Scheduling Variable Rate Links via a SpectrumServer

Chandrasekharan Raman, Roy D. Yates and Narayan B. MandayamWINLAB, Dept of ECE

Rutgers UniversityPiscataway, NJ

{chandru, ryates, narayan}@winlab.rutgers.edu

Abstract We consider a centralized Spectrum Server thatcoordinates the transmissions of a group of links sharing acommon spectrum. Links employ on-off modulation with fixedtransmit power when active. In the on state, a link obtains adata rate determined by the signal-to-interference ratio on thelink. By knowing the link gains in the network, the spectrumserver finds an optimal schedule that maximizes the averagesum rate subject to a minimum average rate constraint for eachlink. Using a graph theoretic model for the network and a linearprogramming formulation, the resulting schedules are a collectionof transmission modes (sets of active links) that are time shared ina fashion that is reminiscent of spatial reuse patterns in cellularnetworks. In the special case when there is no minimum rateconstraint, the optimal schedule results in a fixed dominant modewith highest sum rate being operated all the time. In order tooffset the inherent unfairness in the above solution, we introducea minimum rate constraint and characterize the resulting loss insum rate when compared to the case when there is no minimumrate constraint. We also investigate alternate fairness criteria bydesigning scheduling algorithms that achieve max-min fairnessand proportional fairness. It is shown that the max-min fair rateallocation maximizes the minimum common rate among the links.Simulation results are presented and future work is described.

I. INTRODUCTIONSince the earliest days of radio regulation, spectrum man-

agement has been driven by improvements in technology,from improved filters and frequency stability that allowedmore channels to be created, to sophisticated logic and radiotechniques that created the worldwide phenomenon of cellular.More recently, however, a new paradigm has emerged inwhich regulation has driven technology. A relatively smallregulatory experiment in open spectrum that began in theISM (Industrial Scientific, Medical) bands has spawned animpressive variety of important technologies and innovativeuses, from cordless phones and wireless LANs to toll takers,meter readers and home entertainment products. This obvi-ous success has further energized an already intense debateabout regulatory strategy by introducing a new set of issuesand beliefs, and while this debate displays intensely heldregulatory and economic viewpoints, it inevitably turns onthe old-fashioned fulcrum of technological capability as well.Ultimately, the capacity of the open access bands, and thequality of service they can offer, will depend on the degreeto which radios can be designed to adapt to a wide variety ofconditions.

As a consequence, radios in future wireless systems areenvisaged to be smart and interference aware. Such radios,often referred to as cognitive radios, are expected to have theability to cooperate and dynamically share spectrum amongseveral interfering radios. In addition to the degree of flexibil-ity and adaptability of these radios, the need for global infor-mation regarding signals in space, time and frequency playsa prominent role in successful cooperation and coexistence.In this paper, we introduce the notion of a Spectrum Serverwhich can serve as an information aid to enable coexistenceof radios in a shared environment. Specifically, these radioscould be made to cooperate by the centralized spectrumserver which can determine neighborhood and interferenceinformation from measurements from the radios and enableefficient coordination. The spectrum server could then advisea set of links, so that spectrum can be used efficiently. Thereare many different ways in which the spectrum server cancoordinate a set of radios in a wireless network [1], [2]. In thiswork, we consider the problem of scheduling transmissionsfor a group of links which have a fixed transmission power,under the objective of maximizing the sum rate obtained by thelinks. We also address issues of fairness by deriving schedulingalgorithms that result in max-min fair and proportional fairrate allocations. Max-min fair scheduling of rates have beenstudied extensively in the context of flow control of sources ina network [3]. Proportional fair scheduling has been studied inthe context of multiuser diversity [4] and downlink schedulingfor HDR [5]. But to the best of our knowledge, it has not beenstudied in the context of our framework.

Scheduling transmissions in a wireless network has beenstudied in various contexts. In [6], a joint scheduling andpower control strategy is proposed to maximize networkthroughput and energy efficiency of the system. Their algo-rithm selects candidate subsets of concurrently active links,and applies the distributed power control algorithm [7] to findthe minimal power vector. Another direction in this problemis addressed in [8], [9], where the authors look at the cross-layer issues of routing, scheduling and power control. In [10],a centralized MAC protocol is proposed but the objective isto maximize a utility function. The authors in [11] introducethe concept of transmission modes and develop a frameworkfor integrated link scheduling and power control policies to

Fig. 1. Graph of network showing the nodes and directed links

maximize the average network throughput, when each link issubject to an average power constraint and each node is subjectto a peak power constraint. The authors assume a model inwhich the data rate of a link is a linear function of the signal-to-interference ratio at the receiver.

In contrast, we consider transmitters with a fixed power on-off modulation and devise schedules that maximize the systemthroughput. We assume that we obtain a non-zero rate in thelinks for any non-zero signal-to-interference ratio (SIR). Theoptimization problem, subject to minimum rate constraintsin the individual links, is posed as a linear program. If thelink gains are known to the spectrum server, it can schedulethe transmissions among the links to maximize the systemthroughput. It is shown that when there is no minimum rateconstraint, a fixed set of links (called the dominant mode)which maximizes the sum rate is operated all the time. In orderto offset the inherent unfairness in the above solution, we intro-duce a minimum rate constraint and characterize the resultingloss in sum rate when compared to the case when there is nominimum rate constraint. We also investigate alternate fairnesscriteria by designing scheduling algorithms that achieve max-min fairness and proportional fairness. We show that the max-min fair rate allocation can be obtained in one step by solvinga linear program which maximizes the minimum common rateamong the links. The proportional fair schedule is obtained bysolving a non-linear convex optimization program. The paperis organized as follows. In section II, we describe the systemmodel. The problem formulation and analytical results aredescribed in section III. We present the max-min fair schedulein section IV and the proportional fair schedule in sectionV. The simulation results are presented in section VI. Weconclude in section VII with pointers to future work.

Before we explain the system model, we comment on thenotation of this paper. We use boldface lowercase characters

Fig. 2. Graph of network showing transmission mode corresponding to(1 0 1 0)

for vectors and boldface uppercase for matrices. If a is avector, aT denotes its transpose and aTb =

i aibi represents

the inner product of the vectors a and b. The vector of all zerosand all ones are represented by 0 and 1 respectively.

II. SYSTEM MODELConsider a wireless network with N nodes forming L

logical links sharing a common spectrum. The network canbe represented as a directed graph G(V , E), where the nodesin the network are represented by the set of vertices V ofthe graph and the links are represented by a set of directededges E . Therefore the cardinalities |V| = N and |E| = L. Adirected edge from a node m to node n implies that n wishesto communicate data to node m. We consider the scenariowhere the spectrum server coordinates the activity of the setof L links to share the spectrum efficiently.

Define the set of transmission modes T = {0, 1, . . . ,M 1}, where M = M 1 denotes the number of possibletransmission modes. Then the mode activity vector ti of modei is a binary vector, indicating the on-off activity of the links.If ti = (t1i, t2i, . . . , tLi) is a mode activity vector, then

tli =

{1, link l is active under transmission mode i,0, otherwise.

(1)Note that there are M possible transmission modes includingthe mode in which all links are off. Figure 1 shows a repre-sentative network and Figure 2 shows particular transmissionmode for the set of links.

Let the transmitter power on a link l be Pl. If Glk is thelink gain from the transmitter of link k to the receiver of linkl and 2l is the noise power at the receiver of link l, the SIRli at the receiver of link l in transmission mode i is given by

li =tliGllPl

kE,k 6=l tkiGlkPk + 2

l

. (2)

The link gain between a transmitter and receiver takes intoaccount the path loss and attenuation due to shadow fading.We assume that the link gains between each transmitter andreceiver are known to the spectrum server. The data ratein each link depends on the SIR in that link. We assumethat the transmitter can vary its data rate, possibly through acombination of adaptive modulation and coding. In particular,for a given mode, the transmitter and receiver on a link employthe highest rate that permits reliable communication given thelink SIR in that mode. For purposes of this study, we assumethat the transmission of other links are treated as Gaussiannoise and that a transmission on link l is reliable in a givenmode i with a data rate

cli = log(1 + li). (3)We emphasize here that we do not consider any minimum SIRthreshold required at each receiver, i.e., associated with eachtransmission mode i, a non-zero li defines some rate on thelink l. Let xi be the fraction of time that transmission mode iis active and rl be the average data rate of link l. Each link hasa minimum average data rate requirement rminl . The averagedata rate in link l is the time average of the data rates of allthe transmission modes that include link l. Thus,

rl =i

clixi, (4)

or in vector form,r = Cx, (5)

where C = [cli] is an LM matrix with non-negative entries,such that column i indicates the rate obtained by each link inmode i.

III. MAXIMUM SUM RATE SCHEDULINGWe are interested in maximizing the sum of the average

data rates over all links l = 1, 2, . . . , L, subject to constraintson the minimum rate for each link. The optimization problemcan be posed as the linear program (LP):

max 1T r (6)subject to r = Cx, (6a)

r rmin, (6b)1Tx = 1, (6c)

x 0. (6d)The objective function 1T r = i ri is the sum of averagerates of the individual links. The constraint (6b) representsthe minimum rate constraint and (6c) is the normalization forthe schedule.

The variables in the LP (6) are r and x. Rewriting the LPin terms of the variable x only, we get

copt(rmin) = max 1TCx (7)

subject to Cx rmin, (7a)1Tx 1, (7b)

x 0. (7c)

Since C is a matrix with non-negative entries, the constraint1Tx = 1 can be replaced by the constraint 1Tx 1 since the

optimum x, say xopt, will satisfy 1Txopt = 1. Otherwise, wecould scale xopt up so that the objective function is increased.We denote the optimal value 1TCxopt as copt(0).

A. No minimum rate constraint

We now consider the special case when rmin = 0, i.e., whenthere is no minimum rate requirement for any of the links.

Theorem 1: When rmin = 0, the solution to problem (7) isxopt = [0 0 . . . 1 . . . 0 0]

T, where the position of 1 corresponds

to the transmission mode with the maximum sum rate. Theoptimal objective value is the maximum column sum of therate matrix C. Hence, the optimal strategy is to always operatethe transmission mode with the maximum sum rate.

Proof: The proof of the theorem is straightforward. Sincermin = 0, any x satisfying (7b) and (7c) is feasible, as (7a)is trivially satisfied. Since 1TC represents the row-vector ofcolumn sums of C, the objective function 1TCx is someconvex combination of column sums of the matrix C. Thus,

1TCx =

Ll=1

Mi=1

clixi (8)

=Mi=1

xi

Ll=1

cli (9)

Mi=1

xi maxi

Ll=1

cli (10)

= maxi

Ll=1

cli (11)

where the equality in (11) is true since i xi = 1. Equalityholds in (10) when x = xopt = [0 0 . . . 1 . . . 0 0]T wherethe position of 1 in xopt is i = argmaxi

Ll=1 cli. Hence the

proof.Depending on the geometry of the links, the dominant

transmission mode can be a single active link or a collectionof geographically separated links. However, one implicationof the above theorem is that the links that are not a part ofthe dominant transmission mode are starved. So, the systemis not fair in terms of providing non-zero data rates to all thelinks.

B. Non-zero minimum rate constraint

In the case when rmin is non-zero, any x satisfying (7b) and(7c) may not be feasible. There is an additional constraint in(7a) which has to be met. Hence the optimal objective valuecannot exceed copt(0). We now characterize the loss in sumrate due to the minimum rate constraint. We begin by writingthe dual problem for the LP.

The Lagrangian for the LP (7) is

L(x,u, v) = 1TCx+ uT (Cx rmin) + v(1 1Tx), (12)

where u RL and v R are the dual variables. The Lagrangedual is

g(u, v) = supx0

L(x,u, v) (13)

= uT rmin + v

+ supx0

(1TC+ uTC v1T )x (14)

=

{uT rmin + v, 1

TC+ uTC v1T 0

, otherwise (15)

Thus the dual problem for the LP (7) isminimize rT

minu+ v (16)

subject to CT (1+ u) v1, (16a)u 0, v 0. (16b)

By strong duality [12, Chapter 5], the optimal value of thedual problem in (16) is equal to copt(rmin). Let (u, v) be thesolution of (16). Since by Theorem 1, copt(0) is the maximumcolumn sum of C and u 0, we have according to (16a),v copt(0). Therefore, the optimal value of (16)

copt(rmin) = rTminu

+ v

rTmin

u + copt(0).

Since copt(0) copt(rmin) rTminu, the loss in sum rate is atmost rT

minu. An interpretation of the dual variable u is that

it can be viewed as the amount of rate loss for a unit increasein rT

min. This is analogous to the dual prices interpretation, in

which the dual variables are interpreted as the price paid forusing the limited resources (primal variables), the constraintsof which are specified in the primal problem.

C. Maximum sum rate schedule with high SNR linksWe can examine the special case of high SNR links when

each link transmits with a large power P . Let us define a setof modes

T = {il : tlil = 1, tkil = 0 for all k 6= l} .

In mode il, link l transmits in isolation and thus we call T ={i1, i2, . . . , iL} the set of isolation modes.

When the transmit power P is high, all links have highSNR and a link l achieves a high rate when transmitting in theisolation mode il. However, in a shared (non-isolation) modej / T , links will have interference-limited SIRs and relativelylow data rates. These observations lead to the followingtheorem.

Theorem 2: If the interference gains Glk are all non-zero,then for sufficiently large transmit power P , the solution to(7) is time sharing among the transmission modes in T .

Proof: If P is the transmit power in all links l E , from(2) the SIR lj of link l in transmission mode j is given by

lj =tljGllP

jE,k 6=l tkjGlkP + 2

l

. (17)

For all modes j / T , the nonzero interference gains Glk andthe monotonicity of the fraction P/(cP + 2) imply that

lj < lj =Gll

jE,k 6=l tkjGlk. (18)

We can thus upper bound the SIR lj of any link l in anytransmission mode j / T as

lj < = maxj /T

maxl

lj . (19)

It follows from (3) thatclj c = log(1 + ), j / T . (20)

Note that c serves as an upper bound for the rate that can beobtained by any link l in a shared mode j / T . However, ina mode il T in which only link l is active,

lil =GllP

2l= l(P ), (21)

a monotone increasing function of P . Let us define

cl(P ) = log(1 + l(P )). (22)as the data rate obtained when link l transmits with power P inthe isolation mode il. Since cl(P ) is a monotone increasingfunction of P , there exists a transmit power P , such thatP > P implies cl(P ) > Lc for all links l.

Now, let us suppose that P > P , but x is an optimalschedule for problem (7) with xj > 0 for a shared modej / T . Consider a new schedule x given by

xi =

0 i = j

xi + xj/L i T

xi otherwise(23)

The schedule x reallocates the time xj in mode j equally tothe isolation modes il in T . In particular, an isolation modeil T will now be active for time

xil = xil +xjL. (24)

We now show that every link l receives a positive rate increaseby switching to schedule x. Under schedule x, a link l obtainsrate

rl =i

clixi = cljxj + clilxil +

i/{j,il}

clixi. (25)

Under schedule x, link l obtains rate

rl =i

clixi = clilx

il +

i/{j,il}

clixi. (26)

For link l, the difference in rates is

rl rl = clil(xil xil) cljxj (27)

=(clilL clj

)xj . (28)

However, P > P implies that in the isolation mode il, linkl obtains rate

clil = cl(P ) > Lc. (29)

It follows that rl rl > 0 for all links l. This contradicts theoptimality of schedule x in that every link achieves a strictlyhigher rate under schedule x.

IV. MAX-MIN FAIR RATE SCHEDULINGThe maximum sum rate scheduling is biased towards links

that have the best quality (i.e., least interference) and isunfair to the other links that are not a part of the dominanttransmission mode. To address this, we will consider two otherfairness criteria in deriving scheduling strategies - max-minfair and proportional fair. In this section, we present the max-min fair [3] schedule.

Definition 1: A vector of rates r is said to be max-min fairif it is feasible and for each l E , rl cannot be increasedwhile maintaining feasibility without decreasing rl for somelink l for which rl rl. Formally, for any other feasibleallocation r, with rl > rl, there must exist some l such thatrl < rl rl.

In the context of flow control of sources in a communicationnetwork, iterative algorithms for computing max-min fair ratevectors exist [3]. Such iterative algorithms use a progressivefilling technique that starts with all rates equal to zero andincreases the rates until one or several link capacity limitsare reached. In order to obtain the max-min fair schedule inour setting, we begin by formulating the LP to maximize theminimum common rate in all the links. We will then show thatthe solution to this LP results in the max-min fair solution. TheLP which maximizes the minimum common rate among thelinks is

r = max rmin (30)subject to r = Cx, (30a)

r rmin1, (30b)1Tx = 1, (30c)

x 0. (30d)Before proving that the above LP results in the max-min

fair schedule, we state the following theorem:Theorem 3: If the link gains Glj are all non-zero, then the

LP (30) which maximizes the minimum common rate amongthe links results in all links getting the same rate r, i.e.,r = r1.

The proof of Theorem 3 appears in the appendix. We nowshow that the schedule obtained by solving (30) is max-minfair.

Theorem 4: The solution x obtained by solving the LP(30) results in the max-min fair solution for the maximumsum rate problem (7).

Proof: Our objective is to seek a max-min fair solutionin the set (7a)-(7c). Denote the set by S. Let us consider thesolution x of (30) which results in the rate allocation r =r1. Now, for any feasible x S, there can be only threedifferent possibilities:

1) r such that the rates in all links rl r, l E .2) r such that for some links rl < r and some links rl

r for l, l E .

3) r such that the rates in all links rl > r, l E .The third possibility can be ruled out since it contradicts theoptimality of (30). From Definition 1 of max-min fairness, itfollows that r1 is the max-min fair rate vector when the firsttwo possibilities hold.

V. PROPORTIONAL FAIR SCHEDULINGThe max-min fair schedule derived in the previous section

leads to global fairness. In this section, we discuss a fairnesscriteria which leads to fairness of individual links.

Definition 2: A vector of rates r is proportional fair if it isfeasible, i.e., Cx = r for x such that 1Tx = 1 and x 0, andif for any other feasible vector r, the aggregate of proportionalchange is negative.

i

ri riri

0. (31)In [13], Kelly proposed proportional fairness in the contextof rate control for elastic traffic. It can be shown that theproportionally fair vector is the one that maximizes the sumof logarithms of the utility functions. Hence, to obtain theproportional fair rates, we solve the following non-linearoptimization problem with linear constraints

maxl

log rl (32)

subject to r = Cx, (32a)1Tx = 1, (32b)

x 0. (32c)The objective function of the above non-linear optimization

problem is increasing and strictly concave. The constraint setis linear and hence the problem is a convex optimizationproblem [12]. This implies that the problem has a uniqueglobal maximum over the constraint set. The solution for suchproblems can be found out by gradient search algorithms [12].

VI. SIMULATION RESULTSWe present some simulation results in this section. Though

the analytical results are true for more general cases, wepresent simulation results for some specific cases to illustrateour findings. The simulation set-up is a 5050 grid. The linksare of fixed lengths and placed at random locations in the grid.The interference gain Glj between the transmitter of link j andthe receiver of a link l is given by Glj = d4lj , where dlj isthe separation distance between the transmitter and receiver.The transmit powers are fixed for all transmissions and thelink geometries are characterized through the signal-to-noiseratio (SNR) at the receiver for that link (in the absence ofinterference).

In the case of maximum sum rate scheduling with nominimum rate constraint, the transmission mode with thehighest sum rate is chosen. The links which are not a part ofthis transmission mode are not operated at all. In the specialcase when the noise at the receiver is high, the denominatorin the SIR expression is dominated by the receiver noise. Thisapproximates the case when there is no interference from the

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

40

45

50

12

3

5

4

TransmitterReceiver

Fig. 3. Set of five links each of length d = 10. The dominant transmissionmode at SNR=10 dB is shown in solid lines.

neighboring links. Hence the best policy would be to turn onall the links in order to maximize the sum rate in all the links.

As the SNR in each link increases, the interference fromneighbors also increases. Then the best transmission modeis that which has the highest sum rate among all the othertransmission modes. The set of links chosen follows spatialreuse patterns that are reminiscent of those used in cellularnetworks. Figure 3 shows a set of links and the dominanttransmission mode at SNR = 10 dB. The links in the dominantmode are shown in solid lines.

In the case of maximum sum rate scheduling with non-zero minimum rate constraint, we see that more than onetransmission mode is operated since there is a minimum raterequirement for each link. The best mode is selected for mostof the time and the mode which includes the poorer qualitylinks are turned on for a fraction of time just enough to satisfytheir minimum rate requirement. Most of the transmissionmodes are thus not used at all since it is best to use thedominant mode during all other times except when a minimumrate should be guaranteed to some poor quality links.

We now discuss an illustrative example of this case. Figure 4shows a set of source-destination pairs. When rmin is zeroat 20 dB received SNR, the mode consisting of links {2, 5}is always operated. But as the common minimum rate rminincreases from zero, an additional set of modes is operatedto satisfy the minimum rate requirement for each link. Theschedules of the individual transmission modes as shown inFigure 6 varies with the minimum rate so that the minimumrate constraint in each of links is maintained. Notice that onlyfive different modes are active. When rmin is increased insteps, we observe that the same set of modes is operated.After a certain rmin value, say r, a different set of modes hasto be operated in order to obtain a feasible schedule. Untilthen the rate of all the modes falls linearly with increase inrmin. The break point in the sum rate curve occurs at r. Wezoom in to the graph for rmin values ranging from 1.4 to 1.58

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

40

45

50

2

3

45

1

TransmitterReceiver

Fig. 4. A set of source-destination pairs

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

2

4

6

8

10

12

rmin

Aver

age

Rate

s (bi

ts/se

c/Hz)

Sum rateLink 2Link 5Link 1, 3, 4

Max min rate

Fig. 5. Variation of sum rates and individual rate as a function of rmin

in Figure 7. We observe that mode {1} and mode {3} areactive for equal amounts of time since link 1 and 3 transmit inisolation in this mode, and they require the same minimum rateto transmit. The fraction of time mode {4} is active increasesas the fraction of time mode {2, 4} transmits decreases tocompensate for the increase in rate in link 4. Finally, the ratesin all the links are same at r 1.58.

The rates corresponding to maximum of the minimumcommon rate r is shown in the Figure 5. All the links endup getting the same rates in this case. The schedule and therate allocation vector the same as those obtained when wesolve (7) with rmin = r1. The comparison of schedulingschemes under different optimization settings is shown inFigure 8. From the Figure, we notice that only in the caseof maximum sum rate with no minimum rate constraint, thereexist links with zero obtained rate. In the case of the max-minfair solution, all the links end up getting the same rate. Table Ishows a numerical comparison of the sum rate and individual

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

0

0.2

0.4

0.6

0.8

1

rmin

Frac

tion

of ti

me

the

mod

e is

activ

edominant mode {2,5}mode {2,4}mode {1}mode {3}mode {4}

Fig. 6. Schedule of different transmission modes at various rmin valueswhen maximizing sum rate of links

1.4 1.42 1.44 1.46 1.48 1.5 1.52 1.54 1.56 1.58 1.6

0

0.2

0.4

0.6

0.8

1

rmin

Frac

tion

of ti

me

the

mod

e is

activ

e

dominant mode {2,5}mode {2,4}mode {1}mode {3}mode {4}

Fig. 7. Schedule of different modes - zoomed in for higher rmin values

link rates among the various links in the network.

VII. DISCUSSION AND CONCLUSION

We introduced the notion of a Spectrum Server, whichallocates a schedule for a set of links in a wireless network,which is modelled as a directed graph. We observe that thisproblem formulation can also be applied to the case wherethe links operate in non-overlapping frequency ranges. In thiscase, some of the link gains Glj may be zero. The model canbe easily extended to the case when there are bidirectionallinks between two nodes (if we assume there are separatetransmitters and receivers), in which case the number oftransmission modes will be 22L. In this case, interference atthe receiver which is colocated with the transmitter in anotherlink is very high. This may result in schedules in which one ofthe bidirectional links are active at any given time. If there is arestriction that only one of the bidirectional links can be active

0 1 2 3 4 5 60

1

2

3

4

5

6

Link indices

Rat

es in

bits

/sec

/Hz

max sum rate, rmin=0

max sum rate, rmin=1

proportional fair ratemax min rate

Fig. 8. Comparison between rates of the links under different settings

at any given time (or when each node has a single transceiver).In this case, the number of transmission modes will be only3L.

In our work, the problem of maximizing the sum rate in allthe links subject to minimum rate constraint is posed as a linearprogram. The solution to the linear program gives the optimumschedule for the transmission modes in the network. But fora network involving L links, there is an exponential numberof transmission modes and thus the LP we solve involvesan exponential number of variables. However, we conjecturethat almost always there are only very few active transmissionmodes as corroborated by our simulation results.

In the present work, the whole process is centralized sincethe spectrum server just solves the linear program to computethe schedules for each of the links. A heuristic algorithmto find the best schedule with less measurement overheadwould be an interesting future work to address. This mayinvolve links reporting the interference seen by them fromall the other links. This would also be a first step to finding acompletely distributed scheduling algorithm. The centralizedapproach proposed in this paper would then serve as an upperbound to the performance of such distributed algorithms.

Throughout this paper, we have assumed that the spectrumserver has knowledge of the link gains. This involves mea-surement of link gains by the spectrum server. An interestingissue is how coarse the measurement can be and how it affectsthe scheduling algorithm. If the link gains are modelled by atime-varying fading process, then finer measurements wouldbe very expensive.

The problem formulation in this work yields itself to manyoptimization problems. One such example is to minimize thesum fraction of times the links are on so that the total transmitpower in all the links is minimized. Another example is tomaximize the sum rate in all the links subject to the conditionthat all the links are active for equal amount of time.

In this work, we assume that there is a single hop com-

TABLE ICOMPARISON OF SUM RATE AND INDIVIDUAL RATES

rmin = 0 rmin = 0.5 rmin = 1 rmin = 1.58 Proportional fair(Max-min fair solution) solution

Sum rate 10.8987 10.2035 9.5082 7.9 7.0632Link 1 0 0.5 1.0 1.58 0.8324Link 2 5.3814 4.5640 3.7466 1.58 1.9581Link 3 0 0.5 1.0 1.58 1.3314Link 4 0 0.5 1.0 1.58 1.9516Link 5 5.5173 4.1394 1.0 1.58 0.9896

munication between the source destination pairs. Yet anotherinteresting issue would be to find the solution for the maximumsum rate problem if we have a schedule with multiple hopsbetween the source and destination.

While we have primarily considered links of equal lengthfor the purposes of numerical illustration, it is of interest tostudy the performance of the various scheduling algorithmsfor the case of links of unequal lengths as well.

VIII. ACKNOWLEDGEMENT

This work is supported in part by the NSF under grantnumber NeTS-0434854 and by the Defense Spectrum Office(DSO) of the Defense Information Systems Agency. The firstauthor thanks Jasvinder Singh for useful discussions on thematerial in this paper.

APPENDIX

Proof of Theorem 3: The LP which maximizes the minimumcommon rate is

max rmin (33)subject to r = Cx, (33a)

r rmin1, (33b)1Tx = 1, (33c)

x 0. (33d)

Let r be the optimal value of (33), corresponding to aschedule x and a set of active transmission modes T = {i T : xi > 0}. Note that the idle transmission mode with theall zero activity vector would never be a part of T because,if it were, we can improve the rates of links in L2 and thiscontradicts that r is the optimal solution of (33). It is requiredto prove that at optima, the rate vector Cx = r1. We assumethe contrary that the solution to (33) leads to unequal ratesover the set of L links. We can then partition the sets of linksE into two disjoint non-empty sets

L1 = {l E : rl > r}

andL2 = {l E : rl = r

}.

This in turn induces a partition on the set T of all activetransmission modes for the optimal solution into three disjoint

sets T 1

, T 2

and T 3

such that

T 1

= {i T : til = 0, for all l L2}, (34)T 2 = {i T

: til = 0, for all l L1}, (35)T 3

= T \{T 1 T

2}.

T 1

and T 2

contain active transmission modes which consistof links only from L1 and L2 respectively, and T 3 containstransmission modes which consist of links in both L1 and L2.We consider two cases below.

A. Case (i): T 1

is non-emptyThere exists an active transmission mode i T

1consisting

of links only from L1. Consider the mode i with activityvector ti given by

tli =

{1, for all l L2,0, otherwise. (36)

Mode i consists of all links from L2. Therefore, cli >0 for l L2. In the optimal schedule x, we know that xi > 0but xi may be zero. The rate in link l under schedule x isrl =

k clkx

k. Define for some fixed 1 > 0, the feasible

schedule

x = [x1 . . . xi 1 . . . x

i + 1 . . . x

M1]

T .

For sufficiently small 1, the schedule x will be feasible. Now,for l L2, the rate rl due to schedule x is

rl =k

clkxk = r cli1 + cli1 (37)

Since cli > 0 for l L2,

rl = r + cli1. (38)

Thus, we conclude that rl > r, l L2. Note that 1 needsto be chosen such that for all l L1, rl > r. The choiceof 1 such that cli1 < minlL1 rl r ensures that rl >r for l L1. We can thus improve the rates in all links inL2. This contradicts the optimality of r. We denote this stepas Increase(1).

B. Case (ii): T 1

is empty

In this case, if T 3 is empty, then T = T 2 and hence allrates are equal, and the proof is complete. Thus we consideronly the case of T

3being non-empty. For an active mode

j T 3 , there exist non-empty subsets of L1 and L2, namelyM1 and M2 such that the activity vector tj is given by

tlj =

1, l M1 L1,1, l M2 L2,0, otherwise.

(39)

Consider the mode j for which the activity vector tj is givenby

tlj =

{1, if l M2,0, otherwise. (40)

We have assumed that all link gains Glj are non-zero, that isthere is lesser interference for links in M2 in mode j thanin mode j due to a lesser number of active links in mode j.Thus for links l M2,

clj =GllPl

kM1M2,k 6=ltkjGlkPk + 2l

(41)

>GllPl

kM2,k 6=ltkjGlkPk + 2l

= clj . (42)

Since j T 3

, xj > 0. For some 2 > 0, we define afeasible schedule

x = [x1. . . xj 2 . . . x

j + 2 . . . x

M1]

T .

Under schedule x and x, link l obtains rate

rl =k

clkxk

andrl =

k

clkxk

respectively. Thus,

rl rl = (xj xj )clj + (xj x

j )clj (43)

= 2(clj clj). (44)It follows from (41) that r1 rl > 0. Let us call this stepIncrease(2).

Since L2 is a finite set, repeatedly applying Increase(1) orIncrease(2) on L2\M2, we can increase the rates of all thelinks in L2. This contradicts the optimality of r. The proofis complete since both cases contradict the fact that the optimalsolution leads to unequal rates in the links.

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