Cross-Layer Scheduling and Beamforming in Smart Grid Powered Small-Cell Networks
Yanjie Dong, Md. Jahangir Hossain, Julian Cheng, Victor C. M. Leung
CCross-Layer Scheduling and Beamforming in Smart Grid Powered Small-Cell Networks
Yanjie Dong,
Student Member, IEEE ‡ , Md. Jahangir Hossain, Senior Member, IEEE † ,Julian Cheng, Senior Member, IEEE † , and Victor C. M. Leung, Fellow, IEEE ‡‡ Department of Electrical and Computer Engineering, The University of British Columbia, Vancouver, BC, Canada † School of Engineering, The University of British Columbia, Kelowna, BC, CanadaEmails: { ydong16, vleung } @ece.ubc.ca, { julian.cheng, jahangir.hossain } @ubc.ca Abstract —In the small-cell networks (SCNs) with multiplesmall-cell base stations (ScBSs), the joint design of beamformingvectors, user scheduling and ScBS sleeping is investigatedwith the constraints on proportional rate. A long-term grid-energy expenditure minimization problem is formulated for theconsidered SCNs, which are powered by the smart grid andnatural renewable energy. Since the scheduled user indicatorsare coupled with the beamforming vectors, the formulatedproblem is challenging to handle. In order to decouple thebeamforming vectors from the scheduled user indicators, theLyapunov optimization technique is used. As a result, a practicaltwo-scale algorithm is proposed to allocate the user schedulingindicators and ScBS sleeping variables at the coarse-grainedgranularity (frame) as well as obtain the beamforming vectorsat the fine-grained granularity (slot). Numerical results are usedto verify the performance of the proposed two-scale algorithm.
Index Terms —Beamforming, cross layer design, scheduling,smart grid communications, small-cell networks.
I. I
NTRODUCTION
The wireless data is estimated to exceed , × giga-bytes per month in 2019, and this amount will be doubled overthe next two years [1]. The concept of small-cell networks(SCNs) has prevailed as a promising solution in order to copewith the ever-increasing volume of wireless data. In a typicalSCN, the multiple small-cell base stations (ScBSs) connect tothe core network via optical fibres or digital subscriber lines,and communicate with the user equipments (UEs) over theradio access links. Reducing the link distances of transceivers,the ScBSs can decrease the frequency-reuse factor such thatthe spectrum efficiency of SCN is improved with properinterference mitigation schemes [2]. However, the explosionof ScBSs induces the surging energy bills and carbon footprintfor the operators of wireless communications [3]. Therefore,the research on reducing energy bills, which is the focus ofthis work, becomes imperative and attractive for the operatorsof wireless communications after the ScBSs are deployed.A quantitative study estimated that the radio access linkswill consume around 29% energy of SCNs [3], [4]. Hence,the first research direction to reduce the energy bills focuseson reducing the energy consumption [5], [6] or increasingthe energy efficiency [7] of the ScBSs. For example, theenergy consumption minimization problems with short-termand long-term communication quality of service (QoS) were This work was supported in part by the National Natural Science Founda-tion of China under Grant 61671088, in part by a UBC Four-Year DoctoralFellowship, and in part by the Natural Science and Engineering ResearchCouncil of Canada. respectively studied in [5] and [6] via downlink beamformingof multiple ScBSs in the SCNs. The second research directionleverages the paradigm shift from the traditional power grid tosmart grid [8]–[10]. With the two-way energy trading betweenthe smart grid and SCNs, incorporating the natural renewableenergy (NRE) into the SCNs becomes an ecologically- andeconomically-friendly solution to cut down the energy bills.Due to the volatility of NRE arrival, several research effortshave been made in the design of smart grid powered cellularcommunication frameworks [8], [9] and algorithms [10]–[13].For example, the authors in [10] investigated the impact ofvolatility of NRE arrival on the packet rates. By formulatinga long-term grid-energy (LTGE) expenditure minimizationproblem, the authors in [10] revealed that the LTGE expendi-ture can be reduced by sacrificing the system packet rate. Theauthors in [11] studied the long-term data-rate maximizationproblem with a constraint on the LTGE expenditure in thesmart grid powered communications. Using the dirty papercoding at the multiple-input-multiple-output base station, theauthors in [11] proposed an online beamforming algorithmand proved the asymptotical optimality. While the proposedalgorithms in [10], [11] are only applicable to the single-cellscenario, the authors in [12] considered the NRE coordinationand base station load control problem in a multicell scenario.The authors in [13] investigated the joint content delivery andUE association in a scenario with multiple ScBSs. However,the current works are based on a common assumption that allthe resource allocation actions are performed with a singlegranularity. In other words, all the resources are allocated overslots. Due to the volatility of NRE arrival, several resources(e.g., ScBS sleeping variables and scheduled UE indicators)need to be reallocated at a coarse-grained granularity in prac-tical systems since frequent opening/closing the ScBSs or UEinduces issue of reliability. Limited literature has investigatedthe two-scale resource allocation schemes. The authors in[14] proposed the dynamic beamforming and grid-energymerchandizing algorithm to minimize the LTGE expenditurein a single cell scenario. The authors in [15] investigatedthe joint network selection, subchannel and power allocationin the integrated cellular and Wi-Fi networks. However, theaforementioned literatures [10]–[15] do not consider issueson fairness and scheduling among of the UEs.Different from [14], [15], we minimize the LTGE expendi-ture via the joint design of beamforming vectors, scheduledUE indicators and ScBS sleeping variables among multipleScBSs in a two-scale framework. Moreover, we guarantee a r X i v : . [ c s . I T ] A ug airness among UEs according to the backlog of traffic queuesof UEs. The contributions of this work are summarized asfollows. First, we investigate the LTGE expenditure minimiza-tion problem in the SCN via the joint design of beamformingvectors, scheduled UE indicators and ScBS sleeping variablesin the SCN with multiple ScBSs. The design of beamformingvectors belong to physical layer, and the design of scheduledUE indicators and ScBS sleeping variables belong to theupper layers. Hence, the investigated LTGE expenditure min-imization is a cross-layer problem. Second, we theoreticallyprove that the decisions on scheduled UE indicators andScBS sleeping variables depend on the accumulated effect ofchannel quality. Moreover, we also reveal that the proposedalgorithm can approach the optimal LTGE expenditure viatuning a control parameter. Numerical results are used toverify the performance of our proposed algorithm.II. S YSTEM M ODEL AND P ROBLEM F ORMULATION
We consider an SCN with M ScBSs. The m -th ScBSis associated with N m UEs. Each ScBS is equipped with N T transmit antennas, and each UE is equipped with singleantenna. Moreover, each ScBS connects to the core network(CN) and UEs via the optical-fibre link and wireless links,respectively. Each ScBS is powered by NRE and smart grid.Since the NRE arrival rates and channel coefficient vectorsvary at different time scales in practice [16], a two-scaleframework is considered. Let each frame consist of T slots.The average NRE arrival rates vary over frames, and thechannel coefficient vectors vary over slots. We respectivelydenote the indices for frame and slot as the k -th frameand the t k -th slot with k = 0 , , . . . , ∞ and t k ∈ T k (cid:44) { t k | kT ≤ t k ≤ ( k + 1) T − } . Moreover, each slot has unitduration; therefore, we use the terms “energy” and “power”interchangeably at the scale of slot. A. Traffic Model1) Access Queue:
We consider that the m -th ScBS main-tains N m access queues for the associated UEs, and dynamicequation for the n -th access queue of the m -th ScBS (or the ( m, n ) -th access queue) is given as q A m,n ( t k + 1) = q A m,n ( t k ) − r m,n ( t k ) + ν m,n ( t k ) (1)where q A m,n ( t k + 1) and q A m,n ( t k ) are the backlogs of the ( m, n ) -th access queue at the beginning of the ( t k + 1) -thand the t k -th slot, respectively; ν m,n ( t k ) and r m,n ( t k ) are,respectively, the traffic arrival rate and service rate of the ( m, n ) -th access queue at the t k -th slot. Here, the value of ν m,n ( t k ) is set as ν m,n ( t k ) = (cid:26) λ m,n , t k = kT , otherwise. (2)
2) Processing Queue:
We consider that the n -th UE ofthe m -th ScBS (or the ( m, n ) -th UE) maintains a processingqueue (or the ( m, n ) -th processing queue) for the upperlayer processing. The dynamic equation for the ( m, n ) -thprocessing queue is given as q U m,n ( t k + 1) = q U m,n ( t k ) − s m,n ( t k ) + r m,n ( t k ) (3) where q U m,n ( t k + 1) and q U m,n ( t k ) are the backlogs at thebeginning of the ( t k + 1) -th and the t k -th slot. We considerthe constant service rate of the ( m, n ) -th processing queue.Therefore, the service rate s m,n ( t k ) (cid:44) min (cid:0) ¯ s m,n , q U m,n ( t k ) (cid:1) where ¯ s m,n denotes the average service rate of the ( m, n ) -thprocessing queue. B. Signal Model
Let h m,n ( t k ) be the channel coefficient vector of the linkbetween the n -th UE and the m -th ScBS (or ( m, n ) -th accesslink) at the t k -th slot. Here, h m,n ( t k ) ∼ C N (cid:0) , d − χm,n I N T (cid:1) follows circularly symmetric complex Gaussian distribution(CSCG) where d m,n and χ are, respectively, the distance ofthe ( m, n ) -th access link and pathloss exponent [17].Let a m,n [ k ] be the scheduled UE indicator with a m,n [ k ] =1 denoting that the ( m, n ) -th UE is scheduled at the k -thframe; otherwise, a m,n [ k ] = 0 . Therefore, the receivedsignal and signal-to-interference-plus-noise ratio (SINR) ofthe ( m, n ) -th UE at the t k -th slot are, respectively, denotedas y m,n ( t k ) = (cid:113) a m,n [ k ] h H m,n ( t k ) w m,n ( t k )+ (cid:88) i (cid:54) = n (cid:113) a m,i [ k ] h H m,n ( t k ) w m,i ( t k )+ (cid:88) j (cid:54) = m N j (cid:88) i =1 (cid:113) a j,n [ k ] h H j,n ( t k ) w j,i ( t k ) + z m,n ( t k ) (4)and SINR m,n ( t k ) = a m,n [ k ] (cid:12)(cid:12) h H m,n ( t k ) w m,n ( t k ) (cid:12)(cid:12) I INTRA m,n ( t k ) + I INTER m,n ( t k ) + σ m,n (5)where the term z m,n ( t k ) ∼ C N (cid:0) , σ m,n (cid:1) is the additivewhite Gaussian noise (AWGN) of the ( m, n ) -th UE at the t k -th slot; w m,n ( t k ) denotes the single-stream beamformingvector for the ( m, n ) -th UE at the t k -th slot; and the intra-cellinterference and inter-cell interference terms are, respectively,given as I INTRA m,n ( t k ) = (cid:88) i (cid:54) = n a m,i [ k ] (cid:12)(cid:12) h H m,n ( t k ) w m,i ( t k ) (cid:12)(cid:12) (6)and I INTER m,n ( t k ) = (cid:88) j (cid:54) = m N j (cid:88) i =1 a j,i [ k ] (cid:12)(cid:12) h H j,n ( t k ) w j,i ( t k ) (cid:12)(cid:12) . (7)Hence, the data rate of the ( m, n ) -th access queue at the t k -thslot is given as r m,n ( t k ) = log(1 + SINR m,n ( t k )) .Based on (4), the consumed power of the m -th ScBS isdenoted as P SC m ( t k )= η (cid:80) n ∈ N ACT m [ k ] (cid:107) w m,n ( t k ) (cid:107) F + P CIR m , N m (cid:80) n =1 a m,n [ k ] > , N m (cid:80) n =1 a m,n [ k ] = 0 (8)here P CIR m (cid:44) P SP m (cid:0) .
87 + 0 . N T + 0 . N T (cid:1) [10] with P SP m as the consumed power on baseband processing of the m -thScBS; and η is the power amplifier efficiency of the m -thScBS. Here, N ACT m [ k ] denotes the set of scheduled UEs of the m -th ScBS at the k -th frame. C. Model of Grid-Energy Expenditure
The grid-energy expenditure of SCN at the k -th frame isobtained as G [ k ] = (cid:88) t k ∈ T k (cid:16) α b ( P SG ( t k )) + − α s ( − P SG ( t k )) + (cid:17) = (cid:88) t k ∈ T k (cid:16) ( α b − α s )( P SG ( t k )) + + α s P SG ( t k ) (cid:17) (9)where α b and α s are the electricity prices for purchasing fromand selling to the smart grid with α b > α s to avoid theredundant energy trading; P SG ( t k ) denotes the amount of two-way energy trading between the smart grid and ScBSs at the t k -th slot. Here, P SG ( t k ) takes positive (or negative) valuewhen purchasing from (or selling to) the smart grid. Remark By formulating the grid-energy expenditure ofSCN as in (9), we consider an ideal case where one ScBScan trade NRE with other ScBSs free of charge. The non-ideal case will be considered in the extended version of theconference paper.
D. Problem Formulation
Our objective is to minimize the grid-energy ex-penditure via jointly designing the scheduled UE in-dicators { a m,n [ k ] } ∀ m,n,k in each frame as well asthe beamforming vectors and amount of trading energy { w m,n ( t k ) , P SG ( t k ) , } ∀ m,n,k,t k in each slot. Due to the lack ofknowledge on stochastic arrival of NRE and variations of CSI,we formulate the LTGE expenditure minimization problemwith the following constraints: • Rate-limit constraints: r m,n ( t k ) ≤ q A m,n ( t k ) , ∀ m, n (10)which guarantee that each ScBS does not transmit blankinformation. • Proportional-rate constraints: r m,n ( t k ) r j,i ( t k ) = ψ m,n ψ j,i , n ∈ N ACT m [ k ] , i ∈ N ACT j [ k ] , ∀ m, j. (11)For example, ψ m,n ψ j,i = q A m,n ( t k ) q A j,i ( t k ) guarantees that the UEwith larger backlog obtains better service rate at eachslot. • Slot-level power constraints: (cid:88) n ∈ N ACT m [ k ] (cid:107) w m,n ( t k ) (cid:107) F ≤ P max m , ∀ m (12)where P max m is the maximum transmit power of the m -thScBS. • Power balance constraint: P SG ( t k ) + M (cid:88) m =1 E HAV m [ k ] T = M (cid:88) m =1 P SC m ( t k ) (13) which guarantees that the consumed grid energy is bal-anced with the harvested NRE and merchandized energywith smart grid at the t k -th slot. • Queue-stable constraints: lim sup K →∞ K (14) × K − (cid:88) k =0 E X (cid:8) q A m,n [ k ] + q U m,n [ k ] (cid:9) < ∞ , ∀ m, n where q A m,n [ k + 1] (cid:44) q A m,n ( t k ) (cid:12)(cid:12) t k =( k +1) T and q A m,n [ k ] (cid:44) q A m,n ( t k ) | t k = kT . The queue-stable constraintsindicate that the data of UEs will be served in finitetime.Here, the operator E X {·} denotes the expectation over therandom sources X (cid:44) { h m,n ( t k ) , E HAV m [ k ] } ∀ m,n,k,t k .As a result, the LTGE expenditure minimization problemis formulated as min Y lim K →∞ K K − (cid:88) k =0 E X { G [ k ] } (15a)s.t. (10) − (14) (15b)where resource-allocation-variable set is defined as Y (cid:44) { w m,n ( t k ) , P SG ( t k ) , a m,n [ k ] } ∀ m,n,k,t k .Note that the LTGE expenditure minimization problem (15)is challenging to handle via classical convex optimizationmethods. Since the scheduled UE indicators are coupled withthe beamforming vectors, we are motivated to use the Lya-punov optimization method to obtain a feasible solution to theLTGE expenditure minimization problem (15). Moreover, wealso demonstrate that the LTGE expenditure is asymptoticallyminimized by sacrificing the end-to-end delay of UEs whenscheduling UEs and switching on/off ScBSs are considered.III. J OINT B EAMFORMING , UE S
CHEDULING AND S C BSS
LEEPING
We define the Lyapunov function of LTGE expenditureminimization problem as L [ k ] = 12 M (cid:88) m =1 N m (cid:88) n =1 (cid:16)(cid:0) q A m,n [ k ] (cid:1) + (cid:0) q U m,n [ k ] (cid:1) (cid:17) (16)where q A m,n [ k ] (cid:44) q A m,n ( kT ) and q U m,n [ k ] (cid:44) q U m,n ( kT ) . Then,we introduce the one-frame drift function as [18] ∆ X (cid:44) E X { L [ k + 1] − L [ k ] } . (17)Thus, we obtain the one-frame Lyapunov drift-plus-penaltyfunction as [18] ∆ X + V E X { G [ k ] } . (18) Proposition The one-frame Lyapunov drift-plus-penaltyfunction in (18) is upper-bounded as ∆ X + V E X { G [ k ] } (19) ≤ Ψ + V E X { G [ k ] } + M (cid:88) m =1 N m (cid:88) n =1 q A m,n [ k ] f A m,n [ k ] + M (cid:88) m =1 N m (cid:88) n =1 q U m,n [ k ] f U m,n [ k ] here the functions f A m,n [ k ] and f U m,n [ k ] are, respectively,defined as f A m,n [ k ] (cid:44) E X (cid:40) λ m,n − (cid:88) t k ∈ T k r m,n ( t k ) (cid:41) (20)and f U m,n [ k ] (cid:44) E X (cid:40) (cid:88) t k ∈ T k ( r m,n ( t k ) − s m,n ( t k )) (cid:41) . (21) Proof:
See Appendix.Minimizing the right-hand side (RHS) of (19) subject toconstraints in (10)–(14) results in a feasible solution to theLTGE expenditure minimization problem (15).
A. UE Scheduling Analysis
After some algebraic manipulations on RHS of (19), weobtain the term related to { r m,n ( t k ) } ∀ m,n,k,t k as M (cid:88) m =1 N m (cid:88) n =1 (cid:0) q U m,n [ k ] − q A m,n [ k ] (cid:1) E X (cid:40) (cid:88) t k ∈ T k r m,n ( t k ) (cid:41) . (22)Here, the data rate of the ( m, n ) -th UE is coupled with thescheduled UE indicator. Since our objective is to minimize theterm in (22), we obtain the optimal scheduled UE indicator a ∗ m,n [ k ] as a ∗ m,n [ k ]= (cid:26) , q U m,n [ k ] − q A m,n [ k ] ≥ or q A m,n [ k ] = 01 , otherwise . (23)The motivation of (23) can be justified as follows. The case q A m,n [ k ] = 0 indicates that the backlog of the ( m, n ) -th accessqueue is zero. The data rate (cid:80) t k ∈ T k r m,n ( t k ) of ( m, n ) -thUE at the k -th frame is set to zero, and the ( m, n ) -th UEis not scheduled at the k -th frame. When the case q U m,n [ k ] − q A m,n [ k ] ≥ happens, setting the value of (cid:80) t k ∈ T k r m,n ( t k ) as zero can minimize the term (22). Therefore, the ( m, n ) -thUE is not scheduled at the k -th frame. Remark Note that the m -th ScBS is closed when no UEassociated with the m -th ScBS is scheduled, i.e., N (cid:88) n =1 a m,n [ k ] = 0 . (24) B. Two-way Energy Trading and Beamforming Analysis
Denote the terms related to the beamforming vectors andamount of trading energy in the RHS of (19) as
OB J ( t k ) = V ( α b − α s )( P SG ( t k )) + + V α s P SG ( t k ) (25) + M (cid:88) m =1 (cid:88) n ∈ N ACT m [ k ] (cid:0) q U m,n [ k ] − q A m,n [ k ] (cid:1) r m,n ( t k ) . Based on (25), we minimize the RHS of (19) at the k -thframe as O P T [ k ] = min ¯ Y [ k ] E X (cid:40) (cid:88) t k ∈ T k OB J ( t k ) (cid:41) (26)s.t. (10) − (14)where ¯ Y [ k ] (cid:44) { w m,n ( t k ) , P SG ( t k ) } ∀ m,n,t k . We observe that the challenges in solving the optimiza-tion problem (26) are three-folds: 1) the rate-limit con-straints in (10) are non-convex; 2) the proportional-rateconstraints in (11) are non-convex; and 3) the optimiza-tion problem (26) contains expectation over two-scale ran-dom sources: frame-level source { E HAV m [ k ] } ∀ m,k and slot-levelsource { h m,n ( t k ) } ∀ m,n,k,t k .In order to handle the non-convex proportional-rate con-straints in (11), we introduce auxiliary variables { φ ( t k ) } ∀ t k and relax the proportional-rate constraints in (11) as h H m,n ( t k ) w m,n ( t k ) f m,n ( φ ( t k )) ≥ (cid:113) I INTRA m,n ( t k ) + I INTER m,n ( t k ) + σ m,n , n ∈ N ACT m [ k ] , ∀ m (27) (cid:61) (cid:0) h H m,n ( t k ) w m,n ( t k ) (cid:1) = 0 , n ∈ N ACT m [ k ] , ∀ m (28)where f m,n ( φ ( t k )) (cid:44) (cid:113) exp( ψ m,n φ ( t k )) − (29)and (cid:61) ( · ) denotes the imaginary part of a complex value.Setting the upper bound of φ ( t k ) as min m,n (cid:0) q A m,n ( t k ) /ψ m,n (cid:1) ,the constraints in (10) are satisfied.For fixed values of { φ ( t k ) } ∀ t k , the set of constraints in (27)and (28) are convex. Substituting r m,n ( t k ) = ψ m,n φ ( t k ) and(13) into the objective function (26), we have OB J ( t k ) = V ( α b − α s ) (cid:32) M (cid:88) m =1 (cid:18) P SC m ( t k ) − T E
HAV m [ k ] (cid:19)(cid:33) + + V α s M (cid:88) m =1 (cid:18) P SC m ( t k ) − T E
HAV m [ k ] (cid:19) + M (cid:88) m =1 (cid:88) n ∈ N ACT m [ k ] (cid:0) q U m,n [ k ] − q A m,n [ k ] (cid:1) ψ m,n φ ( t k ) . (30)Replacing constraints in (10) and (11) with (27) and (28),we obtain a relaxed version of optimization problem (26) as O P T [ k ] = min (cid:101) Y E X (cid:40) (cid:88) t k ∈ T k OB J ( t k ) (cid:41) (31)s.t. (12) , (27) , (28)where (cid:101) Y (cid:44) { w m,n ( t k ) } ∀ m,n,k,t k .Given the optimal φ ∗ ( t k ) , the constraints in (27) and (28)constitute a convex hull of the constraints in (11). Hence, weconclude that O P T [ k ] ≤ O P T [ k ] . With slight modification ofthe arguments in [19], we demonstrate that the optimal beam-forming vectors (cid:8) w ∗ m,n ( t k ) (cid:9) ∀ m,n,k,t k make the constraints in(27) active. In other words, we have O P T [ k ] = O P T [ k ] . Thedetailed proof of the activeness of (27) will be provided in theextended version of the conference article. Motivated by [19],the optimal φ ∗ ( t k ) can be obtained via a one-dimensionalsearch method.Now the optimization problem (31) is convex with respectto (cid:101) Y . Let E HAV m [ k ] and { h m,n ( t k ) } ∀ m,n denote the amountof harvested NRE at the m -th ScBS at the k -th frameand the set of random sources at the t k -th slot. Since the X + E X { G [ k ] } ≤ Ψ + E X { G [ k ] } + M (cid:88) m =1 N m (cid:88) n =1 q A m,n [ k ] E X (cid:40) λ m,n − (cid:88) t k ∈ T k r m,n ( t k ) (cid:41) + N m (cid:88) n =1 N m (cid:88) n =1 q U m,n [ k ] E X (cid:40) (cid:88) t k ∈ T k ( r m,n ( t k ) − s m,n ( t k )) (cid:41) . (37)optimization problem (31) contains two-scale random sources,we are motivated to use the principle of opportunisticallyminimizing an expectation [18] with the assumption that thechannel coefficient vectors { h m,n ( t k ) } ∀ m,n are independentand identically distributed (i.i.d.) over different slots.Based on the aforementioned discussions, we summarizethe procedures of our proposed joint beamforming, UEscheduling and ScBS sleeping algorithm as follows. Algorithm 1
Joint Beamforming, UE Scheduling and ScBSSleeping Algorithm At the start of the k -th frame, the CN estimates the harvestedNRE as { E HAV m [ k ] } ∀ m At the start of the k -th frame, the CN updates the set ofscheduled UEs and active ScBSs via (23) and (24) repeat At the start of the t k -th slot, the CN estimates the channelcoefficient vector { h m,n ( t k ) } ∀ m,n With { h m,n ( t k ) } ∀ m,n , the CN solves the following opti-mization problem via CVX [20] min (cid:101) Y OB J ( t k ) s.t. (12) , (27) , (28) (32) At the start of the t k -th slot, the CN performs one dimen-sional search for the optimal φ ∗ ( t k ) until Convergence At the end of the t k -th slot, the CN updates the access queuesand processing queues according to (1) and (3) Proposition Suppose the arrival rate and service rateof the ( m, n ) -th access queue and the ( m, n ) -th processingqueue satisfy the condition T λ m,n < E X { r m,n ( t k ) } < ¯ s m,n .When the proposed joint beamforming, UE scheduling andScBS sleeping algorithm is used, we conclude that • The optimal grid-energy expenditure is asymptoticallyobtained as G ∗ ≤ K K − (cid:88) k =0 E X { G [ k ] } ≤ Ψ V + G ∗ (33)where G ∗ is the optimal grid-energy expenditure. • The constraints in (14) are satisfied.
Proof:
Due to the space limitation, the detailed proof isomitted and it will be provided in the extended version ofthis conference paper.IV. N
UMERICAL R ESULTS
In the section, we use numerical results to verify ourproposed algorithm. We perform the simulations based onpractical data from NASA Remote Sensing Validation Datain Saudi Arabia: Solar Valley at 10:00 am to 10:06 am,on December 2000 . Each ScBS is equipped with an solarenergy harvester with size cm and harvesting efficiency30%. Since NRE arrival rate is updated every 5 minutes, we use the interpolation method to generate NRE arrival rate at0.5 second level. Since the electricity prices vary every hour,the purchasing price and selling price are, respectively, set as . × − cents/slot/mW and × − cents/slot/mW basedon the practical data of Pennsylvania-New Jersey-Marylandmarket . We consider wireless UE has a speed at v UE = 1 . km/h [21]. The Doppler frequency shift f D = v UE C f C = 1 . Hz, where the light speed C is × m/sec. In order toguarantee a slow-fading scenario, we choose the slot durationas . sec [21]. The processing rate at the UEs is s m,n = 3 . nats/slot/Hz. The power of AWGN is set as − dBm. Themaximum transmit power of ScBS is set as P max m = 26 dBm. The consumed power on baseband processing is setas P SP m = 23 dBm. Fig. 1: Variation of the average delay with the control parameter.Fig. 2: Variation of the annualized electricity expenditure with thecontrol parameter.
Figures 1 and 2 reveal the tradeoff between the averageend-to-end delay and grid-energy expenditure. Here, we annu-alize the grid-energy expenditure by considering the mediumcity with × ScBSs. We observe that increasing the valueof control parameter induces an increasing end-to-end delayof UEs (as shown in Fig. 1) and a decreasing grid-energyexpenditure (as shown in Fig. 2). Therefore, the proposedalgorithm provides the wireless operator with flexibility incontrolling the grid-energy expenditure while maintaining a atisfactory level of communication QoS. Moreover, increas-ing the information arrival rate of UEs induces a more effi-cient tradeoff between the end-to-end delay and grid-energyexpenditure. For example, when the information arrival rate λ m,n is . nats/slot/Hz, the wireless operator can tradethe . slot of end-to-end delay for . % grid-energyexpenditure by increasing the value of control parameterfrom . to . When the information arrival rate λ m,n is nats/slot/Hz, the wireless operator can trade the . slotof end-to-end delay for . % grid-energy expenditure withthe same range of control parameter. This is due to the threefacts: 1) the increasing control parameter induces that thegrid-energy expenditure approaches the same optimal valuewith different information arrival rates; 2) a smaller controlparameter indicates a more stringent delay requirement; and3) a larger stringent delay requirement exponentially increasesthe grid-energy expenditure due to the log-concave data rateof UEs. V. C ONCLUSION
We developed a joint beamforming, UE scheduling andScBS sleeping algorithm. Within the proposed algorithm,scheduling UEs and switching ScBSs are performed at thecoarse-grained granularity (frame) while calculating beam-forming vectors are respectively performed at the fine-grainedgranularity (slot). The benefit of the two scale algorithm isto avoid frequently changing scheduled UEs and switchingon/off the ScBSs. Numerical results demonstrate that thewireless operator can trade end-to-end delay for grid-energyexpenditure by tuning a control parameter. Moreover, increas-ing the maximum transmit power of ScBSs can improve thetradeoff efficiency between end-to-end delay and grid-energyexpenditure. A
PPENDIX
Taking the telescoping summation over kT ≤ t k < ( k + 1) T − for q A m,n ( t k ) in (1), we obtain the frame-by-frame dynamic equation of the ( m, n ) -th access queue as q A m,n [ k + 1] = q A m,n [ k ] + λ m,n − (cid:88) t k ∈ T k r m,n ( t k ) . (34)Based on (34), the one-frame drift of the ( m, n ) -th accessqueue is denoted as (cid:0) q A m,n [ k + 1] (cid:1) − (cid:0) q A m,n [ k ] (cid:1) ≤ C A m,n + q A m,n [ k ] (cid:32) λ m,n − (cid:88) t k ∈ T k r m,n ( t k ) (cid:33) (35)where C A m,n (cid:44) λ m,n + T ( r max m,n ) due to the fact that r m,n ( t k ) ∈ (cid:2) , r max m,n (cid:3) .Following a similar argument, we obtain the one-frame driftof the ( m, n ) -th processing queue as (cid:0) q U m,n [ k + 1] (cid:1) − (cid:0) q U m,n [ k ] (cid:1) ≤ C U m,n + q U m,n [ k ] (cid:88) t k ∈ T k ( r m,n ( t k ) − s m,n ( t k )) (36)where C U m,n (cid:44) T s m,n + ( r max m,n ) . Let Ψ (cid:44) (cid:80) Mm =1 (cid:80) N m n =1 (cid:0) C A m,n + C U m,n (cid:1) . Based on (35) and(36), the one-frame Lyapunov drift-plus-penalty function in(18) is derived as in (37).R EFERENCES[1] Cisco,
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