Communicating with Energy HarvestingTransmitters and Receivers

Kaya Tutuncuoglu Aylin YenerWireless Communications and Networking Laboratory (WCAN)

Electrical Engineering DepartmentThe Pennsylvania State University, University Park, PA 16802

kaya@psu.edu yener@ee.psu.edu

Abstract—This paper provides a general framework for utilitymaximization of a wireless network with energy harvesting nodes.The focus is on applying this framework to the single-linkproblem with an energy harvesting transmitter and an energyharvesting receiver. For the general utility maximization problem,it is shown that if the utility of a network can be expressedinstantaneously as a function of the powers of the nodes, thenthe maximum utility achieving power policy for each node canbe found using a water-filling approach for each user. This isachieved by expressing the general utility maximization problemas a pair of nested problems focusing on energy efficiency andadapting to energy harvests separately. The framework extendsthe previous results on offline optimization of energy harvestingtransmitters to networks with all energy harvesting nodes includ-ing receivers and relays as well as any network utility, providedthat the achieved utility is instantaneous and additive in time.The implications of the energy efficiency problem on the energyharvesting problem are demonstrated over an energy harvestingtransmitter-receiver pair, and simulation results are presented toexhibit the performance of the optimal policy along with somealternatives for a range of storage capacities.

Index Terms—Energy harvesting, utility maximization, wire-less networks, optimal scheduling, battery limited nodes.

I. INTRODUCTION

With the recent advances in energy harvesting devicesand an increasing demand for environment friendly mobilecommunications, energy harvesting is emerging as a desirablealternative as a means for wireless networks to function.Whether the objective is to decrease the carbon footprint ofwireless communications or to make nodes energy-wise self-sufficient, harvesting energy from external sources is promis-ing. On the other hand, the availability of ambient energyis time-varying and limited in most applications. Therefore,utilizing this energy efficiently becomes an important problemto match the performances of their battery or grid poweredcounterparts. Recently, there has been a substantial researcheffort on optimal scheduling for energy harvesting transmittersin single and multiple user settings [1]–[12]. In this paper, wepresent a generalized solution that applies to many scenariosand problems considered in the previous work, and beyond.

An offline scheduling problem with a discrete energyharvesting model was first introduced in [1], solving thetransmission completion time minimization problem for atransmitter with infinite energy storage. This work was later

extended in [2] to transmitters with limited energy storage,also showing the duality between transmission completiontime minimization and short-term throughput maximizationproblems in the energy harvesting setting. A directional water-filling algorithm that also considers channel variations wasdeveloped in [3], focusing on a Gaussian fading channelwith an energy harvesting transmitter. This approach wassubsequently extended to other network models including thebroadcast channel [4], [5], multiple access channel [6], andinterference channel [7]. A common theme in these paperswas the achieved instantaneous rate being a function of theinstantaneous transmission power of the nodes.

More involved models also emerged for energy harvestingnetworks in the offline setting, such as the two-hop model withenergy harvesting transmitter and relay [8], and models withinefficient energy storage [9], [10]. Parallel to these efforts,online approaches with causal energy and state informationwere also considered with a queueing theoretic approach in[11], and a Markovian energy model approach in [12].

The aforementioned previous work has focused on an en-ergy harvesting transmitter, ignoring other computational oroperational power costs on both sides of a channel. It is arguedin [13] that receiver decoding power can be significant inspecific applications, and that there is a fundamental trade-offbetween transmitter and decoding power. Arguably receiverpower consumption is even more critical in the energy har-vesting setting.

In this paper, we provide a general solution to the collectionof offline optimization problems maximizing the integral ofany instantaneous utility, for any network model such as in[1]–[7]. The formalization presented enables handling possibletransmitter-receiver power tradeoffs, as well as transmitterand receiver side practicalities including storage inefficiencyand practical transmission schemes, e.g., adaptive modulationtechniques [14], [15]. This is achieved by decoupling theoverall optimization problem to two nested problems focusingon two different aspects of the problem, energy efficiency andadapting to energy harvests. The energy harvesting and net-work models and the general solution is presented in Section IIand Section III respectively. The special case with an energyharvesting transmitter and an energy harvesting receiver isconsidered in Section IV. Numerical results to demonstrate

Fig. 1: Energy harvesting model for the jth network node.

the performance for the energy harvesting transmitter-energyharvesting receiver pair are presented Section V. The paper isconcluded in Section VI.

II. SYSTEM MODEL

We consider M energy harvesting network nodes, each har-vesting and storing their own energy throughout the operationof the network. These nodes can be sensors, transmitters,relays and receivers, each operating within their own energybudget to improve the utility of the network. For simplicity,we assume a time slotted model of slot duration ! , where thechannel can be assumed constant throughout each time slot.The state of the channel is represented by the vector !i in timeslot i. This vector can, for instance, correspond to the fadingstate, in which case our assumption would require that thecoherence time of the channel is longer than a time slot. Westudy a system with an information delivery deadline T = N! .

As the setting suggests, the energy harvesting process isvarying and different for each user. The jth energy harvestingnode harvests an energy packet of Ej,i ! 0 at the beginningof the ith time slot, i.e., at time i! , available immediately forconsumption. The arrival scenario is depicted in Figure 2. Eachnetwork element is assumed to have a battery or high energydensity capacitor of capacity Emax

j for storage. Any arrivalexceeding this capacity is truncated at Emax

j for simplicity,since the overflowing energy is certainly lost.

The offline scheduling problem is considered in this paper,indicating that the energy arrivals Ej,i and the channel param-eters !i are assumed to be known prior to transmission. Thepurpose of this assumption is to quantify the benchmark per-formance, provide insights in development of online policies,and apply to practical cases where these parameters can beestimated in an accurate manner.

Each energy harvesting node utilizes its harvested energytowards the network objective, aiming to improve the overallutility of the network. The rate of energy consumption ateach node is limited by its harvesting process and storagecapacity. Denoting the instantaneous power consumption ofnode j with the non-negative integrable function pj(t), thecumulative energy consumed by node j at any time t fallingin the ith time slot is upper bounded by the harvested energyas ! t

0pj(t) dt "

i"

k=0

Ej,k (1)

Fig. 2: The generalized network model with M nodes andutility f(p,!i) at time slot i.

and lower bounded due to the battery capacity asi"

k=0

Ej,k #! t

0pj(t) dt " Emax

j (2)

for all time instants 0 " t " T and users j = 1, ...,M . Theconstraint in (2) follows from the observation that allowing anyarriving energy to overflow is undesirable. The unlikely casewhere discarding energy at a node is better for the networkwill be handled alternatively in the sequel by manipulating theutility function instead.

We wish to formulate the problem in a general fashionin order to cover a larger set of optimization problems. Inprevious work [1]–[7], the utility to be optimized has beenthe reliable communication rate, or the closely related trans-mission completion time. The achievable rates and capacityfor most channels considered in these papers are well-known,allowing customized solutions to the problem. However thecapacity of an arbitrary network model with M users is anopen problem. We define a general utility function,

f(p,!) = f(p1, p2, ..., pM ,!) (3)

which yields the instantaneous utility of the network given theinstantaneous powers of nodes as pj and the related channelparameter as the vector !. This utility can be replaced withany instantaneous and additive performance metric dependingon the application, some examples of which are instantaneousrate, average distortion, or successful transmission probability.There is no restriction on the utility function in (3) other thanbeing integrable, since the problem will aim to maximize theintegral of this function. Hence the model is applicable to awide range of scenarios, due to the freedom of choice in boththe network topology and objective function.

With the instantaneous utility and energy constraints definedabove, the average utility maximization problem is expressedas

maxp(t)!0

1

T

! T

0f(p(t),!i)dt, i =

#t!

$(4a)

s.t. pj(t) $ Pj , j = 1, ...,M (4b)

where i represents the time slot in which t falls, and Pj is theset of energy-feasible power policies pj(t) for the jth user,

Pj =

!p(t) ! 0

"""""0 "i#

k=0

Ej,k #$ i!

0

pj(t)dt " Emaxj , i " N

%

(5)

Note that verifying the energy feasibility constraints at thebeginning and end of time slots is sufficient. This is dueto consumed powers pi(t) being non-negative by definition,causing the stored energy to reach its local extremum at thebeginning and end of the time-slots. Since harvested energyremains constant within a time slot, feasibility at i!+ and(i + 1)!" implies feasibility throughout time slot i. We nextpresent a general solution to the utility maximization problemin (4).

III. DECOUPLED PROBLEM AND ANALYSIS

In this section, we show that the problem defined in (4) canbe decoupled into two nested problems that can be solvedindependently, with the two resulting problems separatelytargeting the aspects of energy efficient communication andadapting to the energy harvests. We begin by defining theenergy consumed by node j in time slot i as

"j,i =

! (i+1)!

i!pj,i(t) dt. (6)

and the collection of "j,i’s for all nodes at time slot i as thevector "i. This variable can be added to the parameters of themaximization in (4) with the definition in (6) as an equalityconstraint without altering the optimal solution, yielding theexpanded problem

maxp(t)!0,!i

1

T

! T

0f(p(t),!i)dt, i =

#t!

$(7a)

s.t. "j,i =

! (i+1)!

i!pj,i(t) dt, (7b)

pj(t) $ Pj , j = 1, ...,M. (7c)

We present the following observation in order to decouplethe maximization over the two set of variables.

Lemma 1: The optimal total consumed energy values "i inthe ith time slot is sufficient to determine the power policywithin the time slot in consideration, i.e., p(t) for t $ [i!, (i+1)! ]. The optimal policy within this time slot can be found asthe solution to the total energy constraint problem,

maxp(t)

! (i+1)!

i!f(p(t),!i)dt, (8a)

s.t. "j,i !! (i+1)!

i!pj,i(t) dt, i = 1, ...,M, (8b)

which will be referred to as the energy efficiency problem orthe inner problem in the sequel.

The proof of above lemma follows from the fact that unlessthe power policy maximizes the achieved utility within thesaid time slot for a given total energy "i, it can be replaced

with a better one within [i!, (i + 1)! ] without violating theenergy constraints of the original problem given in (5).

We denote by U(",!) the result of the maximization in (8),i.e., the maximum utility that can be achieved in a time slotwith total energy constraint vector " and channel parameter!. Since we know that this will be the achieved utility for theoptimal policy and that it satisfies the constraints in (7b), wecan restate the problem in (7) as

max!i!0

1

T

N"

i=1

U("i,!i), (9a)

s.t. 0 "i"

k=0

Ej,k # "j,k " Emaxj , (9b)

i = 0, ..., N # 1.

We will refer to this problem as the energy allocationproblem or the outer problem on account of being dependenton the values of U(",!). Note that the energy feasibilityconstraints on the consumed power vector in (7c) are nowreplaced with an equivalent set of constraints on "i in (9b).This is possible due to "i being sufficient to determine theavailable energy values at i! for any i.

In order to gain insight on how the energy allocationproblem in (9) can be solved, we focus on characterizing itsobjective function. We start by pointing out that U(",!) isconcave in " for a fixed !. This is a direct consequence of(8); any point violating concavity of U can be improved bytime-sharing, concavifying the function without violating thetotal energy constraint since the constraint is linear. Anotherproperty is that U(",!) is non-decreasing in ", which followsfrom the inequality in (8b), indicating that an energy vector "1must perform at least as good as any " < "1. This means thatif any energy is to be discarded in the optimal policy, this willbe handled by the inner problem by choosing a policy that isnot on the boundary of (8b), justifying the storage capacityconstraint in (2).

Together with (9b) yielding a convex constraint set, it can bestated that the energy allocation problem is convex, and thesolution satisfies the KKT conditions. As found for specialcases in the previous work [4], [6], [7], the solution to thisproblem in each dimension can be obtained by generalizeddirectional water-filling algorithm, with water-levels varyingthroughout the transmission period with

vj,i =#

#"jU(",!i)

%%%"j,i

=N"

k=i

($j # µj)# %i (10)

where $, µ, and % are the non-negative Lagrangian multipliersfor the energy constraints, taking positive values when thestored energy is 0 or Emax

j , and the non-negativity constrainton "i respectively. The water-levels are found by a constrainedwater-flow, the details of which can be found in [3] for a singlelink Gaussian fading channel. This is extended to the multi-node case by alternating the maximization between the energyallocation of each node, which is known to converge due tothe convexity of the problem, as done in [7].

Fig. 3: A single link energy harvesting transmitter and receivermodel.

IV. EXAMPLE: ENERGY HARVESTING TRANSMITTER ANDRECEIVER

An immediate application of the framework described is atwo-node network with an energy harvesting transmitter andenergy harvesting receiver, shown in Figure 3. This case hasnot been studied to date, primarily due to the lack of generalanalytical models on receiver power consumption. Owing tothe ability to handle arbitrary utility functions, our approachin Section III extends to this setting.

Consider the average communication rate as the utility ofthis system. Unlike the previous single-link work with rate-based utility such as in [2], [3] where achieved rate dependson the number of bits departed by the transmitter, we also needto consider how many bits the receiver can decode, process andpossibly store within its own energy budget. Although the ratedependent power consumption of a receiver is not explicitlystudied, recent work on decoding energy has shown that thereis a trade-off between transmitter and receiver power whenachieving a predetermined probability of error [13]. This is dueto better channel coding requiring more energy for decoding,and simpler channel coding requiring more bits to transmitthe same amount of information. Therefore the rate achievedcan be modeled as a function of both transmitter and receiverpowers reflecting this tradeoff. Furthermore, transmitters andreceivers might have additional concerns that affect the utility,or in our case the rate of the system; such as power consumedfor sensing, pre-processing and compressing the data on thetransmitter side, and decompressing, processing, and storageon the receiver side. These factors together yield the utilityor rate that can be achieved with a certain set of powersat the nodes, denoted by the utility function f(pT , pR,!) inFigure 3.

For demonstration purposes, we assume a wireless sensortransmitter and a data center as receiver in a static AWGNchannel. Consider the following power requirements by thenodes for successful operation: For device operation, a con-stant power of ponT and ponR is needed by the transmitter andreceiver respectively to transmit or receive at any nonzerorate. Otherwise the node is assumed to be in a sleep statewith negligible power consumption producing zero utility. Tosuccessfully depart bits at rate rt, the transmitter requires atransmission power Px satisfying rt = B · log(1+kPx) where

0

0

Transmitter power p T

pTonp

Ron

Receiver power p R

Rate

(r T

, r R

, m

in(r

R,r

T))

(a) Instantaneous rates for transmitter, receiver and system

00

!T

!R

U(!

T,!

R)

(b) Resulting solution to the inner problem, U(!)

Fig. 4: Plots of the utility function r(pT , pR) and the solutionto the inner problem U("T , "R).

k depends on received noise at the receiver and the path lossof the channel. On the receiver side, let the receiver require adecoding power exponential in the desired rate, and a circuitpower linear in the rate of processing and storage in additionto ponR . The resulting transmitter power-rate function rT (pT )and receiver power-rate function rR(pR) with some arbitraryparameters is shown in Figure 4a in blue and red respectively,along with the achieved rate-based utility r(p1, p2) given asthe minimum of the two achieved rates for a given powervector (pT , pR).

Next, we solve the inner problem for the utility given inFigure 4a for a time slot of unit length. The inner problemessentially requires finding a possibly infinite set of pointson r(p1, p2) and corresponding weights that, when linearlycombined, consume no more energy than the given energypair ("T , "R) while achieving the maximum rate on aver-age. Conversely, this requires any point on U("T , "R) to beachievable by a linear combination while the overall functionis concave. Thus, if we plot the naively achievable utilitiesas the bits communicated with constant power transmissions,! ·r("1/!," 2/!), the solution is the convex hull of this functionas seen in Figure 4b. The darker region in this plot representsthe values of U("T , "R) for which constant power transmission

is sufficient to solve the inner problem, while the lighter regiondenotes utilities achieved by time-sharing. In effect, this meansthat when the allocated energy in a slot falls on the lightcolored region, the nodes are required to transmit with a powercorresponding to a higher rate for some fraction of the timeslot, and sleep in the remaining fraction to achieve the energyefficient utility.

With the optimal in-slot behavior of transmitter and receivernodes described by the maximum utility function U("T , "R),what remains is to solve the outer problem using generalizeddirectional water-filling alternatingly for the two nodes. Thispart of the problem allows the network to adopt to energyharvests by optimally allocating energy to slots, yielding theaverage rate maximizing power allocation policy. The solutionrequires alternating between the transmitter and receiver onfinding the water levels vT and vR for both users given as,

vT,i =#

#"TU("T , "R)

%%%%"T,i

=N"

k=i

($T,k # µT,k)# %T,i

(11)

vR,i =#

#"RU("T , "R)

%%%%"R,i

=N"

k=i

($R,k # µR,k)# %R,i

(12)

where U("T , "R) is the concavified function in Figure 4b and$, µ and % are non-negative Lagrangian multipliers for eachnode, satisfying the complementary slackness conditions

$j,i(i"

k=0

Ej,k # "j,k) = 0, %j,i · "j,i = 0, (13a)

µj,i(i"

k=0

Ej,k # "j,k # Emaxj ) = 0 j $ {T,R}. (13b)

The two water-filling algorithms interact through U("T , "R),since the partial derivative of this function for one nodechanges with the allocated energy of the other node. Thealternating algorithm converges to yield the optimal policy,for which the water levels for both nodes satisfy equations(11) and (12) simultaneously.

V. SIMULATIONS

In this section, we provide simulation results to accountfor the performance of the optimal and some alternativesuboptimal transmitter and receiver policies. We simulate anenergy harvesting transmitter receiver pair in a static AWGNchannel for 1000 slots and a slot length of ! = 1 sec. Boththe transmitter and the receiver require a constant operatingpower of ponT = ponR = 50mW for being awake. Additionally,the transmitter performs adaptive modulation for power control[14], [15] with a rectangular M-QAM constellation, achievinga discrete set of rates; while the receiver is required toprocess and store the received data, modeled with a constantenergy cost of 5mJ/bit, yielding the following individual ratesachieved:

50 60 70 80 90 100 110 120 130 140 1500

0.5

1

1.5

2

2.5

3

3.5

Storage capacity (mJ)

Achi

eved

ave

rage

rate

(Bits

/s/H

z)

HarvestedHarvested+InnerConstantConstant+InnerIter. Gen. WFUpper Bound

Fig. 5: Average rates achieved using optimal and proposedsuboptimal policies for varying energy storage capacity.

rT (p) =

&''''''(

'''''')

0, [p# ponT ]+ < 1mW (OFF )

1, p# ponT $ [1, 2)mW (BPSK)

2, p# ponT $ [2, 6)mW (QAM)

3, p# ponT $ [6, 10)mW (8#QAM)...

...

bits/s/Hz

(14)

rR(p) =

*p# 50mW

5

++

bits/s/Hz. (15)

We define the system utility as min(rT , rR) when the twonodes achieve rates rT and rR simultaneously. In other words,the utility of this system is the average rate of successfullytransmitted, processed and stored bits. The energy harvestingprocess is assumed to be an i.i.d. process, with ET,i andER,i distributed uniformly in [0, 100]mJ , yielding an averageharvesting rate of 50mW per node.

The system is simulated with various power allocationpolicies, and the average utility achieved against the storagecapacity of the nodes (Emax

T = EmaxR ) is plotted in Figure 5.

The blue lines correspond to algorithms that consume theharvested energy as it is received, labeled Harvested. Thered lines represent a constant power algorithm that waits foran energy threshold and transmits with the optimal constantpower whenever threshold is reached, indicated with Constant.Additionally, the two dashed plots labeled with an additionalTS term are allowed to use the time-sharing rates, i.e., canachieve the utility suggested by the output of the Innerproblem. The performance of the nested optimization in thisproblem is shown by the green plot, with label Iter. Gen.WF representing iterative generalized water-filling used in theouter problem. Finally, an upper-bound on average rate derivedusing a non-energy harvesting model with an average powerof 50mW per node is shown in magenta.

As expected, policies without optimal power allocation ineach time-slot, i.e., without utilization of the time-sharingresults of the energy efficiency problem, perform particularly

worse for smaller storage capacities, since they rarely accumu-late enough energy to just turn their circuitry on for an entiretime-slot. When these algorithms start employing optimaltime-sharing within the slots, shown in dashed lines, theperformance is significantly increased. The optimal algorithm,using both time-sharing and optimal energy allocation withiterative water-filling, performs notably better than all alterna-tives, allowing the system to yield a utility close to the upper-bound, especially when the storage is limited. With increasingstorage capacity, the constant power transmitter is expectedto further approach the proposed optimal policy; however theproposed policy allows using much smaller storage for thesame performance.

VI. CONCLUSION

This paper analyzes an energy harvesting transmitter energyharvesting receiver pair, and provides a method of developingoptimal power policies by demonstrating that the optimizationproblem is an instance of a utility maximization frameworkfor energy harvesting networks whose solution is provided.In particular, we provide a framework for optimizing energyharvesting networks with packet energy arrivals by decouplingthe problem into two nested problems: an inner problemaddressing energy efficiency and an outer problem addressingenergy allocation for energy harvesting nodes. This approachyields a solution that can be extended to any network topologywith an instantaneous, i.e., memoryless, utility function. Forthe application of this framework on the transmitter-receiverpair where both nodes are energy harvesting, numerical resultsreveal that the optimal policy performs notably closer to theupper bound, which is the utility of the non-energy harvestingsystem, i.e., when the nodes have the total energy available tothem at the commencement of the session.

REFERENCES

[1] J. Yang and S. Ulukus, “Optimal packet scheduling in an energy harvest-ing communication system,” IEEE Transactions on Communications,submitted Jun. 2010, available at http://arxiv.org/abs/1010.1295.

[2] K. Tutuncuoglu and A. Yener, “Optimum transmission policiesfor battery limited energy harvesting nodes,” IEEE Transactionson Wireless Communications, submitted Sept. 2010, available athttp://arxiv.org/abs/1010.6280.

[3] O. Ozel, K. Tutuncuoglu, J. Yang, S. Ulukus, and A. Yener, “Transmis-sion with energy harvesting nodes in fading wireless channels: Optimalpolicies,” IEEE Journal on Selected Areas in Communications, vol. 29,no. 8, pp. 1732 –1743, Sep. 2011.

[4] J. Yang, O. Ozel, and S. Ulukus, “Broadcasting with an energy harvest-ing rechargeable transmitter,” IEEE Transactions on Wireless Communi-cations, submitted Oct. 2010, available at http://arxiv.org/abs/1010.2993.

[5] O. Ozel, J. Yang, and S. Ulukus, “Optimal scheduling over fading broad-cast channels with an energy harvesting transmitter,” in Proceedings ofthe The Fourth International Workshop on Computational Advances inMulti-Sensor Adaptive Processing (CAMSAP), San Juan, Puerto Rico,Dec. 2011.

[6] J. Yang and S. Ulukus, “Optimal packet scheduling in a multipleaccess channel with rechargeable nodes,” in Proceedings of the IEEEInternational Conference on Communications, ICC’11, Kyoto, Japan,Jun. 2011.

[7] K. Tutuncuoglu and A. Yener, “Sum-rate optimal power policies forenergy harvesting transmitters in an interference channel,” to appearin JCN Special Issue on Energy Harvesting in Wireless Networks,submitted October 2011, available at http://arxiv.org/abs/1110:6161.

[8] D. Gunduz and B. Devillers, “Two-hop communication with energyharvesting,” to appear in Proceedings of the 4th International Work-shop on Computational Advances in Multi-Sensor Adaptive Processing,(CAMSAP), Dec. 2011, available at arXiv:1109.5484.

[9] B. Devillers and D. Gunduz, “A general framework for the optimizationof energy harvesting communication systems with battery imperfec-tions,” Arxiv preprint arXiv:1109.5490, 2011.

[10] K. Tutuncuoglu and A. Yener, “Optimal power policy for energyharvesting transmitters with inefficient energy storage,” to appear inProceedings of 46th Annual Conference on Information Science andSystems, CISS, Princeton, NJ, Mar 2012.

[11] V. Sharma, U. Mukherji, V. Joseph, and S. Gupta, “Optimal energy man-agement policies for energy harvesting sensor nodes,” IEEE Transactionson Wireless Communications, vol. 9, no. 4, pp. 1326–1336, 2010.

[12] J. Lei, R. Yates, and L. Greenstein, “A generic model for optimizingsingle-hop transmission policy of replenishable sensors,” IEEE Trans-actions on Wireless Communications, vol. 8, no. 2, pp. 547–551, 2009.

[13] P. Grover, K. Woyach, and A. Sahai, “Towards a communication-theoretic understanding of system-level power consumption,” IEEEJournal on Selected Areas in Communications, vol. 29, no. 8, pp. 1744–1755, 2011.

[14] A. Goldsmith and S. Chua, “Variable-rate variable-power mqam forfading channels,” IEEE Transactions on Communications, vol. 45,no. 10, pp. 1218–1230, 1997.

[15] X. Qiu and K. Chawla, “On the performance of adaptive modulationin cellular systems,” IEEE Transactions on Communications, vol. 47,no. 6, pp. 884–895, 1999.