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RESEARCH Open Access

A utility-based approach for secondary spectrumsharingMaxim Dashouk* and Murat Alanyali

Abstract

This paper provides a social welfare framework for coexistence of secondary users of spectrum in the presence ofstatic primary users. We consider a formulation that captures spatial differences in available spectrum whileconsidering general system topologies and utility functions: a collection of wireless sessions is considered under anarbitrary conflict graph that indicates the sessions which cannot transmit simultaneously on a common channel. Itis assumed that each session has a utility associated with its spectrum utilization. A carrier sense multiple access-based randomized channel selection technique is considered to maximize the resulting sum of utilities. Ameasurement-based gradient ascent method is used to improve the channel selection performance and to achievelocal maxima of the social welfare. Distributed versions of the method are discussed and shown to outperformpreviously published work in a variety of simulation scenarios that study effects of primary user presence, varyingsecondary user density, varying total channel availability.

1 IntroductionRecent regulatory proceedings in wireless telecommuni-cations offer tremendous potential for efficient spectrumusage via novel operational models of spectrum access.An important legislative development in the US, for anexample, is the FCC’s recent release [1] of a part ofVHF/UHF band for fixed broadband access systems toaddress the problem of spectrum scarcity. This develop-ment introduces the concept of secondary spectrumusers that are allowed to use the spectrum while avoid-ing conflicts with primary users, which, in this particularcase, are TV broadcast services. Similar primary - sec-ondary usage scenarios are also likely to arise due toregulatory reforms that grant full property rights tospectrum licensees, thereby allowing them to provideservices in secondary markets.While isolation of primary users is challenging due to

the cognitive capabilities imposed on the secondaryusers, yet another technical challenge arises in how theavailable spectrum can be shared among secondaryusers. This latter issue is closely related to the conceptof wireless coexistence. However, it poses further com-plications due to the spatial variability of available spec-trum in the presence of primary users, and due to

possible heterogeneity of secondary users. Resolution ofspectrum access can be addressed by cooperative techni-ques that are based on coordinative messaging, or bynon-cooperative techniques that are based on an eti-quette [2]. The former approach requires over-the-airmessaging or exchange through a backhaul network andit is suitable for homogenous systems. Examples of thisapproach can be found in [3] that describes a distributedhandshake mechanism to share time-spectrum blocks,and in [4,5] that propose spectrum sharing techniquesfor OFDM-based air interfaces. In addition, self-coexis-tence in the IEEE 802.16 WiMax standard [6] and thedeveloping cognitive radio-based IEEE 802.22 standard[7] are based on this approach. Such protocols requiretight network synchronization and capability of directmessage exchange between parties to coordinate spec-trum sharing activities. These assumptions do not holdfor heterogeneous systems of technologically incompati-ble spectrum users. In heterogeneous systems treatingall interference as noise or applying a listen-before-talk(LBT) technique are often the only available options forspectrum sharing [8].A common goal in distributed channel selection is to

achieve minimal collaborative communication amongsecondary users. A game-theoretic view-point isemployed in [9,10] to address the problem of non-coop-erative multi-radio channel allocation. Chen et al. [11]

* Correspondence: [email protected] of Electrical and Computer Engineering, Boston University, 8Saint Mary’s Street, Boston, MA 02446, USA

Dashouk and Alanyali EURASIP Journal on Wireless Communications and Networking 2011, 2011:32http://jwcn.eurasipjournals.com/content/2011/1/32

© 2011 Dashouk and Alanyali; licensee Springer. This is an Open Access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

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employ distributed interference measurements toimprove overall network throughput amid limited coop-eration. Seferoglu et al. [12] elaborate on a slotted Alohamodel to implement a dynamic decentralized multi-channel multiple access algorithm. Opportunistic spec-trum access is studied in [13] to minimize the collisionsbetween cooperating secondary spectrum users andnon-cooperating primary users. Bao et al. [14] presentcollision-free mechanism for distributed channel accessbased on local neighborhood discovery.It is well-recognized that in spatially dispersed systems

channel selection is closely related to graph coloring.This abstraction relies on representing possible conflictsin spectrum access via a graph and interpreting eachchannel as a distinct “color” [15]. Earlier work [16] ana-lyzes channel assignment problem as a graph coloringproblem for special topologies. Computational complex-ity of such application of graph coloring problem isaddressed in [17-19], where distributed coloring algo-rithms are introduced that do not guarantee optimalcoloring but can be applied in a decentralized manner.The coloring approach is overly rigid for cognitive

radio systems for two main reasons: (1) presence of pri-mary users lead to variability of available colors for eachsecondary user, and therefore suggests list coloring pro-blems that are generally harder than graph coloring, and(2) if secondary users are not active continuously,assigning each user a dedicated channel may lead toconsiderable inefficiency. In that case, channels can betime-shared among multiple conflicting users, and it isnatural to seek a basis to determine the allocationamong different users, and mechanisms to implementsuch an allocation.In this work, we adopt a utility-based perspective to

determine channel allocations in systems of generaltopology, and a combination of randomized channelselection and carrier sense multiple access (CSMA)-based techniques to implement such allocations. CSMAhas been considered by Ni et al. [20], Jiang and Walrand[21], and Marbach and Eryilmaz [22] in similar settingsbut for systems that have a single common channel.The main goal in the single channel case is to tunebackoff timers of users so as to achieve throughputoptimality. In setting of this paper, the additional opti-mization dimension due to multiple channels will beshown to lead a nontrivial problem. We will addressthat problem by fixing the backoff rates but dynamicallytuning certain channel access probabilities.In related work, randomized channel access techni-

ques were considered in similar settings by Leith andClifford [23] and Kauffmann and Bacelli [24]. Themechanism presented in [23] ensures that a successfulchannel choice remains unchanged and provides a dis-tributed algorithm that converges to dedicated channel

assignments with no conflicts, provided such assignmentis feasible. Yet, performance of the algorithm remainsunclear if the number of available channels is less thanthe chromatic index of the conflict graph. In [24], a dis-tributed Gibbs-sampling methodology is employed tooptimize the channel selection probabilities, where eachuser autonomously updates its operating channel basedon interference measurements on all available channels.The contribution of the present paper is: (1) a utility-

based formulation of spectrum sharing among secondaryusers is presented. The formulation accounts for thepresence of static primary users, and provides a spec-trum allocation principle that is applicable to all systemtopologies and channel availabilities. (2) a CSMA-baseddynamic channel selection technique is presented as animplementation of the solution. The channel selectiontechnique is a gradient ascent method to adaptively con-verge to local maxima of the utility function. Distributedversions of the method are discussed and justified. (3)extensive numerical experiments indicate that the pre-sented method outperforms the existing algorithms sig-nificantly in a variety of considered scenarios. Theseexperiments address the issues of primary user presence,varying secondary user density, and varying total chan-nel availability.The rest of this paper is organized as follows: Section

2 provides the considered model of secondary spectrumusage and formulates a social welfare optimization pro-blem that is based on channel utilizations of involvedparties. This problem is re-parameterized in Section 3 ina manner that is suitable for CSMA-based randomizedchannel access methods. Although the original problemis convex, the re-parameterization is not, and local max-ima of the latter problem are also characterized in thissection. Section 4 provides a novel gradient ascent algo-rithm to improve the randomized channel accesstowards local maxima, and it identifies distributedapproximations of these algorithms. A detailed numeri-cal and comparative study of the proposed principlesare provided in Section 5. The paper concludes withfinal remarks in Section 6.

2 Utility-based spectrum sharingWe consider a system of M wireless sessions that inter-act through the interference they generate on eachother. This system is represented by an undirectedgraph G = (V, E) where V is the set of nodes and E isthe set of edges. Each node in G represents a wirelesssession; hence, there are exactly M nodes in V. An edgein G indicates that its incident nodes correspond to ses-sions that interfere with each other due to geographicalproximity of transmitters and receivers. This results inthe operational constraint that two such sessions cannotactively use a narrowband channel at the same instant.


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We shall consider that coexistence of M wireless ses-sions on a spectrum band comprised C narrowband chan-nels. Generally, only a subset of these channels may beavailable to each session. This scenario implicitly reflectsthe presence of primary users that hold channels on a sta-tic basis: for example, a TV broadcast in the vicinity of agiven session renders a number of channels unavailablefor that session, while these channels may be available tofar away sessions. Let C = {1, 2, . . . ,C} denote the entireset of channels available in the shared spectrum band. Weshall further denote the set of channels available to node iby Ci ⊆ C. In visualizing the interference relations for eachchannel c Î {1, 2,..., C}, it may be helpful to consider thesubgraph Gc(V c, Ec) that is induced by nodes i for whichchannel c is available. Figure 1 illustrates these subgraphsfor a sample topology with a total of C = 3 channels. Thescope of this paper is general and no assumptions areimposed on the subgraphs Gc. To avoid trivialities wefocus on the case in which there is at least one channelavailable for each node. We shall also assume that eachnode can transmit on one channel at a time.For each node i Î V and channel c ∈ C we define the

quantity μci as

μci = the fraction of time that node i transmits on channel c.

With this definition, it follows that the sum∑C

c=1 μci is

the fraction of time that session i accesses the entire spec-trum, or equivalently, it is the spectrum utilization of ses-sion i.The numbers μc

i depend on the adopted channel allo-cation scheme that determines how the spectrum isshared among the M sessions. However, they have threeinvariant properties:

1. μci = 0 if c �∈ Ci; since channel c is then not avail-

able to session i.

2.∑C

c=1 μci =

∑c∈Ci

μci ≤ 1; since a node can transmit

only on one channel at a time.3. For each channel, c let μc = [μc

i1,μc

i2, . . . ,μc

iN]

where i1, i2,..., iN represent the nodes in the sub-graph Gc. That is, i1, i2,..., iN are the nodes for whichchannel c is available. Let Ic denote the collection ofindependent setsa of the subgraph Gc, and let co {Ic}denote the convex hull of (i.e. all convex combina-tions of the elements in) the set Ic. Then

μc ∈ co{Ic}.

This property is inherited from single-channel sys-tems and it can be verified from [21].

In this paper we consider spectrum sharing so as toachieve channel access rates μ = {μc

i : i ∈ V, c ∈ Ci} thatsolve the following optimization problem:

Problem STATIC: maximize W(μ) =∑i∈V

Ui

( ∑c∈Ci

μci

)

subject to μci ≥ 0, i ∈ V, c ∈ Ci∑

c∈Ci

μci ≤ 1, ∀i ∈ V

μc ∈ co{Ic}, ∀c ∈ C.

(1)

Here each, Ui is a generic utility function and the sum ofthe utility functions amount to social welfare. Particularchoice of the utility functions is dictated by the specificcriteria adopted in spectrum sharing. For example, choos-ing Ui(x) = x amounts to maximization of total spectrumutilization in the entire system. Achieving this goal mayrequire starving some of the sessions. If fairness is also acriterion in addition to efficiency, then the utility functionscan be chosen strictly concave, such as Ui(x) = log(x).The objective function W(μ) is concave in μ, and the

constraints of Problem STATIC identify a convex domainfor μ. Therefore, the problem can be solved using stan-dard optimization methods and optimal μ can thereby beobtained. However, such a solution is descriptive ratherthan prescriptive: it characterizes, the optimal spectrumutilizations for sessions but it does not offer a dynamicperspective that can serve as a basis to develop MAClayer algorithms that achieve such optimality. In the fol-lowing section, we consider a re-parametrization of thisproblem that provides insight on dynamics of optimal,CSMA-based medium access algorithms.

3 CSMA-based utility optimizationCSMA is a fundamental medium access method that isbuilt on the concept of listen-before-transmit. In this

Figure 1 Illustration of an interference graph G with six nodesand three channels. (a) Nodes of G(V, E) are connected if theirtransmissions conflict on the same channel. Not all channels areavailable to all nodes, due to presence of static primary users. Theinterferences graphs associated with each one of the threechannels: (b) G1(V1, E1), (c) G2(V2, E2), (d) G3(V3, E3).


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concept, a transmitter probes the medium at locallydetermined instances. Probing the medium entailsdetecting the possible ongoing transmission activity byother sessions. If there is such activity, then transmis-sion is deferred, otherwise a packet is transmitted. Ineither case, the channel is probed at a later instantunder the same rules.In the present setting, we shall consider the case when

each node adopts CSMA for medium access and probesthe spectrum at random times. It is assumed that eachnode i probes the spectrum ri > 0 times per second, onaverage. The node also maintains a probability vectorpi = [p1i , p

2i , . . . , p

Ci ] that determines which channel to

probe at each probing instant. Here it is understoodthat pci > 0 only if channel c is available to node i; thatis, only if c ∈ Ci. Our goal in this e ort is twofold: (1) toidentify optimal probability vectors pi for all nodes i, tomaximize the social welfare objective, and (2) to charac-terize dynamic algorithms that update these probabilityvectors so as to drive them towards their optimal valuesin time. To keep the focus on the multi-channel aspectof the problem, we shall assume that propagation delaysare negligible and that no hidden terminals exist.A flow chart of the medium-access algorithm at (the

transmitter of) each node is sketched in Figure 2. Here,we assume that transmitters have infinite backlog ofpackets so that they transmit whenever they probe anidle channel. Transmission of a packet may take certaintime that may be different from packet to packet. Eachnode sets up a timeout upon completing transmission ofa packet or upon sensing another transmission on theprobed channel. At the expiration of the timeout, thenode randomly chooses a channel c ∈ Ci according tothe probability distribution pi = [p1i , p

2i , . . . , p

Ci ] Then, the

transmitter senses the chosen channel c and

immediately starts transmission of a packet if none ofthe neighbors currently use this channel. Otherwise, thetransmitter backs off until another timeout expires.We shall analyze the resulting system-wide behavior

using Markovian models. Towards this end, we imposethe following statistical assumptions on packet transmis-sion times and on timeouts: (1) transmission time ofeach packet is exponentially distributed with mean 1,and (2) each timeout period at node i is exponentiallydistributed with mean r−1

i . We shall assume that packettransmission times and timeout values are independentrandom variables. This assumption implies that eachnode i probes the medium at instants of a Poisson clockwith rate ri.Let us proceed further with describing the possible

states of the overall system. Let sci be the binary valuethat is defined as

sci ={1, if node i is transmitting on channel c;0, otherwise.

Clearly, sci is a binary random process whose valuefluctuates according to the probing decisions taken bynodes. For a fixed choice of probability distributions pi,i Î V, the collection of all such binary variabless = [sci ]M×C is an ergodic Markov process. The statespace of this process is SG where

SG = {s : sci scj = 0 for (i, j) ∈ E ∀c ∈ C;∑c∈Ci

sci ≤ 1, ∀i ∈ V}.

A closer inspection of SG yields that for each channelc the vector [sci1 , s

ci2, . . . , sciN ] of transmission activity on

channel c is an independent set of Gc.We define the parameter P = {p1, p2,..., pM} as the col-

lection of all individual probability distributions adoptedat different nodes. For a fixed choice of probability dis-tributions P, we denote the equilibrium distribution ofthe process s by π. Standard state-space truncationarguments [25] can be employed to determine an expli-cit expression for π for any state s Î SG. Namely,

π(s) =

∏i∈V

∏c∈Ci

(ripci )sci

∑s∈SG

∏i∈V

∏c∈Ci

(ripci )sci, s = [sci ]M×C ∈ SG. (2)

Note that expected fraction of time a node i transmitson channel c ∈ Ci can be expressed in terms of π as:

E[sci ] = π(sci = 1) =∑s:sci=1

π(s). (3)

The equilibrium probability π(sci = 1) is the fraction oftime node i transmits on channel c, measured over atime interval that is long enough to allow the process sto settle to equilibrium. In turn, it can be obtained

Figure 2 The algorithm of medium access at every node i. Arandom channel c is chosen according to probability distribution pi.For purposes of analysis, timeouts and durations of packettransmissions are exponentially distributed with mean r−1

i and 1,respectively


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through sufficiently long measurements of channel utili-zation by every node for every available channel. Weshall use π(sci = 1) as a proxy to μc

i , and thereby re-para-metrize Problem STATIC in terms of P. This gives riseto Problem OPT that is defined as:

OPT : maximize W(P) =∑i∈V

Ui

(∑c∈Ci

π(sci = 1)

)

subject to∑c∈Ci

pci = 1, i ∈ V

pci ≥ 0, i ∈ V, c ∈ Ci.

(4)

3.1 Necessary conditions for optimality in OPTWhile the objective of Problem STATIC is concave in μ,the objective of Problem OPT is not necessarily concavein P. Perhaps the easiest way to verify this is to considerthe case when G has two nodes connected by an edge,and there are C = 2 channels both of which are availableto the two nodes. As each utility function Ui(·) isincreasing, it follows that W (P) is maximized if P issuch that the two nodes deterministically choose differ-ent channels at all times. That is, choosing either p1 =[1,0], p2 = [0,1] or p1 = [0,1], p2 = [1,0] maximizes thesocial welfare; however, convex combinations of thesetwo points yield strictly smaller utility since the twonodes would then start blocking each other’s access tothe medium.We therefore turn to the locally optimal solutions of

OPT. Lagrange multiplier method [26] can be utilized todefine candidates for local optimality of the objectivefunction in Equation 4. The Lagrangian function L(P, l,h, x) for the problem Equation 4 with Lagrange multi-pliers l, h, ≥ 0 and a slack variable xis:

L(P,λ−, η−, x−) = W(P) +∑i∈V

λi

⎡⎣∑

c∈Ci

pci − 1

⎤⎦ +

∑i∈V

∑c∈Ci

ηci (p

ci − (xci )

2).

Here, the slack variable x ensures that pci ≥ 0, i Î V,c ∈ Ci. After taking partial derivatives with respect to P,l, h, x, equating the result to zero and discarding theslack variable x one gets.

∂W(P)∂pci

+ λi + ηci = 0,

∑c∈Ci

pci − 1 = 0,

pci ≥ 0, ηci ≥ 0, ηc

i pci = 0,

(5)

for all i Î V; c ∈ Ci. These relations are the well-known Karush-Kuhn-Tucker conditions specifyingnecessary conditions of optimality for Problem OPT.The following theorem presents another representation

of these conditions that is particularly helpful in theconsideration of dynamic optimization algorithms.Theorem 1 If Psolves OPT, then it must satisfy

1)∂W(P)

∂pci=

∂W(P)∂pzi

, for pci > 0, pzi > 0

2)∂W(P)

∂pci≤ ∂W(P)

∂pzi, for pci = 0, pzi > 0

(6)

for every node i Î V ; and c, z ∈ Ci.Proof 1 If pzi > 0then due to the last relation in Equa-

tion 5 ηzi = 0and according to the first relation in Equa-

tion 5:

∂W(P)∂pzi

+ λi = 0. (7)

By the same argument applied to pci , the following istrue

∂W(P)∂pci

+ λi = 0, pci > 0 (8)

∂W(P)∂pci

+ λi + ηci = 0, pci = 0. (9)

Combining Equation 7 and Equation 8 immediatelyyields the first statement of the theorem. Since ηc

i ≥ 0in(5), combining (7) and (9) results in the second statementof the theorem.

4 Updating channel access probabilitiesThe randomized channel selection technique presentedin the previous section needs to be supplied with anadaptation mechanism to update P in a way to solveproblem OPT and maximize its objective. For this task,we shall mimic evolution of the entire collection ofchannel selection probabilities P via differential systemsof the form

P = f (π).

Here, P represents the time derivative of P. In such anexpression, probability distributions P are interpreted tobe updated according to a certain rule f to be deter-mined. This rule depends on the equilibrium distribu-tion π of the system under the current value of P.Hence, implementing the update rule entails measuringthe system state s over time interval that is long enoughto allow equilibrium probabilities to settle in. In turn,we seek update rules that are measurement-based andthat do not require explicit communication of channelprobabilities among nodes.


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The goal of this section is to determine update rules fbased on a gradient ascent method. The idea of thismethod is to change distributions P in the direction ofthe gradient of objective function W (P) in order toreach a local maximum of W (P). A standard gradientascent algorithm in the present context can be charac-terized by

pci =∂W(P)

∂pci, i ∈ V, c ∈ Ci. (10)

However, this differential system cannot be appliedto Problem OPT because it violates the constraint inEquation 4 that each vector pi should be a probabilityvector. For every node i Î V entries of pi must sumup to one; and therefore, the following must be main-tained:

∑c∈Ci

.pci = 0. (11)

We therefore modify the expression Equation 10 inthe following manner:

.pci = pci

⎛⎝∂W(P)

∂pci−

∑c′∈Ci

pc′i∂W(P)

∂pc′i

⎞⎠ , i ∈ V, c ∈ Ci. (12)

Note that in this case

∑c∈Ci

.pci =

∑c∈Ci

pci

⎛⎝∂W(P)

∂pci−

∑c′∈Ci

pc′i∂W(P)

∂pc′i

⎞⎠

=∑c∈Ci

pci∂W(P)

∂pci−

∑c∈Ci

pci∑c′∈Ci

pc′i∂W(P)

∂pc′i

.

Hence if∑c∈Ci

pci = 1 then equality Equation 11 is satis-

fied. In addition, since pci = 0 when pci = 0, the value pcinever changes its sign. This implies that if the initialchoice of pi is a probability vector, then that property issatisfied at all future times.The following theorem proves that, in addition, P con-

verges to a local maximum of the objective function inEquation 4. We note that since P is varying so is W (P);and we denote by W(P) the time derivative of W (P).Theorem 2 Suppose P is updated according to Equa-

tion 12. Then W(P) ≥ 0for any P and the equality holdsif P satisfies conditions Equation 6.Proof 2 The proof is done by expanding time deriva-

tive W(P)using chain rule and by substituting time deri-

vatives.pciwith the right-hand side of the equality

Equation 12.

W(P) =∑i∈V

∑c∈Ci

∂W(P)∂pci

⎧⎨⎩pci

⎛⎝∂W(P)

∂pci−

∑c∈Ci

pci∂W(P)

∂pci

⎞⎠

⎫⎬⎭

=∑i∈V

⎧⎨⎩

∑c∈Ci

pci

(∂W(P)

∂pci

)2

−⎛⎝∑

c∈Ci

pci∂W(P)

∂pci

⎞⎠

2⎫⎬⎭

=∑i∈V

∑c,z∈Ci

pci pzi

2

(∂W(P)

∂pci− ∂W(P)

∂pzi

)2

.

(13)

Since pci ≥ 0, for all i and c ∈ Ci, then Equation 13 isalways nonnegative. Note that any point that makes theright-hand side of Equation 13 equal to zero necessarilysatisfies the Karush-Kuhn-Tucker conditions Equation 6.This completes the proof.The second statement of Theorem 2 shows that if dif-

ferential system Equation 12 is initialized at either a sad-dle point or a local maximum of problem Equation 4,then the value of P never changes from this point. How-ever, the first statement of the theorem practicallyensures that saddle points of problem Equation 4 arenot stable points and rule Equation 12 converges onlyto local maxima. In practice, an implementation of a dif-ferential system Equation 12 is perturbed due to pre-sence of computational error noise and limited timeframe over which equilibrium probabilities π are esti-mated. Therefore, if initialized at a saddle point, anyinfinitesimal change of P will cause the differential sys-tem to further increase the value of the objective func-tion and, thus, to move away from the saddle point.

4.1 Distributed approximations of the methodAlthough gradient ascent method Equation 12 providesa way to converge to a local maxima of Problem OPT, itis not yet clear how it can be applied in a decentralizedfashion. A distributed solution should operate success-fully without global view on measurements in the net-work. In our interpretation of the evolution Equation12, exact distributed implementation would be feasible iffor each node i the partial derivatives

∂W(P)∂pci

, c ∈ Ci

can be obtained via local measurements made by nodei.We start by expressing these partial derivatives in an

explicit form that sheds light on the implementationaspects of the update rule Equation 12. For clarity ofpresentation, we shall restrict attention to the specificobjective of maximizing system-wide spectrum utiliza-tion. That is, we consider the case when Ui(x) = x forall nodes i. The discussion can be readily generalized toother utility functions. The following theorem identifies


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the quantities that should be estimated by each node forstrict implementation of Equation 12:Theorem 3 Suppose that Ui(x) = x for all i. Then

∂W(P)∂pci

=1pci

∑j∈V

∑z∈Cj

[π(sci = szj = 1) − π(sci = 1)π(szj = 1)

], (14)

where i Î V, and c ∈ Ci, pci > 0.Proof 3 Let us express the objective function W(P) by

using explicit expressions for equilibrium distributions πin Equation 2:

W(P) =∑j∈V

∑z∈Cj

π(szj = 1) =∑j∈V

∑z∈Cj

∑s:szj =1

∏Mk=1

∏|Ck|l=1 (rkplk)

slk

∑s∈SG

∏Mk=1

∏|Ck|l=1 (rkplk)

slk.

Therefore,

∂W(P)∂pci

=∑j∈V

∑z∈Cj

⎡⎢⎢⎣

∑S:szj =1

∂

∂pci

∏Mk=1

∏|Ck|l=1 (rkplk)

slk

∑S∈SG

∏Mk=1

∏|Ck|l=1 (rkplk)

slk

−∑

S:szj =1

∏Mk=1

∏|Ck|l=1 (rkplk)

slk

∑S∈SG

∏Mk=1

∏|Ck|l=1 (rkplk)

slk

×∑

S∈SG∂

∂pci

∏Mk=1

∏|Ck|l=1 (rkplk)

slk

∑S∈SG

∏Mk=1

∏|Ck|l=1 (rkplk)

slk

⎤⎥⎥⎦ .

(15)

If pci > 0, then

∂

∂pci

M∏k=1

|Ck|∏l=1

(rkplk)slk = 1{sci = 1} 1

pci

M∏k=1

|Ck|∏l=1

(rkplk)slk .

The theorem follows by substituting the right-hand sideof this expression for the partial derivatives in Equation15, and then by recognizing the fractions in the resultingexpression in terms of the equilibrium probabilitiesEquation 2.We point out that the expression

π(sci = szj = 1) − π(sci = 1)π(szj = 1) (16)

that arises in equality Equation 14 is the covariance ofthe two random variables sci , s

zj (recall that s

ci indicates

transmission of node i on channel c) with respect to theequilibrium distribution π, which is itself induced by P.Every node i Î V that implements system Equation 12

has to estimate partial derivatives of W(P) in Equation12 according to equality Equation 14. However, it ispractically inconvenient for every node to know its cor-relation terms with the rest of the network because,under such a scenario, nodes would require exhaustiveinformation exchange to yield global view of the net-work at every node. Rather, it is desirable to have eachnode operate with regard to only local information. As apossible solution, we propose a distributed version ofEquation 12 that estimates partial derivatives Equation14 based on covariance terms between immediateneighbors.To justify this approach, let us illustrate decay of cov-

ariance Equation 16 as nodes i and j get more distantfrom each other. Consider a rectangular 11-11 grid andthe channels C = 2 for all nodes. The central node ofthe grid has an index 61. Correlation terms between thecentral node and the whole network are presented inFigure 3, where the varying node index is mapped on

Figure 3 Values of covariances between the central node and the nodes of the network. when (a) the central node transmits on thesame channel, (b) the central node transmits on the other channel.


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the actual grid location of the corresponding node. Thisresult is obtained using a simulation frameworkdescribed in Section 5. Probabilities P are chosen at ran-dom, medium probing rate ri is equal to 10 for all nodesi, and 100 experiments are performed to obtain averagevalues. As one can notice, correlation between the cen-tral node and itself has the highest value, followed bycorrelation between the central node and its immediateneighbors. It quickly declines as we move further awayfrom the central node. This observation encourages oneto limit computations of the double sum in Equation 14only to the neighborhood N(i) of node i and yet get apotentially close approximation of the precise value of

partial derivatives∂W(P)

∂pci.

Under the proposed distributed implementation, chan-nel access probabilities are updated as

.pci =

∑j∈N(i)

∑z∈Cj

π(sci = szj = 1) − π(sci = 1)π(szj = 1)

− pci∑k∈Ci

∑j∈N(i)

∑z∈Cj

π(ski = szj = 1) − π(ski = 1)π(szj = 1),(17)

for i Î V and c ∈ Ci. This update rule is inspired byEquation 12, but it is obtained by approximating eachpartial derivative Equation 14 by a sum that is restrictedto a subset N(i) of nodes associated with each node i.The set N(i) defines two different versions of distributedimplementation:

• Local: N(i) is the set of all nodes j such that (i, j) ÎE, and additionally node i,• Greedy: N(i) is comprised only node i.

In practice, the update rule Equation 17 can be imple-mented through periodic exchange of estimated correla-tion terms between immediate neighbors. To calculatethese quantities, a node keeps a time log of channel uti-lization in its immediate neighborhood that alsoincludes the node itself. A sufficiently long measurementinterval would allow the equilibrium probabilities to beestimated by using the time log. To have a flavor of therequired length of this interval, we note that each prob-ability to be estimated is between 0 and 1; so its varia-tion is at most 0.25. Hence, if the samples are takensufficiently apart in time so that they are roughly inde-pendent, Chebyshev’s inequality implies that (4ε2)-1 sam-ples would be enough to estimate each probabilitywithin an error margin of ε, with probability at leastmax(0, 1-ε). Once the measurement interval expires, thenode exchanges its measurements of π-terms in Equa-tion 17 with its immediate neighbors and changes prob-abilities P according to Equation 17. Then, the nodestarts its time log over to get new measurements of

correlation terms. Similarly, the greedy version of Equa-tion 17 would require a node to keep a log of only itsown channel utilization without local informationexchange.How well these distributed versions perform against

each other and centralized Equation 12 version for ran-dom topologies can be seen through extensive numericalsimulations. The results of practical application of thegradient ascent method are presented in the nextsection.

5 Numerical studyPerformance of the CSMA-based channel selection tech-nique presented in Section 3 and now equipped withgradient ascent method Equation 12 and Equation 17can be tested to optimize W(P) in a variety of scenarios.In this section, we choose

Ui(x) = x

for each node i, and thereby aim to maximize theaggregate spectrum utilization in system. Main objectiveof this simulation study is to see how different versionsof this method perform against each other and againstmost relevant published algorithms [23,24] in the com-mon setting. Such comparison is conducted while vary-ing graph connectivity, changing total channelavailability and also introducing spatial variability inavailable channel sets due to presence of primary spec-trum users. Although providing exhaustive experimentsseem difficult in view of the arbitrariness of possibletopologies of G, the reported experimental resultsexpose important aspects of cognitive radio setup andcan serve as benchmarks for practical performancecomparison.

5.1 Simulation frameworkTo start, let us first describe a simulation frameworkthat was implemented to test performance of gradientascent-based methods (12) and (17) and algorithms[23,24] operating within the channel selection techniquedescribed in Section 3.As mentioned earlier, the channel selection technique

is based on iterative estimation of equilibrium probabilitydistributions π followed by changing probability distribu-tions P. Let us describe one such iteration in more detail:

1. Initialize the channel selection technique withinitial choice of probability distributions P. To limiteffect of this step on results, every node has uniforminitial probability of choosing every of its availablechannels.2. Simulate the technique for current P by success-fully transmitting enough packets.


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3. Estimate π by using temporal statistics of chan-nel usage by every node. By using π calculate fullpartial derivatives Equation (12) or partial sumsEquation (17).4. Update P according to: pci (t + 1) = pci (t) + pci , wherepci is either Equation 12 or Equation (17).5. Obtain a new value of the objective function W(P)according to the new estimate of π.6. Terminate the simulation if improvement of theobjective is below a predetermined threshold.7. Otherwise, proceed to step 2.

Medium probing rate ri is 10 for every node i. For faircomparison purposes, channel selection methods in[23,24] need to be embedded in the same simulationframework as described next.

5.2 Benchmark algorithmsTo be clear, the work [23] does not pursue optimizationof an objective function. Rather, it attempts to settleevery node for a deterministic channel choice through arandomized mechanism. In this study, we treat thismechanism in [23] as a way of changing P and we fit itinto the Step 4 of the algorithm in Section 5.1. Namely,at this step, every node i Î V chooses a channel c ∈ Ciaccording to current P. Then, all neighbors comparetheir choices. This comparison is used as an interferencemeasure that [23] do not specify. (Any alternativeshould be chosen considerably to ensure convergence of[23].) If any of neighbors of i happen to choose thesame channel c then pci (t + 1) = 0.5pci (t) andpzi (t + 1) = 0.5pzi (t) + 0.5/(|Ci| − 1) for all other channelsz ∈ Ci. Otherwise, pci = 1 and pzi = 0 for all other z ≠ c.The algorithm in Section 5.1 is then performed basedon the updated P. Let us refer to this modification as“Leith-Clifford” for convenience.The algorithm [24] requires a measure of interference

Fci in the neighbor-hood of every on every of its availablechannels c ∈ Ci. For such measure, we usedFci =

∑j∈{N(i):c∈Cj} π(scj ). Below is a complete procedure

that replaces Step 4:

• Compute parameter T = T0/log2(2+t), where T0 =100 in this studyb.• By using estimated π, compute Fci for every chan-nel c ∈ Ci.• For all c ∈ Ci compute probability distribution

�(c) = exp(−Fci /T)/∑z∈Ci

exp(−Fzi /T)

• Sample channel k according to Θ and assign pki = 1and pci = 0 for c ≠ k, c ∈ Ci

Unlike gradient ascent methods and Leith-Clifford,every node here transmits on a fixed channel that canbe switched only at Step 4. This algorithm is referred toas “Gibbs”.

5.3 Experiments5.3.1 Effect of secondary user densityIn practical scenarios, secondary users can face varyingdegree of competition for limited spectrum in theirneighborhood. In completely connected topologies auser will need to share all of its available channels. Indisconnected topologies, however, a user never has toshare its channels. This experiment studies the effect ofsuch competition by changing connectivity of graph G(V, E). To test this scenario, we fix the number of chan-nels to 11 (due to wide use of IEEE 802.11 hardware inchannel selection literature) and randomly place 30nodes on a unit square. There are 100 different realiza-tions of node placements. Interference radius is intro-duced to generate topologies with varying connectivity.If any two nodes are within the rapdius, then they areconnected on a graph. The radius varies from 0 to

√2

to ensure having a completely connected graph in 30steps. Thus, every node placement yields 30 topologieswith different connectivity. Figure 4 illustrates the per-formance of gradient ascent-based methods againstLeith-Clifford and Gibbs. In general, it is observed thatoverall channel utilization declines as competition for 11channels increases in dense neighborhoods. The gradi-ent ascent-based algorithms are seen to clearly

Figure 4 Performance of the gradient ascent-based methodsagainst Leith-Clifford and Gibbs for different densities of 30nodes. C = 11 channels are considered. The “centralized”, “local”,and “global” versions of the proposed algorithm are represented bythe top curve and their performances are hardly distinguishable.95% confidence intervals are also displayed.


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outperform Leith-Clifford and Gibbs, especially formoderately connected topologies. Performance of Leith-Clifford deteriorates compared with the gradient-basedalgorithms when high node density makes it impossibleto assign channels in a non-conflicting way. The threegradient ascent-based algorithms perform closely toeach other in the given setup.5.3.2 Effect of varying spectrum availabilityDecreasing the total number of available channels alsoincreases competition for limited spectrum. Figure 4indicates that Leith-Clifford and Gibbs significantlyunderperform the rest, for example, at interferenceradius equal to 0.5852. For this radius, 100 differenttopologies are generated. For each topology, a totalnumber of channels available in the network C is chan-ged from 1 to 20. Simulations indicated the absence ofcompetition for channels when C > 20 and thus corre-sponding results are omitted. Figure 5 summarizes theresults for algorithms under consideration. Leith-Cliffordperforms on par with the gradient ascent-based methodswhen the number of channels available is high relativeto moderate node density resulting from the given inter-ference radius. Leith-Clifford’s performance deterioratesfor decreasing channel availability compared with thegradient ascent-based methods which again display closeperformance between each other. Gibbs algorithmunderperforms the others.5.3.3 Effect of primary spectrum user presenceThe goal of this experiment is to test a scenario whenthe presence of primary channel users affect channelavailability throughout the network. Number of ways

exists to model such situations. In this particular case,we generate the same number of topologies in the sameway as in the first experiment for 30 secondary users.Now, a generated topology is combined with 30 fixedprimary users, each of them is assigned one of 11 chan-nels uniformly and at random. Initially, homogeneouschannel availability for the secondary users is nowaffected by primary users. If a primary user and a sec-ondary user are within an interference radius, then theprimary user’s channel is excluded from possible chan-nel choices available to the secondary user. A topologyis not simulated if it contains a secondary user with nochannels to choose from, caused by the presence of ahigh number of primary users in its neighborhood. Theresults of this experiment are displayed in Figure 6which demonstrates the versatility of gradient ascent-based methods (12) and (17) in spectrum sharing pro-blems with spatial spectrum inequality between second-ary users. The methods presented in this workoutperform Leith-Clifford and Gibbs algorithms, espe-cially when connectivity of a topology increases.

6 ConclusionThis work presented a randomized channel selectiontechnique suited for distributed implementation in cog-nitive radio systems. The technique adaptively convergesto a local maximum of a performance objective by usingthe gradient ascent method. Decentralized versions ofthe method were also presented. Reported numericaltests reveal that the proposed gradient ascent-basedmethods outperform benchmark algorithms amid high

Figure 5 Performance of the gradient ascent-based methodsagainst Leith-Clifford and Gibbs for varying number ofchannels C under a moderate node density. The “centralized”,“local”, and “global” versions of the proposed algorithm arerepresented by the top curve and their performances are hardlydistinguishable. 95% confidence intervals are displayed.

Figure 6 Performance of the gradient ascent-based methodsagainst Leith-Clifford and Gibbs for different node densitieswhen primary users are present. Here C = 11 and performancesof “centralized”, “local”, and “global” are very close. 95% confidenceintervals are displayed.


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competition for limited spectrum resources or when pri-mary users are present. For the whole range of topologyconnectivity, performance of greedy and local versionsof the proposed gradient ascent method matches perfor-mance of the centralized version that utilizes all infor-mation available in the network. This result stronglyencourages further research in applications of suchdecentralized methods to spectrum sharing in cognitiveradio systems.

EndnotesaAn independent set of a graph is a subset of nodes thatdoes not include any neighbors in the graph. We shallrepresent each independent set with a binary vector thathas a 1 for each node included in the set, and a 0 foreach of the remaining nodes. Hence convex combina-tions of independent sets are defined accordingly. bTheparameter T represents agility of the algorithm and pro-vides a tradeoff between long term performance (con-vergence to a global maximum) and short termperformance (avoiding getting stuck at local maxima).The reader is referred to [24] for further details andother applications of such parameters.

AbbreviationsCSMA: carrier sense multiple access; LBT: listen-before-talk.

AcknowledgementsThis work was supported in part by NSF through grants CCF-0964652 andCNS-1018154.

Competing interestsThe authors declare that they have no competing interests.

Received: 15 December 2010 Accepted: 9 July 2011Published: 9 July 2011

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doi:10.1186/1687-1499-2011-32Cite this article as: Dashouk and Alanyali: A utility-based approach forsecondary spectrum sharing. EURASIP Journal on Wireless Communicationsand Networking 2011 2011:32.

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