Multi-user Scheduling Schemes for Simultaneous Wireless Information and Power Transfer
MMulti-user Scheduling Schemes for SimultaneousWireless Information and Power Transfer
Rania Morsi † , Diomidis S. Michalopoulos ∗ , and Robert Schober † † Institute of Digital Communications, Friedrich-Alexander-University Erlangen-N¨urnberg (FAU), Germany ∗ Department of Electrical and Computer Engineering, University of British Columbia, Canada
Abstract —In this paper, we study the downlink multi-userscheduling problem for a time-slotted system with simultaneouswireless information and power transfer. In particular, in eachtime slot, a single user is scheduled to receive information, whilethe remaining users opportunistically harvest the ambient radiofrequency (RF) energy. We devise novel scheduling schemes inwhich the tradeoff between the users’ ergodic capacities and theiraverage amount of harvested energy can be controlled. To thisend, we modify two fair scheduling schemes used in information-only transfer systems. First, proportionally fair maximum nor-malized signal-to-noise ratio (N-SNR) scheduling is modified byscheduling the user having the j th ascendingly ordered (ratherthan the maximum) N-SNR. We refer to this scheme as order-based N-SNR scheduling. Second, conventional equal-throughput(ET) fair scheduling is modified by scheduling the user havingthe minimum moving average throughput among the set of userswhose N-SNR orders fall into a certain set of allowed orders S a (rather than the set of all users). We refer to this scheme as order-based ET scheduling. The feasibility conditions required for theusers to achieve ET with this scheme are also derived. We showthat the smaller the selection order j for the order-based N-SNRscheme, and the lower the orders in S a for the order-based ETscheme, the higher the average amount of energy harvested by theusers at the expense of a reduction in their ergodic capacities. Weanalyze the performance of the considered scheduling schemesfor independent and non-identically distributed (i.n.d.) Riceanfading channels, and provide closed-form results for the specialcase of i.n.d. Rayleigh fading. I. I
NTRODUCTION
With the tremendous growth of the number of battery-powered wireless communication devices, the idea of prolong-ing their lifetime by allowing them to harvest energy fromthe environment has drawn a lot of research interest. Wirelesspower transfer is particularly important for energy-constrainedwireless networks such as sensor networks. For such networks,replacing batteries can be costly or even infeasible in scenarioswhere the sensors are deployed in difficult-to-access environ-ments or embedded inside human bodies. However, commonrenewable energy resources such as solar and wind energy areweather dependent and can not be used indoors. On the otherhand, harvesting energy from radio frequency (RF) signals is aviable green solution to supply energy wirelessly to low-powerdevices [1]. Moreover, RF signals can transport informationtogether with energy, which motivates the integration of RFenergy harvesting in wireless communication systems [2–4].Simultaneous wireless information and power transfer(SWIPT) systems were first studied in [2, 3], where theauthors show that there exists a fundamental tradeoff betweeninformation rate and power transfer. This tradeoff can be characterized by the boundary of the so-called rate-energy (R-E) region [4]. However, due to practical circuit constraints,a received signal used for information decoding (ID) cannot be reused for energy harvesting (EH) [4]. One possiblepractical receiver architecture for SWIPT is the time switchingreceiver which switches in time between EH and ID [4].Recently, multi-user systems employing SWIPT have beengaining growing attention in an effort to make SWIPT suitablefor practical networks [4–6]. Multi-user multiple input singleoutput SWIPT systems are studied in [5], where the authorsderive the optimal beamforming design that maximizes thetotal energy harvested by the EH receivers under signal-to-interference-plus-noise ratio constraints at the ID receivers.Moreover, [6] considers a multi-user time-division-multiple-access system with energy and information transfer in thedownlink and uplink channels, respectively. The authors derivethe optimal downlink and uplink time allocation with theobjective of achieving maximum sum-throughput or equal-throughput under a total time constraint.Multi-user scheduling schemes that exploit the independentand time-varying multipath fading of the users’ channelsto create multi-user diversity (MUD) have been extensivelystudied in information-only transfer systems [7, 8]. With suchschemes, the user having the most favourable channel condi-tions is opportunistically scheduled to transmit/receive. Prac-tical opportunistic schedulers aim at maximizing the users’capacities, while maintaining long-term fairness among userswith different channel conditions. For example, scheduling theuser having the maximum instantaneous normalized signal-to-noise ratio (N-SNR) (normalized to its own average SNR)maximizes the users’ capacities while ensuring that the capac-ity of each user is proportional to his channel quality. Anotherway to provide fairness is to guarantee equal throughput(ET) to all users by scheduling the user having the minimummoving average throughput [9]. In SWIPT systems, however,the scheduling rules should be modified to additionally controlthe amount of energy harvested by the users. Multi-userscheduling schemes that exploit MUD and guarantee long-term fairness among users have not been considered in thecontext of SWIPT so far. Thus, in this paper, we devise aproportionally fair order-based
N-SNR scheduler, where theuser having the j th -ordered N-SNR is scheduled and an order-based ET fair scheduler, where the user having the minimummoving average throughput is scheduled among the set of userswhose N-SNR orders fall into a set of allowed orders S a . Ourresults show that the design parameters j and S a can be used a r X i v : . [ c s . I T ] S e p nformation flow Power flow
Fig. 1: Multi-user SWIPT system with time switching re-ceivers. The scheduled user performs ID and the remainingusers perform EH.to control the R-E tradeoff of the order-based N-SNR and ETschedulers, respectively.II. P
RELIMINARIES
In this section, we present the considered system model andthe EH receiver model. We also briefly review round robin(RR) scheduling which serves as a performance benchmarkfor the proposed order-based N-SNR scheduling scheme.
A. System Model
Consider a time-slotted SWIPT system that consists ofone access point (AP) with a fixed power supply and N user terminals (UTs) that are battery-powered. The system isstudied for downlink transmission, where it is assumed that theAP always has a dedicated packet to transmit for every user.We assume that the UT receivers are time switching [4], i.e.,in each time slot, each UT may either decode the informationfrom the received signal or harvest energy from it. In each timeslot, the AP schedules one user for information transmission,while the other idle users opportunistically harvest energy fromthe received signal, as shown in Fig. 1. Furthermore, the APand the UTs are equipped with a single antenna.At time slot t , the AP transmits an information signal to thescheduled user and the received signal at user n is given by r n = (cid:112) P h n e j θ n x + z n , (1)where P is the constant transmit power of the AP, x isthe complex baseband information symbol whose averagepower is normalized to 1, i.e., E [ | x | ]=1 , where E [ · ] denotesexpectation, √ h n and θ n are respectively the amplitude andthe phase of the fading coefficient of the channel from theAP to user n , and z n is zero-mean complex additive whiteGaussian noise with variance σ .The channels from the AP to the users are assumed to beblock fading, i.e., the channel remains constant during onetime slot, and varies independently from one slot to the next.The fading coefficients of the different links are assumed to bestatistically independent. Furthermore, the users’ channels may have different mean channel power gains due to the path lossdifference between users at different distances from the AP.We consider two channel models, namely Ricean and Rayleighfading. The Ricean fading model is relevant for short-range RFEH applications, where a line of sight path may exist togetherwith scattered paths between the transmitter and the receiver.In this case, √ h n is Ricean distributed with Ricean factor K n and mean power gain Ω n , and the instantaneous channel powergain h n is non-central χ -distributed with probability densityfunction (pdf) [10] f h n ( x ) = K n + 1Ω n e − K n − ( Kn +1) x Ω n I (cid:115) K n ( K n + 1)Ω n x , (2)where I ( · ) is the modified Bessel function of the st kind andorder zero. The corresponding cumulative distribution function(cdf) is [10] F h n ( x ) = 1 − Q (cid:112) K n , (cid:115) K n + 1) x Ω n , (3)where Q ( a, b ) is the first-order Marcum Q-function given by Q ( a, b ) = (cid:82) ∞ b xe − ( x a I ( ax ) d x . If a direct link betweenthe transmitter and the receiver does not exist, the Riceanfactor is K n = 0 , ∀ n ∈ { , . . . , N } , and the distribution ofthe AP-UT channels reduces to Rayleigh fading. The resultingchannel power gain of user n is exponentially distributed withmean Ω n , pdf f h n ( x ) = λ n e − λ n x , where λ n = 1 / Ω n , andcdf F h n ( x ) = 1 − e − λ n x . B. Energy Harvesting Receiver Model
We adopt the EH receiver model described in [11]. In thismodel, the average harvested power (or equivalently energy,for a unit-length time slot) is given by [11, Eq. (14)] EH = ηhP, (4)where h is the channel power gain from the AP to the EHreceiver and η is the RF-to-DC conversion efficiency whichranges from 0 to 1. For currently commercially available RFenergy harvesters, η ranges from 0.5 to 0.7 [1]. C. Round Robin (RR) Scheduling
The RR scheduler grants the channels to the users in turn.Hence, no channel knowledge is needed at the AP. Therefore,a user receives information with probability N , and harvestsenergy with probability − N . Thus, user n (denoted byU n ) achieves an ergodic capacity E [ C U n ] that is N times theergodic capacity E [ C U n, f ] that it would have achieved if it had full time access to the channel. That is,E [ C U n ] (cid:12)(cid:12)(cid:12) RR = 1 N E [ C U n, f ] , (5)where E [ C U n, f ] = (cid:82) ∞ log (cid:0) P xσ (cid:1) f h n ( x ) d x is given by[10, Eq. (5)] for Ricean fading. Note that we set the bandwidthin [10, Eq. (5)] to unity since E [ C U n, f ] is in bits/s/Hz. ForRayleigh fading, the ergodic full-time-access capacity of user n is E [ C U n, f ] = e γn E (cid:16) γ n (cid:17) , where ¯ γ n = P Ω n σ is theaverage SNR of user n and E ( x ) is the exponential integralunction of the first order defined as E ( x ) = (cid:82) ∞ e − tx t d t .The resulting average harvested energy of user n for bothRicean and Rayleigh fading channels isE [ EH U n ] = (1 − /N ) ηP Ω n . (6)III. R-E TRADEOFF CONTROLLABLE F AIR S CHEDULING S CHEMES F OR SWIPT S
YSTEMS
Assuming full channel knowledge is available at the AP ,the scheduling decision can be made to control not only thecapacity but also the amount of energy harvested by the users.In this section, we propose two fair scheduling algorithms inwhich the R-E tradeoff can be controlled. A. Order-based Normalized-SNR (N-SNR) Scheduling
In information-only transfer systems, the maximum N-SNRscheme schedules the user having the maximum instantaneousN-SNR, which maximizes the users’ capacities while main-taining proportional fairness among all users [8]. However, forSWIPT systems, such a scheduling rule leads to the minimumpossible harvested energy by the users, since the best statesof the users’ channels are exploited for ID rather than EH.We propose to modify the maximum N-SNR scheme to anorder-based scheme by scheduling the user having the j th ascendingly-ordered N-SNR, where the selection order j ischosen from { , . . . , N } . If j = N , order-based N-SNRscheduling reduces to maximum N-SNR scheduling.
1) Order-based N-SNR Scheduling Algorithm:
Without lossof generality, ordering the N-SNRs in our model is equivalentto ordering the normalized channel power gains h n Ω n since P and σ are identical for all users. Hence, the scheduling rulefor the order-based N-SNR scheme reduces to n ∗ = argorder n ∈{ ,...,N } h n Ω n . (7)where we define “ argorder ” as the argument of the j th ascending order.
2) Performance Analysis:
Next, we analyze the per userergodic capacity and the per user average harvested energyof the considered scheduling scheme. For Ricean fading, thepdf of the metric to be ordered; the normalized channelpower gain X n = h n Ω n , is given by (2) but with unit mean(setting Ω n = 1 ). Thus, X n , n = 1 , . . . , N , are independentand identically distributed (i.i.d.) if all users have the sameRicean factor, i.e., K n = K, ∀ n ∈ { , . . . , N } . This is arealistic assumption since all users exist in the same physicalenvironment. In this case, the pdf and cdf of the normalizedRicean fading channel power gains X n , ∀ n ∈ { , . . . , N } are f X ( x ) = ( K + 1) e − K − ( K +1) x I (cid:16) (cid:112) K ( K + 1) x (cid:17) and F X ( x ) = 1 − Q (cid:16) √ K, (cid:112) K + 1) x (cid:17) , respectively. The pdfof the j th ascendingly ordered random variable X ( j ) is givenby [12, eq. (2.1.6)] f X ( j ) ( x ) = N (cid:18) N − j − (cid:19) f X ( x )[ F X ( x )] j − [1 − F X ( x )] N − j . (8) The channel coefficients of the AP-UT links can be fed back from theusers in the uplink in frequency division duplex systems or can be assumedavailable in time division duplex systems due to channel reciprocity [7].
Thus, the ergodic capacity of user n can be obtained asE [ C j, U n ] = 1 N ∞ (cid:90) log (1 + ¯ γ n x ) f X ( j ) ( x ) d x, (9)where N is the probability that the normalized channel of user n has the j th order. The average harvested energy of user n isE [ EH j, U n ] = ηP Ω n ∞ (cid:90) x (cid:18) f X ( x ) − N f X ( j ) ( x ) (cid:19) d x = ηP Ω n (cid:2) − E [ X ( j ) ] /N (cid:3) , (10)where we used f X ( x ) = N (cid:80) Nj =1 f X ( j ) ( x ) . To the best ofour knowledge, closed-form expressions for (9) and (10) donot exist for Ricean fading channels. Hence, we resort tonumerical integration. For Rayleigh fading, we use (8)–(10)and the pdf and cdf of X n , namely, f X ( x ) = e − x and F X ( x ) = 1 − e − x , to obtain the per user ergodic capacityin closed-form asE [ C j, U n ] = (cid:0) N − j − (cid:1) ln(2) j − (cid:88) l =0 ( − l (cid:0) j − l (cid:1) ( N − j + l + 1) e ( N − j + l +1)¯ γn E (cid:18) N − j + l + 1¯ γ n (cid:19) , (11)and the per user average harvested energy asE [ EH j, U n ] = ηP Ω n − N N (cid:88) l = N − j +1 l . (12)For the special case of j = N , the resulting average systemcapacity (cid:80) Nn =1 E [ C j, U n ] reduces to [8, Eq. (44)]. B. Order-based ET Scheduling
Conventionally, ET fairness can be achieved by schedulingthe user having the minimum moving average throughput [9].However, with such a scheduling rule, neither the resultingET nor the amount of energy harvested by the users can becontrolled. Hence, we devise a new scheme which trades theET level for the amount of energy harvested by the users.
1) Order-based ET Scheduling Algorithm:
First, the users’instantaneous N-SNRs are ascendingly ordered, and thenamong the set of users whose N-SNR orders fall into the setof allowed orders S a , the AP schedules the one having theminimum moving average throughput. Thus, at time slot t ,the scheduler selects user n ∗ that satisfies n ∗ = argmin O n ∈S a r n ( t − , (13)where O n ∈ { , . . . , N } is defined as the order of theinstantaneous N-SNR of user n , and r n ( t − is the throughputof user n averaged over previous time slots up to slot t − .The throughput of the users is updated recursively as r n ( t ) = (cid:26) (1 − β ) r n ( t −
1) + βC n ( t ) if user n is scheduled (1 − β ) r n ( t − otherwise, (14)where C n ( t ) = log (cid:16) P h n ( t ) σ (cid:17) is the feasible rate of user n in time slot t , and β ∈ (0 , is a smoothing factor selectedo asymptotically vanish (e.g., β = 1 /t ) . Confining the set S a to low orders (e.g., S a = { , . . . , (cid:98) N (cid:99)} ) leads potentially to alarger amount of harvested energy compared to conventionalET scheduling (which uses S a = { , . . . , N } ), at the expenseof a reduced ET. This is because a user from the set of lowN-SNR users is scheduled for data reception and the usershaving relatively high N-SNRs are selected for EH. We notethat depending on the choice of S a , ET scheduling may notalways be feasible. This issue is investigated later in Theorem1 in detail.
2) Performance Analysis:
Next, we analyze the ergodiccapacity and the average harvested energy per user. Theergodic capacity of user n can be formulated asE [ C U n ] = E [ C U n | O n ∈ S a ] × Pr ( O n ∈ S a )= |S a | N ∞ (cid:90) log (1 + ¯ γ n x ) |S a | (cid:88) j ∈S a f X ( j ) ( x ) d x × Pr ( U n | O n ∈ S a )= (cid:88) j ∈S a E [ C j, U n ] (cid:12)(cid:12)(cid:12) N-SNR Pr ( U n | O n ∈ S a ) , (15)where | · | denotes the cardinality of a set, |S a | f X ( j ) ( x ) is thelikelihood function that the order of the normalized channel x of user n is j given that j ∈ S a , Pr ( U n | O n ∈ S a ) isthe probability that user n is scheduled given that it hasthe chance to be scheduled, and Pr ( O n ∈ S a ) = |S a | N sincethe probability that a user’s N-SNR takes any order from { , . . . , N } is /N . To write the average capacity of user n in terms of its unconditioned probability of being scheduled p n ∆ = Pr ( n ∗ = n ) , we use p n = Pr ( U n | O n ∈ S a ) |S a | N . (16)Hence, the average capacity of user n reduces toE [ C U n ] = N |S a | (cid:88) j ∈S a E [ C j, U n ] (cid:12)(cid:12)(cid:12) N-SNR p n ! = r, ∀ n ∈ { , . . . , N } , where the average capacities of all users are forced to beequal to r , as required for ET transmission. This requires thescheduling probability of user n to be p n = r N |S a | (cid:80) j ∈S a E [ C j, U n ] (cid:12)(cid:12)(cid:12) N-SNR . (17)Since (cid:80) Nn =1 p n = 1 must hold, the resulting ET reduces to r = 1 N N (cid:80) n =1 1 |S a | (cid:80) j ∈S a E [ C j, U n ] (cid:12)(cid:12)(cid:12) N-SNR . (18)In words, the equal throughput achieved by all users using theorder-based ET scheme can be obtained by first determiningthe arithmetic mean of the order-based N-SNR capacities for An asymptotically vanishing β ensures convergence of the moving averagethroughput r n ( t ) to its ensemble average E [ C U n ] since the considered fadingprocess h n ( t ) is assumed to be stationary [7]. the orders in S a for each user, and then the harmonic mean ofthe resulting quantity for all users. This indicates that the userhaving the worst average channel will have a dominant effecton the resulting ET. Using (17) and (18), the set of schedulingprobabilities required for all users to achieve ET reduces to p n = N (cid:88) i =1 (cid:80) j ∈S a E [ C j, U n ] (cid:12)(cid:12)(cid:12) N-SNR (cid:80) j ∈S a E [ C j, U i ] (cid:12)(cid:12)(cid:12) N-SNR − , ∀ n ∈ { , . . . , N } . (19)As mentioned before, for certain combinations of S a and Ω n , n = 1 , . . . , N , the order-based ET scheduling algorithmmay fail to provide all users with ET. In particular, the set ofscheduling probabilities p n , n = 1 , . . . , N , in (19) required forthe users to achieve ET may be infeasible. In the followingtheorem, we provide necessary and sufficient conditions forthe ET-feasibility of the order-based ET scheduling algorithm. Theorem 1.
The order-based ET scheduling is ET-feasible iff p n ≤ |S a | N , ∀ n ∈ { , . . . , N } , L (cid:88) l =1 p k l ≤ (cid:0) N − |S a |− (cid:1) L + (cid:0) L |S a | (cid:1) (1 − |S a | ) (cid:0) N |S a | (cid:1) , ∀ ( k , . . . , k L ) ∈ C L , ∀ L = |S a | , . . . , N, where C L is the set of all (cid:0) NL (cid:1) combinations ( k , . . . , k L ) of { , . . . , N } .Proof: Please refer to the Appendix.
Remark . It can be verified that the conventional ET scheme,where |S a | = N , is always ET-feasible. Also, the secondcondition is always satisfied for L = N as it reduces to (cid:80) Nn =1 p n ≤ which is satisfied with equality by definition.For the case when |S a | = 1 , the scheme reduces to theorder-based N-SNR scheme discussed in Section III-A whichalways achieves proportional fairness but not ET fairness. Inconclusion, by properly selecting S a with |S a | > , the ET –iffeasible– can be traded for the amount of energy harvested bythe users. Remark . In most practical scenarios, the order-based ETscheduling algorithm is ET-feasible. ET-infeasibility occurswhen the mean channel power gains Ω n of the users dif-fer by many orders of magnitude. For example, a scenariowith 4 users having Rayleigh fading channels with Ω n =1 , , − , and − and with S a = { , } is feasi-ble as the resulting p n = { . , . , . , . } satisfy the conditions in Theorem 1. In contrast, the samescenario but with Ω n = 1 , , − , and − is infea-sible since the required scheduling probability set p n = { . , . , . , . } does not satisfy the secondfeasibility condition in Theorem 1 for L = |S a | = 2 .Next we calculate the average amount of harvested energyper user. Defining S (cid:123) a as the complement of set S a , the average .3 3.4 3.5 3.6 3.7 3.8 3.9 4 4.1 4.200.511.522.533.5 x 10 −5 Average user capacity in bits/channel use A v e r ag e u s e r h a r v e s t e d e n e r g y p e r un i tt i m e ( W a tt s ) RRN-SNR j = 1N-SNR j = 2N-SNR j = 4N-SNR j = 6N-SNR j = 7 user 1to 7R-E curve ofworst-channel user R-E curve ofbest-channel user Fig. 2: Rate-energy curves of the order-based N-SNR and theRR schemes for i.n.d. Ricean fading channels with K = 6 .harvested energy of user n is given byE [ EH U n ] = E [ EH U n | O n ∈ S (cid:123) a ] × Pr ( O n ∈ S (cid:123) a )+ E [ EH U n | O n ∈ S a ] × Pr ( O n ∈ S a )= ∞ (cid:90) ηP Ω n x |S (cid:123) a | (cid:88) j ∈S (cid:123) a f X ( j ) ( x ) d x × |S (cid:123) a | N + ∞ (cid:90) ηP Ω n x |S a | (cid:88) j ∈S a f X ( j ) ( x ) (cid:18) − p n N |S a | (cid:19) d x × |S a | N , where from (16), p n N |S a | is the conditional probability that user n is scheduled given that O n ∈ S a . After simple manipulations,E [ EH U n ] reduces toE [ EH U n ] = ηP Ω n N N (cid:88) j =1 E [ X ( j ) ] − p n |S a | (cid:88) j ∈S a E [ X ( j ) ] = ηP Ω n − p n |S a | (cid:88) j ∈S a E [ X ( j ) ] , (20)where the unit term is obtained using (cid:80) Nj =1 X ( j ) = (cid:80) Nn =1 X n ,thus (cid:80) Nj =1 E [ X ( j ) ] = (cid:80) Nn =1 E [ X n ] = N , since the normalizedchannels X n are unit-mean random variables ∀ n = 1 ,. . ., N .The average user capacities and harvested energies for theorder-based ET scheme in (18) and (20) can be obtained inclosed-form if their order-based N-SNR counterparts in (9) and(10) are in closed-form, which is the case for e.g., Rayleighfading channels. IV. S IMULATION R ESULTS
The investigated scheduling schemes have been simulatedfor an indoor environment with N = 7 users operating in theISM band at a center frequency of
915 MHz , a bandwidth of
26 MHz , and a noise power of σ = −
96 dBm . We adopt theindoor path loss model in [13] for the case when the AP andthe UTs are on the same floor (i.e., a path loss exponent of 2.76 −5 Average user capacity in bits/channel use A v e r ag e u s e r h a r v e s t e d e n e r g y p e r un i tt i m e ( W a tt s ) S a = { , }S a = { , ..., }S a = { , ..., }S a = { , ..., }S a = { , } user 1to 7R-E curve ofworst-channel user R-E curve ofbest-channel user Fig. 3: Rate-energy curves of the order-based ET schedulingscheme for i.n.d. Ricean fading channels with K = 6 .is used, c.f. [13, TABLE I]) and assume the AP-UT channels tobe i.n.d. Ricean fading with Ricean factor K = 6 . We assumean AP transmit power of P = 1 W , an antenna gain of
10 dBi at the AP and at the UTs, and an RF-to-DC conversionefficiency of η = 0 . . The users’ mean channel power gains are Ω n = n × − , which corresponds to an AP-UT distance rangeof .
27 m to . . The simulated results perfectly agree withthe analytical results obtained based on the formulas reportedin Section III for the considered scheduling schemes. Hence,only analytical results are provided in Figs. 2 and 3 for clarityof presentation . In Figs. 2 and 3, every curve represents theergodic capacity and average harvested energy of all users for aspecific scheduling scheme. Also, the R-E curves of the worstand the best-channel users are highlighted.Fig. 2 shows the performance of the order-based N-SNRand the RR scheduling schemes. It is observed that bothschemes achieve proportional fairness in terms of both theergodic capacity and the average amount of harvested energy,since all users are on average scheduled the same number oftimes. Furthermore, RR scheduling performs in-between theorder-based N-SNR curves. This result was expected sincethe RR scheme is neither biased towards power transfer nortowards information transfer. Moreover, by reducing j from N to , the order-based N-SNR scheduling allows the users toharvest more energy at the expense of reducing their ergodiccapacities. For example, for the best-channel user reducing j from N to leads to a . reduction in capacity and a . increase in harvested energy.Fig. 3 shows the performance of the order-based ETscheduling scheme, which is feasible for all considered sets S a .The scheme yields ET for all users and an average harvestedenergy which is proportional to the users’ channel conditions.It is observed that for the same |S a | , the higher the allowedorders in S a , the higher the ET at the expense of a reductionin harvested energy for all users. Hence, sets S a = { , } and S a = { N − , N } provide the extreme ranges of such a tradeoff.In particular, going from S a = { , } to S a = { , } leads ton increase of . and in the amount of harvestedenergy of the best and the worst channel users, respectively,at the expense of only a . reduction in the ET. Remark . We note that for lower data rates (capacities), theclock frequency and the supply voltage of the UT circuitscan be scaled down (a technique known as dynamic voltagescaling). This leads to a cubic reduction in power consumptionbecause dynamic power dissipation depends on the square ofthe supply voltage and linearly on the frequency ( P c ∝ V f )[14]. Hence, when the users can tolerate low data rate, theselection order j of the order-based N-SNR scheduling orthe orders in S a for the order-based ET scheduling can bechosen small to allow the users to harvest more RF energyand simultaneously reduce their power consumption.V. C ONCLUSION
This paper focused on modifying the scheduling objectivesof energy-constrained multi-user communication systems byincluding the RF harvested energy as a performance measure.We presented novel order-based scheduling algorithms whichprovide proportional fairness/ET fairness and allow the controlof the R-E tradeoff. We applied order statistics theory toanalyze the per user ergodic capacity and average harvestedenergy for the considered schemes. Our results reveal that asmaller selection order for the order-based N-SNR schemeand lower orders in the selection set S a for the order-basedET scheme result in a higher average harvested energy for allusers at the expense of reduced average capacities.A PPENDIX
The order-based ET scheduling may fail to provide all userswith ET for one of the following reasons:1) Some user n is required to be scheduled more often thanpossible. That is ∃ n : p n > |S a | N from (16) ≡ Pr ( U n | O n ∈S a ) > . Thus, the first feasibility condition follows.2) For certain combinations of users in S a , the sum ofthe required probabilities that one of them accessesthe channel exceeds one. That is, ∃ a combination ( k , . . . , k |S a | ) in { , . . . , N } , where (cid:80) |S a | l =1 Pr ( U k l |S a = { O k , . . . , O k |S a | } ) > .To find a simple condition for the second case, we firstsynthesize Pr ( U n | O n ∈ S a ) using the law of total probabilityas Pr ( U n | O n ∈ S a ) = 1 (cid:0) N − |S a |− (cid:1) (cid:88) C (cid:48) n Pr ( U n |S a = { O n , O i ,. . ., O i |S a |− } ) , where C (cid:48) n is the set of all (cid:0) N − |S a |− (cid:1) combinations ( i , . . . , i |S a |− ) from { , . . ., n − , n +1 , N } . Thus, from (16), (cid:18) N |S a | (cid:19) p n = (cid:88) C (cid:48) n Pr ( U n |S a = { O n , O i , . . . , O i |S a |− } ) (21)holds ∀ n ∈ { , . . . , N } . In order to check that |S a | (cid:88) l =1 Pr ( U k l |S a = { O k , . . . , O k |S a | } ) = 1 (22) holds ∀ combinations ( k , . . . , k |S a | ) drawn from { , . . . , N } ,we observe that adding |S a | equations of (21) for the userswith indices ( k , . . . , k |S a | ) results in (cid:18) N |S a | (cid:19) |S a | (cid:88) l =1 p k l = |S a | (cid:88) l =1 Pr ( U k l |S a = { O k , . . . , O k |S a | } ) + . . . . Hence, applying (22) and limiting every remaining probabilityterm to , the whole summation will be limited to (cid:32) N |S a | (cid:33) |S a | (cid:88) l =1 p k l ≤ (cid:32) N − |S a | − (cid:33)(cid:124) (cid:123)(cid:122) (cid:125) total number ofprobabilities pereq. in (21 ) |S a | (cid:124)(cid:123)(cid:122)(cid:125) number ofeqs. added + (1 − |S a | ) (cid:124) (cid:123)(cid:122) (cid:125) replacing the numberof probability termsthat add to 1 by 1 . Moreover, adding
L > |S a | equations of (21) for the users withindices ( k , . . . , k L ) , and applying (22) for every |S a | -lengthcombination from the set { k , . . . , k L } , the whole summationwill be limited to (cid:18) N |S a | (cid:19) L (cid:88) l =1 p k l ≤ (cid:18) N − |S a | − (cid:19) L + (cid:18) L |S a | (cid:19) (1 − |S a | ) . which must hold for every combination ( k , . . . , k L ) in { , . . . , N } and for every L = |S a | , . . . , N . Hence, the secondfeasibility condition follows.R IEEE Intern. Symp. Inform. Theory (ISIT) , July 2008, pp. 1612–1616.[3] P. Grover and A. Sahai, “Shannon Meets Tesla: Wireless Informationand Power Transfer,” in
IEEE Intern. Symp. Inform. Theory (ISIT) , June2010, pp. 2363–2367.[4] R. Zhang and C. K. Ho, “MIMO Broadcasting for Simultaneous WirelessInformation and Power Transfer,” in
IEEE Global Telecommun. Conf.(GLOBECOM) , Dec. 2011.[5] J. Xu, L. Liu, and R. Zhang, “Multiuser MISO Beamforming forSimultaneous Wireless Information and Power Transfer,”
ArXiv e-prints ,Mar. 2013, arxiv:1303.1911.[6] H. Ju and R. Zhang, “Throughput Maximization for Wireless PoweredCommunication Networks,”
ArXiv e-prints , Apr. 2013, arxiv:1304.7886.[7] X. Wang, G. Giannakis, and A. Marques, “A Unified Approach toQoS-Guaranteed Scheduling for Channel-Adaptive Wireless Networks,”
Proceedings of the IEEE , vol. 95, no. 12, pp. 2410–2431, 2007.[8] L. Yang and M.-S. Alouini, “Performance analysis of multiuser selectiondiversity,”
IEEE Trans. on Vehicular Technology, , vol. 55, no. 6, pp.1848–1861, 2006.[9] A. Fernekess, A. Klein, B. Wegmann, and K. Dietrich, “Analysis ofCellular Mobile Networks using Fair Throughput Scheduling,” in
IEEE20th Intern. Symp. on Personal, Indoor and Mobile Radio Commun. ,Sept. 2009, pp. 2945 –2949.[10] N. Sagias, G. Tombras, and G. Karagiannidis, “New Results for theShannon Channel Capacity in Generalized Fading Channels,”
IEEECommun. Letters , vol. 9, no. 2, pp. 97 – 99, Feb. 2005.[11] X. Zhou, R. Zhang, and C. Keong Ho, “Wireless Information andPower Transfer: Architecture Design and Rate-Energy Tradeoff,”
ArXive-prints , May 2012, arXiv:1205.0618.[12] H. A. David and H. N. Nagaraja,
Order Statistics , 3rd ed. John Wiley& Sons, Inc., August 2003.[13] S. Seidel and T. Rappaport, “914 MHz Path Loss Prediction Modelsfor Indoor Wireless Communications in Multifloored Buildings,”
IEEETrans. on Antennas and Propagation , vol. 40, no. 2, pp. 207–217, 1992.[14] L. Wang, Y. Chang, and K. Cheng,