On Quality of Monitoring for Multi-channel WirelessInfrastructure Networks

Arun Chhetri and HuyNguyen

Dept. of Computer ScienceUniversity of Houston

Houston, Tx, USAnahuy@cs.uh.edu

Gabriel ScalosubDept. of Communication

Systems EngineeringBen-Gurion University

Beer-Sheva, Israelsgabriel@bgu.ac.il

Rong ZhengDept. of Computer Science

University of HoustonHouston, Tx, USArzheng@uh.edu

ABSTRACTPassive monitoring utilizing distributed wireless sniffers isan effective technique to monitor activities in wireless in-frastructure networks for fault diagnosis, resource manage-ment and critical path analysis. In this paper, we introducea quality of monitoring (QoM) metric defined by the ex-pected number of active users monitored, and investigatethe problem of maximizing QoM by judiciously assigningsniffers to channels based on knowledge of user activities ina multi-channel wireless network. Two capture models areconsidered. The first one, called the user-centric model as-sumes frame-level capturing capability of sniffers such thatthe activities of different users can be distinguished. Thesecond one, called the sniffer-centric model only utilizes bi-nary channel information (active or not) at a sniffer. For theuser-centric model, we show that the implied optimizationproblem is NP-hard, but a constant approximation ratio canbe attained via polynomial complexity algorithms. For thesniffer-centric model, we devise a stochastic inference schemethat transforms the problem into the user-centric domain,where we are able to apply our polynomial approximationalgorithms. The effectiveness of our proposed scheme andalgorithms is further evaluated using both synthetic data aswell as real-world traces from an operational WLAN.

Categories and Subject DescriptorsC.2.1 [Computer-communication Networks]: Networkarchitecture and design—distributed networks, wireless com-munication

General TermsAlgorithms, theory

KeywordsQuality of monitoring, wireless networks, approximation al-gorithms, binary independent component analysis

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1. INTRODUCTIONDeployment and management of wireless infrastructure

networks (WiFi, WiMax, wireless mesh networks) are oftenhampered by the poor visibility of PHY and MAC charac-teristics, and complex interactions at various layers of theprotocol stacks both within a managed network and acrossmultiple administrative domains. In addition, today’s wire-less usage spans a diverse set of QoS requirements frombest-effort data services, to VOIP and streaming applica-tions, making the task of managing the wireless infrastruc-ture even more difficult, due to the additional constraintsposed by QoS sensitive services. Monitoring the detailedcharacteristics of an operational wireless network is criticalto many system administrative tasks including, e.g., faultdiagnosis, resource management, and critical path analysisfor infrastructure upgrades.

Passive monitoring is a technique where a dedicated setof hardware devices called sniffers, or monitors, are used tomonitor activities in wireless networks. These devices cap-ture transmissions of wireless devices or activities of inter-ference sources in their vicinity, and store the information intrace files, which can be analyzed distributively or at a cen-tral location. Wireless monitoring [23, 24, 19, 6, 7] has beenshown to complement wire side monitoring using SNMP andbasestation logs since it reveals detailed PHY (e.g., signalstrength, spectrum density) and MAC behaviors (e.g, col-lision, retransmissions), as well as timing information (e.g.,backoff time), which are often essential for wireless diag-nosis. The architecture of a canonical monitoring systemconsists of three components, 1) sniffer hardware, 2) sniffercoordination and data collection, and 3) data processing andmining.

Depending on the type of networks being monitored andhardware capability, sniffers may have access to differentlevels of information. For instance, spectrum analyzers canprovide detailed time- and frequency- domain information.However, due to the limit of bandwidth or lack of hard-ware/software support, it may not be able to decode thecaptured signal to obtain frame level information on the fly.Commercial-off-the-shelf network interfaces such as WiFicards on the other hand, can only provide frame level in-formation1. The volume of raw traces in both cases tendsto be quite large. For example, in the study of our campus

1Certain chip sets and device drivers allow inclusion ofheader fields to store a few physical layer parameters in theMAC frames. However, such implementations are generallyvendor and driver dependent.

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WLAN, 4 million MAC frames have been collected per snif-fer per channel over an 80-minute period resulting in a totalof 8 million distinct frames from four sniffers. Furthermore,due to the propagation characteristics of wireless signals, asingle sniffer can only observe activities within its vicinity.Observations of sniffers within close proximity over the samefrequency band tend to be highly correlated. Therefore, twopertinent issues need to be addressed in the design of pas-sive monitoring systems; 1) what to monitor, and 2) how tocoordinate the sniffers to maximize the amount of capturedinformation.

This paper assumes a generic architecture of passive mon-itoring systems for wireless infrastructure networks, whichoperate over a set of contiguous or non-contiguous channelsor bands2. To address the first question, we consider twodifferent models for the capturing capability of the system.The first model, called the user-centric model, assumes avail-ability of frame-level information such that activities of dif-ferent users can be distinguished. The second model, calledthe sniffer-centric model, only assumes binary informationregarding channel activities, i.e., whether some user is activein a specific channel near a sniffer. Clearly, the latter modelimposes minimum hardware requirements, and incurs mini-mum cost for transferring and storing traces. In some cases,due to hardware constraints (e.g., in wide-band cognitiveradio networks) or security/privacy considerations, decod-ing of frames to extract user level information is infeasible.

We further characterize theoretically the relationship be-tween the two models. To address the second question, weintroduce a quality-of-monitoring (QoM) metric defined asthe total expected number of active users detected, where auser is said to be active at time t, if it transmits over one ofthe wireless channels. The basic problem underlying all ofour models can be cast as finding an assignment of sniffersto channels so as to maximize the quality-of-monitoring.

We note that the problem of sniffer assignment, in an at-tempt to maximize the QoM metric, is further complicatedby the dynamics of real-life systems such as 1) the user popu-lation changes over time (churn), 2) activities of a single useris dynamic, and 3) connectivity between users and sniffersmay vary due to changes in channel conditions or mobility.These practical considerations reveal the fundamental inter-twining of “learning”, where the usage pattern of wirelessresources is to be estimated online based on captured in-formation, and “decision making”, where sniffer assignmentsare made based on available knowledge of the usage pattern,and in turn affect the QoM. In this paper, we do not addressthe learning problem. Rather, we focus on designing algo-rithms that aim at maximizing the QoM metric with differ-ent granularities of a priori knowledge. The usage patternsare assumed to be stationary during the decision period.

Our Contribution.In this paper, we make the followingcontributions toward the design of passive monitoring sys-tems for multi-channel wireless infrastructure networks,

• We provide a formal model for evaluating the qualityof monitoring.

• We study two models that differ in the capturing ca-pability of passive monitoring systems. For each of

2A channel can be a single frequency band, a code in CDMAsystems, or a hopping sequence in frequency hopping sys-tems.

these models we provide algorithms and methods thatoptimize the quality of monitoring.

• We unravel interactions between the various models.

More specifically, we show that in both the user- and sniffer-centric models considered, a pure strategy where a sniffer isassigned to a single channel suffices in order to maximize theQoM. In the user-centric model, we show that our problemcan be formulated as a covering problem. The problem isproven to be NP-hard, and constant-approximation polyno-mial algorithms are provided. In the sniffer-centric model,we characterize the expressiveness of such a model depend-ing on the amount of a priori information available. Weshow that somewhat surprisingly, although the only infor-mation retrieved by the sniffers in this model is binary (interms of channel activity), it still maintains much of the“structure” of the underlying processes. Discovery of thestructure using binary adaptation of the Independent Com-ponent Analysis (ICA) technique [16] allows mapping thesniffer assignment problem to the user-centric model. Wecomplete our study by extensive evaluation using both syn-thetic data as well as real-world traces from an operationalWLAN to demonstrate the effectiveness of our schemes inboth models.

The paper is organized as follows. We first provide anoverview of related work in Section 2. In Section 3, we for-mally introduce the QoM metric and the user-centric andsniffer-centric models for a passive monitoring system. TheNP-hardness and polynomial-time algorithms for the maxi-mum effort coverage problem which underlies two variants ofthe user central model are discussed in Section 4. The rela-tionship between the user-centric and sniffer-centric modelsis established in Section 5, where we also describe our schemefor solving the QoM problem under the sniffer-centric. Fi-nally, we present the results of our evaluation study usingboth synthetic and real traces in Section 6, and conclude thepaper in Section 7.

2. RELATED WORKWireless monitoring is an active area of research, that has

received much attention from several perspectives. Therehas been much work done on wireless monitoring from asystem-level approach, in an attempt to design complete sys-tems, and address the interactions among the componentsof such systems. The work in [2, 14] uses AP, SNMP logsand wired side traces to analyze WiFi traffic characteristics.Passive monitoring using multiple sniffers was first intro-duced by Yeo et al. in [23, 24], where the authors articu-late the advantages and challenges posed by passive mea-surement techniques, and discuss a system for performingwireless monitoring with the help of multiple sniffers, syn-chronization and merging of the traces via broadcast beaconmessages. The results obtained for these systems are mostlyexperimental. Rodrig et al. in [19] used sniffers to capturewireless data, and analyze the performance characteristics ofan 802.11 WiFi network. One key contribution was the in-troduction of a finite state machine to infer missing frames.The Jigsaw system, that was proposed in [6], focuses on largescale monitoring using over 150 sniffers.

A number of recent works focused on the diagnosis of wire-less networks to determine causes of errors. In [3], Chandraet al. proposed WiFiProfiler, a diagnostic tool that uti-

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lizes exchange of information among wireless hosts abouttheir network settings, and the health of network connec-tivity. Such shared information allows inference of the rootcauses of connectivity problems. Building on their monitor-ing infrastructure, Jigsaw, Cheng et al. [5] developed a setof techniques for automatic characterization of outages andservice degradation. They showed how sources of delay atmultiple layers (physical through transport) can be recon-structed by using a combination of measurements, inferenceand modeling. Qiu et al. in [18] proposed a simulation basedapproach to determine sources of faults in wireless mesh net-works caused by packet dropping, link congestion, externalnoise, and MAC misbehavior.

All the afore-mentioned work focuses on building moni-toring infrastructure, and developing diagnosis techniquesfor wireless networks. The question of optimally allocat-ing monitoring resources to maximize captured informationremains largely untouched. In [20], Shin and Bagchi con-sider the selection of monitoring nodes and their associatedchannels for monitoring wireless mesh networks. The opti-mal monitoring is formulated as maximum coverage problemwith group budget constraints (denoted MC-GBC), whichwas previously studied by Chekuri and Kumar in [4]. In thisproblem, we are given a ground set of n elements U , and aset S of m sets of subsets of U . Given an integer boundd ≤ m, we are required to find a subset T ⊆ S of size atmost d, and for each s ∈ T a unique subset cs ∈ s, such thatthe number of elements covered by

⋃s∈T cs is maximized.

One can view each element u ∈ U as a user, each s ∈ S asa sniffer, and each c ∈ s as the set of users using a uniquechannel, that are within the sensing range of s. The problemis then to choose at most d sniffers, and assign each one toa channel, so as to maximize the number of users covered.The standard maximum coverage problem is the special casewhere for each s ∈ S, |s| = 1. Polynomial-time algorithmsare devised and are shown to achieve optimal approximationratio. The user-centric model results in a problem formu-lation that is similar to (albeit different from) the one ad-dressed in [20]. On the one hand, we assume all sniffers maybe used for monitoring (hence parting with our problem be-ing akin to the classical maximum-coverage problem whichis trivial for d = m), while on the other hand we focus onthe weighted version of the problem, where elements to becovered have weights. One should note that all the lowerbounds mentioned in [4, 20] do not apply to our problem.

Independent component analysis (ICA) is a computationalmethod for separating a multivariate signal into additivesubcomponents supposing the mutual statistical indepen-dence of the non-Gaussian source signals. Most ICA meth-ods assume linear mixing of continuous signals [16]. A spe-cial variant of ICA, called binary ICA (BICA), considersboolean mixing (e.g., OR, XOR etc.) of binary signals, andhas been applied in the context of multi-assignment cluster-ing for boolean data [22], and medical diagnosis [8], etc. Ex-isting solutions to BICA mainly differ in their assumptions ofprior distribution of the mixing matrix, noise model, and/orhidden causes. For instance, in [8], an infinite number ofhidden causes following the same Bernoulli distribution areassumed. Accordingly, reversible jump Markov chain MonteCarlo and Gibbs sampler techniques are applied. In con-trast, in our sniffer-centric model, the hidden causes mayfollow different distribution and the mixing matrix tends tobe sparse.

3. PROBLEM FORMULATION

3.1 Network model and QoM metricConsider a system of m sniffers, and n users, where each

user u operates in one of K channels, c(u) ∈ K = {1, . . . ,K}.The users can be wireless (mesh) routers, access points ormobile users. At any point in time, a sniffer can only monitorpacket transmissions over a single channel. We assume thepropagation characteristics of all channels are similar. Werepresent the relationship between users and sniffers usingan undirected bi-partite graph G = (S,U,E), where S isthe set of sniffer nodes and U is the set of users. An edgee = (s, u) exists between sniffer s ∈ S and user u ∈ U if scan capture the transmission from u. If transmissions from auser cannot be captured by any sniffer, the user is excludedfrom G. For every vertex v ∈ S ∪ U , we let N(v) denotevertex v’s neighbors in G. For users, their neighbors aresniffers, and vice versa. Abusing the notation slightly, wealso refer to G as the binary adjacency matrix of graph G.

We will consider sniffer assignments of sniffers to chan-nels, a : S → K. Given a sniffer assignment a, we considera partitioning of the set of sniffers S =

⋃Kk=1 Sk, where

Sk is the set of sniffers assigned to channel k. We fur-ther consider the corresponding partition of the set of usersU =

⋃Kk=1 Uk, where Uk is the set of users operating in

channel k. Let Gk = (Sk, Uk, Ek) denote the bipartite sub-graph of G induced by channel k. Given any sniffer s, welet Nk(s) = N(s)∩Uk, i.e., the set of neighboring users of sthat use channel k.

A monitoring strategy determines the channel(s) a sniffermonitors. It could be a pure strategy, i.e., the channel asniffer is assigned to is fixed, or a mixed strategy wheresniffers choose their assigned channel in each slot accordingto a certain distribution. Formally, let A = {a | a : S → K}be the set of all possible assignments. Let π : A → [0, 1] bea probability distribution over the set of sniffer assignments.We refer to such a distribution as a mixed strategy. Clearly,a pure strategy a that selects a single channel per sniffer isa special case of mixed strategies, namely, π(a) = 1.

We measure the quality of monitoring (QoM) by the totalexpected number of active users monitored by the set ofsniffers. This measure is called the QoM metric.

3.2 Models for Observing User Access PatternsIn this section, two parametric models are proposed to

describe the observability of usage patterns. We assumetime is slotted, and that all channel and users’ statisticsremain stationary for a period of time T .

User-centric model.First, we consider transmission eventsin the network from the user’s viewpoint. We assume thatthe bipartite graph G is known by inspecting the packetheader information from each sniffer’s captured traces.

In the user-centric model, the transmission probabilitiesof the users p = (pu)u∈U are known and assumed to beindependent 3 (pu denotes the transmission probability ofuser u)4. Clearly, if pu is set to 1, user u always has packets

3The assumption that user activities are independent hasbeen widely adopted in literature, examples are [10] and[17].4The proposed optimization framework is valid for both IIDand non-IID user processes.

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to transmit. This can be used to model scenarios whereworst-case traffic load is assumed.

Sniffer-centric model.The user-centric model requires de-tailed knowledge of each user’s activities. This necessitatesframe-level capturing capability by the passive monitoringsystem. In the sniffer-centric model, only binary informa-tion (on or off) of the channel activity at each sniffer isassumed.

Let x = [x1, x2, . . . , xm] be a vector of m binary ran-dom variables, where xi denotes whether or not sniffer sicaptures communication activities in its associated channel.We further denote by xk the binary vector of observationsfor sniffers Sk (operating on channel k). We will sometimesabuse notation and let x = {xk | k = 1, . . . ,K}. We assumethat sniffers’ observations in different channels are indepen-dent. However, dependency exists among observations ofsniffers operating in the same channel (as a result of trans-missions made by the same set of users). Given an assign-ment a, a complete characterization of the sniffers’ observa-tions is given by the joint probability distribution Pa(xk),k = 1, . . . ,K. By independence of different channels we havePa(x) =

∏Kk=1 Pa(xk).

Clearly, the sniffer-centric model is not as expressive as theuser-centric model (formally proven in Section 5.1). How-ever, it has the advantage of being based on aggregated statis-tics, which are likely to remain stationary in the presenceof moderate user-level dynamics, such as joining and leav-ing the networks, or changes in transmission activities (e.g.,busy or thinking time). Furthermore, obtaining such binaryinformation is less costly in both hardware requirements andcommunication/storage complexity.

4. QOM UNDER THE USER-CENTRIC MODELUnder the user-centric model, the goal is to maximize the

expected number of active users monitored. Recall that puis the transmission probability of user u. This problem canbe formulated formally by:

max∑

u∈U pu ×∑

a∈A(u) π(a)

s.t. π(a) ∈ [0, 1]∑a∈A π(a) = 1,

(1)

where

A(u) = {a | ∃s ∈ N(u) s.t. a(s) = c(u)} ,

i.e., A(u) is the set of assignments that monitors user u.The objective function can be rewritten as∑

a∈A

πa

∑u∈U

pu · 1(a ∈ A(u)), (2)

where 1(·) is an indicator function. From Eq. (2) it is clearthat a pure strategy can be adopted, i.e., an optimal assign-ment is given by

a∗ = arg max∑u∈U

pu · 1(a ∈ A(u)).

4.1 Problem formulationLet MAX-EFFORT-COVERAGE (MEC) denote the prob-

lem of finding the largest (weight) set of users that can bemonitored by a set of sniffers, where each sniffer can monitorone of a set of k channels. Note that in MEC the weightscan in fact be any non-negative values and are not limited

to [0, 1]. The MEC problem can be cast as the followinginteger program (IP):

max∑

u∈U puyu

s.t.∑K

k=1 zs,k ≤ 1 ∀s ∈ Syu ≤

∑s∈N(u) zs,c(u) ∀u ∈ U

yu ≤ 1 ∀u ∈ Uyu, zs,k ∈ {0, 1} ∀u, s, k.

(3)

Each sniffer is associated with a set of binary decisionvariables, zs,k = 1 if the sniffer is assigned to channel k; 0,otherwise. yu is a binary variable indicating whether or notuser u is monitored, and pu is the weight associated withuser u.

One should first note that the problem is trivial if k =1, since all sniffers would simply be assigned to the soleavailable channel. We can therefore assume that k ≥ 2.

The MEC problem can be viewed as a special case of theMC-GBC (described in Section 2), where we have d = m.One should note that previous hardness results for MC-GBC(both NP-hardness, as well as hardness of approximation)were based on a reduction to the standard maximum cov-erage problem for d < m (the maximum coverage problembecomes trivial for d = m). It follows that none of theseproofs are applicable to the MEC problem. Surprisingly,there has not been any work done explicitly on the MECproblem, which seems to be a natural and important vari-ant of the maximum coverage problem.

4.2 Hardness of MECIn what follows we show that the MEC problem is NP-

hard for k ≥ 2, even for the unweighted case (i.e., wherepu = 1 for all u ∈ U). Our proof shows that the hardness ofthe MEC problem actually follows from the choices availableto the different sniffers. It is inherently different from thehardness suggested for the MC-GBC problem, which followsfrom limiting the number of sniffers one is allowed to use.

The proof uses a reduction from the problem of Monotone-3SAT (MON3SAT), which is known to be NP-hard (see [12,11]). In MON3SAT we are given as input an instance of3SAT where every clause consists of either solely positivevariables, or solely negated variables. The goal is to decidewhether or not there exists an assignment which satisfies allclauses.

Theorem 1. The unweighted MEC problem is NP-hard,even for k = 2.

Proof. Let C = {C1, . . . , Cn} be a set of 3SAT clauses,where each clause Ci is either the disjunction of 3 positivevariables, or the disjunction of 3 negated variables, over aground set of variables X = {x1, . . . , xm}. We construct thefollowing instance to MEC with k = 2 channels: for everyvariable xi we define a sniffer si, and for every clause Cj

we define a user uj . We let c(uj) = 1 if Cj consists solelyof positive variables, and c(uj) = 2 if Cj consists solelyof negated variables (note that this definition is consistentsince we start with an instance of MON3SAT, where in ev-ery clause all variables agree in sign). We now define thebipartite graph G = (U, S,E) which defines which user is inthe range of which sniffer. We do this by defining the neigh-boring users of every sniffer and every channel. For everyi = 1, . . . ,m we define

N(si) = {uj | xi ∈ Cj} ∪ {uj | ¬xi ∈ Cj}

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The set of edges E is therefore defined by

E = {(uj , si) | uj ∈ N(si)} .

Given a channel assignment a : S → {1, . . . ,K}, we definea truth assignment φ for the variables in X as follows:

φ(xi) =

{T a(si) = 1F a(si) = 2

Clearly φ is well defined. We now show that assignment ais able to monitor all the users if and only if all the clausesare satisfied by truth assignment φ. Assume a is able tomonitor all the users, and let Cj be a clause in C. By theassumption, uj must be monitored by at least one sniffer sisuch that a(si) = c(uj) and uj ∈ N(si). Assume Cj consistsof solely positive variables. It follows that si is assignedto channel c(uj) = 1. It follows that φ(xi) = T . Sinceby the definition of N(si) we have that xi ∈ Cj (since Cj

is a clause of solely positive variables), we are guaranteedthat truth assignment φ satisfies Cj . The case where Cj

consists of solely negative variables is symmetric. Assumenow that a does not monitor all the users, and let uj be anun-monitored user. Assume Cj consists of solely negativevariables, which in turn implies that c(uj) = 2. Since uj isun-monitored, it follows that for every sniffer si such thatuj ∈ N(si), a(si) = 1. By the definition of the reduction,and the definition of φ, this implies that for every variablexi which appears in Cj , φ(xi) = T . Since all these variablesappear in their negated form in Cj , Cj is not satisfied byassignment φ. Again, the case where Cj consists of solelypositive variables is symmetric.

Theorem 1 implies that one would have to settle for ap-proximate solutions to MEC. We first note that Guruswamiand Khot show in [13] that MON3SAT is NP-hard to ap-proximate within a factor of 7/8 + ε for every ε > 0. Thefollowing is a corollary of the above fact, and the proof ofTheorem 1:

Corollary 2. The MEC problem is NP-hard to approx-imate to within a factor of 7/8 + ε for every ε > 0.

Proof. By closely examining the reduction appearing inthe proof of Theorem 1, it follows that the number of sat-isfied clauses given the truth assignment φ is exactly thenumber of users that are covered by the channel assignmenta. It therefore follows that the reduction is approximation-preserving (i.e., any α-approximation for MEC implies anα-approximation for MON3SAT). Combining this with thefact that it is NP-hard to approximate MON3SAT to withina factor of 7/8 + ε for every ε > 0 ([13]), the result fol-lows.

4.3 Algorithms for MECAs previously stated, since MEC is a special case of the

MC-GBC problem, we can use the available approximationalgorithms for MC-GBC (e.g., [4, 20]) to solve our problemin the user-centric model. In what follows we give a briefoverview of the algorithms we use. These algorithms wouldserve as a crucial component in maximizing QoM in the moreoblivious settings of the sniffer-centric model (where no apriori knowledge of the problem’s structure is available), aswe discuss in Section 5.

The greedy algorithm.The greedy algorithm Greedy iter-atively assigns sniffers to users, where at each step it choosesthe sniffer and the assignment that (locally) maximizes theweight of coverage of those not yet monitored users.

It is proven in [4] that in the unweighted case, i.e., whereall users have the same weight, Greedy guarantees to pro-duce a 1

2-approximate solution, and that this is tight. The

following theorem shows that the same holds also for theweighted case, which generalizes the MEC problem.

Theorem 3. Greedy is a 12

-approximation algorithm forthe weighted MC-GBC problem.

Proof. The proof follows closely the proof appearingin [4], and is omitted due to space limit.

Note that the example provided in [4] showing that thisanalysis is tight naturally also holds for the weighted case.

LP-based algorithm.This algorithm is based on solvingthe LP-relaxation of the IP formulation for MEC appear-ing in Eq. (3). Once we have an optimal solution to theLP-relaxation, we round the fractional solution into an inte-gral solution, with e.g., the probabilistic rounding techniqueof Srinivasan [21]. We next sketch the basic idea of thisprobabilistic rounding technique. Let z∗ be an optimal so-lution to the LP relaxation of Eq. (3), and let s be anysniffer. If

∑k z∗s,c > 0, one can view the induced solution

z∗s : C → [0, 1] as a probability measure over the differentchannels (via normalization). The goal is to decide on anintegral channel assignment for s, namely, setting each z∗s,cto a value in {0, 1} such that exactly one variable out of thek variables corresponding to sniffer s is set to the value 1.The algorithm builds a binary tree whose leaves correspondsto the k variables zs,k associated with sniffer s, and pairs un-set variables in a bottom-up fashion. The pairing is madesuch that an internal node sets at least one of the variablescorresponding to its children. This is done while adjustingthe (probability) value of the (other) unset variable. Thisapproach is proven to produce a valid assignment in lineartime [21]. We refer to the above algorithm as ProbRand.

As mentioned in [4] (and later made explicit in [20]), thismethod produces a (1 − 1/e)-approximate solution, for thecase where all users have the same weight. As is the casewith the greedy algorithm, the analysis for the unweightedcase (e.g., appearing in [20]) can be extended to provide thesame guarantee also for the weighted case, as demonstratedby the following theorem (the proof is omitted due to spaceconstraints).

Theorem 4. ProbRand is a (1 − 1/e)-approximation al-gorithm for the weighted MC-GBC problem.

One could also use the pipage LP-based technique sug-gested by Ageev and Sviridenko [1], as an alternative toProbRand. This approach has the exact same approxima-tion guarantee as ProbRand.

We note that the approximation guarantee of the LP-based algorithms are best possible for the MC-GBC prob-lem, again, due to a reduction from maximum coverage, andthe fact that it is hard to approximate the maximum cov-erage problem to within a factor better than (1 − 1/e) [9].However, this lower bound does not necessarily hold for theMEC problem, for which the previous lower bound of 7/8+εfor every ε > 0 is the best available.

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5. QOM UNDER THE SNIFFER-CENTRICMODEL

Recall that in the sniffer-centric model, given an assign-ment a ∈ A,

∏Kk=1 Pa(xk) is the probability distribution of

binary observations x = {xk, k = 1, . . . ,K} from m sniffers.Let w(xk) be the expected number of active users moni-tored by sniffers in channel k. The MEC problem under thesniffer-centric model is defined as follows.

max∑

a∈A π(a)∑K

k=1 w(xk)Pa(xk)s.t. π(a) ∈ [0, 1]∑

a∈A π(a) = 1,

Clearly, a pure strategy suffices, i.e., there exists an optimalassignment such that,

a∗ = arg max

K∑k=1

w(xk)Pa(xk). (4)

Note that in contrast to the user-centric model, wheretransmission activities from different users are independent,observations of sniffers are correlated. As a result, Pa(xk)cannot be simplified as a product form. This motivates usto exploit the underlying (though not directly observable)independence among users, and map the problem to QoMunder the user-centric model.

5.1 Relationship between the user-centric andsniffer-centric models with known G andunknown p

In the sniffer-centric model, each sniffer only reports bi-nary output regarding the channel activities, and thus theaccess probability of the users as well as the bipartite graphG, are both hidden. In this section we derive the sufficientand necessary conditions for unraveling the access probabil-ities of the users given G and P(x).

Let y = [y1, y2, . . . , yn] be a vector of n binary randomvariables, where yj = 1 if user uj transmits in its asso-ciated channel, and yj = 0 otherwise. yc is the vectorof activities for users transmitting in channel k (i.e., usersin Uk). The joint distribution of y is given by P(y) =∏

yj=1 pj∏

yj=0(1 − pj). The product form is due to the

independence among users’ activities. The main questionwe aim to answer is the following: Given the vector xk ofsniffers’ observations, what knowledge can be obtained re-garding yc? Throughout this section, unless otherwise spec-ified, we limit the discussion to users and sniffers in a fixedchannel k, and drop the subscript. We will also denote bygij the entry in the i’th row and j’th column of G.

Using the adjacency matrix, we have the following,

xi =

n∨j=1

gij ∧ yj , i = 1, . . . , I, (5)

where ∧ is Boolean AND and ∨ Boolean OR. Define the set

Y (x) = {y |n∨

j=1

gij ∧ yj = xi, ∀i}.

Therefore,

P(x) = P(y ∈ Y (x)) =∑

y∈Y (x)

P(y) (6)

The necessary and sufficient conditions that uniquely de-termine p using G and P(x) is characterized in the followingtheorem.

Theorem 5. Given G = (S,U,E), p can be uniquely de-termined by P(x) iff ∀uj 6= uj′ ∈ U , N(uj) 6= N(uj′).

Proof. The proof is omitted due to space limit.

The above theorem essentially shows that in the sniffer-centric model, if the adjacency matrix is known, then onecan effectively determine the transmission probabilities ofthe users from the joint distribution of sniffers’ observa-tions. In presence of measurement noise, methods such asExpectation-Maximization can be applied. We therefore ob-tain an instance of the problem corresponding to our user-centric model, which can be solved efficiently using the al-gorithms described in Section 4.3.

5.2 Inference of G and unknown p using bi-nary ICA

In this section we deal with the sniffer-centric model, andthe problem of QoM in a scenario where both the user accessprobabilities and the adjacency matrix are unknown. Insuch a scenario, the question is whether knowledge regardingG and p can still be obtained from P(x). The answer ispositive. Consider the simple example, where two sniffers s1and s2 observe the activity of a single user u. In this case,x1 = x2. Therefore,

P(x) = P(x1)1(x1 = x2)= P(yu = x1)

=

{pu, x1 = 11− pu, x1 = 0

.

Therefore, if the joint distribution of xk is the product ofa marginal distribution with an indicator function, and thetwo marginal distributions are identical, we can infer thatboth sniffers observe the same set of users. Generally, thejoint distribution of x preserves a certain stochastic “struc-ture” of the user’s activities. We will formalize this obser-vation in the subsequent section by devising an inferencemethod to estimate G and p from P(x).

Estimation of G.This problem is similar to the problemaddressed by the Independent Component Analysis (ICA)scheme [16], where the observed data is expressed as a lineartransformation of latent variables that are non-Gaussian andmutually independent. ICA is widely used in applicationssuch as blind source separation, feature extraction, noisereduction in imaging data etc. In the classic ICA model,continuous-value variables are mixed linearly:

x = Gy,

where x = (x1, x2, . . . , xm)T is the vector of observed ran-dom variables, y = (y1, y2, . . . , yn)T is the vector of indepen-dent latent variables (the “independent components”), andG is an unknown constant matrix, called the mixing matrix.The problem is then to estimate both the mixing matrix Gand the distribution of the latent variables yi, using obser-vations of x alone. ICA can be solved by casting it as anoptimization problem which aims at maximizing the non-gaussianity of estimates y (thus aiming at preserving thelevel of independence in y). (see [16]).

ICA assumes that both y and x are continuous randomvariables and linear mixing of y, and thus is not directly

116

P (x)LinearICA

GL

Quantization

G, P (x)Quadratic

Programming

P (y)MEC

zs,c

Figure 1: Channel selection algorithm under the sniffer-centric model

applicable to our problem. In Eq. (5), x and y are binaryrandom variables, and Boolean operations are used. In [15]Himberg et al. show that with a proper transformation,extensions of well-known ICA algorithms work well when thedata is sparse enough. We adopt the algorithm presented in[15] with some modifications. The basic idea is as follows.

First, Eq. (5) is simplified using linear mixing and a (coordinate-wise) unit step function.

x = U(Gy), (7)

where U(·) is unit step functions defined by U(r) = 1(r > 0).By applying the standard ICA, an estimation of the linear

mixing matrix GL can be found by ignoring the step func-tion. Then, the binary mixing matrix G can be estimatedby a quantization operation, defined by

G = U(Λ−1GL − T ). (8)

The diagonal scaling matrix Λ has

λii = signmax(gLi),

where

signmax(r) =

{max(r) if |max(r)| > |min(r)|min(r) otherwise.

Λ scales the elements in the mixing matrix to the maximumvalue 1. The matrix T contains thresholds, such that thehigher the threshold value, the sparser G is. We note thatour definition of the signmax is different than the one usedin [15].5

Estimation of P(y).Once G is determined, P(y) needs tobe estimated. From xi = U(giyi), where gi is the ith row of

G, we have,

p(xi = 0) =∏

gij=1

p(yj = 0).

The product is due to independence of yi’s. Taking log(·)on both sides, we have

log(p(xi = 0)) =∑gij=1

log(p(yj = 0)).

Let αi = log(p(xi = 0)), and βi = log(p(yj = 0)). Defineα = [α1, α2, . . . , αm]T , and β = [β1, β2, . . . , βn]T . Therefore,we formulate the following optimization problem.

min ‖ α− Gβ ‖2

s.t β < 0, (9)

where ‖ · ‖ is the norm of a vector. Clearly, this is aconstrained quadratic programming problem with a positivesemi-definite matrix (i.e., all eigenvalues are non-negative),and can be solved in polynomial time.

5We believe the form of signmax in [15] is not correct.

Channel selection.With the estimates y and G at hand,we effectively transform the sniffer-centric model to the user-centric model. The method described in Section 4.3 can thenbe applied to determine the channel assignment of each snif-fer. The complete channel assignment scheme in the sniffer-centric model is illustrated in Figure 1.

6. EVALUATIONIn this section we evaluate the performance of different

algorithms under the user-centric and sniffer-centric modelsusing both synthetic and real traces. Synthetic traces allowus to control the parameter settings by varying the numberof users, the number of channels as well as the traffic load ofusers, and investigate their effects on the performance of dif-ferent algorithms. The real-world traces are collected froman operational WLAN. They provide insights on the per-formance under realistic traffic loads and user distributionsover space.

In addition to the greedy and LP-based algorithm, twobaseline algorithms are considered:

• Random channel assignment (Rand), where thesniffer channels are assigned randomly. Rand assumesno prior information regarding either user activities orsniffers’ observations. It is independent of whether theuser or sniffer centric model is assumed. However, forconciseness of presentation, we only compare it side-by-side with results of user-centric models.

• Max Sniffer Channel (Max), where a sniffer is as-signed to its busiest channel. This scheme is the mostintuitive approach in the sniffer-centric model whensniffers decide their channel assignment non-cooperativelybased on local observations. Note it is easy to con-struct scenarios where Max performs arbitrarily bad.Thus, its worst case performance is unbounded.

For the inference scheme in the sniffer-centric model, we usedthe FastICA algorithm [16] to compute the linear mixing

matrix GL.

6.1 Synthetic tracesA toy example.We begin this section by considering a toyexample, which provides insight as to the operation of ourproposed scheme, as well as to the limitation of both ourscheme and the other algorithms considered.

Consider four sniffers and 5 users operating in two chan-nels, as depicted in Figure 2(a). The center user operates inchannel 1, and the remaining ones in channel 2. The trans-mission probabilities of the users are marked, with the centernode having higher transmission probability. The marginalprobabilities of sniffers observing the first and the secondchannel busy are [0.5 0.5 0.5 0.5] and [0.49 0.49 0.49 0.49],respectively. The inference algorithm yields the inferred bi-partite graph G and the access probabilities p as specified in

117

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���������������

0.29

0.29 0.29

0.29

0.5

1

2

3 4���������������

���������������

0.49

0.49

1

2

3 40.5

(a) Original (b) Inferred

Figure 2: A toy example. Users are shown in circles with theiractive probabilities. Sniffers are shown as solid circles. An edgebetween a sniffer and a user exists if the sniffer can capture theuser’s transmissions. The center user operates in channel 1, andthe remaining ones in channel 2.

Table 1: Comparison of QoM under user and sniffer-centric mod-els in the toy example

Rand/Max Greedy LP-Round

User-centric 1.08 (Rand) 1.66 1.16Sniffer-centric 0.50 (Max) 1.37 1.66

Figure 2(b) (note that the inferred graph is a strict subgraphof the original graph).

Table 1 summarizes the performance of the various algo-rithms with respect to the QoM under both the user-centricmodel and the sniffer-centric model. The Rand algorithm isapplied in the user-centric model; and the Max algorithm isapplied in the sniffer-centric model. In both cases the op-timal QoM is 1.66. Although the inference algorithm doesnot yield exactly the same bipartite graphs, nor the sameuser transmission probabilities (by comparing Figures 2(a)and 2(b)), the resulting sniffer assignment still provides goodmonitoring quality. In contrast, the Max algorithm, whichgreedily selects the busiest channel for each sniffer individu-ally can lead to significant performance degradation. Inter-estingly, the greedy algorithm sometimes outperforms theLP rounding algorithm.

Large scale networks.In this set of simulations, 1000 wire-less users are placed randomly in a 500x500 square meterarea. The area is partitioned into hexagon cells with cir-cumcircle of radius 86 meters. Each cell is associated witha base station operating in a channel (and so are the usersin the cell). The channel to base station assignment ensuresthat no neighboring cells use the same channel. 25 Sniffersare deployed in a grid formation separated by distance 100meters, with a coverage radius of 120 meters. A snap shot ofthe synthetic deployment is shown in Figure 3. The trans-mission probability of users is selected uniformly from [0,0.06], resulting in an average busy probability of 0.2685 ineach cell. We vary the total number of orthogonal channelsfrom 3 to 9.6 The results shown are the average of 20 runswith different seeds.

Figure 4(a) shows the simulation results under the user-centric model. One can see that the performance of thegreedy algorithm and LP-based algorithm with random round-ing are comparable to the LP upper bound. When the num-

6In 802.11a networks, there are 8 orthogonal channels in5.18-5.4GHz, and one in 5.75GHz.

ber of channels is small, a random assignment can yield areasonable monitoring quality. However, as the number ofchannels increases, some channels of a sniffer do not haveany activity leading to poorer performance of the randomassignment.

Figure 4(b) summarizes the simulation results for the sniffer-centric model. In this part, we compare the QoM using theMax algorithm, with that attained by the greedy and LP-based channel assignment, applied to the mixing matrix Gand user access probabilities p inferred by our scheme de-picted in Figure 1. We observe that the algorithms basedon the inferred G and y outperform the Max algorithm.Recall that according to the Max algorithm, a sniffer non-cooperatively decides its own channel assignment and selectsthe most active channel. Clearly, the Max algorithm doesnot take into account the correlations among the observa-tions of neighboring sniffers in the same channel. In contrast,the proposed inference algorithm can indeed extract such acorrelative structure from the binary observations. We fur-ther note that by comparing Figure 4(a) and Figure 4(b),one can detect QoM degradations due to uncertainty in Gand p. Such degradations are expected due to the infor-mation loss in having merely the binary observations of thesniffers.

6.2 Real tracesIn this section, we evaluate our proposed schemes using

real traces collected from the campus wireless network using21 WiFi sniffers deployed in our building. Over a periodof 6 hours, between 12pm and 6pm, each sniffer capturedapproximately 300,000 MAC frames. Altogether, 655 uniqueusers are observed operating over three channels.7

The number of users observed on channels 1, 2, 3 are 382,118, and 155, respectively.

The histogram of user active probability (calculated asthe percentage of 20µs slots that a user is active) is shown

7Our measurements used the campus IEEE 802.11g WLAN,which has three orthogonal channels.

100 150 200 250 300 350 400100

150

200

250

300

350

400

Figure 3: Hexagonal layout with users (‘+’), sniffers (solid dots),and channels of each cell (in different colors)

118

3 6 90

5

10

15

20

25

30

35

40

# of channel

Ave

rage

num

ber

of a

ctiv

e us

ers

mon

itore

d

RandGreedyLP−RoundLP−UP

3 6 90

5

10

15

20

25

30

35

40

# of channel

Ave

rage

num

ber

of a

ctiv

e us

ers

mon

itore

d

MaxGreedyLP−RoundUser Centric LP−UP

(a) User-centric model (b) Sniffer-centric model

Figure 4: QoM under user-centric and sniffer-centric models with synthetic traces

in Figure 5. Clearly, most users are active less than 1% ofthe time except for a few heavy hitters. The average activeprobability is 0.0014.

Figure 6 gives the average number of active users moni-tored under the user-centric and sniffer-centric models. Thenumber of sniffers in the experiments varies from 5 to 21 byincluding only traces from the corresponding sniffers. Thenumber of channels is fixed at 3. Except for the case with21 sniffers, all data points are averages of 5 scenarios withdifferent sets of sniffers, chosen uniformly at random. Recallthat the average active probability is 0.0014. Thus, the num-ber of users active in a slot is around 1. In the user-centriccases (Figure 6(a)), both the greedy and the LP-round algo-rithms with random rounding significantly outperform therandom assignment. Moreover, their performance is compa-rably with the LP upper bound on the optimal performancepossible. As the number of sniffers increases, the averagenumber of users monitored increases but tends to flattenout since most users have been monitored. The performancegain of the greedy/LP-round over random also reduces from31.6% at 5 sniffers to about 10% at 21 sniffers. In the sniffer-centric case, both the greedy and the LP-round algorithmsoutperform Max. However, we observe performance degra-dation due to the uncertainty in G and p, as would be ex-pected due to the loss of information, when compared to theperformance in the user-centric model.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.140

100

200

300

400

500

600

700

User active probability

Num

ber

of u

sers

Figure 5: Histogram of user active probability measured as thepercentage of active 20µs slots. The average active probability is0.0014.

7. CONCLUSIONIn this paper, we formulate the problem of maximizing

QoM in multi-channel infrastructure wireless networks withdifferent a priori knowledge. Two different models are con-sidered, which differ by the amount (and type) of informa-tion available to the sniffers. We show that when completeinformation of the underlying cover graph and the accessprobabilities of users is available, the problem is NP-hard,but can be approximated within a constant factor. We fur-ther show that when only binary information about chan-nel activities is available to the sniffers, one can map theproblem to the the one where complete information is athand using the statistics of the sniffers’ observations. Eval-uations demonstrate the effectiveness of our proposed infer-ence method and optimization techniques.

There are a few fundamental open questions that remainto be addressed, including 1) design and analysis of binaryinference schemes, with provable performance bounds, and2) resolving the gap between the upper and lower bounds onthe approximation ratio of the MEC problem.

•AcknowledgmentThe work of the third author is partially supported by theCORNET consortium, sponsored by the Magnet Programof Israel MOITAL. The work of the fourth author is fundedin part by the National Science Foundation (NSF) underaward CNS-0832089.

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