Dynamic spectrum allocation for uplink users with heterogeneous utilities

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 3, MARCH 2009 1405

Dynamic Spectrum Allocation for Uplink Userswith Heterogeneous Utilities

Joydeep Acharya, Student Member, IEEE, and Roy D. Yates, Senior Member, IEEE

Abstract—Dynamic allocation of spectrum prior to trans-mission is an important feature for next generation wirelessnetworks. In this work, we develop and analyze a model fordynamic spectrum allocation, that is applicable for a broadclass of practical systems. We consider multiple service providers(SPs), in the same geographic region, that share a fixed spectrum,on a non-interference basis. This spectrum is allocated to theircustomer end users for transmission to the SPs. Assuming that auser can obtain service from all the SPs, this work develops anefficient algorithm for spectrum allocation. The quality of servicedepends on system parameters such as number of users and SPs,the channel conditions between the users and SPs and the totaltransmit power of each user. The SPs have different efficienciesof reception. We adopt a user utility maximization frameworkto analyze this system. We develop the notion of spectrumprice that enables a simple distributed spectrum allocation withminimal coordination among the SPs and users. Given the userutility functions and the system parameters, we characterize thespectrum price and the users’ optimal bandwidth allocations.Our work provides theoretical bounds on performance limits ofpractical operator to user based dynamic spectrum allocationsystems and also gives insights to actual system design.

Index Terms—Dynamic spectrum allocation, optimization.

I. INTRODUCTION

WE are witnessing a large growth in the scope of wirelesscommunications services. In future new broadband

applications like Worldwide Interoperability for MicrowaveAccess (WiMAX) and Third Generation Partnership Project-Long Term Evolution (3GPP-LTE) will co-exist with tradi-tional technologies like WLAN and 2G cellular. Spectrum al-location among different wireless systems is thus important forensuring fairness and efficiency for end-to-end applications.

The traditional regulatory process for spectrum allocationhas been largely non-responsive to application requirements.This has lead to an artificial scarcity of spectrum and reducedQoS for the users who are being serviced. This has moti-vated the development of dynamic spectrum allocation (DSA)techniques that take into account the application requirements,presence of other devices in the region and link gains betweenthe transmit-receive pairs.

In 2004, the IEEE set up a working group to developthe 802.22 cognitive radio standard to employ the unusedspectrum in the VHF and UHF TV bands to offer wirelessbroadband services [1]. It has been decided that fixed wirelessaccess will be provided in these bands by professionally

Manuscript received January 18, 2008; revised October 3, 2008; acceptedOctober 14, 2008. The associate editor coordinating the review of this paperand approving it for publication was T. Hou.

The authors are with WINLAB, Rutgers University, NJ (e-mail: {joy,ryates}@winlab.rutgers.edu).

This work was supported via NSF grants CNS-0721826 and CNS-0434854.Digital Object Identifier 10.1109/TWC.2009.080073

installed Wireless Regional Area Network (WRAN) basestations to WRAN user terminals. A service provider (SP)operating a base station will share the total spectrum withother SPs in the region and further allocate this spectrum tousers efficiently.

Motivated by this SP-user model of DSA, we propose andanalyze a dynamic spectrum allocation algorithm based onlimited coordination among devices in this paper. We considera two tiered spectrum allocation scheme as shown in Figure 1.There is some total spectrum C Hz available in a geographicarea which is allocated to the users through the SPs. Theusers are permitted to obtain spectrum from all the SPs.We assume that the users obtain non-overlapping chunks ofspectrum from the SPs to avoid interference. Assuming thateach user application has an associated utility which is concaveand increasing as a function of spectrum obtained, we adoptan utility maximization framework [2] to analyze the system.Given user utility functions, channel coefficients between usersand SPs and user power constraints, our aim is to derive howmuch spectrum should a user obtain from a SP and how powershould be subsequently allocated for sending information tothe SPs. We assume that the spectrum utilized by a SP is thespectrum it has to allocate to the users and allow for simple SPinteraction to share the spectrum C. We facilitate SPs sharingthe spectrum C by a spectrum clearing house (SCH), akin toan FCC-controlled regional spectrum broker [3]. Based on ouranalysis, we develop the notion of a spectrum price and useit to propose a simple distributed allocation algorithm.

A. Related Work and Our Contribution

Our work lies in the domain of systems with non-strategicusers who follow a common spectrum allocation protocolwithout greedily trying to maximize their objectives. Such sys-tems could be centralized or distributed. Upcoming OFDMAbased cellular systems such as 3GPP-LTE fall in the central-ized category. User to subcarrier assignment and power alloca-tion for OFDMA have been considered for the downlink [4]and for uplink [5]. These works mostly consider weightedsum rate maximization while we generalize this to concaveutility functions of rates. Also prior work had considered asingle SP with fixed frequency bins (OFDM tones) whereasin our work we allow for multiple SPs and treat spectrumas a continuous resource. Treating spectrum as continuous isjustified for systems where the subcarrier spacing is smalland the number of subcarriers is large. An example systemis LTE which can operate with 15 KHz spacing and 2048subcarriers [6]. In addition, we also allow the different SPs tohave different efficiencies which is defined as the fraction ofShannon capacity that the SP can reliably deliver.

1536-1276/09$25.00 c© 2009 IEEE

1406 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 3, MARCH 2009

In the distributed non-strategic regime, each user followsa distributed spectrum sharing algorithm. Reference [3] in-troduce the notion of Coordinated Access Bands of spec-trum to achieve distributed coordination for spectrum sharing.Spectrum etiquette protocols that act as simple overlays overinterfering devices such as bluetooth and 802.11a/b/g deviceshave been studied in [7]. While [3], [7] have mostly focusedon the network architecture and protocol signaling, this paperintroduces an analytical price based-distributed algorithm.

The assumption of non-strategic users is valid when theusers are transmitter-receiver pairs of a single system or whendifferent systems in a geographic region are jointly designedwith a common objective [8]. This would require the usersand SPs operating in a region to agree on simple spectrumetiquettes which precludes strategic user behavior. In passing,we note that strategic behavior in wireless systems has beenstudied for centralized mechanism design systems in [9] andfor decentralized Nash power games in [8], [10].

We also assume that the SPs do not maximize profits andsimply provide diversity to the users in their choice of channelsfor transmission. This is reasonable for many cases such asthe spectrum being allocated by a single SP who sets upmultiple access points in a region. In practice, SPs couldengage in competition to attract users [11]. In such situationsour work also provides a baseline case for understanding theperformance loss and changes in pricing structure as a resultof competition.

Our model precludes spectrum overlap as we assume thatthe various users are close to each other and to the SPsand thus spectral overlap would cause significant interference.As noted in [8], for regimes when the cross gains betweentransmit receive pairs are stronger than direct gains, onlyorthogonal spectrum allocation guarantees Pareto efficiency.Because of orthogonal allocation, our model has similaritieswith network flow control models [2], [12]. The notion ofSP efficiency and the usage of the Shannon rate function(defined later) in the user utilities distinguishes our work.Practical transmitters might employ directional antennae toachieve frequency reuse but in this work, we limit ourselvesto finding the fundamental limits on gain possible with onlybandwidth allocation. The result will serve as a baseline casefor understanding the additional benefits if multiple antennaeare deployed.

II. SYSTEM MODEL

The network topology is shown in Figure 1. There areN SPs and L end users and a central Spectrum ClearingHouse (SCH). Based on the demand for spectrum, ServiceProvider i provides Xi units to the L users or a subset ofthem. Let xij be the amount of spectrum obtained by user jfrom SP i. The users and SPs are assumed to be capable oftransmitting and receiving over any spectrum band xij whichlies within C. This could be achieved using non-contiguousOFDM technology [13]. Subsequently, user j transmits hisdata to SP i over spectrum xij at rate rij and with power pij .Each user has a total transmit power constraint. The channelbetween SP i and user j is characterized by the link gaincoefficient hij , which remains constant during the period of

Fig. 1. The network topology.

spectrum allocation and subsequent transmission to the SP. Weassume that hij is flat over frequency and hence is same nomatter in which band xij lies. The coefficients hij are assumedto be known by both users and SPs. The background additiveGaussian noise is assumed of unit power spectral density.We first introduce some notations. A source transmitting withpower p, over a flat channel of bandwidth x and link gain hhas signal to noise ratio snr(x, p, h) = hp/x and achieves therate

r(x, p, h) = x log (1 + snr(x, p, h)) . (1)

In terms of r(x, p, h) the rate rij is given by

rij = ηir(xij , pij , hij), (2)

where ηi is the fraction of the Shannon capacity that can bereliably guaranteed by SP i to a user. A possible examplewould be SP i, who has invested in a better decoder (a Turbodecoder with more iterations or better interleaver design) hasa higher ηi than an SP with a conventional Viterbi decoder.Thus the total rate at which user j can transmit reliably is

Rj = R(xj ,pj ,hj) =N∑

i=1

rij , (3)

where xj = [x1j · · ·xNj ], hj = [h1j · · ·hNj ] and pj =[p1j · · · pNj].

There is a utility function Uj(Rj) associated with user jwhich is concave and increasing in Rj . The operating principleof the network is to maximize social welfare or the sum utilityof the users. The optimization problem is

maxxij≥0,pij≥0,Xi≥0

L∑j=1

Uj (Rj) (4a)

s.t.L∑

j=1

xij ≤ Xi, 1 ≤ i ≤ N, (4b)

N∑i=1

pij ≤ Pj , 1 ≤ j ≤ L, (4c)

N∑i=1

Xi ≤ C. (4d)

As shown in (4b), Xi is the spectrum utilized by SP i whichis equal to the spectrum it has to allocate to the users. User jtransmits with power pij to SP i and as (4c) shows there is a

ACHARYA et al. YATES: DYNAMIC SPECTRUM ALLOCATION FOR UPLINK USERS WITH HETEROGENEOUS UTILITIES 1407

constraint Pj on the total transmit power. The total amount ofavailable spectrum is C. User j optimizes over xij and pij . InAppendix A, we show that the objective is concave in thesevariables and since the constraints are linear, the problem canbe solved efficiently.

A. Distributed Solution and Pricing

In this section we give a distributed implementation of thespectrum allocation problem (4). First we relax the constraints(4b) and (4d) in the objective function to form the partialLagrangian L [14]

L(xij , pij , Xi, λ, μ) =L∑

j=1

Uj (Rj) +N∑

i=1

λi

⎛⎝Xi −

L∑j=1

xij

⎞⎠

+μ

(C −

N∑i=1

Xi

), (5)

where λ = [λ1, · · · , λN ]T . The stationarity conditions w.r.t.Xi can be expressed as,

∂L∂Xi

= λi − μ ≤ 0, (6)

with equality holding iff Xi > 0. Interpreting μ as the pricethe SCH charges to the SPs and λi as the price that SP icharges to its users [2] we see that each SP i that providesnon-zero spectrum (Xi > 0) charges the same price λi = μ.This is because the SPs have no objectives of their own.

Thus we form the Lagrangian

L(xij , pij , μ) =L∑

j=1

Uj (Rj) − μN∑

i=1

L∑j=1

xij + μC (7)

for the new optimization problem and the dual

D(μ) = maxxij≥0,pij≥0

L(xij , pij , μ)

s.t.N∑

i=1

pij ≤ Pj , 1 ≤ j ≤ L. (8)

The spectrum price μ is set jointly by the SPs and the SCHby minimizing the dual

minμ>0

D(μ). (9)

From (7) the optimization in (8) decomposes into separateoptimization problems for the users [14]. The optimizationsubproblem for user j is

Uj = maxxij≥0,pij≥0

Uj (Rj) − μ

N∑i=1

xij (10a)

s.t.N∑

i=1

pij ≤ Pj . (10b)

Quantity Uj , called the user surplus in microeconomics [15,Chapter 14], is the residual utility of user j after paying thespectrum cost. In the context of a spectrum price, this is thepayment in terms of the utility function that has to be given to

user j to persuade him to give up his consumption of spectrum.The price μ is set by a distributed price update for (9)

μ(t + 1) =

⎡⎣μ(t) − αμ(t)

⎛⎝C −

N∑i=1

L∑j=1

xij(μ(t))

⎞⎠⎤⎦+

(11)where xij(μ(t)) is the spectrum obtained by user j fromSP i, for a given value of μ(t) and αμ(t) is a positivestep size. From (11) we see that if the spectrum is under-utilized, C −∑N

i=1

∑Lj=1 xij(μ(t)) is positive and thus the

price decreases to facilitate greater utilization of spectrum.Similarly if spectrum is over-utilized, the price increases. Thisis summarized in the following theorem

Theorem 1: The global spectrum price μ charged by all theSPs is set such that the entire spectrum is utilized.

The distributed spectrum allocation mechanism is given inTable I. From [16, Proposition 3.4], μ(t) given in Table Iconverges to the equilibrium price μ for proper choice of stepsize αμ(t).

III. CHARACTERIZING THE SPECTRUM ALLOCATION

We will denote the first and second derivatives of the utilityfunction by

Uj(Rj) � ∂Uj

∂Rj, Uj(Rj) � ∂2Uj

∂R2j

. (12)

The derivatives of the rate function r(x, p, h), in (1), are

Γp(x, p, h) � ∂r

∂p=

hx

x + hp=

h

1 + snr(x, p, h)(13a)

Γx(x, p, h) � ∂r

∂x= log

(1 +

hp

x

)− hp

x + hp

= log (1 + snr(x, p, h)) − snr(x, p, h)1 + snr(x, p, h)

(13b)

It follows from (2) and (3) that the derivatives of Rj wrt xij

and pij can be expressed as,

∂Rj

∂pij= ηiΓp(xij , pij , hij), (14a)

∂Rj

∂xij= ηiΓx(xij , pij , hij). (14b)

To arrive at the optimal solution for the user subproblem(10), we first write its Lagrangian

Lj = Uj (Rj) −N∑

i=1

μxij + γj

(Pj −

N∑i=1

pij

), (15)

where all Lagrange multipliers are positive. The stationarityconditions for the Lagrangian are

∂Lj

∂xij≡ ηiUj(Rj)Γx(xij , pij , hij) ≤ μ, (16a)

∂Lj

∂pij≡ ηiUj(Rj)Γp(xij , pij , hij) ≤ γj , (16b)

with equality holding for users with xij > 0 and pij > 0respectively.

Theorem 2: In the optimal solution of (10) only one SP isactive per user almost surely.


Distributed Spectrum Allocation Mechanism1) At time t, SPs broadcast price μ(t).2) Each user j solves (10) and calculates xij(μ(t)) and pij(μ(t)) for all i SPs.3) All users pass xij(μ(t)) to each SP i.4) The SPs calculate μ(t + 1) from (11).

TABLE IDISTRIBUTED UPDATE OF SPECTRUM AND POWER

Proof: Consider user j and SP i and assume xij > 0 andpij > 0. Thus (16a) and (16b) are satisfied with equality.Dividing (16a) by (16b) and after some manipulation weobtain,(

1 +hijpij

xij

)log(

1 +hijpij

xij

)− hijpij

xij= κjhij , (17)

where κj = μ/γj . Now consider the function Ψ(snr) = (1 +snr) log(1 + snr)− snr, which can be shown to be one-to-oneand increasing in snr. Substituting for snr = hijpij/xij =Ψ−1(κjhij) in (13a) and then substituting for Γp(·) in (16b)we obtain

ηiUj(Rj)[

hij

1 + Ψ−1(κjhij)

]= γj . (18)

We prove the rest by contradiction. Let user j obtainspectrum from SPs i and k. From (18)

ηihij

1 + Ψ−1(κjhij)=

ηkhkj

1 + Ψ−1(κjhkj). (19)

Now since hij is a continuous random variable the probabilityof event (19) is zero. Thus each user obtains spectrum fromone SP almost surely.Various flavors of Theorem 2 are also observed in [17], [18].If instead of a net spectrum constraint ((4b) and (4d) together)there were individual spectrum constraints at each SP (only(4b)), then the problem of ties would occur [4], [5].

Let the active SP of user j be denoted by i∗j . Denotexi∗j j , hi∗j j and ηi∗j by x∗

j , h∗j and η∗

j respectively. The useroptimization in (10) can be re-written by considering onlyi = i∗j . The rate Rj given in (3) has contribution only fromri∗j j and is denoted by R∗

j = η∗j x∗

j log(1 + h∗

jPj/x∗j

).

A. Insights to SP User Assignment

Since user j is attached to SP i∗j , from (4), it means that ifit were allocated the optimum spectrum x∗

j from any other SPi �= i∗j , it would have still obtained a lower utility. Since theutility Uj(Rj) is an increasing function of Rj this implies thatuser j obtains the highest rate from SP i∗j , for given spectrumx∗

j . Define the signal to noise ratio, snrij = hijP/x∗j . Thus

i∗j = arg maxi

ηix∗j log (1 + snrij) (20a)

= arg maxi

(1 + snrij)ηi . (20b)

Actually if user j were to be associated with SP i �= i∗j , theallocated spectrum would be different from x∗

j , but this doesnot affect our result.

Observation 1: The following observations can be made

a) Low snrij regime: Use (1 + x)n � 1 + nx in (20b) toobtain i∗j = arg max

iηihij .

b) High snrij regime: Use the approximation (1 + x)n �xn in (20b). For a better insight consider 2 SPs withSP 1 being more efficient. Thus η = η1/η2 > 1. Thecondition for which SP 1 is the active SP for user jturns out to be

x∗j < P

(hη

1j

h2j

)1/(η−1)

. (21)

Thus user j attaches to the more efficient SP when theoptimal bandwidth allocation x∗

j is less than a threshold.That is, user j will use the more efficient SP whenbandwidth becomes scarce.

Corollary 1: If all SPs have the same efficiency, then eachuser obtains spectrum from the SP to which it has the highestlink gain.

Proof: Follows from condition (20b) with ηi = η.Lemma 1: The following facts hold,

a) U(R) for R = r(x, P, h), as defined in (1), is a decreasingfunction of x.b) Γx(x, P, h) is a strictly decreasing function of x and ispositive for all values of v = [x, P ] for fixed h.

Proof: a) Since U(R) is concave, U(R) < 0. This meansU(R) is decreasing in R. But R increases in x from (1).Combining we get the desired result.b) It can be verified that r(x, P, h) is concave and increasingin v = [x, P ] for fixed h and thus concave and increasingin x for fixed P and h. From concavity of r(x, P, h) wrt x,Γx(x, P, h) is monotonic decreasing in x and since R(x, P, h)is increasing, we conclude that Γx(x, P, h) > 0.In the next Theorem, we verify that each user obtains a strictlypositive spectrum allocation. Intuitively this makes sense as ifa user is not allocated spectrum then the potential increase tothe sum utility due to his transmit power is wasted. The proofappearing in the Appendix B shows that when a new userL + 1 joins the system of L users, a new allocation in whicheach of the original L users forfeits spectrum ε and user Lobtains spectrum Lε provides higher sum utility for small ε.

Theorem 3: In the optimal allocation each user j obtainsspectrum x∗

j > 0.

B. Dependence on Marginal Utility and Received Power

Theorem 4: When two users have the same channel gains,transmit powers and active SP efficiencies, the optimal alloca-tion of spectrum favors the user with a higher marginal utilityof spectrum i.e. whose utility function has a higher rate ofincrease with spectrum.

Proof: Consider users j and k with utility functions satis-fying Uj(R) > Uk(R) for all R and for whom h∗

k = h∗j = h,

Pk = Pj = P and η∗k = η∗

j . Let the allocated spectrum for


users j and k be x∗j and x∗

k respectively. We have to showthat x∗

j > x∗k.

Assume the contrary i.e. x∗k ≥ x∗

j . Now consider (16a) forboth users

Uj(R∗j )Γx(x∗

j , P, h) = Uk(R∗k)Γx(x∗

k, P, h) = μ. (22)

Consider x∗k ≥ x∗

j . Let R∗j = Rj(x∗

j , P, h) and R∗k =

Rj(x∗k, P, h). This implies

1) Uj(R∗j )

(a)> Uk(R∗

j )(b)

≥ Uk(R∗k) where (a) is given

in the statement of the problem and (b) is true fromLemma 1(a)

2) Γx(x∗j , P, h) ≥ Γx(x∗

k, P, h) from Lemma 1(b)

Thus Uj(R∗j )Γx(x∗

j , P, h) > Uk(R∗k)Γx(x∗

k, P, h) from points1) and 2), which contradicts (22).This is because a unit of spectrum Δx yields a highercontribution to sum utility when allocated to user j than touser k. This has also been observed in [12] for a networkflow control problem.

We can illustrate this phenomenon with the class of expo-nential utilities given by

Uj(R) = Γj

(1 − e−R/Γj

), (23)

where Γj is the target rate of user j. For example, Γj =106 b/s might be appropriate for a file transfer while Γj =104 b/s would be adequate for a voice application. SinceUj(R) = e−R/Γj is increasing in Γj for all R, the high targetrate users are allocated more spectrum than those with lowones. As R → ∞, these utilities become flat, i.e. Uj(R) → Γj .

Another class of utilities used to model elastic applicationsare α utilities [19], given by

Uα(R) =1α

Rα, 0 < α ≤ 1, U0(R) = log(R). (24)

α = 1 gives rate as the utility and for lower values of α, theutility increases sub-linearly for rates above a threshold. Thushigh α models applications with high rate requirements.

Lemma 2: For α utilities, U(R)Γx(x, P, h) is a strictlyincreasing function of P for fixed x.

Proof: Refer to Appendix C.Note that U(R(x, P, h)) is actually decreasing in P whileΓx(x, P, h) is increasing in P . For α utilities, we show, inAppendix C, that their product increases with P . This neednot be true for any arbitrary increasing concave function, suchas the exponential utilities in (23) as they flatten out at Γj .

Theorem 5: If all users have α utilities and the receivedpower of one user increases and user to SP assignments remainthe same or the user switches to a SP with same efficiency,then that user obtains more spectrum and the spectrum priceincreases.

Proof: Consider user j and let k �= j be any other user.Let the price be μ and users j and k obtain spectrum x∗

j andx∗

k. Let user j increases his power from Pj to Pj > Pj . Let thenew allocations be x∗

j and x∗k for users j and k. The spectrum

price changes from μ to μ and the rates from R∗j and R∗

k toR∗

j and R∗k for users j and k. By Theorem 3, all spectrum

allocations are strictly positive and relation (16a) holds with

equality for the old and new allocations and

η∗j Uj(R∗

j )Γx(x∗j , Pj , hj) = η∗

kUk(R∗k)Γx(x∗

k, Pk, hk) = μ.(25a)

η∗j Uj(R∗

j )Γx(x∗j , Pj , hj) = η∗

kUk(R∗k)Γx(x∗

k, Pk, hk) = μ.(25b)

We have to show that x∗j > x∗

j . Assume the contrary that theevent A ≡ x∗

j ≤ x∗j holds. Since there is a sum spectrum

constraint, A ⇒ B, where B ≡ x∗k ≥ x∗

k for some user k �= j.From (25a), the old allocation for user k satisfies

η∗kUk(R∗

k)Γx(x∗k, Pk, hk) = μ. From B, x∗

k ≥ x∗k and applying

Lemma 1 we obtain,

η∗kUk(R∗

k)Γx(x∗k, Pk, hk) ≤ μ. (26)

For user j, there are two changes: a decrease in allocatedspectrum and an increase in transmit power. Let us see theireffects in isolation. First keep transmit power unchanged.From A, x∗

j ≤ x∗j and using Lemma 1 we get

η∗j Uj(Rj(x∗

j , Pj , hj))Γx(x∗j , Pj , hj)︸︷︷︸

∂Uj/∂x∗j

≥ μ. (27)

Next we keep the spectrum fixed and consider the increase intransmit power. From Lemma 2

η∗j Uj(Rj(x∗

j , Pj , hj))Γx(x∗j , Pj , hj)︸︷︷︸

∂Uj/∂Pj

> μ. (28)

Recall that R∗j = Rj(x∗

j , Pj , hj). Since Uj(·) is jointlyconcave in x∗

j and Pj from Appendix A we conclude from(27) and (28) that,

η∗j Uj(R∗

j )Γx(x∗j , Pj , hj) > μ, (29)

But (26) and (29) taken together contradict (25b). Hence ouroriginal assumption, events A and B are wrong. Thus x∗

j > x∗j

which implies x∗k < x∗

k for some user k �= j. Hence,

μ(a)= η∗

kUk(R∗k)Γx(x∗

k, Pk, hk)(b)> μ, (30)

where (a) follows from relation (25b) and (b) follows fromLemma 1. Hence proved.Thus user j demands more spectrum as his transmit powerincreases. This leads to a higher price and all other users obtainless spectrum.

Corollary 2: The user with increased power derives ahigher utility and surplus and the sum utility also increases.

Proof: The utility of user j, Uj(R∗j ) increases as it is

an increasing function of both Pj and x∗j . In Appendix D we

show that the surplus, Uj(x∗j ) = Uj(R∗

j )−μx∗j increases with

x∗j . The increase in sum utility can be proved indirectly as

follows: consider the suboptimal allocation where each userl is retained at x∗

l . Since the power of user j increases, thisallocation will still increase the utility of user j and thus thesum utility. The optimal utility can not be worse.


C. Dependence on number of SPs and Users

Theorem 6: As more users are added to the system, thespectrum price increases.

Proof: Assume that the system is in equilibrium with Lusers who have been allocated spectrum and user L + 1 userjoins in with link gain h∗

L+1 and transmit power PL+1. FromTheorem 3, in the new equilibrium, he is allocated non-zerospectrum. This will reduce the allocated spectrum for all otherusers j, 1 ≤ j ≤ L. Since h∗

j and Pj stay the same, thismeans that the price of spectrum goes up from Lemma 1 and(16a) considered with equality at the new price. A new userincreases the demand for spectrum thus raising the price.

Theorem 7: If all SPs are equally efficient and users haveα utilities then the addition of an SP either increases thespectrum price or keeps it unchanged.

Proof: Assume that the system is in equilibrium and SPN +1 joins in the system. If it offers no better channel to anyof the users than their existing ones, i.e. if h∗

j > h(N+1)j forall j, then no user engages itself to the SP and the optimalsolution (spectrum price, spectrum allocated etc) is the sameas before.

However, if for user j, the new SP provides a better channelcoefficient, i.e h∗

j < h(N+1)j , then user j engages itself to SPN + 1 and adjusts its engaged SP index to i∗j = N + 1 andchannel coefficient to h∗

j = h(N+1)j . Thus user j’s channelcondition to his active SP has improved and as per Theorem 5,the price goes up.

As more SPs join the system, a subset of them offerbetter link gains to users resulting in better access to thespectrum. This increases demand for spectrum and hence theprice increases. To understand this consider an analogy frombeachfront property: There exist beach-houses (analogous tospectrum) and they are in demand from vacationers. If goodroads are built so that these houses become easily accessible(analogous to improving link gains or transmit power) thentheir demand goes up and so do their prices.

IV. LINEAR UTILITY FUNCTIONS, Uj(Rj) = Rj

This is the sum rate maximization problem and gives anindication of the capacity of the user-SP vector channel. Wepresent the results and the reader is referred to our previouswork [20] for the details.

Theorem 8: For given link gain h∗j , power Pj and efficiency

η∗j , user j operates at a unique signal to noise ratio, snr∗j which

is given by the solution of

Φ(snr∗j ) = log(1 + snr∗j

)− snr∗j1 + snr∗j

=μ

η∗j

. (31)

From (31) we can also interpret SP efficiency as a scalingfactor of spectrum price μ, i.e. a SP with higher efficiencyhas a smaller effective price μ/η∗

j .Corollary 3: If all SPs are equally efficient, allocated spec-

trum and user surplus are given by

x∗j =

h∗jPj∑L

k=1 h∗kPk

C, (32a)

U(x∗j ) =

h∗jPj

1 +∑L

k=1 h∗kPk/C

. (32b)

500m

SP1, �1 SP2, �2

User 1User 1 User 2User 2

variable fixed

Fig. 2. The linear network with two SPs and two users.

It can be shown that (32b) is an increasing function of h∗j thus

validating Theorem 5. From (32a), the spectrum allocation isdirectly proportional to the received signal power and hencecan be very unfair if the users have wide variations in linkgains and transmit powers. The use of exponential and αutilities mentioned in Section III lead to more fair allocationof spectrum as the allocation now depend on the marginalutilities which have a lesser variation than the link gains. Wewill explore this in Section V via numerical experiments.

V. NUMERICAL RESULTS

The spectrum allocation algorithm has the following basicsteps

1) SP selection by users: The atomic setting is a networkwith one user and two SPs with different efficiencies.

2) Spectrum allocation to users: The atomic setting isa network with one SP and two users with differentreceived powers.

We consider a network of two users and two SPs whichincorporates both steps. We believe that insights from thisnetwork will be applicable to bigger networks as well. Weconsider two SPs in a linear cell with inter-base distanceof 500 meters as shown in Figure 2. For path loss, wechoose the COST-231 propagation model for outdoor WiMAXenvironments [21] at an operating frequency of 2.4 GHz. Letthe noise power spectral density of N0 = −174 dBm/Hz.Denote the SP i to user j distance by dij and the link gain,that incorporates N0, by hij ,

hij,dB = Ploss − N0 = −31.5 − 35 log(dij) − N0. (33)

The distances are measured with SP 1 located at the origin.User 2 is fixed at a distance of d22 = 100 m from the SP2 and the location of user 1 is varied from d11 = 1 m tod11 = 499 m from SP 1 in steps of 1 m. The total spectrumis 50 KHz. The following classes of utilities are consideredbased on the required rates of a user,

a) low required rate: For α utilities U(R) = log(R) andfor exponential utilities Γ = 1 Kbps.

b) high required rate: For α utilities U(R) = R and forexponential utilities Γ = 1 Mbps.

We first consider the spectrum allocation for users withexponential utilities. Let user 2 have a high required rate.SP efficiency ratios of η2/η1 = 1 and 10 are considered.Figure 3 shows the fraction of the spectrum allocated to user1. It also indicates the active SP of user 1. The term SP Switchat distance d = dS means that for d < dS user 1 is attached


0 50 100 150 200 250 300 350 400 450 50010

−4

10−3

10−2

10−1

100

SP 1 to User 1 distance (meters)

Use

r 1

Spe

ctru

m F

ract

ion

η2/η

1 = 10, high, high

η2/η

1 = 1 high, high

η2/η

1 = 1, low, high

η2/η

1 = 10, low, high

SP SwitchSP Switch

SP SwitchAlways SP 2

Fig. 3. Fraction of total spectrum allocated to user 1 as a function of distancefor different target rates and SP efficiencies. Both users have exponentialutilities and user 2 is fixed at 100m from SP 2

0 50 100 150 200 250 300 350 400 450 5000

2

4

6

8

10

12


Spe

ctru

m P

rice

η2/η

1 = 10 high, high

η2/η

1 = 10, low, high

η2/η

1 = 1, high, high

η2/η

1 = 1, low, high

Fig. 4. The spectrum price μ as a function of user 1 distance from SP 2 fordifferent target rates and SP efficiencies. Both users have exponential utilitiesand user 2 is fixed at 100m from SP 2

to SP 1 and for d > dS it switches to SP 2. First considerthat user 1 has a high required rate. Note that the switch toSP 2 occuers earlier when it is more efficient. The spectrumratio is mostly increasing in the link gain to the active SP, h∗

1,as the rate function in (1) is increasing in h∗

1 and spectrumx∗

1 and if h∗1 improves then the rate achieved is increased

even more by allocating more spectrum. Also an increase inR for low/medium R increases the utility U(R). However x∗

1

becomes constant in the region V defined by η2/η1 = 10 andd21 > 400 m. This is because the exponential utility U1(R)flattens near the value of Γ1 at high R. In region V user 1 hasa very high h∗

1 (to SP 2) and SP 2 is more efficient. So user1 achieves a high rate and his utility is near Γ1. This can beseen in Figure 5. Thus as user 1 gets closer to SP 2, any extraspectrum would increase its rate but not its utility. Anotherway to interpret this is to look at the prices in Figure 4. Forregion V both users are close to the flat regions of utilities andhence demand for additional spectrum is less. Consequentlythe prices are initially constant and then falls slightly.

0 50 100 150 200 250 300 350 400 450 50010

2

103

104

105

106


Use

r U

tiliti

es User 2, low, highUser 2, high, highUser 1, high, highUser 1, low, high

Fig. 5. The utilities for both users as a function of distance for differenttarget rates. Both users have exponential utilities and user 2 is fixed at 100mfrom SP 2. The efficiency ratio is η2/η1 = 10

0 50 100 150 200 250 300 350 400 450 50010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100


Use

r 1

Spe

ctru

m F

ract

ion

η2/η

1 = 1, high, high

η2/η

1 = 10 high, high

η2/η

1 = 1, low, high

η2/η

1 = 10, low, high

SP Switch

Always SP 2

Always SP 2

SP Switch

Fig. 6. Fraction of total spectrum allocated to user 1 as a function of distancefor different target rates and SP efficiencies. Both users have α utilities anduser 2 is fixed at 100m from SP 2

Figures 3-5 also show results when user 1 has low requiredrate. Allocation x∗

1 is much less as per Theorem 4. Howeverx∗

1 is enough to satisfy user 1’s utility. Since x∗1 is less, user

1 always attaches to the more efficient SP as per observa-tion 1(b). The prices are almost invariant to changes in d21.This is because user 2 gets majority of the spectrum and thussets the demand. Since it is stationary the prices change onlywith SP efficiencies.

The corresponding results when users have α utilities areshown in Figures 6-8. The same trends of exponential utilityresults are observed but the disparities between the users interms of spectrum allocated and utilities are much more severefor dissimilar link gains. Comparing Figures 3 and 6 we seethat when user 1 has low required rate, allocation x∗

1 forα utilities is significantly less than x∗

1 for the exponentialutilities. The user j with a stronger h∗

j has a much largerimpact on the prices for α utilities. From Figures 4 and 7 wesee that when h∗

2 > h∗1, the α prices vary much less with

d21 than the exponential prices. The unbounded nature of αutilities also mean that there is always demand for spectrum.


0 50 100 150 200 250 300 350 400 450 5000

10

20

30

40

50

60

70


Spe

ctru

m P

rice

η2/η

1 = 10 high, high

η2/η

1 = 10, low, high

η2/η

1 = 1, high, high

η2/η

1 = 1, low, high

Fig. 7. The spectrum price μ as a function of user 1 distance from SP 2 fordifferent target rates and SP efficiencies. Both users have α utilities and user2 is fixed at 100m from SP 2

0 50 100 150 200 250 300 350 400 450 50010

0

101

102

103

104

105

106

107


Use

r U

tiliti

es User 2, low, highUser 2, high, highUser 1, high, highUser 1, low, high

Fig. 8. The utilities for both users as a function of distance for differenttarget rates. Both users have α utilities and user 2 is fixed at 100m from SP2. The efficiency ratio is η2/η1 = 10

Accordingly Figures 4 and 7 for the (high,high) case showthat α prices in region V keeps on increasing unlike theexponential prices. Overall exponential utilities yield moreequitable spectrum allocation than α utilities.

VI. DISCUSSIONS AND CONCLUSION

Dynamic spectrum allocation is important both for central-ized broadband access networks and decentralized cognitiveradio systems. Efficient networks are often designed for non-strategic behavior either by a central command and controlplane or by adherence to a distributed protocol. In this workwe have developed and analyzed a two tier allocation systemfor non-strategic users who obtain spectrum from multipleSPs. We model the system from user welfare maximizationframework. We show that in the optimal policy each userobtains spectrum only from one service provider given by afunction of the link gains and provider efficiency. Based on ouranalysis we develop the notion of a spectrum price to facilitatedistributed allocation. For two general classes of concave

utility functions namely exponential and α, we analyticallycharacterize the spectrum allocation and price. We show thatour results are consistent with basic economics principles. Ourwork provides theoretical bounds on performance limits ofpractical operator to user based dynamic spectrum allocationsystems and also gives insights to actual system design.

We have assumed that system parameters such as hij ,N and L stay constant during the optimization operationand the subsequent transmission. Whenever they change theoptimization needs to be re-done. While we have not addressedsuch timescale issues, it is safe to say that proposed pricebased allocation is ideal for static outdoor settings with astrong Line-of-Sight component between users and SPs. Formore mobile environments the average values of link gainscan be used to derive reasonable allocations.

APPENDIX APROPERTIES OF THE UTILITY FUNCTION

Lemma 3: If U(R) is an increasing and concave functionin R then U(ηR) for R = x log(1 + hP/x) is an increasingand concave function of the vector v = [x, P ]

Proof: U(R) is increasing and concave in R. It can beshown that R(x, P, h) is concave in v = [x, P ]. The restfollows from [14, Section 3.2.4].

Theorem 9: If Uj(Rj) is increasing and concave then

Uj

(∑Ni=1 ηixij log(1 + hijpij/xij)

)is increasing and con-

cave in the vector v(N) = [v1, · · · ,vN] where vi = [xij , pij ].Proof: Let rij(vi) = ηixij log(1 + hijpij/xij) and

Rj

(v(N)

)=∑N

i=1 rij(vi). Thus ∇2Rj = diag[D1, · · · , DN ]is the hessian. Consider any vector z ∈ R2N . zT (∇2Rj)z =∑N

i=1 ziT Dizi where zi = [z2i−1, z2i]. Since each Di is

negative definite from Lemma (3), the sum is negative.

APPENDIX BPROOF OF THEOREM (3)

Define ∇Uj(x∗j ) = ∂Uj/∂x∗

j = Uj(Rj)Γx(x∗j , P, h) and

M = maxj ∇Uj(x∗j − ε) for some ε > 0. From Lemma 1,

∇Uj(x∗j ) is decreasing. The decrease in sum utility is

ΔUdec =L∑

j=1

∫ x∗j

x∗j−ε

∇Uj(x) dx ≤L∑

j=1

∇Uj(x∗j −ε)ε < MLε.

(34)However the utility of user L + 1 is

ΔUinc =∫ Lε

0

∇UL+1(x)dx ≥ ∇UL+1(Lε)Lε. (35)

From (34) and (35), we have to show existence of ε > 0such that ∇UL+1(Lε) > M . Now M is increasing in ε while∇UL+1(Lε) is decreasing in ε. As ε → 0, ∇UL+1(Lε) → ∞due to Γx while M → maxj ∇Uj(x∗

j ). Thus at ε = 0, thedecreasing function is above the increasing function and sothey are sure to intersect at some x = xs. So for ε satisfying0 < ε < xs there is a net increase in sum utility by allocatingspectrum to user L + 1.


APPENDIX CPROOF OF LEMMA 2

We have to show that U(R(x, p, h))Γx(x, p, h) is a strictlyincreasing function of p for fixed x when U(R) = Rα/α for0 ≤ α ≤ 1. Alternatively substituting z = hp/x we have toshow that the following is strictly increasing in z,

(x log (1 + z))α−1

[log (1 + z) − z

1 + z

]

=xα−1 (log (1 + z))α

[1 − z/(1 + z)

log (1 + z)

]. (36)

Since (log (1 + z))α is strictly increasing in z, a sufficientcondition is to show that f(z) = (1 + z) log(1 + z)/z isstrictly increasing in z, which is proved by evaluating f(z)and using the fact that z − log(1 + z) > 0 for all z > 0.

APPENDIX DUSER SURPLUS IN COROLLARY 2

We have to show that U = U(R(x, p, h)) − μx forμ = U(R(x, p, h))Γx(x, p, h) is increasing in p. A sufficientcondition is to show that U(x, p, h) is increasing in both xand p for fixed h, since Theorem 5 proved that increasing pincreases x. Define Rxx(x, p, h) = ∂Γx/∂x. We can show

∂U∂x

= −x[U(R)Rxx(x, p, h) + Uj(R)Γ2

x(x, p, h)]. (37)

Since U(R) is increasing and concave, U(R) > 0 and U(R) <0. Since R(x, p, h) is concave in x, Rxx(x, p, h) < 0. Usingall these we can show that ∂U/∂x > 0. Differentiating

∂U∂p

= U(R)Γp − x

[U(R)ΓpΓx + U(R)

∂Γx

∂p

]= U(R)

[Γp − x

∂Γx

∂p

]− xU(R)ΓpΓx.

It can be shown that,

Γp − x∂Γx

∂p=

hx2

(x + hP )2> 0. (38)

With this information we can also show that ∂U/∂p is positive.

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Joydeep Acharya received his B.Tech. degree inElectronics and Electrical Communications fromIndian Institute of Technology, Kharagpur in 2001and M.S. in Electrical Engineering from RutgersUniversity in 2005. He is currently pursuing hisPhD in Electrical Engineering from Rutgers Uni-versity. From 2001 - 2002, he worked as a researchconsultant in GS Sanyal School of Telecommuni-cations, IIT Kharagpur on Physical Layer designof WCDMA. Since 2003, he has been a graduateassistant at Wireless Information Networks Labora-

tory (WINLAB), Rutgers University. His research interests include spectrumregulation for wireless systems, resource allocation and microeconomicsprinciples as applied to wireless communications, MIMO and OFDM systems.

Roy Yates received the B.S.E. degree in 1983 fromPrinceton and the S.M. and Ph.D. degrees in 1986and 1990 from MIT, all in Electrical Engineering.Since 1990, he has been with the Wireless Informa-tion Networks Laboratory (WINLAB) and the ECEdepartment at Rutgers University. Presently, he isan Associate Director of WINLAB and a Professorin the ECE Dept. He is a co-author (with DavidGoodman) of the text Probability and StochasticProcesses: A Friendly Introduction for Electricaland Computer Engineers (John Wiley and Sons). He

is a co-recipient (with Christopher Rose and Sennur Ulukus) of the 2003 IEEEMarconi Prize Paper Award in Wireless Communications. His research inter-ests include power control, interference suppression and spectrum regulationfor wireless systems.

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Dynamic spectrum allocation for uplink users with heterogeneous utilities