLANs either. It follows that WLANs must be provided
with some kind of adaptability to improve both the individual throughput and fairness with acceptable certainty.

7

C ONCLUSIONS

In this work we show the effect of DCB in spatially distributed scenarios, where WLANs are not required to be
within the carrier sense range of each other. We study
different types of DCB policies including a new approach
that selects the transmission channel width probabilistically.
We also present the SFCTMN analytical framework for
modeling WLAN scenarios through CTMNs, and use it to
depict the phenomena that occur in several toy scenarios. In
this regard, we prove that i) the feasible states in the CTMNs
highly depend on the DCB policy, ii) always selecting the
widest available channel found free does not always maximize the own throughput (short vs. long-term throughput
dilemma), iii) cumulative interference from hidden nodes
may cause important throughput losses in spatially distributed scenarios due to packet losses or inadmissible large
contention periods, and iv) often, there is not an optimal
global policy to be applied to each WLAN, but different
policies are required, specially in non-fully overlapping
scenarios where chain reaction actions are complex to foresee
in advance.
Besides, we analyze via simulations the impact of the
node density on the individual throughput and system’s
fairness in 11ax WLANs. Results corroborate that, while
DCB is normally the best policy to maximize the individual
short and long-term throughput, there are cases, specially in
high density scenarios, where other policies like stochastic
PU perform better both in terms of individual throughput
and throughput fairness among WLANs.
Therefore, we conclude that the performance of DCB
can be importantly improved by implementing adaptive
policies capable of harnessing the knowledge gathered from
the medium and/or via information distribution. In this
regard, our next work will focus on studying machine learning based policies to enhance WLANs performance in high
density scenarios. We also aim to capture non-saturation
regimes in the SFCTMN framework, as the reduction of
overlapping situations may have an important impact on
the performance of the policies.

A PPENDIX A
11 A X HDWLAN S S I M & 11 AX PARAMETERS
For the high-density results presented in Section 6, we have
simulated multiple 11ax scenarios9 in 11axHDWLANsSim.
This simulator is a particular release of Komondor,10 a
wireless networks simulator built on top of COST library
[30]. COST facilitates the development of discrete event simulation using CompC++, a component-oriented extension
to C++. COST provides a collection of components that
interact with each other by exchanging messages through
communication, and a simulation engine that is responsible
for synchronizing components.
The value of the parameters considered in the simulations are shown in Table 5. Regarding the path loss,
we use the simple but accurate partition loss model for 5
GHz indoor environments with corridors proposed in [31].
Specifically, the path loss in dB experienced at a distance d
is defined by PL(d) = PLFree (d) + αd, where PLFree (d) is the
well-known free space path loss at distance d and α = 0.44
dB/m is the constant attenuation per unit of path length
that fits the best to the measurements collected in an office.
TABLE 5: Parameters considered in the presented scenarios.
Parameter

Description

Value

CWmin
m
CCA
Ptx
Gtx
Grx
Ldata
LBACK
LRTS
LCTS
nagg
CE
N
Tslot
SIFS
DIFS
PIFS
η
fc
Tofdm
Tphy
nss
HE
Tphy
Lsf
Ldel
Lmac
Ltail
ACS

Min. contention window
Backoff stage
CCA threshold
Transmission power
Transmitting gain
Reception gain
Length of a data packet
Length of a block ACK
Length of an RTS packet
Length of a CTS packet
Num. data packets aggregated
Capture effect threshold
Background noise level
Slot duration
SIFS duration
DIFS duration
PIFS duration
Packet error rate
Central frequency
OFDM symbol duration
Legacy PHY header duration
SU spatial streams
HE header duration
Length of MAC’s service field
Length of MAC’s MPDU delimiter
Length of MAC header
Length of MAC’s tail
Single-channel carrier sense area

16
5
-82 dBm
15 dBm
0 dB
0 dB
12000 bits
240 bits
160 bits
112 bits
64
20 dB
-95 dBm
9 µs
16 µs
34 µs
25 µs
0.1
5 GHz
16 µs
20 µs
1
32 µs
16 bits
32 bits
272 bits
6 bits
5384.6 m2

The data rate (r) used in the data packet transmissions
among all the nodes has always been the highest allowed
by the MCS in the 11ax depending on the distance between
AP and STA. Specifically, r = Nsc Ym Yc nss , where Nsc is
9. It is important to note that the 11ax amendment is still in an
early development stage and newer version may include changes with
respect the parameters considered in this work.
10. All of the source code of Komondor is open, encouraging sharing
of algorithms between contributors and providing the ability for people
to improve on the work of others under the GNU General Public
License v3.0. The release used in this work and all the output files of the
performed simulations can be found at https://github.com/wn-upf/
Komondor.

15

the number of subcarriers (dependent on the transmission
channel width), Ym is the number of modulation bits and Yc
is the coding rate. The latter two parameters depend on the
MCS. Nsc can be 234, 468, 980 or 1960 for 20, 40, 80, and 160
MHz channel widths, respectively. For instance, for the MCS
providing highest data rate, Ym = 10 bits and Yc = 5/6. Accordingly, the data rate for 20 MHz channel transmissions is
defined by r20MHz = 52Ym Yc . With such parameters we can
define the duration of the different packets transmissions,
and the duration of a successful transmission accordingly:

[9]

[10]

[11]

[12]




Lsf + LRTS + Ltail
Tofdm ,
r20MHz


Lsf + LCTS + Ltail
Tofdm ,
tCTS = Tphy +
r20MHz
HE
tdata = Tphy + Tphy
+


Lsf + nagg (Ldel + Lmac + Ldata ) + Ltail
+
Tofdm ,
r


Lsf + LBACK + Ltail
tBACK = Tphy +
Tofdm ,
r20MHz
tsuc = tRTS + SIFS + tCTS + SIFS+
+ tdata + SIFS + tBACK + DIFS + Tslot .

tRTS = Tphy +

ACKNOWLEDGMENT
This work has been partially supported by the Spanish
Ministry of Economy and Competitiveness under the Maria
de Maeztu Units of Excellence Programme (MDM-20150502), and by a Gift from the Cisco University Research Program (CG#890107, Towards Deterministic Channel Access
in High-Density WLANs) Fund, a corporate advised fund
of Silicon Valley Community Foundation. The work done
by S. Barrachina-Muñoz is supported by a FI grant from the
Generalitat de Catalunya.

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Sergio Barrachina-Muñoz obtained his B.Sc.
degree in Telematics Engineering and his M.Sc.
in Intelligent Interactive Systems in 2015 and
2016, respectively, both from Universitat Pompeu Fabra (UPF), Barcelona. Currently, he is a
PhD student and teacher assistant in the Department of Information and Communication Technologies (DTIC) at Universitat Pompeu Fabra
(UPF). His main research interests are focused
on developing autonomous learning methods
and techniques for improving the performance of
next-generation wireless networks.

Francesc Wilhelmi holds a BSc degree in
Telematics Engineering from the Universitat
Pompeu Fabra (2015) with a focus on Broadband and Wireless Communications. With the
aim of applying new techniques for solving
many well-known problems in communications,
Francesc obtained his MSc degree in Intelligent
and Interactive Systems also from the UPF in
2016. He is now a PhD Student in the Wireless Networking Group (WN) of the Department
of Information and Communication Technologies
(DTIC) at the UPF. The main topics of his PhD Thesis are related
to spatial reuse in high-density wireless networks through power and
sensitivity adjustment by taking advantage of Reinforcement Learning
(RL) techniques.

Boris Bellalta is an Associate Professor in
the Department of Information and Communication Technologies (DTIC) at Universitat Pompeu Fabra (UPF). He obtained his degree in
Telecommunications Engineering from Universitat Politècnica de Catalunya (UPC) in 2002
and the Ph.D. in Information and Communication
Technologies from UPF in 2007. His research
interests are in the area of wireless networks,
with emphasis on the design and performance
evaluation of new architectures and protocols.
The results from his research have been published in more than 100
international journal and conference papers. He is currently involved
in several international and national research projects, including the
coordination of the ENTOMATIC FP7 collaborative project. At UPF he
is giving several courses on networking, queuing theory and wireless
networks. He is co-designer and coordinator of the interuniversity (UPF
and UPC) masters degree in Wireless Communications.

