On the Interactions between Multiple Overlapping WLANs using

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5IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, 2015. 1

On the Interactions between MultipleOverlapping WLANs using Channel Bonding

Boris Bellalta, Alessandro Checco, Alessandro Zocca, Jaume Barcelo

Abstract—Next-generation WLANs will support the use ofwider channels, which is known as channel bonding, to achievehigher throughput. However, because both the channel centerfrequency and the channel width are autonomously selected byeach WLAN, the use of wider channels may also increase thecompetition with other WLANs operating in the same area forthe available channel resources. In this paper, we analyse theinteractions between a group of neighboring WLANs that usechannel bonding and evaluate the impact of those interactionson the achievable throughput. A Continuous Time MarkovNetwork (CTMN) model that is able to capture the coupleddynamics of a group of overlapping WLANs is introduced andvalidated. The results show that the use of channel bonding canprovide significant performance gains even in scenarios witha high density of WLANs, though it may also cause unfairsituations in which some WLANs receive most of the transmissionopportunities while others starve.

Index Terms—WLANs, CSMA/CA, channel bonding, channelallocation, dense networks, IEEE 802.11ac, IEEE 802.11ax

I. I NTRODUCTION

The number of multimedia devices, including smartphones,laptops and High Definition (HD) audio/video players, thataccess the Internet through deployed WLAN Access Pointsis increasing every day and everywhere. To improve the per-formance of WLANs, the use of wider channels —comparedto a single or basic20 MHz channel— has been consideredrecently. This technique is commonly known as channel bond-ing [1].

The use of channel bonding in WLANs was introduced inthe IEEE 802.11n amendment [2], where two basic20 MHzchannels can be aggregated to obtain a 40 MHz channel. TheIEEE 802.11ac amendment [3] further extends this feature byallowing the use of80 and160 MHz channels by grouping 4and 8 basic channels, respectively. It is expected that futureWLAN amendments, such as the IEEE 802.11ax, will continueto develop the use of wider channels [4].

However, the use of channel bonding also increases theprobability that WLANs operating in the same area will over-lap (i.e., two WLANs overlap if they share at least one basic

Copyright (c) 2013 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to [email protected]. Bellalta and J. Barcelo are with Universitat Pompeu Fabra, Barcelona,Spain; A. Checco is with Hamilton Institute, Ireland; A. Zocca is withEindhoven University of Technology, The Netherlands. The research of the 1stand 4th authors was partially supported by the Spanish government (projectTEC2012-32354), and by the Catalan Government (SGR2009#00617). The re-search of the 2nd author was financially supported by the Science FoundationIreland (grant 11/PI/11771). The research of the 3rd authorwas financiallysupported by The Netherlands Organization for Scientific Research (NWO)through the TOP-GO grant 613.001.012. Corresponding author: B. Bellalta.e-mail: [email protected]

channel), which may cause severe performance degradation forsome or all of them. This performance degradation is causedby the coupled dynamics that occur between the overlappingWLANs due to the listen-before-talkcharacteristic of theCSMA/CA protocol. This effect may be particularly relevantin urban areas, where the high density of WLANs may impactthe suitability of this approach.

To better understand the coupled dynamics during theoperation of overlapping WLANs using channel bonding andto evaluate their effects in terms of performance, we model thedescribed scenario using a Continuous Time Markov Network(CTMN) [5]. We show that the CTMN model is able toaccurately capture the operation and the achievable throughputof each WLAN, despite considering a continuous backofftimer instead of the slotted backoff counter that is used in theIEEE 802.11 Distributed Coordination Function (DCF). Notethat models of the DCF that assume that all nodes are able tolisten all transmissions from other nodes, such as the modelpresented in [6], are not valid for the scenarios consideredinthis paper because this requirement does not hold in general.

The contributions of this paper are as follows:1) We introduce a CTMN model that captures the cou-

pled dynamics of multiple overlapping non-saturatedWLANs. It allows to configure at each node the trafficload, the packet size, the backoff contention window(CW), the channel position and width, and the trans-mission rate.

2) To improve the computational efficiency when solvingthe CTMN model, we reduce its number of states byaggregating the activity of all nodes that belong to thesame WLAN. We refer to it as the WLAN-centric model.

3) We describe, model and categorise the interactions thatoccur between multiple overlapping WLANs, as well ascapture their coupled operation using the WLAN-centricmodel. We also show that some of the interactions aresimilar to those that appear in single-channel CSMA/CAmulti-hop networks.

4) We formulate the optimal proportional fair channel al-location for WLANs when they use channel bonding,which gives us the upper bound performance for a groupof overlapping WLANs in saturated conditions.

5) We evaluate numerically the performance achieved by agroup of neighboring WLANs that use channel bondingas a function of the number of overlapping WLANs,the number of available basic channels and the set ofchannel widths when WLANs randomly choose boththe channel center frequency and the channel width.We then compare the results with those obtained usingthe proposed optimal proportional fair channel allocation

http://arxiv.org/abs/1412.0929v2

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, 2015. 2

scheme.

The paper is structured as follows. First, we introduce somerelated work in Section II. In Section III, we describe thesystem model and all of the assumptions that are made. InSection IV, we present and validate the analytical model.Section V characterises the potential interactions betweenWLANs. It also describes the extension of the node-centricthroughput model to a WLAN-centric model in order toimprove the computational efficiency when solving it, andprovide a more compact characterisation of the overall systemas well. In Section VI, we introduce both the centralised anddecentralised channel allocation schemes considered in thiswork. In particular, for the centralised case, we propose awaterfilling algorithm for allocating channels to a group ofoverlapping WLANs, as well as the hypothesis that result inthe optimal proportional fair allocation. We present the resultsin Section VII, studying the effect that the quantity of availablebasic channels and of WLANs has on the system performance.Finally, the most important results of the paper are summarisedin the conclusions, and several recommendations about the useof channel bonding in next-generation WLANs are provided.

II. RELATED WORK

Since most previous studies only focused on channelbonding, channel selection algorithms or continuous timeCSMA/CA throughput models, we present the related work inthree separate sections. To the best of our knowledge, only [7]and [8] simultaneously consider the channel center frequencyand channel width selection.

A. Channel Bonding

The performance gains and drawbacks of channel bondingin IEEE 802.11n WLANs are analysed experimentally in [1],[7], where the authors show that channel bonding results in:i) a lower SINR (Signal to Interference and Noise Ratio)due to the reduction of the transmission power per Hz eachtime the channel width is doubled,ii) a lower coveragerange because wider channels require higher sensitivity,iii)a greater chance to suffer from and create interference, andiv) more competition with other WLANs operating in thesame area. However, they also show that channel bonding canprovide significant throughput gains when those issues can beovercome by adjusting the transmission power and rate.

The same considerations as in IEEE 802.11n are validfor channel bonding in IEEE 802.11ac. However, because itextends the channel bonding capabilities of IEEE 802.11nby allowing the use of80 and 160 MHz channels, boththe negative and positive aspects are accentuated. Therefore,there is much interest in developing effective solutions atboth PHY and MAC layers to get the most benefit fromchannel bonding. The performance of channel bonding inIEEE 802.11ac WLANs has been investigated by simulationin [9], [10], where both SBCA (Static Bandwidth ChannelAccess) and DBCA (Dynamic Bandwidth Channel Access)schemes are considered. The results presented in [9], [10]show that channel bonding can provide significant through-put gains, but also corroborate the fact that these gains

are severely compromised by the activity of the overlappingwireless networks. The impact of hidden nodes on the networkperformance in a specific scenario is evaluated in [9], wherea protection mechanism based on the exchange of RTS/CTSframes is proposed. The sensitivity of the secondary basicchannels and how the position of the primary basic channelaffects the system performance are evaluated in [10]. However,neither [9] nor [10] present any analytical model. Finally,channel bonding for short-range WLANs is considered in [11],where the impact of other WLANs on the system performanceis evaluated.

B. Channel Selection Algorithms

Channel selection algorithms in wireless networks havebeen the subject of numerous investigations. The first studieson this topic focused on either centralised or distributedschemes that rely on message passing (see for instance [12],[13], [14], [15] and references therein).

These schemes are not applicable in our case, however,because different WLANs generally have different adminis-trative domains: indeed, as such, they are independent andautonomous systems. Channel bonding complicates the anal-ysis even more because different groups of basic channelsare used, which potentially makes communication betweenWLANs more difficult.

Several solutions, based mainly on graph theory, have beenproposed trying to consider these constraints on communi-cation. This solution requires decentralised algorithms forchannel selection, see [16], [17], [18].

Channel selection when wider channels are used has beenconsidered only in [7] and [8]. In [7], the authors proposean algorithm to dynamically select the channel center fre-quency and to dynamically switch between a 20 or a 40MHz channel width to maximise the throughput. However,the authors assume that the access points (APs) are able toexchange information (i.e., the achieved throughput on eachchannel) or that a central authority provides such information.A decentralised algorithm is proposed in [8] to select both thechannel center frequency and the channel width by sensing theinterference that is caused by the other neighboring WLANs.

C. Continuous Time CSMA/CA Models

The use of CTMN models for the analysis of CSMA/CAnetworks was originally developed in [5] and was furtherextended in the context of IEEE 802.11 networks in [19],[20], [21], [22], [23], among others. Although the modellingof the IEEE 802.11 backoff mechanism is less detailed thanin Bianchi [6], it offers greater versatility in modelling abroad range of topologies. Moreover, the experimental resultsof [20], [21] demonstrate that CTMN models, even if stylized,provide remarkably accurate throughput estimates for actualIEEE 802.11 systems. A comprehensible example-based tuto-rial of CTMN models applied to different wireless networkingscenarios can be found in [24].

Boorstyn et al. [5] introduce the use of CTMN modelsto analyse the throughput of multi-hop CSMA/CA networksand study several network topologies, including a simple


chain, a star and a ring network. Wang et al. [23] extendthe work in [5] by considering also the fairness betweenthe throughput achieved by each node, as well as providingseveral approximations with the goal of reducing the modelcomplexity by using only local information. In addition, theyrelate the parameters of the CTMN model with those definedby the IEEE 802.11 standard, such as the contention windowand the use of RTS/CTS frames. Durvy et al. [19] alsouse CTMN models to characterise the behaviour of wirelessCSMA/CA networks and investigate their spatial reuse gain.Nardelly et al. [21] extend previous models to specificallyconsider the negative effect of collisions and hidden terminals.They evaluate several multi-hop topologies and compare theresults with experimental data to show that CTMN modelscan be very accurate. Liew et al. [20] validate the accuracy ofCTMN to model CSMA networks using both simulations andexperimental data. They also introduce a simple but accuratetechnique to compute the throughput of each node based onidentifying the maximum independent sets of transmittingnodes. Recently, Laufer et al. [22] extended such CTMNmodels to support non-saturated nodes and flow-based analysisof multi-hop networks. Finally, the CTMN model presentedin [22] is used in [25] to evaluate the performance of avehicular video surveillance system.

III. SYSTEM MODEL

In the following, we will say that a group of WLANsare neighbors when all of the WLANs are within the carriersense range of the others. A WLAN may belong to severalgroups of neighboring WLANs, and therefore, those groups ofneighboring WLANs may also interact between them throughit (see Figure 1). Table I summarizes the notation used in thispaper.

A. Network description

We consider a system withM WLANs spatially distributedover a certain area, where WLANi containsUi nodes, i.e., theAP andUi−1 STAs. A set ofN predefined basic channels areat the disposal of allM WLANs. When WLAN i is initiated,or switches to a new channel, it selects a channelCi of widthWi, which is a contiguous subset ofci = |Ci| basic channels.If a basic channel has a width of 20 Mhz, then the widthof channelCi is given byWi = 20 · ci. The global set ofchannel allocations for theM WLANs is C = C1, . . . , CM.We say that WLANsi and k overlap if Ci andCk share atleast one basic channel, i.e., ifCi ∩Ck 6= ∅, given that bothiandk are inside the carrier sense range of the other. In casetwo WLANs overlap, we assume they are outside the datacommunication range of the other, which makes the adjacentchannel interference negligible. Finally, we also assume thatthe propagation delay between any pair of nodes is zero.

B. Node operation

The traffic load of nodej in WLAN i isαi,j packets/second.When a node has a packet ready for transmission, it checksthe state of the channelCi that it has allocated. Once the

a

b

c1

c2

d

WLAN A

WLAN B

WLAN C WLAN D

AP

STA

Fig. 1. Two groups of neighboring WLANs (WLANs A, B and C in theone hand; and WLANs C and D in the other). The Data CommunicationRange (continuous line) and Carrier Sense Range (dashed line) are indicatedin the plot. The two groups of neighboring WLANs interact because WLANC belongs to both of them. Nodesa, b, c1, c2 andd are transmitting a dataflow.

channel has been sensed as being free for the duration ofa DIFS (Distributed InterFrame Space), the node starts thebackoff procedure by randomly initializing a timer. Everytime a portion of the channel is detected as busy during thebackoff interval, the backoff countdown is frozen until theentire channel widthWi is detected as free again for theduration of a DIFS interval. This counter is decremented untilit reaches zero, at which time the node starts transmittinga packet using the entire channel widthWi. Note that allnodes belonging to a group of neighboring WLANs will defertheir backoff countdown accordingly if they share at least abasic channel with the transmitting node. Figure 2 shows theoperation of the channel access for the specific case in whichthe target node uses four basic channels.

We assume that the backoff countdown at each node isin continuous time and has an average duration ofE[Bi,j ]seconds for nodej in WLAN i. Therefore, when nodej haspackets waiting for transmission, the attempt rate for everynode is equal toλi,j = E[Bi,j ]

−1.The duration of a transmission of a packet by nodej in

WLAN i is denoted byTi,j(ci, γi,j , Li,j) and depends on thenumberci of basic channels used, on the Signal-to-Noise Ratio(SNR) observed at the receiver side for that transmission,γi,j ,and on the payload sizeLi,j . Therefore, the packet departurerate, i.e., the rate at which packets depart from a node, isµi,j = E[Ti,j(ci, γi,j , Li,j)]

−1. The probability that a packetis successfully received isηi,j . We assume that the maximumnumber of retransmissions per packet is infinite. In this case,the effective number of packets per second that nodej has totransmit to successfully deliver its traffic load isα′

i,j =αi,j

ηi,j.

C. Implications

We discuss now the assumptions we have made on thenode operation, and their implications for the results andconclusions in this work.

1) No collisions with neighboring nodes: Due to thechoice of using a continuous-time backoff timer and tothe fact that the propagation delay is assumed to be


Notation Meaning

N Number of available basic channelsM Number of neighboring WLANsUi Number of nodes in WLANiCi Channel selected by WLANiWi Width of the channel used by WLANi

C = C1, C2, . . . , CM Global channel allocationci Number of basic channels inCi

γi,j SNR observed at the receiver of nodej in WLAN i transmissionsLi,j Size of the packets that nodej in WLAN i transmits

Ti,j(ci, γi,j , Li,j) Packet transmission duration from nodej in WLAN iηi,j Probability that a packet transmitted by nodej in WLAN i is not received correctlyµi,j Packet departure rate from nodej in WLAN iBi,j Duration of the backoff of nodej in WLAN iλi,j Backoff rate of nodej in WLAN i given it has packets waiting for transmissionθi,j Activity ratio for nodej in WLAN iρi,j Stationary probability that nodej in WLAN i has packets to transmit when the channelCi is sensed idleΩ(C) Collection of all feasible network statesπs Steady-state probability of the network states ∈ Ω(C)

TABLE INOTATION USED IN THE SYSTEM AND ANALYTICAL MODEL .

T

time

Ch. 1

Ch. 2

Ch. 3

Ch. 4

Ch. 5

Ch. 6

Ch. 7

Ch. 8

ACK

DATA

DIFS

Backoff

Transmissions from other WLANs

Fig. 2. Temporal evolution of the considered channel accessscheme.

negligible, the probability of packet collisions betweentwo or more nodes within the carrier sense range of theother nodes becomes zero. Therefore, the results wepresent could be considered as optimistic. However, forstandard operating conditions and configurations, thecollision probability in IEEE 802.11-based WLANs isalso low, which makes this assumption very reasonable.The accuracy of such approximation has beenextensively validated in previous works such as [20],[22], and we will further validate it in Section V.Finally, it is worth to mention that this assumptionallows us to easily model the interactions between nodesthat are outside their carrier sense range because of thedistance between them, or because they are operatingin different channels. Other widely used IEEE 802.11-based WLANs analytical models such as those basedon the works of Bianchi [6] and Cali et al. [26] requirethat all nodes in the network are able to listen thetransmissions from the others, and therefore they cannot be applied in the scenarios considered in this work.

2) No hidden nodes: One key characteristic of IEEE

802.11 devices is that their carrier sense range is atleast two times greater than their data range [27]. In thissituation, the impact of hidden nodes is very low, as agiven transmission can be only interfered by other trans-missions from very distant nodes, with energy levels nothigher than the noise floor. However, in specific deploy-ments, where obstacles play also an important role onthe propagation effects, hidden nodes may appear, andmay severely affect the network performance [28], [29].

3) Infinite Retransmissions: In terms of the WLAN per-formance, allowing an infinite maximum number ofretransmissions per packet does not affect much thefinal result because the probability that a packet isretransmitted more than few times is very low [30].However, such an assumption simplifies the analyticalmodel as we do not need to keep track of the numberof on-going retransmissions per packet.

IV. T HROUGHPUTMODEL

In this section, we introduce the Markovian model of theglobal system. In order to model the system as a Markovnetwork, we assume that the durations of both the back-off and packet transmissions are exponentially distributed.Successively, we illustrate that, thanks to the insensitivityproperty of the Markov network, the results remain valid formore general probability distributions. Indeed, the insensitivityproperty guarantees that the throughput is insensitive to thedistribution of the backoff and of the packet transmissionduration, as it only depends on their expected value.

A. Continuous Time Markov Networks

Suppose that a global channel allocationC = (C1, . . . , CM )for the M WLANs is given. A feasible network stateis asubset of nodes that can transmit simultaneously, i.e., such thatthe WLANs to which they belong do not overlap. LetΩ(C)be the collection of all feasible network states. Note that anychange in the global channel allocationC results in a differentcollectionΩ(C) of feasible network states.


Denote byui,j nodej in WLAN i, with j = 1, . . . , Ui. Thelocal dynamics at every node described in previous sectionimply that the backoff rate of nodeui,j is ρi,jλi,j , with ρi,jthe long-run stationary probability that nodej in WLANi has packets ready for transmission when the channelCi

is sensed empty, and therefore the node is decreasing itsbackoff counter. The transmission rate of nodeui,j is µi,j =1/E[Ti,j(ci, γi,j , Li,j)]. Then, the transition rates between twonetwork statess, s′ ∈ Ω(C) are

q(s, s′) =

ρi,jλi,j if s′ = s ∪ ui,j ∈ Ω(C),

µi,j if s′ = s \ ui,j,

0 otherwise.

(1)

Denote bySt ∈ Ω(C) the network state at timet. Thanks to theassumption on the backoff and transmission durations,(St)t≥0

is a continuous-time Markov process on the state spaceΩ(C).This Markov process is aperiodic, irreducible and thus positiverecurrent, since the state spaceΩ(C) is finite. Hence, it has astationary distribution, which we denote byπss∈Ω(C).

Let θi,j be theactivity ratio of nodeui,j , defined by

θi,j :=ρi,jλi,j

µi,j

=ρi,jE[Ti,j(ci, γi,j , Li,j)]

E[Bi,j ].

Note that θi,j depends on the number of basic channelciassigned to WLANi, sinceµi,j does. The process(St)t≥0 hasbeen proven to be a time-reversible Markov process in [31].In particular, detailed balance applies and the stationarydistri-butionπss∈Ω(C) of the process(St)t≥0 can be expressed asa product form. The detailed balance relation for two adjacentnetwork states,s ands ∪ ui,j, reads

πs∪ui,j

πs

=ρi,jλi,j

µi,j

= θi,j . (2)

This relation implies that for anys ∈ Ω(C)

πs = π∅ ·∏

ui,j∈s

θi,j , (3)

where∅ denotes the network state where none of the nodes istransmitting. The last equality, together with the normalizingcondition

∑

s∈Ω(C) πs = 1, yields

π∅ =1

∑

s∈Ω(C)

∏

ui,j∈s θi,j(4)

and

πs =

∏

ui,j∈s θi,j∑

s∈Ω(C)

∏

ui,j∈s θi,j, s ∈ Ω(C). (5)

Note that the normalizing constantπ∅ and the stationarydistributionπss∈Ω(C) depend on the state spaceΩ(C), andhence, they depend implicitly on the global channel allocationC.

Since the process(St)t≥0 is irreducible and positive recur-rent onΩ(C), it follows from classical Markov chains resultsthat πs is equal to the long-run fraction of time the systemspends in the network states ∈ Ω(C).

B. Packet Errors, Hidden Nodes & External Interferers

Packets can be received with errors. Errors are generallycaused by the presence of ambient noise and interference.The sources of interference are diverse. We can define twomain categories based on the use or not of the CSMA/CArules by the interferer. If the interferer is operating under theCSMA/CA rules, we will refer to it either as a contender(i.e., the interferer is inside the carrier sense range of thetransmitter) or as a hidden node (i.e., the interferer is outsidethe carrier sense range of the transmitter). Otherwise, we willsimply classify it as an external interferer.

The characterisation of the interference created by hiddennodes is complicated because of the coupled dynamics with theother nodes in the network, including also the one that suffersfrom the interference. In case of an external interferer, tocharacterise it we simply require its activity pattern. Assumingall those sources of errors are independent between them, wecan define the probability that a packet transmitted by nodejin WLAN i is successfully received as:

ηi,j = (1− pi,j(γi,j))(1 − phi,j)(1 − pexti,j) (6)

wherepi,j(γi,j) is the probability that a packet is corrupteddue to ambient noise,phi,j is the probability that a packet iscorrupted by a hidden node, andpext

i,j the probability that it iscorrupted by an external interferer.

There are several works that already consider the analysis ofhidden nodes in Markov-based CSMA/CA network analysis,e.g., see [32], [21] for further details.

C. Performance Metrics

From the stationary distribution we compute the followingperformance metrics:

• Throughput : the throughputxi,j(C) of nodej in WLANi for a given channel allocationC is

xi,j(C) := ηi,jE[Li,j ]µi,j

∑

s∈Ω(C) :ui,j∈s

πs

. (7)

• Proportional Fairness: The proportional fairness of thecurrent channel allocation with respect to the throughputis

f(C) :=M∑

i=1

Ui∑

j=1

log xi,j(C). (8)

• Jain’s Fairness index: The Jain’s Fairness Index (JFI)of the current channel allocation with respect to thethroughput is

J (C) :=

(

∑Mi=1

∑Ui

j=1 xi,j(C))2

(

∑Mi=1 Ui

)(

∑Mi=1

∑Ui

j=1 x2i,j(C)

) . (9)


D. Computing the stationary distribution of the Markov net-work

To compute the stationary distribution of the Markovnetwork we need to compute all theρi,j values, i.e.,πss∈Ω(C) = f(ρ), whereρ is a vector with allρi,j valuesrespectively. However, in turn, their value depend also onthe stationary distribution of the Markov network, i.e.,ρ =g(πss∈Ω(C)). Thus, we have a set of non-linear equations,and in general, without a close-form solution.

To solve this set of non-linear equations, we have usedan iterative fixed-point approach in which we update all theρi,j values until the throughput of all nodes converge to thesolution. Note that if a node is not able to carry a load equalto its traffic load, i.e.,xi,j(C)/E[Li,j] = αi,j , it will becomesaturated (i.e.,ρi,j = 1).

E. Solving the Model

To solve the throughput model in a general scenario, wefollow the next steps:

1) We fix a global channel allocationC, possibly generatedat random.

2) Starting fromC, we compute all the overlapsCi ∩ Ck

between any two WLANsi andk.3) We construct the collectionΩ(C) of all feasible network

states.4) We calculate the stationary probabilityπs for every

network states ∈ Ω(C).5) We calculate the throughputxi,j(C) for every nodej in

WLAN i using the stationary distributionπss∈Ω(C).6) We compute the proportional fairnessf(C) and Jain’s

Fairness index using the throughputsxi,j(C), i =1, . . . ,M andj = 1, . . . , Ui.

F. Numerical Example

Let us consider the four neighboring WLANs shown inFigure 1, and the following channel allocation for each one:CA = 1, 2, 3, 4, CB = 4, 5, CC = 5, 6, 7, 8 andCD = 5. The network states in this scenario areΩ(C) =s∅, sa, sb, sc1 , sc2 , sd, sa,c1 , sa,c2 , sa,d, sb,d, wheres∅ is thenetwork state in which none of the nodes is transmitting,sa, sb, sc1 , sc2 and sd are the network states in which onlynodea, b, c1, c2 or d is transmitting, respectively, and lastlysa,c1 , sa,c2 , sa,d, and sb,d are the network states in whichthe two indicated nodes are simultaneously transmitting. Notethat nodesb andd can transmit at the same time because theyare outside the carrier sense area of the other though theyhave overlapping channels. Likewise, nodesa and c1 or c2can transmit simultaneously because they use non-overlappingchannels in spite of being inside the carrier sense range ofthe other. A given snapshot of the temporal evolution ofthe five nodes is depicted in Figure 3, where the differentnetwork states are separated by vertical dotted lines. The blueareas represent the time a node is transmitting and the whiteareas the time a node is in backoff. In case all nodes haveexponentially distributed backoff and transmission times, thisscenario can be modeled by the CTMN shown in Figure 4.

sa sbsc1 sc2 sa,c1sd sb,d

a

b

c1

c2

d t

Y (t)

Fig. 3. Snapshot of the temporal evolution of the system considered in theexample of Section IV-F. In the vertical axis,Y (t) represents the amountof remaining backoff (white area) or transmission duration(blue area). Thearrows inside the plot represent new packet arrivals.

∅

a

b

c1

c2

d

a, c1

a, c2

a, d

b, d

ρaλa, µa

ρbλb, µb

ρc1λc1 , µc1

ρc2λc2 , µc2

ρdλd, µd

ρbλb, µb

ρc2λc2 , µc2

ρdλd, µd

ρdλd, µd

ρaλa, µa

ρaλa, µa

ρaλa, µa

ρc1λc1 , µc1

Fig. 4. CTMN model for the example of Section IV-F.

The stationary distribution from previous exampleis given by: πa = θaπ∅, πb = θbπ∅, πc1 = θc1π∅,πc2 = θc2π∅, πd = θdπ∅, πa,c1 = θaθc1π∅,πa,c2 = θaθc2π∅, πa,d = θaθdπ∅, πb,d = θbθdπ∅, with π∅ =(1 + θa + θb + θc1 + θc2 + θd + θaθc1 + θaθc2 + θaθd + θbθd)

−1.In order to validate the correctness of the presented analysis,

we evaluate the described system considering the parametervalues shown in Table II, and compare the analysis resultswith simulations. In both cases, the backoff and the packettransmission duration are exponentially distributed. Therest ofthe considered parameters and their values, with the exception


Parameters Computation Throughput per node [Mbps]

Node αL [Mbps] E[T (c, γ, L)] [msecs] p(γ) ρ Analysis Simulation

Example 1

a 18 0.1790 0.01 0.3673 18.00 17.97b 8 0.2070 0.10 0.3662 8.00 7.98c1 10 0.2150 0.05 0.6466 10.00 9.99c2 22 0.1790 0.02 1.0000 15.95 15.75d 12 0.2630 0.15 0.6333 12.00 11.99

Example 2

a 4 0.1790 0.10 0.0744 4.00 3.9b 12 0.2070 0.10 0.3845 12.00 12.00c1 20 0.2150 0.15 1.0000 11.18 11.06c2 5 0.1790 0.20 0.4752 5.00 5.00d 24 0.2630 0.05 1.0000 19.00 19.06

TABLE IIVALIDATION OF THE ANALYSIS IN NON -SATURATION CONDITIONS. A SINGLE SIMULATION EXECUTION WITH A DURATION OF 1000SECONDS IS

CONSIDERED FOR EACH EXAMPLE.

that in this example we are not considering packet aggregation(Na = 1), are shown in the Appendix A, as well as informationabout the simulation tool used. The results are also shown inTable II.

G. Insensitivity

For the Markov networks considered in this work, it turnsout that the stationary distributionπss∈Ω(C) (and thus anyanalytic performance measure linked to it, such as the through-put) is insensitive to the distributions of backoff countdownsand transmission times, in the sense that it depends on theseonly through the ratios of their averages, i.e.θi,j . The proofof the insensitivity result can be found in [20], [33]. Theinsensitivity property is crucial since back-off and transmissiontimes may be not exponentially distributed in a real network.

V. I NTERACTIONS BETWEENOVERLAPPING WLAN S

The goal of this section is to characterise the differentexisting interactions between multiple overlapping WLANs. Inorder to do this, we first simplify the node-centric analyticalmodel described in Section IV by aggregating states. Inthis way, we reduce the total number of resulting networkstates and make its resolution more efficient. Moreover, thisnew point of view provides a more compact description ofthe interactions between neighboring WLANs. Secondly, wecategorize the different types of interactions between WLANs,discussing how they impact on the performance of each one.We also show that some of those interactions are similarto those that appear in single-channel CSMA/CA multi-hopnetworks (see for example [5], [22]). Lastly, we validate theuse of the WLAN-centric analytical model by comparing itsthroughput predictions with the throughput values obtainedfrom a detailed simulation of the same considered scenarios.Also, the presented numerical results give us some moreinsights about the interactions between WLANs.

A. WLAN-centric Throughput Analysis

Considering several WLANs with some active nodes in eachresults in a large number of network states, which requireslarge computation resources to solve the analytical model.Therefore, to make it more efficient, we simplify here the

node-centric analytical model described in previous sectionby aggregating all those states in which the nodes of a givenWLAN participate. We make also the following assumptions:

1) We assume that all nodes in WLANi are close to eachother and to the AP, and they observe similar SNRvalues. Therefore, they have a similar behavior from thepoint of view of a node belonging to another WLAN.

2) Considering non-saturation conditions, it is difficult toassess if the obtained results are due to the interactionbetween the different WLANs or the actual traffic loadconfiguration of each node. To avoid such uncertainty,we assume from now on that all nodes are saturated,which can be considered as a worst-case scenario andwill allow us to obtain more clear conclusions. More-over, it also simplifies the development of the WLAN-centric model.

Therefore, since the activity of each WLAN is the sum ofthe activity of itsUi nodes, the WLAN-centric model is builtbased on the following considerations:

1) A network state is now defined as the set of WLANs thatare active simultaneously, instead of the set of nodes.

2) The backoff rate of a WLANi is the sum of the backoffrates of all nodes in it, i.e.,λi =

∑Ui

j=1 λi,j .3) The duration of a packet transmission in WLANi

is Ti(ci, γi, Li), where γi is the SNR observed byall packet transmissions inside WLANi. Similarly, allnodes in WLANi transmit packets of sizeLi and havethe same probabilityηi to receive a packet correctly.

4) Since all nodes in WLANi are assumed to be saturated,WLAN i is also saturated andρi = 1.

5) The activity ratio of WLAN i is given by θi =λiE[Ti(ci, γi, Li)].

To solve the model, the same approach as presented inSection IV is considered. Also, because using the WLAN-centric model we can only compute the fraction of time aWLAN is active, the performance metrics previously describedare modified accordingly. In Table III we compare both nodeand WLAN-centric models in terms of computational cost.Both models are executed in the same computer and usingthe same version of Matlab. The number of basic channelsis set to N = 16. Statistics are obtained by executing200 times each case. Each WLAN selects a channel width


uniformly at random from the set of available channel widths20, . . . ...,Wmax MHz. The position of the selected channelwithin the available channels is also picked uniformly atrandom. The throughput values converge by increasing thenumber of executions, which increases also the computationdelay. It can be observed that both models give the samethroughput but the node-centric one requires much more timeand computational resources.

B. Cases of interest

To illustrate the different cases of interest, we considerthree neighboring WLANs,A, B andC. All three WLANstransmit packets of fixed size (L), have the same numberof nodes (U ), have backoffs with the same average duration(E[B] = λ−1), and use the same modulation and coding rateregardless of the number of basic channels selected by eachWLAN. Therefore, if two WLANs use the same number ofbasic channels, the duration of a transmission is the same inboth cases. Thus, for clarity, in the notation of time durationsand activity ratios in this subsection, we will drop the subscripti (which distinguishes the WLANs) and instead explicitlywrite the number of basic channelsci assigned to WLANi.

1) To overlap or not to overlap:In the first example, weshow that in terms of the throughput, the best option for allneighboring WLANs is to use non-overlapping channels. Toillustrate this, we first consider the case in which all threeWLANs use the same basic channels, namelyCA = CB =CC = 1, 2, 3, 4, 5, 6. Therefore, the set of feasible networkstates isΩ(C) = ∅, sA, sB, sC. The throughput achieved byWLAN A is

xA =L

E[T (6)]πsA =

LE[T (6)]θ(6)

1 + 3θ(6)=

UλL

1 + 3θ(6).

By symmetry, the throughput achieved by each WLAN isidentical and therefore

xA = xB = xC =UλL

1 + 3θ(6).

Now consider a different scenario in which each WLANuses two non-overlapping channels, namelyCA = 1, 2,CB = 3, 4, and CC = 5, 6. For this new channelallocation, the set of feasible network states isΩ(C) =∅, sA, sB, sC , sAB, sAC , sBC , sABC. In this case, eachWLAN is completely independent of the others and thenetwork can therefore be modelled as three different systems.The throughputs achieved by the WLANs are again equal andare given by

x′A = x′

B = x′C =

UλL

1 + θ(2).

Therefore, using WLANA as a reference, we can studythe cases in which the achieved throughput when all WLANsoverlap is better than the case in which each WLAN uses anon-overlapping set of channels. BecausexA andx′

A have thesame numerator in both cases, the case in which all WLANsoverlap will be better if1+3θ(6) < 1+ θ(2), or, equivalentlyT (6) < T (2)/3. Due to the channel access protocol defined in

Section III, the latter inequality will never hold, becausetheduration of some headers and other protocol overheads is notaffected by the channel width.

2) Performance Anomaly:The performance anomaly inmulti-rate WLANs is well known [34]. Due to the channelaccess mechanism, which is fair in terms of transmissionopportunities, all nodes are able to transmit the same numberof packets on average per unit of time, and therefore the nodesthat are able to transmit at a fast rate are severely affectedbynodes that can only transmit at a low rate. A similar resultis observed when several WLANs overlap if they are usingdifferent number of basic channels.

Consider three overlapping WLANs,A, B andC, with thefollowing channel allocations:CA = 1, 2, 3, 4, CB = 4, 5andCC = 4. Despite the different channel widths, all threeWLANs achieve the same throughput, which is given by:

xA = xB = xC =UλL

1 + θ(4) + θ(2) + θ(1)

which confirms the performance anomaly that was previouslydescribed.

The performance anomaly can be solved in several ways.For instance, the WLANs that use a wider channel can beallowed to transmit larger packets, so the overall transmissionduration in all WLANs is the same. Alternatively, a differentbackoff duration can be assigned to each WLAN to guaranteethat the WLANs that use more basic channels transmit moreoften.

3) Non-direct interactions:In this last example, we con-sider the case in which the performance of two WLANsthat do not overlap is affected by the presence of a thirdWLAN. Suppose again that there are three WLANs,A, BandC, and that the channel allocation isCA = 1, 2, 3, 4,CB = 5, 6, 7, 8 andCC = 4, 5. In this scenario, the set offeasible network states isΩ(C) = ∅, sA, sB, sC , sAB. Thethroughput achieved by WLANA is

xA =L

E[T (4)](πsA + πsAB

) =

LE[T (4)]

(

θ(4) + θ(4)2)

1 + θ(2) + 2θ(4) + θ(4)2

=UλL · (1 + θ(4))

1 + θ(2) + 2θ(4) + θ(4)2,

and, becausexB = xA by symmetry, the throughput achievedby WLANs B and C is

xB = xA =UλL · (1 + θ(4))

1 + θ(2) + 2θ(4) + θ(4)2

xC =UλL

1 + θ(2) + 2θ(4) + θ(4)2.

WLAN A benefits from the existence of WLANB, andvice versa, because they implicitly cooperate to starve WLANC in the competition for the channel resources. WLANC canonly transmit when WLANsA andB are both silent.

C. Numerical Example

Let us consider the network that is composed of fourneighboring WLANs shown in Figure 5, and the four dif-ferent channel allocations shown in Figure 6, which represent


Parameters States Agg. Throughput [Mbps] Computation Delay [seconds]

Model M U Wmax [MHz] Mean St. Dev. Mean St. Dev. MeanNode 6 4 160 185.46 135.71 766 177 3.1WLAN 6 4 160 30.53 9.85 774 168 0.4Node 8 3 80 1195.4 855.9489 945 169 20.2WLAN 8 3 80 106.0 36.4014 966 177 1.6Node 12 2 40 20704.0 17967.0 1141.8 1646.2 395.4WLAN 12 2 40 738.7 356.3 1124.5 176.68 14.7

TABLE IIICOMPARISON BETWEEN THE NUMBER OF STATES AND THE COMPUTATION DELAY TO OBTAIN THE STATIONARY DISTRIBUTION BETWEEN THE

NODE-CENTRIC AND WLAN- CENTRIC APPROACHES.

AP

STA

WLAN AWLAN B

WLAN C WLAN D

Fig. 5. A group of four neighboring WLANs. Arrows represent active trafficflows.

the non-overlapping (Scenario 1), fully overlapping (Scenario2), WLAN in the middle (Scenario 3) and random channelselection (Scenario 4) scenarios, respectively. The number ofavailable basic channels is set toN = 10.

The throughput achieved by each WLAN is plotted inFigure 7 (all WLANs have two active nodes: the AP and oneSTA) and Figure 8 (each WLAN has a different number ofactive STAs, exactly as shown in Figure 5). Comparing thesetwo cases allows us to visualise the effect of a different numberof active STAs in each WLAN on the system performance andto determine if modelling the aggregated operation of a WLANinstead of the operation of every node is a valid approach.

f

f

WLAN A

WLAN A

WLAN B

WLAN B

WLAN C

WLAN C

WLAN D

WLAN D

1 1

11

2 2

22

3 3

33

4 4

44

5 5

55

6 6

66

7 7

77

8 8

88

9 9

99

10 10

1010

Scenario 1 Scenario 2

Scenario 3 Scenario 4

Fig. 6. Channel allocations.

The throughput for the scenario in which all WLANs havethe same number of nodes, and the scenario in which they do

not are shown in Figures 7 and 8, respectively. Four curvesare plotted for each WLAN: the throughput computed usingthe WLAN-centric analytical model (bars), the throughputobtained from the simulator when the capture effect is consid-ered (Sim 1), the throughput from the simulator when captureeffect is not considered (Sim 2), and the throughput from thesimulator when the same assumptions used for the analysisare considered (Sim 3).

The results of the throughput model and Sim 3 matchperfectly, which validates again the correctness of the resultsand shows that the insensitivity property indeed holds. Sincethe throughput model does not allow two or more nodes totransmit simultaneously, it does not benefit from concurrentpacket receptions when the capture effect is enabled. There-fore, in some cases when the number of overlapping WLANsis high, the capture effect causes a higher throughput than themodel (Sim 1). Otherwise, if packet capture is not considered,the achieved throughput is lower than the predicted by theanalytical model due to the negative effect of collisions (Sim2). The impact of each of the four channel allocations isdiscussed next.

Figure 7 shows the throughput achieved by each WLAN inthe four scenarios. In Scenario 1, the WLANs do not overlapbecause they use different groups of basic channels. Therefore,the throughput achieved by each WLAN only depends on thenumber of basic channels it uses. In Scenario 2, all WLANsoverlap because they all use8 basic channels. In this case, allWLANs compete with all of the others for the channel, whichresults in the same throughput for all of them. A comparisonof the results of Scenarios 1 and 2 indicates that unless thepacket capture effect is enabled, using a single basic channelis better than using8 basic channels if there is overlap with theother three neighboring WLANs. In Scenario 3, the channelsof WLANs B and C are located between WLANs A andD, and they all use 4 basic channels. This situation benefitsWLAN A and D because they only overlap with WLANs Band C, respectively, which are also competing for the channelresources. Lastly, Scenario 4 represents a random channelallocation. It is remarkable that WLAN A, which uses morebasic channels, achieves nearly zero throughput. This occursbecause it has to compete with the other three WLANs, whichare in two independent groups that do not compete. WLANsB and D have the same throughput despite using differentchannel widths due to the performance anomaly.

Figure 8 shows the throughput achieved by each WLAN


Scenario Same Number of Nodes Different Number of Nodes

1 0.9135 0.916432 1 0.833333 0.90167 0.949874 0.75199 0.60826

TABLE IVJAIN ’ S FAIRNESSINDEX

in the same four scenarios as in Figure 7 but with differentnumbers of active STAs in each WLAN. WLAN A has a singleSTA, WLAN B has three STAs, WLAN C has two STAs, andonly the AP transmits in WLAN D. Increasing the numberof STAs in a WLAN is equivalent to increasing its activityfactorθ, which also affects its throughput and how it interactswith the other networks. It is worth mentioning that a similareffect would be achieved by keeping the number of nodes perWLAN constant, but reducing the backoff duration.

WLAN A WLAN B WLAN C WLAN D0

50

100

150

200

250

300

350

Thr

ough

put (

Mbp

s)

AnalysisSim 1 [Prob. capture = 1]Sim 2 [Prob. capture = 0]Sim 3 [Analysis assump.]

(a) Scenario 1


50

100

150

200

250

300

350

Thr

ough

put (

Mbp

s)


(b) Scenario 2


50

100

150

200

250

300

350

Thr

ough

put (

Mbp

s)


(c) Scenario 3


50

100

150

200

250

300

350

Thr

ough

put (

Mbp

s)


(d) Scenario 4

Fig. 7. Throughput achieved by each WLAN when all of them have2 activenodes (i.e., the AP and one STA) in the four channel allocations considered.Each simulation result comes from a single simulation run ofduration 10000seconds.

In terms of fairness, we compute the JFI with respect to thethroughput achieved by each WLAN. The results are shownin Table IV, where a low JFI value indicates that the fourWLANs achieve very different throughputs.

Figure 9 shows the throughput achieved by each one of thefour WLANs in Scenario 4 when the CW increases from8to 8192. We consider the case in which WLANs have twonodes active. The continuous time backoff mechanism is ableto capture the same dynamics as when a discrete backoffmechanism (i.e., as in IEEE 802.11 WLANs) is considered.Only when the effect of collisions is significant, the continuoustime backoff mechanism offers optimistic results. Moreover, itcan be observed that almost exact values are achieved in bothcases when the CW value is optimal for the discrete backoffscheme (i.e., the CW value that maximizes the throughput)


50

100

150

200

250

300

350

Thr

ough

put (

Mbp

s)

AnalysisSim 1 [Prob. capture = 1]Sim 2 [Prob. capture = 0]Sim 3 [Model assump.]

(a) Scenario 1


50

100

150

200

250

300

350

Thr

ough

put (

Mbp

s)


(b) Scenario 2


50

100

150

200

250

300

350

Thr

ough

put (

Mbp

s)


(c) Scenario 3


50

100

150

200

250

300

350

Thr

ough

put (

Mbp

s)


(d) Scenario 4

Fig. 8. Throughput achieved by each WLAN in the four channel allocationsconsidered when WLAN A has 1 active STA, WLAN B has3 active STAs,WLAN C has two active STAs and in WLAN D only the AP is transmittingpackets. Each simulation result comes from a single simulation run of duration10000 seconds.

32 128 1024 81920

50

100

CW

Mbp

s | W

LAN

A

SimAnalysis

32 128 1024 81920

50

100

150

CWM

bps

| WLA

N B

SimAnalysis

32 128 1024 81920

50

100

150

CW

Mbp

s | W

LAN

C

SimAnalysis

32 128 1024 81920

50

100

150

CW

Mbp

s | W

LAN

D

SimAnalysis

Fig. 9. Throughput achieved by each WLAN in Scenario 4 when eachone has two active nodes. The probability of capturing a packet in case ofcollisions is set to0, and therefore we are considering the worst case in termsof the negative effect of collisions.

since it is the value at which the negative effect of collisionsbecomes marginal. Besides that, Figure 9 also shows thatincreasing the CW value we can reduce the starvation sufferedby WLAN A. The downside is that we reduce severely thethroughput achieved by the other three WLANs.

VI. CHANNEL ALLOCATION SCHEMES

Neighboring WLANs operating in the Industrial, Scientificand Medical (ISM) band may belong to different adminis-trative domains, and therefore they may select the channelto use autonomously and, in most of the cases, withoutany information about the current spectrum occupancy. This


situation is equivalent to select the channel to use uniformlyat random by each WLAN.

In this section, we describe such random channel selectionapproach, considering two channelisation cases:i) any groupof basic channels can be selected, andii) only the channelsspecified by the IEEE 802.11ac amendment can be selected.In order to determine the network capacity that is lost becauseof the absence of a controlled channel allocation, we alsointroduce an optimal centralised proportional fair channelallocation strategy.

A. Decentralised approaches

Channel allocation in autonomous WLANs is done in adecentralised way. That is, each WLAN chooses the groupof basic channels to use independently. In this category weconsider two cases:

1) Random Channel Selection:In this scheme, WLANiuniformly selectsci consecutive basic channels at randomfrom theN available basic channels.

2) IEEE 802.11ac channelisation:IEEE 802.11ac chan-nelisation tries to prevent that WLANs using the same numberof basic channels partially overlap. This is achieved by explic-itly defining the groups of basic channels that can be selectedwhen ci channels are going to be used. Namely, given that aWLAN is going to useci basic channels, it can only select⌊Nci⌋ different channels. Once one of these channels is selected,

the first basic channel in it isci(Z1−1)+1, and the last one isciZ1, whereZ1 = U([1, . . . , ⌊N

ci⌋) is an uniformly distributed

random value between1 and ⌊Nci⌋. Note that the available

basic channels are numbered from1 to N .

B. Centralised approach

When all WLANs are mutually within carrier sense range,we characterise the optimalproportional fair channel al-location as a linear combination ofwaterfilling solutions,assuming that WLANs can alternate periodically between dif-ferent allocations. This optimal allocation is the best trade-offbetween maximising throughput and fairness, in the sense that,starting from the optimal allocation, a proportional increaseof the throughput for any set of WLANs would result in abigger proportional decrease of throughput for the remainingWLANs.

Then, we will show how to relax the two assumptions wemade in the following way:

• when not all WLANs are mutually within carrier senserange, we present a technique to devise a sub-optimalsolution;

• if the WLANs cannot alternate periodically betweendifferent allocations, we show that a single waterfillingsolution is a reasonable sub-optimal choice.

This is an idealised approach, where a central server withknowledge of the WLANs topology is needed. However, thecomputation required only depends on the number of WLANsthat mutually interfere and on the number of basic channelsavailable. Moreover it is easy to compute in an efficient way,and such computation can be done preemptively.

1) Proportional Fair Channel Allocation:Let K be thecollection of all possible sets of channels, i. e.C ∈ K. Wecall x(C) the corresponding aggregate throughput of a set. Forexample, each of the four channel allocations represented inFigure 6 has a differentC ∈ K and a corresponding aggregatethroughputx(C).

We want to characterise the optimalC ∈ K. However,this problem is in general hard to solve because of thecombinatorial structure of the discrete collection of setsK. Tosimplify the analysis, we need to allow WLANs to switch theirchannel configurationC at any time and look for an optimaltime schedule for the network along a time period. In otherwords, we allow WLANs to switch to a differentC and keepthat configuration for a certain period of time.

We define a(global) schedulep(C) : K 7→ [0, 1] as theportion of time the network spends on each channel con-figurationC. Since we are including all the possible channelconfigurations inK, including those in which some WLANsare not transmitting at all (i. e.Ci = ∅), the schedule vectormust sum to one, i.e.,

∑

C∈K p(C) = 1.For example, considering Figure 6, a possible (although

clearly not optimal) schedule would be the one in which thesystem uses Scenario 1 for half of a time period, and noWLAN is transmitting (Ci = ∅ for all i) for the rest of thetime.

To determine the proportional fair global scheduling, weneed to solve the following utility optimisation problem:

Problem 1 (Proportional Fairness).

maxp

M∑

i=1

log∑

C∈K

p(C)xi(C)

s.t.∑

C∈K

p(C) = 1,

p(C) ∈ [0, 1], for all C ∈ K.

The quantity∑

C∈K p(C)xi(C) is the throughput achievedby WLAN i using the schedulep(C), and it is computed asthe weighted average of the different throughputsxi(C) forthe variousC ∈ K.

a) Properties of Problem 1:This problem requires themaximisation of a concave function in a convex set; thus,it is easy to solve in principle. The objective function isconcave because whenp(C) is a vector with |K| entries,fi(p(C)) =

∑

C∈K p(C)xi(C) is affine, so log fi(p(C)) isconcave because is composed of an affine function, and thesum of concave functions is concave. Moreover, the constraintsare clearly convex. This formulation is broad enough to includethe case when not all WLANs are mutually within carrier senserange.

Unfortunately, the size ofK grows exponentially withM ,which makes the computation of the throughput functionx(·)challenging. To overcome this issue we will now characterisemore in detail the optimal and sub-optimal solutions, to be ableto derive them without explicitly solving the convex problem.

b) Waterfilling Solution:We can define the waterfillingsolution when all WLANs are mutually within carrier senserange. We will relax this assumption later. Given the number


Algorithm 1 Waterfilling Algorithm1: assign to each WLAN a single basic channel, i. e.ci = 1

for all i = 1, . . . ,M .2: loop3: for i = 1, . . . ,M do4: if 2ci +

∑

j 6=i cj ≤ N then5: ci ← 2ci6: else7: goto 118: end if9: end for

10: end loop11: For each WLAN i, select the basic

channels as the contiguous set [1 +∑

j<i min(cj ,Wmax),∑

j≤i min(cj ,Wmax)] moduloN .

of basic channelsN , we can easily build a mapping from thenumber of WLANsM to the allocation that minimises thenumber of overlaps between WLANs.

Algorithm 1 shows the pseudo-code to build such a map-ping, f(M,N) (similarly as [35] for the case of free-disposalproperty). The number of basic channels used is doubled oncefor each WLAN until the number of available basic channelsallows it. The first channel positions are then chosen such thatthe spectrum is evenly used.

This procedure always produces an allocation that min-imises the number of overlaps per channel. Moreover, theobtained allocation is such that either all WLANs have thesame width, or there are only two sets of widths. In the lattercase we can split the WLANs into two setsG1 andG2, suchthat ci = 2 · cj for eachi ∈ G1, j ∈ G2.

Such waterfilling allocation plays a key role in the pro-portional fair allocation, even in the relaxed cases of whennot all WLANs are mutually within carrier sense range andwhen WLANs cannot alternate periodically between differentallocations, we show in the following that a single waterfillingsolution is a good sub-optimal choice.

c) Proportional Fairness and Waterfilling:We nowpresent a conjecture regarding the relationship between thewaterfilling configuration and the proportional fair configura-tion.

Conjecture 1. The proportional fair solution to Problem 1with the throughput function that was defined in Section IVis a linear combination on waterfilling configurations only.

This means that to have a proportional fair configuration, theWLANs should change roles in turn between the (non-unique)waterfilling solutions.

If such a time slicing function is not available, any so-lution from the waterfilling configuration is an acceptablesub-optimal solution. We corroborate this claim by means ofsimulation in Section VII, see Figure 15 in particular.

We simulated different scenarios, and solved Problem 1using the Matlab CVX framework. Conjecture 1 was neverconfuted in our simulations.

WLAN A

WLAN B

WLAN C

WLAN D

WLAN E

WLAN F

WLAN GWLAN H

Fig. 10. Eight WLANs distributed in four groups of neighboring WLANs:WLANs A, B and C in group 1; WLANs C, D and H in group 2; WLANsD, E and H in group 3; and WLANs E, F and G in group 4.

d) Interactions between multiple groups of neighboringWLANs: We present a technique to devise a sub-optimalsolution when not all WLANs are mutually within carriersense range. We need such a technique because, althoughProblem 1 would still represent such scenarios and would stillbe convex, Conjecture 1 is not valid anymore. Consequently,characterising the optimal solution becomes very hard ingeneral.

First we need to consider the interference graph of thenetworkG = (V,E), whereV is the set of WLANs, and theedges are defined ase = (i, j) ∈ E if WLAN i can interferewith WLAN j. Interference is assumed to be symmetric andthusG is an undirected graph.

We can compute the chromatic numberχ of this graph,i.e., the minimal number of colors necessary to have theproperty that no neighbours share the same color. A coloringwith χ colors represents an equivalence relation of minimumcardinality such that all WLANs that share the same color donot interfere, and thus can choose the same set of channels.

Therefore, we can consider the collection ofχ groups ofWLANs with same color as a collection of virtual WLANs,and use the mappingf(χ,N) obtained using Algorithm 1 overthese virtual WLANs, i.e., we use a waterfilling allocationwhere all WLANs that share same color will have the sameallocation.

As an example, let us consider the scenario depicted inFigure 10. The number of available basic channels is set toN = 19. WLANs select both the width and the position ofthe selected channel uniformly at random. The set of availablechannel widths is20, 40, 80, 160 MHz.

In this case we haveχ = 3 and the groupsthat share the same color areI1, I2, I3 =A,D, F, C,E, B,H,G. If we run the waterfillingalgorithm on I1, I2, I3 we get the same solution of acomplete graph with three WLANs, so one WLAN will havewidth equal to 8 and two WLANs will have width equal to4. A possible solution obtained with this technique is thefollowing: CA = 1− 8, CB = 13− 16, CC = 9− 12,CD = 1 − 8, CH = 13 − 16, CE = 9 − 12,CF = 1 − 8, and CG = 13 − 16, which as shownin Figure 11 results in a higher throughput than using therandom channel allocation scheme. In case of all WLANs areinterfering with each other, then the setI corresponds simply


Wmax=160 WF0

100

200

300

400

500

600T

hrou

ghpu

t [M

bps]

WLAN AWLAN BWLAN CWLAN DWLAN HWLAN EWLAN FWLAN G

Fig. 11. Expected throughput achieved by each WLAN in the scenario shownin Figure 10. Random channel allocation versus waterfillingalgorithm.

with the set of WLANs.If WLANs can alternate between channel allocations, then

all WLANs will have on average the same throughput, alter-nating the roles amongst the color groups. But even in thiscase the solution is in general sub-optimal, because a veryunbalanced interference graph could allow more aggressivesolutions, i.e., it could happen that some nodes belonging to acertain color group would be allowed to select a wider widththan the other nodes in the same group.

VII. PERFORMANCEEVALUATION

In this section, we evaluate the impact to the system of adifferent number of neighboring WLANs, a different numberof available basic channels and the set of channel widths thatare available for each WLAN.

A. Increasing the number of channels

Figure 12 shows the expected spectrum utilisation (Fig-ure 12(a)), the expected throughput of a single WLAN (Fig-ure 12(b)) and the expected throughput fairness (Figure 12(c))for 4 and 8 neighboring WLANs when the number of basicchannels increases from1 to 100 and all WLANs use the samechannel widthW . The spectrum utilisation is computed as thefraction of basic channels that are occupied by one or moreWLANs versus the total number of basic channels, i.e.,

v(C) :=1

N

N∑

k=1

I(k). (10)

where the functionI(k) returns1 if the basic channelk isfound occupied by one or more WLANs.

The results from Figure 12 show that the waterfillingalgorithm is able to maximise the spectrum utilisation whiledistributing the available basic channels evenly among theneighboring WLANs. As a consequence, it provides the high-est WLAN network throughput and fairness. The JFI valuesthat are less than 1 are obtained when not all WLANs in the

resulting channel allocation have allocated the same channelwidth.

When each WLAN randomizes the group of basic channelsto be used without any information about the spectrum occu-pancy or the number of neighbours, selecting a large group ofbasic channels only guarantees a higher throughput when thenumber of neighboring WLANs is small or when the numberof available basic channels is very large. For example, whenthere are only four neighboring WLANs, selectingW = 160MHz only gives a higher throughput thanW = 80 MHz ifmore than50 basic channels are available. Finally, in terms offairness, the use of a largeW also accentuates the differencesin the throughput achieved by each WLAN. The fairnessis therefore low because most of the neighboring WLANsoverlap, which results in a significantly lower throughput thanthe few that do not.

To obtain more insight into the system dynamics, Fig-ure 13 shows the histogram of the achieved throughput bya single WLAN when there are6 neighboring WLANs andtwo numbers of basic channels:N = 8 and N = 24. Weused the throughput of10000 randomly generated scenariosto obtain these histograms. The histogram shows all of thepossible throughput values and the probability of achievingeach one. ForN = 8 (Figure 13(a)) andW = 20 MHz, thethroughput achieved by a single WLAN is higher than100Mbps in approximately 50% of the cases, which correspondsto the case in which none of the WLANs overlap. Increasingthe channel width toW = 40 MHz increases the chancesthat the WLANs overlap, which reduces the expected WLANthroughput. However, in approximately 20% of the cases, theWLANs randomly select40 MHz non-overlapping channels,which results in a higher throughput. Similar observationscanbe made forW = 80 MHz, where a maximum throughput ofapproximately300 Mbps can be achieved in only a few cases,as it is more likely to obtain a lower throughput than whenusingW = 40 MHz due to the higher overlapping probability.Finally, there is a single throughput value forW = 160 MHzbecause all WLANs overlap. Similar observations can be madefor N = 24 basic channels (Figure 13(b)). In this case, it isclear that the presence of more basic channels allows morecombinations and improves the overall system performance.In this example, the optimal value ofW , which is theWvalue that results in the highest average throughput, is40MHz (103.21 Mbps) and80 MHz (170.33 Mbps) forN = 8and N = 24, respectively. Note that the expected WLANthroughput achieved in each case is shown in the caption ofFigure 13.

B. Increasing the number of WLANs

In Figure 14 we show the system performance when thenumber of neighboring WLANs increases. The number ofavailable basic channels is set toN = 19. We also evaluatethe case in which each WLAN randomly chooses the value ofW given a maximum value,Wmax (i.e.,W is a random valuethat is uniformly distributed between the feasible values of 20Mhz andWmax).

The results show that increasing the number of WLANsresults in a higher spectrum utilisation (Figure 14(a)), lower


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Fig. 12. Throughput, Spectrum Utilization and Fairness when the number of basic channels increases.

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(a) N = 8, M = 6. The expected throughput values are:89.847, 103.21,78.965, 69.031 Mbps forW = 1, 2, 4, and8 respectively.

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(b) N = 24, M = 6. The expected throughput values are:110.23, 166.28,170.33, 130.7 Mbps forW = 1, 2, 4, and8 respectively.

Fig. 13. Histogram of the aggregate throughput for different N and Mvalues

throughput (Figure 12(b)) and generally lower fairness (Fig-ure 14(c)). Note that the effect of randomly selectingW

increases the ways that different WLANs interact becausemore combinations are feasible, which for anyWmax valueand a large number of WLANs results in a throughput similarto the achieved whenWmax = 20 MHz. However, the fairnessdecreases withWmax.

Figure 15 shows the average per-WLAN ratio distribution1M

∑Mi=1

x∗

i

xibetween the optimal throughput and the sampled

throughput of random allocations withWmax = 160MHz. Thesame quantity for the waterfilling solution is shown. The boxrepresents the samples inside the interquartile rangeQ3−Q1,the crosses represent the average and the notches representthe medians. The black dots represents outliers (samples morethan 1.5 times the interquartile range). Any solution that isdifferent than the proportional fair solution makes the sumof the proportional gain negative (and thus also the average).When Wmax = 160MHz, rare events of starvation effectoccur, especially when the number of WLANs increase.

C. IEEE 802.11ac channelisation

In Table V we show the expected aggregate throughputand throughput fairness for the two decentralised channelallocation schemes considered in this work. The considerednumber of available basic channels is set toN = 16 for a faircomparison between both schemes. Similar results betweenboth channel selection schemes are obtained. However, sincethe IEEE 802.11ac channelisation prevents the negative effectsof partial overlaps between WLANs, and non-direct interac-tions as well, it results in a slightly better aggregate throughputand throughput fairness for largeW .

D. Final Remarks

Figure 14 shows that a random channel selection greatlyaffects the overall system performance. For instance, with10WLANs, the expected throughput achieved by a single WLANusing the waterfilling algorithm is almost100 % higher thanthe expected throughput achieved withWmax = 40 MHz,which is the channel width that gives the best throughputwhen the channel position is selected randomly. Similarly,the JFI value is significantly lower than that obtained by thewaterfilling algorithm. The spectrum utilisation is also lowerbecause several WLANs use the same basic channels, while


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Fig. 14. Throughput, Spectrum Utilisation and Fairness when the number of WLANs increases.

Parameters Aggregate Throughput [Mbps] JFI

Wmax [MHz] Wmax [MHz]Channelisation N M 20 40 80 160 20 40 80 160

Random 16 8 789.1 897.6 909.2 844.0 0.95 0.95 0.93 0.91802.11ac 16 8 794.2 936.4 966.5 928.2 0.95 0.95 0.95 0.93Random 16 12 1058.4 1119.7 1092.9 974.0 0.96 0.95 0.92 0.89802.11ac 16 12 1058.4 1156.6 1157.1 1081.2 0.96 0.96 0.94 0.92Random 16 16 1264.7 1276.7 1212.2 1065.3 0.97 0.94 0.91 0.87802.11ac 16 16 1263.0 1310.7 1288.8 1157.0 0.97 0.95 0.93 0.90

TABLE VCOMPARISON BETWEEN PURE RANDOM ANDIEEE 802.11AC CHANNEL ALLOCATION SCHEMES. THE VALUE OF W IS SELECTED UNIFORMLY AT

RANDOM BETWEEN20 AND Wmax .

1 2 4 6 8 10 12 14 16 18 2010

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rag

e p

er−

sta

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atio

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WLANs

Fig. 15. Average per-WLAN ratio distribution1

M

∑Mi=1

x∗

i

xibetween

optimal throughput and sampled throughput of random channel allocationswith Wmax = 160MHz. Blue dotted line represents the waterfilling solution.

others remain empty. Similar observations can be made fromFigure 12.

Therefore, there is an important gap between the perfor-mance of the centralized channel allocation algorithm andthe performance when each WLAN randomly selects thechannel to use. There are several possible solutions to reducethis gap for autonomous WLANs: 1) Use a database in thecloud to store information about the channels that are used ineach geographical area. This database could be used to findempty channels for new WLANs. However, there is no way

to force already existing WLANs to adapt to an increasingdemand in that area and reduce their channel width. 2) Usea decentralised channel selection algorithm that is able toadapt to the spectrum occupancy based on the instantaneousinformation that it is able to infer from the behaviour of theneighboring WLANs.

VIII. C ONCLUSIONS

In this paper, we have introduced, described and charac-terised the interactions that occur in the operation of multipleneighboring WLANs when they use channel bonding. Tocapture these interactions, we have developed and validatedan analytical framework based on a CTMN model. Thisframework was then used to evaluate the system performancein terms of the number of neighboring WLANs, the number ofbasic channels available and the set of channel widths that eachWLAN is allowed to use. We have also proposed a centralisedwaterfilling algorithm that provides a proportional fair globalchannel allocation, or at least the best suboptimal allocation,because we are dealing with a discrete state space.

The results obtained when WLANs select the channel centerfrequency and the channel width randomly, which is a goodrepresentation of what occurs in real deployments, show thatthe throughput, the spectrum utilisation and the fairness aresignificantly lower than the values obtained using the cen-tralised algorithm. This indicates the need to develop smarterdecentralised channel selection algorithms to make an efficientuse of the available spectrum for autonomous WLANs.


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APPENDIX ASYSTEM PARAMETERS AND SCENARIO CONSIDERATIONS

We assume that all WLANs operate using the IEEE802.11ac amendment [3]. Therefore, the WLANs operate inthe 5 GHz ISM band, where each basic channel has a widthof 20 MHz. W can take values from20, 40, 80, 160 MHz.In other words, the channelCi, selected by WLANi can becomposed ofci ∈ 1, 2, 4, 8 basic channels. The parametersconsidered are shown in Table VI. Moreover, we consider thatfor each value ofc, WLANs use a different modulation andcoding rate, as shown in Table VII. Unless otherwise stated,all WLANs have two nodes.

Parameter Notation Value

Packet Length Ld 12000 bitsNumber of aggregated packets Na 64 packetsNumber of SU-MIMO spatial streams Nsu 2 packetsBackoff Contention Window CW 16 slotsSlot Duration Tslot 9 µsAverage time decreasing the backoff E[B] CW

2· Tslot

DIFS - 34 µs

TABLE VIPARAMETER VALUES BASED ON IEEE 802.11AC

All nodes are equipped with at least two antennas, whichthey use to transmit two spatial streams in single-user MIMO


c Data Subcarriers, ξ(c) Modulation, Nm Coding Rate,Nc Tx Rate (Mbps) Min. Sensitivity (dBm)

1 52 64-QAM, 6 bits 5/6 65 −642 108 64-QAM, 6 bits 3/4 121.5 −624 234 16-QAM, 4 bits 3/4 175.5 −618 468 16-QAM, 4 bits 1/2 232 −65

TABLE VIITRANSMISSIONRATES FOR EACH NUMBER OF BASIC CHANNELS. THIS VALUES ARE COMPUTED FOR A PACKET ERROR PROBABILITY LESS THAN 10 %

FOR 4096 BYTES PACKETS[3]

mode. Packet aggregation is also considered, and64 packetsare included in each transmission. Under these conditions,thetime required to transmit a packet by nodej in WLAN i isTi,j(ci, γi,j , Li,j), which is computed as follows

Ti,j(ci, γi,j , Li,j) =

=

TPHY +

⌈

SF+Na(MD + MH + Li,j) + TBNsuLDBPS(ci, γi,j)

⌉

Ts

+

+ SIFS+

TPHY +

⌈

SF+ LBA + TBLDBPS(1, γ)

⌉

Ts

+ DIFS+ Tslot,

(11)

whereTPHY = 40µs is the duration of the PHY-layer preambleand headers,Ts = 4 µs is the duration of an OFDM(Orthogonal Frequency Division Multiplexing) symbol. SF isthe service field(16 bits), MD is the MPDU Delimiter (32bits) MH is the MAC header(288 bits), TB is the numberof tail bits (6 bits), andLBA is the Block-ACK length (256bits).LDBPS(c, γ) = Nm(γ)Nc(γ)ξ(c) is the number of bits ineach OFDM symbol, whereNm(γ) is the number of bits permodulation symbol,Nc(γ) is the coding rate, andξ(c) is thenumber of data subcarriers whenc basic channels are bondedtogether.Nsu = 2 is the number of single-user MIMO streams,andNa = 64 is the number of packets that are aggregated ineach transmission.

Lastly, we consider that in all WLANs the STAs are locatednear the AP. In that situation, the packet error probabilityisassumed to be negligible.

In all the plots, unless otherwise stated, each point is theaverage result of2000 different randomly generated scenarios,where each scenario represents a single generated globalchannel allocationC. This number of realisations guaranteesthat the standard deviation of the error in the sample meanrelative to the true mean of the computed throughput is lessthan10 Mbps, which is considered an acceptable error.

A. Simulation Tool

To validate the analytical model, a simulator of the de-scribed scenario was built based on the COST (ComponentOriented Simulation Toolkit) libraries [36]. The simulatoraccurately reproduces the described scenario and the operationof each node, including the slotted backoff mechanism that isconsidered in IEEE 802.11 WLANs, the presence of collisionsand the capture effect when multiple packets are simultane-ously received with very different power levels [27]. In thesimulator, when the capture effect is enabled, we assume that

collisions between packets from nodes that belong to differentWLANs do not cause the loss of the transmitted packets atthe corresponding receiver.

By comparing the results obtained from the simulator withthose obtained from the analytical model, we can assess themodel’s accuracy and the impact of the assumptions that wererequired to construct it.

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