Energy Management and Cross Layer Optimization for Wireless Sensor Network Powered by Heterogeneous Energy Sources
aa r X i v : . [ c s . N I] N ov Energy Management and Cross Layer Optimizationfor Wireless Sensor Network Powered byHeterogeneous Energy Sources
Weiqiang Xu, Yushu Zhang, Qingjiang Shi, Xiaodong Wang
Abstract —Recently, utilizing renewable energy for wireless sys-tem has attracted extensive attention. However, due to the instableenergy supply and the limited battery capacity, renewable energycannot guarantee to provide the perpetual operation for wirelesssensor networks (WSN). The coexistence of renewable energy andelectricity grid is expected as a promising energy supply mannerto remain function of WSN for a potentially infinite lifetime. Inthis paper, we propose a new system model suitable for WSN,taking into account multiple energy consumptions due to sensing,transmission and reception, heterogeneous energy supplies fromrenewable energy, electricity grid and mixed energy, and multi-dimension stochastic natures due to energy harvesting profile,electricity price and channel condition. A discrete-time stochasticcross-layer optimization problem is formulated to achieve theoptimal trade-off between the time-average rate utility andelectricity cost subject to the data and energy queuing stabilityconstraints. The Lyapunov drift-plus-penalty with perturbationtechnique and block coordinate descent method is applied to ob-tain a fully distributed and low-complexity cross-layer algorithmonly requiring knowledge of the instantaneous system state. Theexplicit trade-off between the optimization objective and queuebacklog is theoretically proven. Finally, through the extensivesimulations, the theoretic claims are verified, and the impacts ofa variety of system parameters on overall objective, rate utilityand electricity cost are investigated.
Index Terms —Wireless sensor networks, energy harvesting,electricity grid, heterogeneous energy, cross-layer optimization,Lyapunov optimization, drift-plus-penalty, block coordinate de-scent.
I. I
NTRODUCTION
Wireless sensor network (WSN) consist of a lot of spatiallydistributed autonomous sensor nodes with limited energy,computation and sensing capabilities, to monitor physicalphenomena, and to cooperatively transmit their data to asink. WSN have a variety of potential applications, rangingfrom multimedia surveillance, environmental monitoring, andadvanced health care delivery to industrial process control.Traditionally, sensor nodes are powered by a non-rechargeablebattery with limited energy storage capacities. However, a lotof applications are expected to operate over a virtually infinite
Weiqiang Xu, Yushu Zhang, and Qingjiang Shi are with School of Informa-tion Science & Technology, Zhejiang Sci-Tech University, Hangzhou, 310018,P. R. China. Email: [email protected] Wang is with the Department of Electrical Engineering, ColumbiaUniversity, New York, NY, 10027, USA. Email: [email protected]: We are very grateful to Prof. Michael J. Neely at theUniversity of Southern California, Dr. Cristiano Tapparello at University ofPadova, and Prof. Osvaldo Simeone at New Jersey Institute of Technology,for very helpful discussions. lifetime. The energy scarcity represents one of the majorlimitations of WSN. Indeed, the post-deployment replacementof the sensors batteries is generally not practical or evenimpossible. Thus, a variety of hardware optimizations, energymanagement policies and energy-aware network protocolshave been proposed to carefully manage the limited energyresources and thus to prolong the lifetime of a WSN [1]–[3].Recent advances in hardware design have made energy har-vesting (EH) technology possibly applied in wireless systems.Sensor node equipped with EH device replenishes energyfrom renewable sources with a potentially infinite amount ofavailable energy [4]–[6]. Since EH technology is essentiallydifferent from the traditional non-rechargeable battery, a newenergy management policy is expected to well-match withthe energy replenishment process. As such, a great deal ofresearch efforts have been devoted to investigate the energymanagement and data transmission in the EH powered sce-nario. Some efforts proposed the optimal schemes to achievethe maximum throughput, the minimum transmission comple-tion time, and/or the minimum information distortion for asingle EH node with finite or infinite data buffer and finiteor infinite battery capacity [7]–[13]. However, for wirelessmultihop network powered by EH, different nodes may havequite different workload requirements and available energysources. Due to the fact that the network performance is tightlycoupled with energy management policy and mechanisms atthe physical, MAC, network, and transport layers, a limitedamount of works investigated the cross-layer optimizationschemes in [14]–[18]. In particular, some works of cross-layer optimization leveraged Lyapunov optimization tech-niques. Gatzianas et al. in [19] applied Lyapunov techniquesto design an online adaptive transmission scheme for wirelessnetworks with rechargeable battery to achieve total systemutility maximization and the data queue stability. Huang etal. in [20] applied Lyapunov optimization techniques withweight perturbation [21] to achieve a close-to-optimal utilityperformance in finite energy buffer. The proposed techniqueobtains an explicit and controllable tradeoff between optimal-ity gap and queue sizes. Similarly, by adopting perturbation-based Lyapunov techniques, Tapparello et al. in [22] pro-posed the joint optimization scheme of source coding andtransmission to minimize the reconstruction distortion cost forthe correlated sources measurement. All the above-mentionedworks showed that network-wide cross layer optimization ishelpful for achieving the performance gain. However, theworks mentioned above are still not suitable to efficiently deal with the energy scarcity limitation of WSN. There are stillseveral technical challenges, including:
A. Multiple energy consumption
A sensor node isequipped with a sensing module for data measurements andprocessing, and a communication module for data transmissionand data reception. Almost all of the works mentioned aboveonly account for the energy consumed in data transmission.Traditionally, energy consumption is known to be dominatedby the communication module. However, this is not alwaystrue. In [23], it was shown that communication-related taskswere possibly less energy consumption than intensive pro-cessing, and data transmission is only a slight more energyconsumption than data reception. There exists a very limitedworks in [11], [13], [24] to investigate the problem of energyallocation accounting for the energy requirement of datatransmission and sensing together, only suitable for a singleEH nodes. To the best of our knowledge, so far, almost noworks, except [22], studied the joint energy allocation forcommunication module and sensing module together in themultihop scenario.
B. Hybrid energy supply
Due to the low rechargingrate and the time-varying profile of the energy replenishmentprocess, sensor nodes solely powered by harvested energy cannot guarantee to provide reliable services for the perpetualoperation. They may currently be suitable only for very-low duty cycle devices. Other complementary stable energysupplies should be required to remain a perpetual operationfor WSN. As the electricity grid (EG) is capable of providingpersistent power input, the coexistence of renewable energyand electricity grid is considered as a promising technologyto tackle the problem of simultaneously guaranteeing thenetwork operation and minimizing the electricity grid energyconsumption, which had been confirmed in single-hop wirelesssystem [25] [26]. However, as far as we know, no prior workaddressed to cross-layer optimization for WSN powered byheterogeneous energy sources in multihop scenario.
C. Fully distributed implementation
In WSN, the entiresystem state is characterized by channel condition, energyharvesting profile, electricity price, data queue size and energyqueue size. Therefore, the centralized solution requiring theentire system state will lead to heavy signaling overheadand high computational complexity in the central optimizer.Furthermore, this information about the entire system statemay be hard to obtain or even unattainable in practical imple-mentation. It is desirable to have the distributed optimizationbased on local information only. Some existing works designedthe partly distributed optimization solution in WSN poweredby EH. For instance, in [20] [22], the power allocation problemstill requires centralized optimization. However, a partly dis-tributed optimization solution is still impractical, or too costlyin large-scale networks. Fully distributed optimization solutionis particularly attractive.This motivates us to address a novel energy managementand cross-layer optimization for WSN powered by heteroge-neous energy sources. The key contributions of this paper aresummarized as follows:(1) We propose a more realistic energy consumption model,which takes the energy consumption of sensing, transmission and reception into account. We propose a new heterogenousenergy supply model suitable for the node powered by renew-able energy or/and electricity grid. We also consider the multi-dimension stochastic natures from channel condition, energyharvesting profile and electricity price. For such a model, weformulate a discrete-time stochastic cross-layer optimizationproblem in WSN with the goal of maximizing the time-averageutility of the source rate and the time-average cost of energyconsumption in electricity grid subject to the data and energyqueuing stability constraints.(2) To obtain a distributed and low-complexity solution, weapply the Lyapunov drift-plus-penalty with perturbation tech-nique [21] to transform the stochastic optimization probleminto a series of iterations of the deterministic optimizationproblems. Furthermore, by exploiting the special structure,we design a fully distributed algorithm—Energy mAnagementand croSs laYer Optimization (EASYO) which decomposesthe deterministic optimization problem into the energy man-agement (including energy harvesting and energy purchasing),source rate control (implicitly including energy allocationfor sensing/processing), routing selection (implicitly includingenergy allocation for data reception), session scheduling andtransmission power allocation. EASYO is a fully distributedalgorithm which makes greedy decisions at each time slotwithout requiring any statistical knowledge of the channelstate, of the harvestable energy state and of the electricityprice state. Note that our proposed fully distributed algorithmis different from the cross-layer optimization algorithms in[20] [22], where the transmission power allocation problemis optimized in the centralized manner, leading to the hugechallenging in practical implementation.(3) We analyze the performance of the proposed distributedalgorithm EASYO, and show that a control parameter V enables an explicit trade-off between the average objectivevalue and queue backlog. Specifically, EASYO can achievea time average objective value that is within O (1 /V ) of theoptimal objective for any V > , while ensuring that theaverage queue backlog is O ( V ) . Finally, through the extensivesimulations, the theoretic claims are verified, and the impactsof a variety of system parameters on overall objective, rateutility and electricity cost are investigated.Throughout this paper, we use the following notations. Theprobability of an event A is denoted by Pr ( A ) . For a randomvariable X , its expected value is denoted by E [ X ] and itsexpected value conditioned on event A is denoted by E [ X | A ] .The indicator function for an event A is denoted by A ; itequals 1 if A occurs and is 0 otherwise. [ x ] + = max( x, .The remainder of the paper is organized as follows. In Sec-tion II, we give the system model and problem formulation. InSection III, we present the distributed cross-layer optimizationalgorithm. In Section IV, we present the performance analysisof our proposed algorithm. Simulation results are given inSection V. Concluding remarks are provided in Section VI.II. S YSTEM M ODEL AND P ROBLEM F ORMULATION
We consider a general interconnected multi-hop WSN thatperfect CDMA-based medium access, and operates over time
TABLE IS
UMMARY OF KEY NOTATIONS
Notation Description N The set of sensor nodes N H The set of nodes powered by EH N G The set of nodes powered by EG N M The set of nodes powered by both EH and EG N s The set of source nodes N d The set of destination nodes F The set of all information sessions L The set of communication links O ( n ) The set of nodes m with ( n, m ) ∈ LI ( n ) The set of nodes m with ( m, n ) ∈ L r f The source rate of f -th information session p Tnb
The transmission power of link ( n, b ) b γ nb The signal to interference plus noise ratio (SINR) oflink ( n, b ) x fnb ( t ) The data transmission rate of the session f overlink ( n, b )˜ C nb The link capacity of link ( n, b )˜ P Sf The energy consumption per data of the f -th sessionfor data sensening/processing ˜ P Rn The energy consumed when node n receiving one unitdata from the neighbor nodes in the network P Gn The cost per unit of electricity drawn from the electricitygrid at node n ∈ N G ∪ N M p Totaln
The total energy consumption of node nθ En The limited battery capacity of node n . e n The harvested energy at node ng n The energy supplied by the electricity grid at node nS Cnb
The channel state of link ( n, b ) S H The harvestable energy state of node n ∈ N H ∪ N M S Gn The electricity price state at node nh n The available amount of harvesting energy at node nE n The energy queue size for n ∈ N Q fn The data backlog of the f -th session at node n slots t ∈ T = { , , , . . . } . WSN is modeled by a directgraph G = ( N , L ) . N = N H ∪ N G ∪ N M = { , , , . . . , N } denotes the set of sensor nodes in the network, N H is the set ofnodes powered by EH, called EH nodes, N G is the set of nodespowered by EG, called EG nodes, and N M is the set of Mixedenergy (ME) nodes powered by both EH and EG, respectively. N s ⊂ N denotes the set of all source nodes which measurethe information source(s). Each source node n ∈ N s hasmultiple sensor interfaces, such that it can measure multipleinformation sources F n = { , , , . . . , F n } at the same time .We use F = S n ∈N s F n = { , , . . . , F } to denote the set of allinformation sources in the network. The source node transmitsthe data to the corresponding destination node through multi-hop routing. L = { ( n, m ) , n, m ∈ N } represents the set ofcommunication links. O ( n ) denotes the set of nodes m with ( n, m ) ∈ L , and I ( n ) denotes the set of nodes m with ( m, n ) ∈ L . Fig. 1 describes the composition of a single nodesystem. The key notations of our system model are shown inTABLE I. A. Source Rate and Utility
At time slot t , the node n measures F n independent parallelinformation sources F n . The measured samples of the session In the following, we use the terms information source, flow and sessioninterchangeably.
RxSensor TxData QueueRelayTrafficLocal TrafficSensor DataProcessing
To Next Hop
Energy SupplySub-systemSensorNodeSub-system
Renewable Energy GridEnergy
To NextHop
Battery
Fig. 1. Diagram of a single node system. f ∈ F n is compressed with rate r f ( t ) before putting into thedata queue , where r f ( t ) denotes the source rate of the session f at time slot t . We assume that ≤ r f ( t ) ≤ r max f , ∀ f ∈ F (1)where r max f ≤ R max for all f with some finite R max at alltime. We assume that each session f is associated with autility function U f ( r f ( t )) , which is increasing, continuouslydifferentiable and strictly concave in r f ( t ) with a boundedfirst derivative and U f (0) = 0 . We use β fU ( t ) to denote themaximal first-order derivative of U f ( r f ( t )) w.r.t. r f ( t ) , denote β U = max f ∈F ,t ∈T β fU ( t ) . B. Data Transmission
We assume that the links in the network may interfere witheach other when they transmit data simultaneously. We define p T ( t ) = (cid:0) p Tnb ( t ) , ( n, b ) ∈ L (cid:1) as the transmission power allo-cation matrix for data transmission at slot t , where p Tnb ( t ) isthe transmission power allocated of link ( n, b ) , and then thefollowing inequality should be satisfied: ≤ X b ∈O ( n ) p Tnb ( t ) ≤ P max n , n ∈ N . (2)where P max n is a finite constant to denote the maximal trans-mission power limitation at node n .We use b γ nb ( t ) to denote the signal to interference plus noiseratio (SINR) of link ( n, b ) : b γ nb ( t ) ∆ = b γ nb (cid:0) p T ( t ) , S C ( t ) (cid:1) = S Cnb ( t ) p Tnb ( t ) N b + P a ∈J n,b P ( a,m ) ∈L S Cab ( t ) p Tam ( t ) , where N b is the noise spectral density at node b , and S Cnb ( t ) represents the link fading coefficient from n to b at theslot t . J n,b is the set of nodes whose transmission mayinterfere with the receiver of link ( n, b ) , excluding node n .We assume that S Cnb ( t ) may be time varying and indepen-dent and identically distributed (i.i.d.) at every slot. Denote We measure time in unit size slots, for simplicity, and thus we suppressthe implicit multiplication by 1 slot when converting between data rate anddata amount. S C ( t ) = (cid:0) S Cnb ( t ) , ( n, b ) ∈ L (cid:1) as the network channel statematrix, taking non-negative values from a finite but arbitrarilylarge set S C .The link capacity is defined as ˜ C nb ( t ) = log (1 + K nb b γ nb ( t )) . Here, K nb denotes the processing gain of the CDMA system.Note that the dependence of ˜ C nb ( t ) on p T ( t ) and S C ( t ) isimplicit for notational convenience. Let x fnb ( t ) denote the datatransmission rate of the session f over link ( n, b ) , b ∈ O ( n ) .Because of the total rates of all sessions cannot exceed the linkcapacity, so, ≤ P f ∈F x fnb ( t ) ≤ ˜ C nb ( t ) , ∀ n ∈ N , ∀ b ∈ O ( n ) .Due to the fact that K nb is typically very large in CDMAnetworks, C nb ( t ) = log γ nb ( t ) is a good approximation of ˜ C nb ( t ) = log(1 + γ nb ( t )) , where γ nb ( t ) = K nb b γ nb ( t ) . Thus,we make a stricter bound by the following constraint: ≤ X f ∈F x fnb ( t ) ≤ C nb ( t ) , ∀ n ∈ N , ∀ b ∈ O ( n ) . (3)Without loss of generality, we assume that for all time overall links under any power allocation matrix and any channelstate, there exists some finite constant X max . C. Energy Consumption Model
At every time slot t , each node n allocates power to accom-plish its tasks, including data sensening/processing, data trans-mission and data reception. We define a function p Sf ( r f ( t )) todenote the energy consumption of sensing/processing mod-ule for acquiring the data at a particular rate r f ( t ) of thesession f at node n . Inspired by [22], we also assume alinear relationship between the rate r f ( t ) and p Sf ( r f ( t )) , i.e., p Sf ( r f ( t )) = ˜ P Sf r f ( t ) , where ˜ P Sf denotes the energy consump-tion per data of the f -th session for data sensening/processing.Thus, the total energy consumption p T otaln ( t ) of node n at slot t is: p T otaln ( t ) ∆ = X f ∈F n ˜ P Sf r f ( t )+ X b ∈O ( n ) p Tnb ( t ) + ˜ P Rn X a ∈I ( n ) X f ∈F x fan ( t ) (4)where ˜ P Rn is the energy consumed when node n receiving oneunit data from the neighbor nodes in the network. D. Energy Supply Model
First, we describe the energy supply model of ME nodeshown in Fig. 1. Each ME node is equipped with a batteryhaving the limited capacity θ En . As depicted in Fig. 1, theharvested energy e n ( t ) at time t for ME node n is stored inthe battery. On the other hand, the energy supplied by theelectricity grid at time t for ME node n is denoted with g n ( t ) .Different from the ME node, the EH node only stores theharvested energy e n ( t ) and the EG node only stores the energy g n ( t ) supplied by the electricity grid. We measure time in unit size slots, for simplicity, and thus we suppress theimplicit multiplication by 1 slot when converting between power and energy.
We assume each n knows its own current energy availability E n ( t ) denoting the energy queue size for n ∈ N at time slot t . We define E ( t ) = ( E n ( t ) , n ∈ N ) over time slots t ∈ T as the vector of the energy queue sizes. The energy queuingdynamic equation is E n ( t + 1) = E n ( t ) + n ∈N H ∪N M e n ( t )+ n ∈N G ∪N M g n ( t ) − p T otaln ( t ) (5)with E n (0) = 0 . At any time slot t , the total energyconsumption at node n must satisfy the following energy-availability constraint: E n ( t ) ≥ p T otaln ( t ) , ∀ n ∈ N . (6)At any time slot t , the total energy volume stored in batteryis limited by the battery capacity, thus the following inequalitymust be satisfied E n ( t ) + n ∈N H ∪N M e n ( t ) + n ∈N G ∪N M g n ( t ) ≤ θ En (7)We assume the available amount of harvesting energy at slot t is h n ( t ) with h n ( t ) ≤ h max for all t . The amount of actuallyharvested energy e n ( t ) at slot t , should satisfy ≤ e n ( t ) ≤ h n ( t ) , ∀ n ∈ N H ∪ N M , (8)where h n ( t ) is randomly varying over time slots in an i.i.d.fashion according to a potentially unknown distribution andtaking non-negative values from a finite but arbitrarily largeset S H . We define the harvestable energy state S H ( t ) =( h n ( t ) , n ∈ N H ∪ N M ) .The energy supplied by the electricity grid g n ( t ) of thebattery of node n at slot t should satisfy: ≤ g n ( t ) ≤ g max n , ∀ n ∈ N G ∪ N M , (9)with some finite g max n . E. Electricity Price Model
The cost per unit of electricity drawn from the electricitygrid at node n ∈ N G ∪ N M at slot t is denoted by P Gn ( t ) . Ingeneral, it may depend on both g n ( t ) , the total amount of elec-tricity from the electricity grid at slot t , and an electricity pricestate variable S Gn ( t ) , which represents such as both spatial andtemporal variations, etc. For example, the per unit electricitycost may be higher during daytime, and lower at late night. Weassume that S Gn ( t ) is randomly varying over time slots in ani.i.d. fashion according to a potentially unknown distributionand taking non-negative values from a finite but arbitrarilylarge set S G . Denote S G ( t ) = (cid:0) S Gn ( t ) , n ∈ N G ∪ N M (cid:1) asthe electricity price vector. Similarly in [29], we assume that P Gn ( t ) is a function of both S Gn ( t ) and g n ( t ) , i.e., P Gn ( t ) = P Gn ( S Gn ( t ) , g n ( t )) Note that the dependence of P Gn ( t ) on S Gn ( t ) and g n ( t ) is im-plicit for notational convenience in the sequel. For each given S Gn ( t ) , P Gn ( t ) is assumed to be a increasing and continuousconvex function of g n ( t ) . Let β G and β G denote the maximumand minimum unit electricity price in any slot in any node,respectively. F. Data Queue Model
For f ∈ F at node n , we use Q fn ( t ) to denote the databacklog of the f -th session at time slot t . We define Q ( t ) = (cid:0) Q fn ( t ) , n ∈ N , f ∈ F (cid:1) over time slots t ∈ T as the dataqueue backlog vector. Then the data queuing dynamic equationis Q fn ( t +1) = Q fn ( t ) − X b ∈O ( n ) x fnb ( t )+ X a ∈I ( n ) x fan ( t ) + f ∈F n r f ( t ) . (10)with Q fn (0) = 0 . In any time slot t , the total data output atnode n must satisfy the following data-availability constraint: ≤ X b ∈O ( n ) x fnb ( t ) ≤ Q fn ( t ) , ∀ n ∈ N , f ∈ F . (11)To ensure the network is strongly stable, the followinginequality must be satisfied: lim T →∞ T T − X t =0 X n ∈N X f ∈F E (cid:8) Q fn ( t ) (cid:9) < ∞ . (12) G. Optimization Problem
The goal is to design a full distributed algorithm thatachieves the optimal trade-off between the time-average util-ity of the source rate and the time-average cost of energyconsumption in electricity grid, which subject to all of theconstraints described above. Specifically, we define O ( t ) = ̟ X f ∈F U f ( r f ( t )) (13) − (1 − ̟ ) X n ∈N G ∪N M ̟ P Gn ( t ) g n ( t ) where ̟ ( ≤ ̟ ≤ ) is a weight parameter to combinethe objective functions together into a single one, and ̟ isa mapping parameter to ensure the objective functions at thesame level.Mathematically, we will address the stochastic optimizationproblem P1 as follows:maximize { χ ( t ) ,t ∈T } O = lim T →∞ T T − X t =0 E { O ( t ) } (14)subject to (1) , (2) , (3) , (6) , (7) , (8) , (9) , (11) , (12)with the queuing dynamics (5) for ∀ n ∈ N and (10) for ∀ n ∈N , ∀ f ∈ F . χ ( t ) ∆ = ( e ( t ) , g ( t ) , p T ( t ) , r ( t ) , x ( t )) is the set of theoptimal variables of the problem P1 , where e ( t ) , g ( t ) , p T ( t ) , r ( t ) , x ( t ) are the vector of e n ( t ) , g n ( t ) , p Tnb ( t ) , r f ( t ) , x fnb ( t ) ,respectively.III. D ISTRIBUTED C ROSS - LAYER O PTIMIZATION A LGORITHM : EASYOIn this section, we propose an Energy mAnagement andcroSs laYer Optimization algorithm (EASYO) for the problem P1 . Based on the Lyapunov optimization with weight perturba-tion technique developed in [21], [27] and [28] , EASYO willdetermine the energy harvesting, and the energy purchasing,source rate control, energy allocation for sensing/processing,transmission and reception, routing and scheduling decisions.EASYO is a fully distributed algorithm which makes greedydecisions at each time slot without requiring any statisticalknowledge of the harvestable energy states, of the electricityprice states and of the channel states. A. Lyapunov optimization
First, we introduce the weight perturbation θ E = (cid:0) θ En , n ∈ N (cid:1) . Note that the weight perturbation θ En is thelimited battery capacity of node n defined in Section II-D.Then we define the network state at time slot t as Z ( t ) ∆ = ( S C ( t ) , S H ( t ) , S G ( t ) , Q ( t ) , E ( t )) (15)Define the Lyapunov function as L ( t ) = 12 X n ∈N X f ∈F (cid:0) Q fn ( t ) (cid:1) + 12 X n ∈N (cid:0) E n ( t ) − θ En (cid:1) . (16) Remark 3.1
From (16), we can see that when minimizing theLyapunov function L ( t ) , the energy queue backlog is pushedtowards the corresponding perturbed variable value, and thedata queue backlog is pushed towards zero, which ensure thestrong network stability constraint (12). Furthermore, as longas we choose appropriate perturbed variables according to (40)in Theorem 1 at the next section, the constraint (6) will alwaysbe satisfied due to (44) in Theorem 1 at the next section. Thus,we can get rid of (12) and (6) in the sequel.Now define the drift-plus-penalty as ∆ V ( t ) ∆ = E ( L ( t + 1) − L ( t ) − V O ( t ) | Z ( t )) (17)where V is a non-negative weight, which can be tuned to con-trol O arbitrarily close to the optimum with a correspondingtradeoff in average queue size.We have the following lemma regarding he upper bound ofthe drift-plus-penalty ∆ V ( t ) : Lemma 1 : Under any feasible energy management, sourcerate control, transmission power allocation, routing andscheduling actions that can be implemented at time t , we havethe upper bound of ∆ V ( t ) as follows ∆ V ( t ) ≤ B + E (cid:16) e ∆ V ( t ) | Z ( t ) (cid:17) , (18)where B = N F B Q + X n ∈N B E (19)with B Q = l X + ( R max ) , where l max denotesthe largest number of the outgoing/incoming linksthat any node in the network can have. B E = ( n ∈N H ∪N M h max + n ∈N G ∪N M g max n ) + (cid:0) P T otaln, max (cid:1) , P T otaln, max = P f ∈F n ˜ P Sf r max f + P max n + ˜ P Rn l max X max . The core idea of Lyapunov optimization theory can be shortly acquiredfrom the two following linkage: http : // en . wikipedia . org / wiki / Drift plus penaltyhttp : // en . wikipedia . org / wiki / Lyapunov optimization . e ∆ V ( t ) = X n ∈N (cid:2)(cid:0) E n ( t ) − θ En (cid:1) n ∈N H ∪N M e n ( t ) + ( D n ( t ) + E n ( t ) − θ en ) n ∈N G ∪N M g n ( t ) (cid:3) (20) − X n ∈N s X f ∈F n h V ̟ U f ( r f ( t )) − Q fn ( t ) r f ( t ) + A n ( t ) ˜ P Sf r f ( t ) i − X n ∈N X b ∈O ( n ) X f ∈F W fnb ( t ) x fnb ( t ) + A n ( t ) p Tnb ( t ) e ∆ V ( t ) is given in (20), where D n ( t ) ∆ = V (1 − ̟ ) ̟ P Gn ( t ) , (21) A n ( t ) ∆ = E n ( t ) − θ En , (22)and W fnb ( t ) ∆ = Q fn ( t ) − Q fb ( t ) + A b ( t ) ˜ P Rb . (23) Proof : See Appendix A.
B. Framework of EASYO
We now present our algorithm EASYO. The main designprinciple of EASYO is to minimize the right hand side (RHS)of (20) subject to the constraints (1), (2), (3), (7), (8), (9) and(11).The framework of EASYO is described in Algorithm 1summarized in TABLE II.
TABLE IIA
LGORITHM
1: EASYO1 Initialization: The perturbed variables θ E and the penaltyparameter V is given.2 Repeat at each time slot t ∈ T :3 Observe Z ( t ) ;4 Choose the set χ ∗ ( t ) of the optimal variables as theoptimal solution to the following optimization problem P2 :minimize χ ( t ) e ∆ V ( t ) subject to (1) , (2) , (3) , (7) , (8) , (9) , (11)5 Update the energy queues and data queues according to(5) and (10), respectively. Remark 3.2
Note that the algorithm EASYO only requiresthe knowledge of the instant values of Z ( t ) . It does not requireany knowledge of the statistics of these stochastic processes.The remaining challenge is to solve the problem P2 , which isdiscussed below. C. Components of EASYO
At each time slot t , after observing Z ( t ) , all componentsof EASYO is iteratively implemented in distributed manner tocooperatively solve the problem P2 . Next, we describe eachcomponent of EASYO in detail.(1) Energy Management
For each node n ∈ N , combiningthe first term of the RHS of (20) with the the constraint (7), (8) and (9), we have the optimization problem of e n ( t ) and g n ( t ) as follows:minimize e n ( t ) ,g n ( t ) (cid:0) E n ( t ) − θ En (cid:1) n ∈N H ∪N M e n ( t )+( D n ( t ) + E n ( t ) − θ en ) n ∈N G ∪N M g n ( t ) subject to ≤ e n ( t ) ≤ h n ( t ) (24) ≤ g n ( t ) ≤ g max n (25) n ∈N H ∪N M e n ( t ) + n ∈N G ∪N M g n ( t ) ≤ θ En − E n ( t ) (26) Remark 3.3
Energy management component is composedof energy harvesting and energy purchasing. Furthermore,since P Gn ( t ) is increasing and continuous convex on g n ( t ) for each S Gn ( t ) , it is easy to verify that energy managementcomponent is a convex optimization problem in ( e n ( t ) , g n ( t )) , which can be solved efficiently. Remark 3.4
From (26), we can see that all the incomingenergy is stored if there is enough room in the energy bufferaccording to the limitation imposed by θ En , and otherwise itstores all the energy that it can, filling up the battery size of θ En . Hence, E n ( t ) < θ En for all t , which means that EASYOcan be implemented with finite energy storage capacity θ En atnode n ∈ N .(2) Source Rate Control
For each session f ∈ F n at sourcenode n ∈ N s , combining the second term of the RHS of (20)with the constraint (1), we have the optimization problem of r f ( t ) as follows:maximize r f ( t ) V ̟ U f ( r f ( t )) − (cid:16) Q fn ( t ) − A n ( t ) ˜ P Sf (cid:17) r f ( t ) subject to ≤ r f ( t ) ≤ r max f (27)Let r ∗ f be the unique maximizer. By the Kuhn-Tucker theorem, r ∗ f is given by r ∗ f = h U ′ − f (cid:16) Q fn ( t ) − A n ( t ) ˜ P Sf (cid:17)i r max f (28)where [ z ] ba = min { max { z, a } , b } , U ′ − f ( · ) is the inverse ofthe derivative of U f ( · ) .(3) Joint Optimal Transmission Power Allocation, Rout-ing and Scheduling
Combining the third term of the RHSof (20) with the constraints (2), (3) and the data-availabilityconstraint (11), we have the optimization problem of x ( t ) and p T ( t ) as follows:maximize x ( t ) , p T ( t ) X n ∈N X b ∈O ( n ) X f ∈F W fnb ( t ) x fnb ( t ) + (cid:0) A n ( t ) p Tnb ( t ) (cid:1) subject to ≤ X f ∈F x fnb ( t ) ≤ C nb ( t ) , ∀ n ∈ N , ∀ b ∈ O ( n )0 ≤ X b ∈O ( n ) p Tnb ( t ) ≤ P max n , ∀ n ∈ N ≤ X b ∈O ( n ) x fnb ( t ) ≤ Q fn ( t ) , ∀ f ∈ F (29)Now, we will solve the optimization problem (29). Definethe weight of the session f over link ( n, b ) as: ˜ W fnb ( t ) ∆ = h W fnb ( t ) − σ i + , (30)where σ = l max X max + r max f (31)denotes the data amount of the session f which the node n can receive at most at time slot t . Transmission Power Allocation Component
For eachnode n , find any f ∗ nb ∈ arg max f ˜ W fnb ( t ) . Define ˜ W ∗ nb ( t ) =max f ˜ W fnb ( t ) as the corresponding optimal weight of link ( n, b ) . Observe the current channel state S C ( t ) , and selectthe transmission powers p T ∗ by solving the following opti-mization problem:maximize p T X n ∈N X b ∈O ( n ) (cid:16) ˜ W ∗ nb ( t ) C nb ( t ) + A n ( t ) p nb ( t ) (cid:17) subject to ≤ X b ∈O ( n ) p Tnb ( t ) ≤ P max n , ∀ n ∈ N (32) Routing and Scheduling Component
The data of session f ∗ nb is selected for routing over link ( n, b ) whenever ˜ W f ∗ nb ( t ) > . That is, if ˜ W f ∗ nb ( t ) > , set x f ∗ nb nb ( t ) = C nb (cid:0) p T ∗ , S ( t ) (cid:1) . Remark 3.5 : If we set σ = 0 , the joint transmission powerallocation, routing and scheduling component is to minimizethe third term of the RHS in (20). Inspired by [27] and [28], weset a non-zero σ in (30), leading to a easy way to determine theupper bound of all queue sizes shown in Theorem 1. Also thedefinition (31) of σ can ensure the constraints (11) will alwaysbe satisfied. The detailed proof will be given in Theorem 1.Thus, we can get rid of this constraint (11) in (32). Remark 3.6 : Our proposed EASYO is designed to minimizethe RHS of (20). Each component contributes to minimizingthe part of the RHS of (20). Taking all components together,EASYO contributes to minimize the whole RHS of (20), andthus to minimize ∆ V ( t ) . Because the whole RHS of (20)incorporates the Lyapunov drift, EASYO is stable. Meanwhile,since it also incorporates the objective of the problem P1 ,EASYO is optimal. Remark 3.7 : The former two components of EASYO arecomputed in closed form or numerically solved through asimple convex optimization problem, only based on the localinformation. The unique challenge of distributed implementa-tion of EASYO is to distributedly solve the transmission powerallocation problem (32). Next, we will develop the distributedalgorithm.
D. Distributed Implementation of Transmission Power Alloca-tion
After implementing a variable change ˆ p Tnm ( t ) =log (cid:0) p Tnm ( t ) (cid:1) , and taking the logarithm of both sides of theconstraint in problem (32), the problem (32) can be equiva-lently transformed into the problem P3 max ˆ p T ( t ) X n ∈N X b ∈O ( n ) (cid:16) ˜ W ∗ nb ( t )Ψ nb (cid:0) ˆ p T ( t ) (cid:1) + A n ( t ) e ˆ p Tnb ( t ) (cid:17) s.t. log X b ∈O ( n ) e ˆ p Tnb ( t ) − log P max n ≤ , ∀ n ∈ N (33)where ˆ p T ( t ) = (ˆ p Tnb ( t ) , n ∈ N , b ∈ O ( n )) , Ψ nb (cid:0) ˆ p T ( t ) (cid:1) isdefined in (34). Ψ nb (cid:0) ˆ p T ( t ) (cid:1) (34) ∆ = log ( γ nb ( t )) = log S Cnb ( t ) + ˆ p Tnb ( t ) − log N b + X a ∈J n,b X ( a,m ) ∈L exp (cid:0) ˆ p Tam ( t ) + log S Cab ( t ) (cid:1) . It is not difficult to prove that Ψ nb (cid:0) ˆ p T (cid:1) is a strictly concavefunction of a logarithmically transformed power vector ˆ p T ( t ) [30]. Due to (7) or (26), we have E n ( t ) ≤ θ En , so A n ( t ) ≤ ,thus A n ( t ) e ˆ p Tnb ( t ) is a strictly concave function of ˆ p Tnb ( t ) . Tosum up, the objective of P3 is a strictly convex in ˆ p T ( t ) .Furthermore, since log P b ∈O ( n ) e ˆ p Tnb ( t ) is a strictly concave in ˆ p T ( t ) , P3 is a strictly convex optimization problem, whichhas the global optimum.To distributedly solve P3 , we propose a distributed iter-ative algorithm based on block coordinate descent (BCD)method whereby, at every iteration, a single block of vari-ables is optimized while the remaining blocks are held fixed.More specifically, at iteration t i , which represents the i -thiteration at the time slot t , for each node n ∈ N , theblocks ˆ p Tn = (ˆ p Tnb , b ∈ O ( n )) are updated through solvingthe following optimization problem (35), where ˆ p T − n ( t i ) =( ˆ p T ( t i ) , · · · , ˆ p Tn − ( t i ) , ˆ p Tn +1 ( t i ) , · · · , ˆ p TN ( t i )) are held fixed.maximize ˆ p Tn X n ∈N X b ∈O ( n ) ( ˜ W ∗ nb ( t )Ψ nb (cid:0) ˆ p Tn , ˆ p T − n ( t i ) (cid:1) + A n ( t ) e ˆ p Tnb ) subject to log X b ∈O ( n ) e ˆ p Tnb − log P max n ≤ . (35)The global rate of convergence for BCD-type algorithm hasbeen studied extensively when the block variables are updatedin both the classic Gauss-Seidel fashion and the randomizedupdate rule [31] [32]. Since the optimization problem P3 isstrongly convex in ˆ p T ( t ) , our proposed BCD-based distributediterative algorithm can converge to the global optimum of P3 .IV. P ERFORMANCE ANALYSIS
Now, we analyze the performance of our proposed algorithmEASYO. To start with, we assume that there exists δ > suchthat C nb (cid:0) p T ( t ) , S C ( t ) (cid:1) ≤ δp Tnb ( t ) , ∀ n ∈ N , ∀ b ∈ O ( n ) . (36) Theorem 1 : Implementing the algorithm EASYO with anyfixed parameter
V > for all time slots, we have the followingperformance guarantees:( A ). Suppose the initial data queues and the initial energyqueues satisfy: ≤ Q fn (0) ≤ Q max , n ∈ N , f ∈ F (37) ≤ E n (0) ≤ θ En , n ∈ N (38)where the queue upper bounds are given as follows: Q max = ̟ β U V + r max f , (39) θ En = δ̟ β U V + P T otaln, max . (40)Then, the data queues and the energy queues of all nodes forall time slots t are always bounded as ≤ Q fn ( t ) ≤ Q max , n ∈ N , f ∈ F (41) ≤ E n ( t ) ≤ θ En , n ∈ N , (42)( B ). The objective function value of the problem P1 achieved by the proposed algorithm EASYO satisfies thebound O ≥ O ∗ − ˜ BV (43)where O ∗ is the optimal value of the problem P1 , and ˜ B = B + N F σl max X max .( C ). When node n ∈ N allocates nonzero power for datasensing, data transmission and/or data reception, we have: E n ( t ) ≥ P T otaln, max , n ∈ N . (44)( D ). For node n ∈ N , when any data of the f -th session istransmitted to other node, we have: Q fn ( t ) ≥ l max X max . (45) Proof : Please see Appendix B-E.
Remark 4.1 : Theorem 1 shows that a control parameter V enables an explicit trade-off between the average objectivevalue and queue backlog. Specifically, for any V > ,the proposed distributed algorithm EASYO can achieve atime average objective that is within O (1 /V ) of the optimalobjective shown in (43), while ensuring that the average dataand energy queues have upper bounds of O ( V ) shown in (41)and (42), respectively. In the section V, the simulations willverify the theoretic claims. Remark 4.2 : The inequality (44) guarantees that the energy-availability constraint (6) is satisfied for all nodes and alltimes. Similarly, the inequality (45) ensures that the dataavailability constraint (11) is always satisfied.V. S
IMULATION R ESULTS
In this section, we provide the simulation results of thealgorithm EASYO for the network scenario shown in Fig.2.In this scenario, we consider a multi-channel WSN with20 nodes, 78 links, 6 sessions transmitted on 14 differentchannels. Throughout, the form of the rate utility function isset as U f ( r f ( t )) = log(1 + r f ( t )) , so β U = 1 . The form ofthe electricity cost function is set as P Gn ( t ) = S Gn ( t ) . Fig. 2. Network topology.
Set several default values as follows: δ = 2 ; r max f =3 , ˜ P Sf = 0 . , ∀ f ∈ F ; g max n = 2 , ∀ n ∈ N G ∪ N M ; X max = 2 , l max = 6 , P max n = 2 , ˜ P Rn = 0 . , ∀ n ∈ N ; N b = 5 × − , ̟ = 0 . , ̟ c = 0 . . We set all the initialqueue sizes to be zero.The channel-state matrix S C ( t ) has independent entriesthat for every link are uniformly distributed with interval [ S C min , S C max ] × d − , S C min = 0 . , S C max = 1 . as default valuesand d denotes the distance between transmitter and receiverof the link, while the energy-harvesting vector S H ( t ) hasindependent entries that are uniformly distributed in [0 , h max ] ,with h max = 2 as default value. The electricity price vector S G ( t ) has independent entries that are uniformly distributedin [ S G min , S G max ] with S G min = 0 . , S G max = 1 as default values,so β G = S G max , β G = S G min . All statistics of S C ( t ) , S H ( t ) and S G ( t ) are i.i.d. across time-slots.We simulate V =[100,300,500,700,1000,1500]. In all simu-lations, the simulation time is time slots. The simulationresults are depicted in Fig. 3. From Fig. 3 (a), we see that as V increases, the time average optimization objective value keepincreasing and converge to very close to the optimum. Thisconfirms the results of (43). From Fig. 3 (b), we see that as V increases, the average data queue length keeps increasing.From Fig. 3 (c)-(e), we observe that the battery queue sizeincreases as V increases. A closer inspection of the resultsalso reveals a linear increase of the time average data andenergy queue size with respect to V . This shows a good matchbetween the simulations and Theorem 1.For better verification of the queueing bounds, we alsopresent the data queue process of node for session , ofnode for session and of node for session under V = 1000 in Fig 4(a), and the energy queue processes for EHnode , , , for EG node , , and for ME node , , under V = 1000 in Fig. 4(b)-(d), respectively. It can be verifythat all queue sizes can quickly converge with the upper boundgiven in Theorem 1.Transmission power allocation problem P3 is the mostcomplex component in our proposed EASYO. We proposed aBCD-based distributed iterative algorithm to solve the problem P3 . During the implementation of EASYO, we catch four O b j e c t i v e v a l ue (a) 0 1000 200000.511.52 x 10 V D a t a ba ck l og (b)0 1000 2000010,00020,000010,000 V E ne r g y ba ck l og (c)EH node 0 1000 20000200040006000800010000 V E ne r g y ba ck l og (d)EG node 0 1000 20000200040006000800010000 V E ne r g y ba ck l og (e)ME node Fig. 3. Verification of Theorem 1. D a t a queue p r o c e ss e s (a) 0 0.5 1 1.5 2x 10 E ne r g y queue p r o c e ss e s (b)EH nodes 0 0.5 1 1.5 2x 10 E ne r g y queue p r o c e ss e s (c)EG nodes 0 0.5 1 1.5 2x 10 E ne r g y queue p r o c e ss e s (d)ME nodes Q Q Q E E E E E E E E E Fig. 4. Detailed verification of the queueing bounds. different snapshots of the iterative procedure of BCD-basedalgorithm, shown in Fig. 5. From Fig. 5, we can see that BCD-based algorithm can quickly converge to the global optimum.Thus, our proposed EASYO is a low-complexity distributedalgorithm.Next, we investigate the impacts of a variety of systemparameters on the objective value, rate utility and electricitycost. Fig. 6 shows the impact of the node power supply mannerand the maximum available harvested energy h max on theobjective value. From Fig. 6, we can see that the lowestobjective value is achieved at all EH nodes scenario with h max = 0 . much smaller than g max = 2 and the highestobjective value is achieved at all EH nodes scenario with h max = 2 equal to g max = 2 . Due to the expense of the highestenergy cost, all EG nodes scenario achieves the objective valuelower than all EH nodes scenario or default node scenario with h max = 2 . In contrast, all EG nodes scenario achieves theobjective value higher than all EH nodes scenario or defaultnode scenario with h max = 0 . , which results in the lowenergy supply and low data transmission.We investigate the impact of the electricity price on therate utility and energy cost. We set three different electricityprices as S G max = 0 . , S G max = 1 and S G max = 10 , respectively. t i O b j e c t i v e v a l ue o f P O b j e c t i v e v a l ue o f P t i O b j e c t i v e v a l ue o f P t i O b j e c t i v e v a l ue o f P t i Global optimumBCD−based iterative
Fig. 5. Convergence of BCD-based distributed iterative algorithm.
100 200 300 400 500 600 700 800 900 100000.511.522.533.54 V O b j e c t i v e v a l ue All EH nodes scenario with h max =2Default nodes scenario with h max =2All EG nodes scenarioDefault nodes scenario with h max =0.2All EH nodes scenario with h max =0.2
Fig. 6. The impact of node power supply manner and maximum availableharvested energy on objective value.
Fig. 7 shows that the electricity cost increases along withthe increase of the electricity price. In order to reduce theelectricity cost, EASYO reduce the energy consumption, andthus the corresponding rate utility decreases.We investigate the impact of the weight parameter on therate utility and energy cost. We set three the weight parametersas ̟ = 0 . , ̟ = 0 . and ̟ = 0 . , respectively.When the weight parameter ̟ is chosen as a large value,EASYO focuses on the rate utility maximization rather thanthe electricity cost minimization. The results of Fig. 8 verifythis situation, where the rate utility increases and the electricitycost also increases under a large value ̟ = 0 . .Fig. 9 shows the impact of ˜ P Sf on the rate utility and energycost. The larger ˜ P Sf , the more energy is required to supply fordata sensing/processing, leading to the less energy used in datatransmission, and the lower rate utility.VI. C ONCLUSIONS
Because of the instable energy supply and the limitedbattery capacity in EH node, it is very difficult to en-sure the perpetual operation for WSN. In this paper, we R a t e u t ili t y
200 400 600 800 10000.10.20.30.40.50.60.70.80.91 V E l e c t r i c i t y c o s t S maxG =0.2S maxG =1S maxG =10 Fig. 7. The impact of different electricity prices on rate utility and energycost.
200 400 600 800 100022.533.544.555.566.5 V R a t e u t ili t y ϖ =0.9 ϖ =0.6 ϖ =0.3 200 400 600 800 100000.20.40.60.811.21.41.61.82 V E l e c t r i c i t y c o s t Fig. 8. The impact of different weight parameters on rate utility and energycost. consider heterogeneous energy supplies from renewable en-ergy and electricity grid, multiple energy consumptions andmulti-dimension stochastic natures in the system model, andformulate a discrete-time stochastic cross-layer optimizationproblem to optimize the trade-off between the time-averagerate utility and electricity cost. To the end, we propose afully distributed and low-complexity cross-layer algorithmonly requiring knowledge of the instantaneous system state.The theoretic proof and the extensive simulation show thata parameter V enables an explicit trade-off between theoptimization objective and queue backlog. In the future, weare interested in two aspects of delay reduction by utilizingthe shortest path concept, and by modifying the queueingdisciplines. A PPENDIX AP ROOF OF L EMMA B defined in (19).Plugging the definition (4) of p T otaln ( t ) into (48), andrearranging all terms of the RHS in (48), e ∆ V ( t ) is changedinto (20). (cid:3)
200 300 400 5001.522.533.544.555.566.5 V R a t e u t ili t y
200 300 400 5000.10.20.30.40.50.60.70.80.911.1 V E l e c t r i c i t y c o s t P fs =0P fs =0.1P fs =1 Fig. 9. The impact of different sensing energy consumption on rate utilityand energy cost. A PPENDIX BP ROOF OF P ART (A) IN T HEOREM t = 0 , we can easily have (41), then we assume (41) ishold at time slot t , next we will show that it holds at t + 1 . Case 1 : If node n doesn’t receive any data at time t , wehave Q fn ( t + 1) ≤ Q fn ( t ) ≤ ̟ β U V + r max f . Case 2 : If node n receives the endogenous data from othernodes a ∈ I ( n ) , we can get from (30) that W fan ( t ) − σ ≥ .By plugging (23) and (31), we have Q fa ( t ) − Q fn ( t ) + A n ( t ) ˜ P Rn − (cid:16) r max f + l max X max (cid:17) ≥ . Then, Q fn ( t ) ≤ Q fa ( t ) − (cid:16) r max f + l max X max (cid:17) + A n ( t ) ˜ P Rn . Due to A n ( t ) ≤ and ˜ P Rn > , we have Q fn ( t ) ≤ Q fa ( t ) − (cid:0) r max f + l max X max (cid:1) . (49)Plugging (41) into (49), we have Q fn ( t ) ≤ ̟ β U V + r max f − (cid:0) r max f + l max X max (cid:1) = ̟ β U V − l max X max . (50)At every slot, the node can receive the amount of data at most r max f + l max X max . So Q fn ( t + 1) ≤ Q fn ( t ) + l max X max + r max f . (51)Combing (50) and (51), we have Q fn ( t + 1) ≤ ̟ β U V + r max f . Case 3 : If node n only receives the new local data, ac-cording to (27), the optimal value r ∗ f will met V ̟ U ′ f ( r ∗ f ) = Q fn ( t ) − A n ( t ) ˜ P Sf , where U ′ f ( r f ( t )) denotes the first derivativeof U f ( r f ( t )) . So we have Q fn ( t ) − A n ( t ) ˜ P Sf ≤ V ̟ β U , andthen Q fn ( t ) ≤ ̟ β U V . At every time, the new local datareceived at most is r max f , so, Q fn ( t + 1) ≤ Q fn ( t ) + r max f ≤ ̟ β U V + r max f .To sum up the above, we complete the proof of (41).From Remark 3.5 we can have (42). (cid:3) A PPENDIX CP ROOF OF P ART (B) IN T HEOREM h(cid:0) Q fn ( t +1) (cid:1) − (cid:0) Q fn ( t ) (cid:1) i ≤ f ∈F n r f ( t ) + X a ∈I ( n ) x fan ( t ) + 12 ( X b ∈O ( n ) x fnb ( t )) + Q fn ( t ) f ∈F n r f ( t ) + X a ∈I ( n ) x fan ( t ) − X b ∈O ( n ) x fnb ( t ) ≤ ( R max ) + 32 ( l max X max ) + Q fn ( t ) f ∈F n r f ( t ) + X a ∈I ( n ) x fan ( t ) − X b ∈O ( n ) x fnb ( t ) (46) h(cid:0) E n ( t + 1) − θ En (cid:1) − (cid:0) E n ( t ) − θ En (cid:1) i ≤ (cid:16) ( n ∈N H ∪N M e n ( t ) + n ∈N G ∪N M g n ( t )) + (cid:0) p T otaln ( t ) (cid:1) (cid:17) + (cid:0) E n ( t ) − θ En (cid:1) (cid:0) n ∈N H ∪N M e n ( t ) + n ∈N G ∪N M g n ( t ) − p T otaln ( t ) (cid:1) ≤ (cid:16) ( n ∈N H ∪N M h max + n ∈N G ∪N M g max n ) + (cid:0) p T otaln, max (cid:1) (cid:17) + (cid:0) E n ( t ) − θ En (cid:1) (cid:0) n ∈N H ∪N M e n ( t ) + n ∈N G ∪N M g n ( t ) − p T otaln ( t ) (cid:1) (47) e ∆ V ( t ) = X n ∈N X f ∈F Q fn ( t ) f ∈F n r f ( t ) + X a ∈I ( n ) x fan ( t ) − X b ∈O ( n ) x fnb ( t ) + X n ∈N (cid:0) E n ( t ) − θ En (cid:1) (cid:0) n ∈N H ∪N M e n ( t ) + n ∈N G ∪N M g n ( t ) − p T otaln ( t ) (cid:1) − V ̟ X f ∈F U f ( r f ( t )) − (1 − ̟ ) X n ∈N G ∪N M ̟ P Gn ( t ) g n ( t ) (48)A PPENDIX DP ROOF OF P ART (C) IN T HEOREM C ab (cid:0) p T ( t ) , S ( t ) (cid:1) : C ab (cid:0) p T ( t ) , S ( t ) (cid:1) ≤ C ab (cid:16) p T ′ ( t ) , S ( t ) (cid:17) (52)where p T ′ ( t ) obtained by setting p Tnm ( t ) of p T ( t ) to zero, ( a, b ) ∈ L and ( a, b ) = ( n, m ) .For link ( n, m ) , plugging (23) into (30), we get ˜ W fnm ( t ) = h Q fn ( t ) − Q fm ( t ) + A m ( t ) ˜ P Rm − σ i + ≤ (cid:2) Q fn ( t ) − σ (cid:3) + (53)By plugging (31) and (41) into (53), we have ˜ W fnm ( t ) ≤ (cid:2) ̟ β U V + r max f − l max X max − r max f (cid:3) + = [ ̟ β U V − l max X max ] + (54)Since (54) holds for any session f through link ( n, m ) , wehave ˜ W ∗ nm ( t ) ≤ [ ̟ β U V − l max X max ] + (55) We assume that E n ( t ) < P T otaln, max , when node n ∈ N allocatesnonzero power for data sensing, compression and transmission.Furthermore, we assume that the power allocation controlvector p T ∗ ( t ) is the optimal solution to (32), and withoutloss of generality, there exists some p T ∗ nm ( t ) > . By setting p T ∗ nm ( t ) = 0 in p T ∗ ( t ) , we get another power allocation controlvector p T ( t ) . We denote G (cid:0) p T ( t ) , S ( t ) (cid:1) as the objectivefunction of (32). In this way, we get: G (cid:0) p T ∗ ( t ) , S ( t ) (cid:1) − G (cid:0) p T ( t ) , S ( t ) (cid:1) = X n ∈N X b ∈O ( n ) [ C nb ( p T ∗ ( t ) , S ( t )) − C nb ( p T ( t ) , S ( t ))] ˜ W ∗ nb ( t )+ (cid:0) E n ( t ) − θ En (cid:1) p T ∗ nm (56)From (52), we have C nb ( p T ∗ ( t ) , S ( t )) − C nb ( p T ( t ) , S ( t )) ≤ for b = m . 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