Achievable Throughput of Energy Harvesting Fading Multiple-Access Channels under Statistical QoS Constraints

Achievable Throughput of Energy Harvesting Fading Multiple-Access Channels under Statistical QoS Constraints Deli Qiao and Jingwen Han Abstract This paper studies the achievable throughput of fading multiple-access channels, where the transmitters harvest random amounts of energy from the environment and the data transmissions are subject to statistical quality of service QoS) constraints in the form of limitations on the buffer overflow probability. Effective capacity, which characterizes the maximum constant arrival rate that a given process can support while satisfying the QoS constraints is employed as the performance metric. Perfect channel state information CSI) and energy arrivals are assumed to be available at both the transmitters and the receiver. With the assumption of naive power control scheme, in which the transmission power level is irrespective of the CSI and decided by the instantaneous harvested energy, the effective throughput regions of different transmission strategies are characterized, namely time division multiple access TDMA) and superposition coding with fixed/variable decoding. In the two-user case, the optimal energy and channel aware decoding strategy is determined for the scenario in which the two users have the same QoS constraints. With the assumption that the channel states and harvested energy in all time slots are known at the transmitters, the point-to-point link is first revisited to obtain the effective capacity expression with the optimal offline optimal power control policy. Then, for a given decoding order strategy in two-user multiple access channels, the conditions that the optimal offline power control policies must satisfy are determined, and an algorithm to compute the optimal policies is provided. This work has been supported in part by the National Natural Science Foundation of China 667205, 6672238) and the Shanghai Sailing Program 6YF402600). The authors are with the School of Information Science and Technology, East China Normal University, Shanghai, China 20024. D. Qiao is also with the Key Laboratory of Wireless Sensor Network & Communication, Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, 865 Changning Road, Shanghai China 200050. Email: dlqiao@ce.ecnu.edu.cn, 5524025@ecnu.cn.

2 I. INTRODUCTION Energy harvesting EH) systems, in which the transmitters harvest energy from the environment and store in their rechargeable batteries, have attracted much interest recently see, e.g., []-[] and references therein). The harvested energy is generally generated by renewable resources, such as solar, wind, motion, etc. Utilizing the available energy in the batteries, the transmitters can transmit messages to the receivers through power adaptation to achieve better performance. There have been numerous works on the power allocation problems in EH systems under the energy harvesting constraints, i.e., the total consumed energy at any time cannot be greater than the harvested energy so far. There are generally two types of power control policies: online and offline. The online power control policies only require the causal energy and channel state information CSI), and hence can be implemented for all EH systems. However, obtaining the optimal online power control policies is extremely complicated even for the point-to-point links with finite time slots [2]. Suboptimal power control policies that can achieve the average rate within constant additive and multiplicative gap of the channel capacity have been proposed in [6]. On the other hand, the offline power control policies are identified based on the assumption of non-causal energy and channel state information CSI), and hence are applicable to highly predictable systems. For instance, in [], the authors have characterized the optimal offline power control policies for point-to-point fading channels, which have been shown to be piecewise water-filling in case of infinite energy buffers. In [2], the authors have also derived the optimal offline power control policies for point-to-point links and shown that the water-levels are non-decreasing staircase functions. They have also designed an algorithm to determine the optimal power control policy. Meanwhile, multiple-access channel MAC) in which multiple users communicate with a single user e.g., a base station) is one of the important scenarios of energy harvesting systems, especially considering the wireless sensor networks [3]. Through energy harvesting operations, massive wireless connections with sensors and objects are expected to be enabled for the next generation wireless systems. The design and analysis of efficient transmission schemes in this model have been of significant interest recently. For instance, in [7], the offline optimal power control policy that maximizes the sum rate in energy harvesting Gaussian MAC has been identified. The authors have designed a dynamic iterative water-filling algorithm that maximizes

3 the sum rate of fading MAC with energy harvesting users in [8]. The capacity region of the energy harvesting MAC and the inner and outer bounds that differ by a constant gap have been characterized in [9]. In [0], the authors have investigated the performance of time division multiple access TDMA) with energy harvesting users and showed that in case of unknown instantaneous energy arrivals of each other, each user should transmit at the same power level decided by the average energy arrivals. In [], the authors considered the Gaussian MAC with energy cooperation as well and derived the capacity region, which is shown to coincide with the traditional Gaussian MAC with energy cooperation. In this paper, we investigate the achievable throughput of fading MAC with energy harvesting transmitters under statistical quality of service QoS) constraints, in the form of limitations on the buffer violation probabilities. For this analysis, we employ the concepts of effective bandwidth [2] and effective capacity [3] to determine the maximum throughput in such settings. Recently, we have investigated the throughput regions of fading MAC under QoS constraints without energy harvesting constraints in [4]. We note that there have been some works on the energy harvesting systems under statistical QoS constraints. For instance, the optimal power control policies for point-to-point links under statistical QoS constraints have been characterized in [5]. The energy efficient time slot allocation for energy harvesting MAC with TDMA has been identified in [6]. In this work, we assume that both the data and energy buffers at the transmitters are infinite and that both the energy arrivals and CSI are independent and identically distributed IID). Initially, we assume that the causal energy and CSI information is available at both the transmitters and the receiver. Suppose that the users use all harvested energy in each time slot, i.e., naive power control. We identify the effective capacity for three different transmission strategies: TDMA and superposition coding with fixed/variable decoding order. Specifically, for the two-user MAC case, we design an optimal energy and channel aware decoding strategy that maximizes the weighted sum rate of the users with the same QoS constraints. Then, we assume that the noncausal energy and CSI information of all time slots are available at both the transmitters and the receiver. We first revisit the point-to-point links and show that the offline optimal power control policy can achieve the same performance as that of traditional systems without energy harvesting constraints with a large number of time slots. Motivated by this finding, we derive the effective capacity expression with the associated offline optimal power control policy in case of infinite number of time slots. We propose a suboptimal online power control policy that requires only the statistics

4 Fig. : System Model and causal energy and CSI information, which can achieve almost the same performance as the offline optimal one. Subsequently, we consider the two-user MAC with superposition coding with variable decoding order strategy, and identify the optimality conditions that the offline optimal power control policy should satisfy. Incorporating the algorithm proposed in [2], we also design an algorithm that can determine the allocated power under such policies. The organization of this paper is as follows. In Section II, the system model and preliminaries on the channel rate and effective capacity are briefly discussed. Performances of naive power control policies are investigated in Section III with numerical results provided as well. In Section IV, the offline optimal power control policies for point-to-point links and the two-user MAC case are studied. Finally, Section V concludes this paper, with some length proofs in Appendices.

5 II. SYSTEM MODEL AND PRELIMINARIES A. System Model As shown in Fig., we consider a multiple-access fading channel, in which M energy harvesting users transmit to a single receiver. It is assumed that the transmitters generate data sequences which are divided into frames of duration T and initially stored in the buffers before transmitted over the wireless channels. Each user is subject to certain statistical QoS constraints in order to limit the buffer overflow probability. The transmitters randomly harvest energy from the environment and store them in the rechargeable batteries. We assume that both the data and energy buffers are of infinite length. We consider flat fading channels and the channel input-output relationship can be described by y = M g j x j + n, ) j= where x j is the channel input of the j-th user and y is the output at the receiver. n denote the zero-mean circularly-symmetric, complex Gaussian noise at the receiver with E{ n 2 } = N 0. g j denotes the fading coefficient of the channel between user j and the receiver, and we denote the magnitude-square of the fading coefficients by h j = g j 2. We consider a block fading scenario, and hence the fading coefficients stay constant over the frame duration T and change independently for each frame and each user. Let b j [t] denote the amount of energy present at the beginning of frame t at transmitter j. Then, b j [t] evolves as b j [t + ] = b j [t] e j [t] + E j [t + ], j =,..., M, 2) where e j [t] is the energy consumed by user j in frame t, and E j [t + ] denotes the amount of energy harvested by user j at the beginning of frame t +. Assuming that the symbol rate is B complex symbols per second, we can see that the energy for the channel input of the j-th user is e j [t]/t B. Let the instantaneous transmitted SNR level of user j be µ j [t] = e j[t] N 0. It is assumed T B that the harvested amounts of energy {E j [t]} are independent and identically distributed IID) with mean Ēj for all frames and users. The average energy harvesting rates are then given by

6 Ē j T J/s Watt), respectively. Obviously, we must have the energy harvesting constraints e j [t] b j [t], t, 3) i.e., the transmitters can consume at most all the current available energy. B. Channel Rate We assume that the CSI of the network is available at both the transmitters and the receiver. We can obtain the instantaneous channel rate for different transmission strategies. ) Time Division Multiple Access TDMA): We first consider the time division scheme in which the users send their signals in non-overlapping time intervals. Denote τ j as the fraction of time allocated to user j. Note that we have τ j 0 and M j= τ j =. Then, the instantaneous channel rate of user j is given by R j = τ j B log 2 + µ jh j τ j ), bits/s. 4) Note that user j is assumed to transmit with the amount of energy e j [t] in the allocated τ j fraction of the time, resulting in higher energy for each input e j τ j T B. 2) Superposition Coding with Fixed Decoding Order SC-FDO): We assume that the transmitters simultaneously send data to the receiver and the receiver decodes the received signal in a fixed order in each frame. Denote τ k as the fraction of time allocated to decoding order π k, k =,..., M!. Note that we have τ k 0 and M! k= τ k =. In τ k fraction of the time, the instantaneous channel rate of user j at a given decoding order π k is given by ) R π k j) = B log µ j h j 2 + + π k i)>π j) µ, 5) ih i where π k is the inverse trace function of π k. From the above expression, the transmitted signal of user i that is decoded later than user j in decoding order π k is seen by user j as interference. Then, the transmission rate of user j for each frame is M! R j = τ k B log 2 + k= + π k µ j h j k i)>π k j) µ ih i ). 6)

7 The instantaneous rate region is given by [20] { R MAC = R,..., R M ) : RS) B log 2 + ) } µ j h j, S {,..., M} j S 7) 3) Superposition Coding with Variable Decoding Order SC-VDO): We also consider the case in which the decoding order can vary over different frames. We assume that the receiver varies the decoding order depending on the channel states h = h,..., h M ). Assume that the vector space R M + of the possible values for h is partitioned into M! disjoint regions denoted by {H k } M! k=. When h H k, the receiver decodes the information in the order π k, and the instantaneous channel rate of user j is then given by 5) for this frame. C. Effective Capacity We employ effective capacity to measure the throughput under statistical queueing constraints. In [3], effective capacity is defined as the maximum constant arrival rate that a given service process can support in order to guarantee a statistical QoS requirement specified by the QoS exponent θ. If we define Q as the stationary queue length, then θ is the decay rate of the tail distribution of the queue length Q log PrQ q) lim q q = θ. 8) Therefore, for large q max, we have the following approximation for the buffer violation probability: PrQ q max ) e θq max. Hence, while larger θ corresponds to more strict QoS constraints, smaller θ implies looser QoS guarantees. The effective capacity is given by C E θ) = lim t θt log E{e θs[t] } bits/s, 9) where the expectation is with respect to S[t] = t i= r[i], which is the time-accumulated service process. {r[i], i =, 2,...} denotes the discrete-time stationary and ergodic stochastic service process. Throughout the text, logarithm expressed without a base, i.e., log ), refers to the natural logarithm log e ).

8 In case of IID service rates for different frames under the block fading scenario, the effective capacity can be expressed as [8] C E θ) = θt log E{e θt R } bits/s 0) where R is the instantaneous transmission rate. Throughout this paper, we use the effective capacity normalized by bandwidth B, which is denoted by C E θ) = C Eθ) B = θt B log E{e θt R } bits/s/hz. ) Let Θ = θ,..., θ M ) be the vector composed of the QoS constraints of M users and C E Θ) = C E, θ ),..., C E,M θ M )) denote the vector of the normalized effective capacities of the users. The effective throughput region is described as { C MAC Θ) = C E Θ) 0 : R R MAC s.t. 3) C E,j θ j ) θ j T B log E { e θ jt R j } where R = [R,..., R M ] represents the vector composed the instantaneous transmission rate of M users in the instantaneous achievable rate region R MAC satisfying the energy harvesting constraints 3). Similar to [4, Theorem ], we can show that the effective throughput region is convex. Remark : Throughout this text, we generally obtain the effective capacity values on the boundary surface for simplicity and brevity. Note that effective capacity regions can be decided immediately using the boundary points. } 2) III. TRANSMISSIONS WITH NAIVE POWER CONTROL In this section, we assume that the energy used by each user for transmission is exactly the harvested energy at the beginning of each frame, i.e., e j [t] = E j [t]. Then, we have µ j [t] = E j [t] N 0. We consider the effective throughput region of TDMA and superposition coding with T B fixed/variable decoding order. Note that the energy harvesting and channel fading processes can be different in time scale. For instance, the energy arrivals are typically slower than the variations in channel [4], in which the transmission power of different frames becomes correlated. In case of naive power control policy, we can show the following result.

9 Proposition : Assume that the incoming energy stays constant for m Z, m > frames and varies independently each m frames. Then, the effective capacity can be expressed as Proof: See Appendix A for details. C E θ) = θt log E { e θt R}. 3) The above result tells us that slower variations in the harvested energy will not alter the expression for effective capacity. As can be seen in the proof, the expectation is now taken over the energy arrivals E and the channel state h. Henceforth, we consider the scenario of very fast change of the incoming energy, where the energy varies in the same time scale as that of channel fading coefficients. A. TDMA Considering the transmission strategy described in Section II-B and 0), we can obtain the maximum effective capacity for user j { C E,j θ) = θ j T B log E )} e θ jτ j T B log 2 + µ j h j τ j. 4) The optimal time allocation policy that maximizes the sum rate can be obtained through the optimization problem max {τ j } s.t. M j= { )} θ j T B log E e θ jτ j T B log 2 + µ j h j τ j M τ j =, τ j 0. 5) j= It has been shown in [4] that the above optimization problem is concave for constant transmission power levels, i.e., µ j is constant. Following a similar way, we can show that the above optimization problem is concave with IID energy arrivals E j, i.e., random variable µ j. Then, we can solve the above problem through Lagrangian maximization method. Although obtaining the closed-form solutions seems unlikely, we can employ convex optimization tools to derive the optimal values of τ j numerically.

0 B. SC-FDO In this scheme, the time sharing strategy is independent of the energy arrivals and the channel states, and hence is fixed in all blocks. Combining 6) and 0), we have C E,j θ) = θjt B θ j T B log E e M! k= τ k log 2 + + π k µ j h j i)>π k j) µ i h i ). 6) Note that the rate 5) is the maximum instantaneous rate achieved with superposition coding and a particular decoding order. Therefore, the effective capacities specify the maximum throughput under the statistical QoS constraints. The optimal time-sharing strategy that maximizes the sum rate can be obtained similarly to the discussions for TDMA. C. SC-VDO Given the channel state partition {H k } M! k=, we have the effective capacity for user j as C E,j θ) = θjt B θ j T B log E e M! k= log 2 + + π k µ j h j i)>π k j) µ i h i ) {h H k }, 7) where { } is the indicator function. Obtaining the optimal partition of the fading states in a general scenario seems intractable. We consider a simplified two-user MAC case and the users have the same QoS constraints θ = θ 2 = θ. As a significant departure from the results in [4] for fixed power transmissions, we denote h 2 = gh, E, E 2 ) as the partition function that depends on both the energy arrivals and the CSI. Now, users are decoded in the order,2) if h 2 < gh, E, E 2 ) and 2,) if h 2 > gh, E, E 2 ). The effective capacity expressions for the two users can be expressed as C E, θ) = θt B log + gh,e,e 2 ) e θt B log 2 0 0 0 0 0 0 0 gh,e,e 2 ) ) + µ h +µ 2 h 2 p h h, h 2 )p E E, E 2 )dh dh 2 de de 2 e θt B log 2 +µ h ) p h h, h 2 )p E E, E 2 )dh dh 2 de de 2 ). 8)

C E,2 θ) = θt B log + gh,e,e 2 ) 0 0 0 0 e θt B log 2 0 0 0 gh,e,e 2 ) e θt B log 2 +µ 2h 2 ) p h h, h 2 )p E E, E 2 )dh dh 2 de de 2 ) + µ 2 h 2 +µ h p h h, h 2 )p E E, E 2 )dh dh 2 de de 2 ), where p h h, h 2 ) and p E E, E 2 ) are the joint distribution of the fading gains and harvested energies, respectively. Implicitly, gh, E, E 2 ) should be greater than zero in 8) and 9). If this condition is not satisfied, we should have the partition function h = fh 2, E, E 2 ) instead. Then, we have the following result. Theorem : The optimal joint energy and channel aware decoding strategy that maximizes the weighted sum rate for a specific QoS constraint θ in the two-user case is characterized by ) + E h N 0 λ T B gh, E, E 2 ) =, if λ 20) fh 2, E, E 2 ) = where λ > 0 is some constant. Proof: See Appendix B for details. E 2 N 0 T B + E 2h 2 N 0 T B E N 0 T B 9) ) λ, if 0 λ < 2) Remark 2: Note that the optimal decoding strategy derived in this paper depends on both the channel states and the harvest energy. Through numerical evaluations, we find that the policy that maximizes the throughput is opposite to the opportunistic scheduling. When one user s channel or harvested energy is stronger, it can be decoded first to offer a more fair treatment of users. In this way, the variance of the channel rate of the users can be smaller leading to a higher achievable throughput, which is in consistent with the findings in [9]. When the users have the same number of QoS constraints, we also propose a suboptimal decoding order for an arbitrary number of users given by κ π) h π) κ π2) h π2) κ πm) h πm), 22) where κ is the weight of the weighted sum with κ j 0, M j= κ j =.

2 0.35 0.3 0.25 User 2 0.2 0.5 0. Variable decoding w/ optimal order Variable decoding w/ suboptimal order TDMA Time sharing 0.05 0 0 0.05 0. 0.5 0.2 0.25 0.3 0.35 User Fig. 2: Effective throughput region. θ = θ 2 = 0.0. Ē T = Ē2 T = 0 db. D. Numerical Results In this part, we perform numerical analysis for independent poisson energy arrivals and Rayleigh fading with E{h} =. We assume T = 2 ms, B = 00 KHz, and N 0 = 0 5. In Fig. 2, we plot the effective throughput regions of the transmission strategies. In the figure, we assume θ = θ 2 = 0.0 and Ē T = Ē2 T = 0 db. In the figure, Time sharing represents the superposition coding with fixed decoding, and Variable decoding is the superposition coding with the optimal decoding strategy characterized in Theorem and suboptimal decoding strategy 22). Obviously, the superposition coding with the joint energy and channel aware decoding strategy achieves the largest region. It is interesting that the suboptimal decoding strategy can achieve performance very close to the optimal one. Ē T In Fig. 3, we assume θ = θ 2 = 0.0 and plot the sum rate as the average arrival energy = Ē2 T varies. The sum rate increases as the average arrival energy increases for all transmission strategies. Again, we can find that superposition coding with the proposed energy and channel aware decoding achieves the best performance. It is interesting that TDMA can achieve better performance than superposition coding with fixed decoding when the mean arrival energy is larger. In Fig. 4, we plot the sum rate as θ = θ 2 = θ varies. Now, the curves of different

3 Sum ratebps/hz) 2.8.6.4.2 0.8 0.6 0.4 0.2 Variable decoding w/ optimal order Variable decoding w/ suboptimal order TDMA Time sharing 0 0.5.5 2 2.5 3 3.5 4 4.5 5 Average arrival energy J/s) Fig. 3: Sum rate as a function of average arrival energy. θ = θ 2 = 0.0. strategies converge as θ increases, which implies that orthogonal transmission strategies can be efficient in terms of sum rate under strict queueing constraints. In addition, we can find that TDMA can achieve better performance than the superposition coding with fixed coding order as θ increases, i.e., more stringent QoS constraints. IV. OPTIMAL OFFLINE POWER CONTROL In this section, we first revisit the point-to-point links and derive the achievable effective capacity associated with the optimal offline power control policy. Then, we consider the twouser MAC case and assume that the CSI and energy arrivals of all time slots are available at the transmitters, and identify the conditions that the offline optimal power control schemes should satisfy for superposition coding with variable decoding with a given decoding order strategy, from which we design an algorithm to determine the optimal power allocated to each time slot with any large number of time slots. A. Point-to-Point Links Note that in absence of delay constraints, the offline optimal power control policies for a point-to-point link have been characterized for any finite number of time frames in [], [2].

4 4 3.5 3 Variable decoding w/ optimal order Variable decoding w/ suboptimal order TDMA Time sharing Sum ratebps/hz) 2.5 2.5 0.5 0 0 3 0 2 0 0 0 θ Fig. 4: Sum rate as a function of QoS exponent θ. Ē T = Ē2 T = 0 db. Specifically, the optimization problem can be written as where µ[t] = e[t] N 0 T B max µ[t] 0, t s.t. B log 2 + µ[t]h[t]) 23) t= t e[k] k= t E[k], t, 24) k= denotes the instantaneous transmitter SNR level with the consumed energy e[t] in the t-th frame. The optimal power control policy is given in the form of µ opt [t] =, h[t] α[t] α[t] h[t] 0, else. where α[t] = log 2 B 25) N k=t λ[k] with λ[k], k =,..., N representing the Lagrange multipliers associated with energy harvesting constraints 24). Without the energy harvesting constraints, the optimal power control policy is the well-known water-filling policy [ µ[t] = ] + 26) α 0 h[t]

5 with constant water level α 0 satisfying the average power constraints E{µ} = lim N N N t= µ[t] Ē N 0 T B with equality. In this work, we consider the case of infinite transmission frames, i.e., N. Then, we immediately have the following result. Theorem 2: As the number of frames N goes to infinity, lim α[n] = α 0. 27) N That is, the optimal power control policy is converging to the water-filling scheme without energy harvesting constraints for infinite frames. Proof: See Appendix C for details. Remark 3: This is in general due to the fact that the energy buffer is at the boundary of an absorbing and a non-absorbing queue and the event that 25) cannot take the value of 26) has almost zero probability [5]. Then, we can also show that the average rate of the point-to-point energy harvesting system with the optimal offline power control policy is the same as that of the equivalent system with average power constraints only for infinite number of frames. Note that for power control policies other than the naive scheme, the available energy for each frame b[t] can be viewed as a first-order stationary Markov process over time t [2]. In this case, obtaining the exact expression for the effective capacity seems intractable. Fortunately, inspired by the above characterizations of the optimal power control policy in absence of QoS constraints, we can show the following result. Proposition 2: In a point-to-point link with IID harvested energy process and channel fading process described in Section II-A, the effective capacity is given by C E θ) = θt B log E { e θt B log +µ[t]h[t])} 2 28) = log lim + µ[t]h[t]) β θt B N N 29) t= with the associated optimal power control policy β, h[t] α[t], µ opt [t] = α[t] β+ h[t] h[t] β+ 0, else, 30)

6 where α[t] = N β k=t λ[k] with λ[k], k =,..., N representing the Lagrange multipliers associated with constraints 24). Proof: See Appendix D for details. Remark 4: With the above characterization, it is possible for us to approach the effective capacity in energy harvesting systems in the form of 29) given by C E θ) = θt B log N + µ[t]h[t]) β, 3) t= for any large but finite N number of time slots, which has been adopted in [5] to characterize the performance of point-to-point links. Note that the above expression holds only for the offline optimal power control policy. In case of other power control policies, the effective capacity expression is still difficult to obtain [7]. ) Suboptimal Online Power Control Policy: We also propose a suboptimal online power control policy given by b[t] µ[t] = min N 0 T B, α β+ 0 h[t] β β+ + h[t], 32) where b[t] is the available energy {[ in the current ] time slot, and α 0 is the threshold to satisfy the + } average power constraint E β β+ α0 h[t] β+ h[t] = Ē N 0. Due to Remark 3, we can show T B that the proposed online power control policy can achieve almost the same performance as the offline optimal one. B. Two-User MAC In this section, we study the optimal power allocation policy for the two-user MAC employing superposition coding with variable decoding order with respect to the channel fading states described in Section II-B3. Denote H = H, H 2 ) as a particular partition of the space of the positive values of h = h, h 2 ). In particular, when h[t] H, the decoding order at the receiver is given by, 2), i.e., signal from user is decoded first in the presence of interference from

7 the signal from user 2. We know R [t] = B log 2 + µ ) [t]h [t], 33) + µ 2 [t]h 2 [t] R 2 [t] = B log 2 + µ 2 [t]h 2 [t]). 34) On the other hand, when h[t] H 2, the decoding order at the receiver is 2, ), and R [t] = B log 2 + µ [t]h [t]), 35) R 2 [t] = B log 2 + µ ) 2[t]h 2 [t]. 36) + µ [t]h [t] Given the power control policies µ = µ [t], µ 2 [t]) and H, the associated effective capacity of each user can be expressed as C µ, H) = θ T B log N C 2 µ, H) = θ 2 T B log N t= + µ ) β [t]h [t] {h[t] H } + µ 2 [t]h 2 [t] ), 37) + + µ [t]h [t]) β {h[t] H 2 } + t= + µ 2 [t]h 2 [t]) β 2 {h[t] H } + µ 2[t]h 2 [t] + µ [t]h [t] ) β2 {h[t] H 2 }). 38) The optimal power control policy can be obtained by solving the convex optimization problem max µ s.t. κc µ, H) + κ)c 2 µ, H) 39) t µ [k] k= t µ 2 [k] k= t k= t k= E [k], t =,..., N, 40) N 0 T B E 2 [k], t =,..., N, 4) N 0 T B where κ 0, ). The following analysis is conducted from given channel state partition H, and the notation C j µ, H) is replaced by C j µ) for brevity. We identify the conditions that the optimal power control policies should satisfy.

8 Let us express the Lagrangian of the convex optimization problem as J = κc µ, H) + κ)c 2 µ, H) t ) t E [k] λ [t] µ [k] N N t= 0 T B k= k= t ) t E 2 [k] λ 2 [t] µ 2 [k], 42) N N 0 T B t= k= where κ 0, ), and {λ [t]} N t=, {λ 2 [t]} N t= are the Lagrangian multipliers associated with the energy harvesting constraints 40) and 4), respectively. Define ϕ = N ϕ 2 = N t= t= + µ ) β [t]h [t] {h[t] H } + + µ [t]h[t]) β {h[t] H 2 }), + µ 2 [t]h 2 [t] 43) + µ 2 [t]h 2 [t]) β 2 {h[t] H } + + µ ) β2 2[t]h 2 [t] {h[t] H 2 }). + µ [t]h [t] Taking the derivative of J with respect to µ and µ 2 in H and H 2, respectively, we obtain the optimality conditions as follows: ) k= 44) κ + µ ) β [t]h [t] h [t] ϕ log 2 + µ 2 [t]h 2 [t] + µ 2 [t]h 2 [t] λ [k] = 0, h[t] H 45) k=t κ 2) + µ ) β [t]h [t] µ [t]h [t]h 2 [t] ϕ log 2 + µ 2 [t]h 2 [t] + µ 2 [t]h 2 [t]) 2 + κ h 2 [t] ϕ 2 log 2 + µ 2 [t]h 2 [t]) λ β+ 2 [k] = 0, h[t] H 2 46) k=t

3) κ + µ ) β 2[t]h 2 [t] µ 2 [t]h 2 [t]h [t] ϕ 2 log 2 + µ [t]h [t] + µ [t]h [t]) 2 κ h [t] λ [k] = 0, h[t] H 47) + ϕ log 2 + µ [t]h [t]) β+ κ 4) + µ 2[t]h 2 [t] ϕ 2 log 2 + µ [t]h [t] k=t ) β h 2 [t] + µ [t]h [t] λ 2 [k] = 0, h[t] H 2, 48) k=t where 45) and 46) are obtained by differentiating the Lagrangian with respect to µ [t] and µ 2 [t], respectively, over h H. Similarly, 47) and 48) are obtained by differentiating with respect to µ and µ 2, respectively, over h H 2. Due to the convexity, whenever µ j, j =, 2 is negative valued, we set µ j = 0, j =, 2 [2]. N k=t Let us first define ζ [t] = λ [k]ϕ log 2, ζ κ 2 [t] = ζ 2 [t] = N k=t λ [k]ϕ 2 log 2 κ N k=t λ 2[k]ϕ 2 log 2 κ, ζ 2 [t] = N k=t λ 2[k]ϕ log 2 κ 9, and, where {λ [t]} N t=, {λ 2 [t]} N t= are the Lagrange multipliers whose values are chosen to satisfy the energy harvesting constraints constraints 40) and 4) with equality, and ϕ and ϕ 2 are defined in 43) and 44). Now, consider 47) and 48). The channel state h[t] lies in H. Through a simple computation using 48), we can derive µ 2 [t] = + µ [t]h [t]) β2 β 2 + β α 2 [t] 2 + h 2 [t] β 2 β 2 + + µ [t]h [t] h 2 [t] 49) which tells us that µ 2 [t] = 0 if h 2 [t] + µ [t]h [t] < α 2[t]. 50) If µ 2 [t] = 0, we have from 47) that which gives us that κ ϕ log e 2 + µ [t]h [t]) β h [t] λ [k] = 0 5) k=t µ [t] = β α [t] + h [t] β β + h [t] 52) which implies that µ [t] = 0 if h [t] < α [t]. 53)

20 Now, if we substitute 49) into 47), we obtain the following additional condition for having µ [t] = 0: the equation h [t] α [t] + µ [t]h [t]) β +) h [t]α 2 [t] h 2 [t]α 2 [t] h 2 [t] α 2 [t] + µ [t]h [t]) ) β 2 + ) = 0 54) has a solution that returns a negative or zero value for µ [t]. The above discussion enables us to characterize the regions in which one user transmits while the other one is silent. We also have a closed-form formula in 52) for the optimal power adaptation policy when only one user transmits, which is the optimal power control policy derived in Proposition 2 for a point-to-point link. When both users transmit, the power control policies µ [t], µ 2 [t]) are given directly by the non-negative solution of 47) and 48). Similar analysis can be conducted to obtain µ [t], µ 2 [t]) when h[t] H. With the above characterizations, we can derive an algorithm to determine the optimal power allocated for each time slot t with given parameters as shown in Table I. Note that now the threshold values may vary for different time slots. In the algorithm, the optimal parameters {λ [t]} N t=, {λ 2 [t]} N t= with given ϕ and ϕ 2 can be determined through [2, Algorithm 2], while the power values allocated to the users in different time slots are determined by the algorithm in Table I instead. The next step is to find the optimal ϕ and ϕ 2. We propose an algorithm in Table II to find the optimal ϕ and ϕ 2. Note that we above have not specified how the values of ϕ and ϕ 2 are updated for each iteration in order to keep the algorithm generic. In our numerical computations, we have updated {λ [t]} N t= and {λ 2 [t]} N t= using the bisection search algorithm. The values of ϕ and ϕ 2 are updated in Step 7 of the algorithm by assigning them the values evaluated in Step 4. Hence, the most recent values are carried over to the new iteration. by ) Online Power Control: We also consider an online power control algorithm that is given { } bj [t] µ j [t] = min N 0 T B, µ j,0[t], 55) where µ j,0 [t] is the allocated power with channel states h[t] under the average power constraints only according to [4, Section V].

2 TABLE I: The Power Allocation Algorithm. Given κ 0, ), the partition H, ϕ, ϕ 2, {λ [t]} N t=, {λ 2 [t]} N t=; 2 Determine ζ [t] = N k=t λ [k]ϕ 2 log 2 κ ; N k=t λ[k]ϕ log 2 κ, ζ 2 [t] = ζ 2 [t] = 3 for t = : N 4 do if h[t] H 2 5 then if h 2 [t] > ζ 2 [t] 6 then µ 2 [t] = N k=t β 2 ζ 2 [t] β 2 + h 2 [t] β 2 + λ2[k]ϕ2 log 2 κ, ζ 2 [t] = h 2[t] ; 7 if h[t] ζ [t] + µ [t]h [t]) β+) ) h[t]ζ2[t] ) h 2[t] β 2 + h 2 [t]ζ 2 [t] ζ 2 [t]+µ [t]h [t]) = 0 returns nonpositive µ [t] 8 then µ [t] = 0; 9 else if h 2[t] ζ 2 [t] < h [t] ζ [t] 0 then µ 2 [t] = 0, µ [t] = [ β ζ [t] β + h [t] β + ) β + h [t]] + ; else Compute µ [t], µ 2 [t] from 47) and 48); [ 2 else µ 2 [t] = 0, µ [t] = 3 if h[t] H 4 then if h [t] > ζ [t] 5 then µ [t] = β ζ [t] β + h [t] β + β ζ [t] β + h [t] β + h [t] ; 6 if h2[t] ζ 2 [t] + µ 2[t]h 2 [t]) β 2+) h2[t]ζ[t] h [t] h [t]ζ 2[t] ζ [t]+µ 2[t]h 2[t]) returns nonpositive µ 2 [t] 7 then µ 2 [t] = 0; ) 8 else if h β ζ < h2 2 + ζ 2 9 then µ [t] = 0, [ µ 2 [t] = β 2 ζ 2 [t] β 2 + h 2 [t] β 2 + ) β + h 2 [t]] + ; h [t]] + ; ) = 0 20 else Compute µ [t], µ 2 [t] from 45) and 46); [ 2 else µ [t] = 0, µ 2 [t] = β 2 ζ 2[t] β 2 + h 2[t] β 2 + h 2 [t]] + ; N k=t λ2[k]ϕ log 2 κ,

22 TABLE II: Finding Optimal ϕ and ϕ 2. Given κ 0, ), the partition H; 2 Initialize ϕ and ϕ 2 ; 3 Incorporate the algorithm in Table I into [2, Algorithm 2] to determine the power allocated to the users {µ [t]} N t=, {µ 2 [t]} N t= with given ϕ and ϕ 2 ; 4 Evaluate ϕ and ϕ 2 according to 43) and 44); 5 Check if the new values of ϕ and ϕ 2 agree up to a certain margin) with those used in Step 3; 6 if do not agree 7 then update the values of ϕ and ϕ 2, and return to Step 3; 8 else declare the power allocation policies {µ [t]} N t= and {µ 2 [t]} N t=, and ϕ and ϕ 2 as the ones obtained with the given parameters..8 Effective capacity bps/hz).6.4.2 0.8 Average power constraints only Offline optimal Proposed online 0.6 0.4 0.5.5 2 2.5 3 3.5 4 4.5 5 Average arrival energy J/s) Fig. 5: Effective capacity as average arrival energy varies. C. Numerical Results We first consider the point-to-point link to validate our analysis. We assume N = 0 5 in the following results. We assume θ = 0.0. In Fig. 5, we plot the effective capacity as the average arrival energy varies. The curve of average power constraints only is computed numerically, while the other curves are obtained through Monte Carlo simulations over N frames of IID poisson energy arrivals and Rayleigh fading channels with unit mean. We can see from the

23 0 0 Simulation Theoretical PrQ>q max ) 0 0 2 0 0.5.5 2 2.5 3 3.5 4 q max bps/hz) Fig. 6: The queue overflow probability. figure that both the offline optimal power control policy and the proposed online power control policy almost achieve the same performance as the one achieved with average power constraints 2. We also would like to validate that the approximation of the effective capacity under the offline optimal power control policy is correct. In Fig. 6, we let the constant arrival rate to the source to be C E θ) = 0.626 bps/hz with Ē T = 0 db and θ = 0.0, and the source employs the offline optimal power control policy in each time slot. We plot the buffer overflow probability Pr{Q > q max } of the queue at the transmitter as the queue length threshold q max varies. We are interested in the decay exponent of the buffer overflow probabilities. Note that according to 8), the logarithm of the buffer overflow probability is linear in q max with the slope given by θ. Computing the derivatives of the logarithms of the simulated curves, we obtain θ from the simulation results, which are very close to the theoretical values, verifying the theoretical analysis. Then, we consider the two-user MAC case. We assume Ē/T = Ē2/T = 0 db. Due to the 2 Note that the performance gap in larger average arrival energy is due to the fact that N = 0 5 for Monte Carlo simulations. When N is larger, this gap diminishes. However, the computation complexity for finding the offline optimal power control policy increases dramatically, and becomes unaffordable. Therefore, we only consider N = 0 5 in the numerical results.

24 2.6 2.4 2.2 Sum throughput bps/hz) 2.8.6.4.2 Average power constraints only Offline optimal Online suboptimal 0.8 0.5.5 2 2.5 3 3.5 4 4.5 5 Average arrival energy J/s) Fig. 7: Sum rate as average arrival energy varies. symmetric settings, we consider the decoding strategy given by H = {h : h > h 2 }. In Fig. 7, we plot the sum rate as the average arrival energy varies. Again, we can see from the figure that both the offline optimal power control policy and the proposed suboptimal online power control policy can achieve almost the same performance as the one achieved with the optimal power control policy in [4] without energy harvesting constraints. This tells us that the proposed online power control policy is sufficient enough to achieve performance close to the optimal one in large number of time slots. V. CONCLUSION In this paper, we have investigated the performance of fading MAC with energy harvesting transmitters under statistical QoS constraints. We have employed the effective capacity as a performance metric of the throughput. We have considered different transmission strategies for the case of causal energy and CSI information at the transmitters and the receiver. With the naive power control policy, we have derived the effective capacity for TDMA and superposition coding with fixed/variable decoding order. Regarding the two-user MAC case with the same QoS constraints, we have derived the optimal energy and channel aware decoding strategy for superposition coding. We have noted that when the arrival energy or the channel state is stronger

25 for one user, the optimal strategy is to decode this user first and subsequently decode the other user such that the other user does not experience any interference. We have also found that TDMA can achieve better performance in comparison with superposition coding with fixed decoding order in larger mean arrival energy or more stringent QoS constraints. Regarding the case of noncausal energy and CSI information at the transmitters and the receiver, we have revisited the point-to-point links. We have shown that the offline optimal power control policy can achieve the same performance as the traditional water-filling scheme in infinite number of time slots. Afterwards, we have derived the expression for the effective capacity with the associated offline optimal power control policy. We have also proposed a suboptimal online power control policy based on causal energy and channel information and shown that this policy can achieve performance close to the optimal one in case of large number of time slots. Then, for the two-user MAC employing superposition coding, we have identified the offline optimal power control control policies with given decoding order. We have designed an algorithm to determine the values of the power allocated to the users in each time slot. A. Proof of Proposition According to 9), we have APPENDIX { C E θ) = lim t θt log E e θ } t i= r s[i] = lim n = lim n = lim n = lim n θnt log E {E[i]} n i=,{h[i]}n i= θnt log E {E[i]} n i=,{h[i]}n i= n/m θnt log i= {e θt n i= R[i]} {e θt n/m m R[im+j]} i= j= 56) E {E[im]},{h[i )m+j]} m j= { e θt m j= R[i )m+j]} 57) { θnt log E E,{h[j]} m j= e θt m R[j]}) n/m j= = θmt log E E,{h[j]} m j= = θmt log E E,h { e θt R[j]}) m = θt log E E,h 58) {e θt m j= R[j]} 59) 60) { e θt R } 6) where E denotes the expectation taken over the random variables { }, E and h are the random variables for the harvested energy and channel state, respectively, and we have divided the frames

26 into intervals of mt such that in each interval the harvest energy stays constant and varies independently in 56) to obtain 57), IID assumptions of the harvested energy and channel states are incorporated into 57) to derive 58) in which we take the expectation over the harvested energy in the intervals and the resulted expected values are the same for the intervals, and 60) is due to the IID channel fading processes. B. Proof of Theorem Assume that the optimal partition function is h 2 = gh, E, E 2 ). We define J ĝh, E, E 2 )) = κc θ, ĝh, E, E 2 )) + κ)c 2 θ, ĝh, E, E 2 )), 62) where κ 0, ) and ĝh, E, E 2 ) = gh, E, E 2 ) + sηh, E, E 2 ). s is any constant, and ηh, E, E 2 ) represents arbitrary perturbation. A necessary condition that needs to be satisfied is [22] We define ϕ = + ϕ 2 = + gh,e,e 2 ) ) e θt B log 2 + µ h +µ 2 h 2 0 0 0 0 0 0 0 gh,e,e 2 ) gh,e,e 2 ) 0 0 0 0 e θt B log 2 + µ 2 h 2 0 0 0 gh,e,e 2 ) By noting that dĝh,e,e 2 ) ds 0 0 0 κ + ϕ log 2 d ds J ĝh, E, E 2 ))) = 0. 63) s=0 p h h, h 2 )p E E, E 2 )dh 2 dh de de 2 e θt B log 2 +µ h ) p h h, h 2 )p E E, E 2 )dh 2 dh de de 2, 64) e θt B log 2 +µ 2h 2 ) p h h, h 2 )p E E, E 2 )dh 2 dh de de 2 +µ h )p h h, h 2 )p E E, E 2 )dh 2 dh de de 2. 65) = ηh, E, E 2 ), and from 63) 65), we can derive ) β ) µ h + µ h ) β + µ 2 gh, E, E 2 ) κ + µ 2 gh, E, E 2 )) β ϕ 2 log 2 + µ ) β ) ) 2gh, E, E 2 ) p h h, gh, E, E 2 ))p E E, E 2 )ηh, E, E 2 )dh de de 2 = 0, 66) + µ h

27 where β = θt B is the normalized QoS exponent. Since the above equation holds for any log 2 ηh, E, E 2 ), it follows that ) ) β κ µ h + + µ h ) β ϕ log 2 + µ 2 gh, E, E 2 ) κ ϕ 2 log 2 which after rearranging yields Defining λ = κ)ϕ κϕ 2 + + µ 2 gh, E, E 2 )) β ) β µ h +µ 2 gh,e,e 2 ) + µ h ) β + µ2gh,e,e2) +µ h ) β + µ2 gh, E, E 2 )) β = ) β equation after simple computations + µ ) β ) 2gh, E, E 2 ) = 0 67) + µ h κ)ϕ κϕ 2. 68) > 0 and noting that µ j = E j N 0, j =, 2, we can obtain the following T B + E N 0 T B h + E 2 N 0 T B gh, E, E 2 ) ) = λ, 69) which leads to 20) after rearrangement. Note here that if λ <, gh, E, E 2 ) < 0 for h < /λ E. We need to find the optimal partition function given by h = fh 2 ) instead. Following a N 0 T B similar approach as shown in 8) through 69) yields 2). C. Proof of Theorem 2 Note that it has been shown in [2, Theorem 3] that the optimal power allocation policy for point-to-point links is staircase water-filling, where the threshold α[t] is non-increasing. First, we can show that α[n] 0 is finite as N. Otherwise, µ[t] = 0 for all t. Then, we have N N t= µ[t] = 0 violating 24) at t = N. Then, we know that {α[t]}n t= are lowered bounded. So there must be a limit for α[t]. Denote ξ = lim N α[n]. We must have ξ = α 0. Otherwise, the average power constraint 24) at t = N cannot be satisfied. We can show this by contradiction. Assume that ξ < α 0. Then, given ϵ = α 0 ξ 2 > 0, there exists N ξ such that for all t N ξ, α[t] ξ < ϵ, 70)

28 or equivalently, α[t] ξ < ϵ α[t] < ξ + ϵ = ξ + α 0 2 < α 0. 7) Then, we have where N N t= [ N µ[t] = N t= α 0 h[t] = N > N = N N ξ t= N ξ t= N ξ t= N ξ t= ] + = Ē N 0 T B µ[t] + µ[t] 72) N t=n ξ [ α[t] ] + + h[t] N [ α[t] ] + + h[t] N t=n ξ t=n ξ [ α[t] ] + [ ] ) + + h[t] α 0 h[t] [ α[t] ] + 73) h[t] [ ] + 74) α 0 h[t] Ē N 0 T B, 75) is incorporated for 75). Note that as N, the first term in 75) approaches zero since N ξ is finite. Then, we have N N k= µ[k] > Ē N 0. That is, T B 24) cannot be satisfied at t = N. Similarly, we can show the contradiction if ξ > α 0. Therefore, we must have 27). D. Proof of Proposition 2 The idea of the proof is similar to the proof of Theorem. First, we can solve the following optimization problem min µ[t] 0, t t= s.t. + µ[t]h[t]) β, 76) t e[k] k= k= t E[k], t, 77) and obtain the optimal µ[t] given in 30). Then, similar to [2, Theorem 3], we can show that α[t] is a staircase like function, where it stays constant for certain time intervals and changes when all available energy has been consumed up. Denote t, t 2,..., t m as the time slot when

29 α[t] changes and let t m+ = N +. Since the harvested energies are IID, we can show that { C E θ) = lim t θt log E e θ } t i= rs[i] = lim N θnt log E {E[i]} N i=,{h[i]}n i= = lim N θnt log E {E[i]} N i=,{h[i]}n i= = lim N = lim N m θnt log i= m θnt log i= E t i+ j=t i t i+ j=t i { { e θt B } N i= log 2 +µ[i]h[i]) e θt B m t i+ i= j=t log 2 +µ[j]h[j]) i E[j],{h j } t i+ j=t i E t i+ j=t i { e } t i+ θt B j=t log 2 +µ[j]h[j]) i } 78) 79) E j,h j { e θt B log 2 +µ[j]h[j]) }, 80) where we have divided the frames into intervals specified by {t i } such that the threshold α[j] stays constant for µ[j] of each interval [t i, t i+) in 78). IID assumptions of the harvested energy and channel states are incorporated into 78) to derive 79) such that the consumed energy for each interval t i+ j=t E j is independent of each other. 80) is due to the IID channel fading i processes within each interval such that the allocated power in each frame 30) only varies with the instantaneous channel state and hence is independent of each other. Now, similar to Theorem 2, we can show that α[n] in 30) approaches to some finite value determined by the average power constraints only, and hence the effective capacity 80) can be reduced to the form 29) as the number of frames N approaches to infinity, which is the value achieved by the power control policies with average power constraints only. Above, we have shown the achievability, i.e., the rate 29) with the associated power control policy 30) can be achieved. Note also that 28) with the optimal power control policy satisfying the average power constraints only is indeed the maximum value of the effective capacity achievable in the energy harvesting system, since it only considers the average energy constraints, i.e., t = N in 77). Therefore, the effective capacity of the point-to-point energy harvesting system is given by 29) with the optimal power control policy given by 30). REFERENCES [] O. Ozel, K. Tutuncuoglu, J. Yang, S. Ulukus, A. Yener, Transmission with energy harvesting nodes in fading wireless channels: optimal policies, IEEE J. Sel. Areas Commun., vol. 29, no. 8, pp. 732-743, Sep. 20.

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