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Long-term scheduling and power control for wirelessly powered cell-free IoT

We investigate the long-term scheduling and power control scheme for a wirelessly powered cell-free Internet of Things (IoT) network which consists of distributed access points (APs) and large number of sensors. In each time slot, a subset of sensors are scheduled for uplink data transmission or downlink power transfer. Through asymptotic analysis, we obtain closedform expressions for the harvested energy and the achievable rates that are independent of random pilots. Then, using these expressions, we formulate a long-term scheduling and power control problem to maximize the minimum time average achievable rate among all sensors, while maintaining the battery state of each sensor higher than a predefined minimum level. Using Lyapunov optimization, the transmission mode, the active sensor set, and the power control coefficients for each time slot are jointly determined. Finally, simulation results validate the accuracy of our derived closed-form expressions and reveal that the minimum time average achievable rate is boosted significantly by the proposed scheme compare with the simple greedy transmission scheme.

the efficiency of WPT [12], the performance of cell-boundary users is still poor.
Recently, cell-free massive MIMO with distributed and cooperative APs has been proposed to improve both the spectral and energy efficiencies. Compared with cellular massive MIMO, the heavy path-loss of the cell boundary terminals can be avoided since the distances between the terminals and served APs are smaller. With the help of cell-free massive MIMO with a user-centric architecture [13], [14], Wang et al. proposed a wirelessly powered cell-free IoT scheme with jointly optimized downlink and uplink power control coefficients [15]. Compared with WPT in small cells and collocated massive MIMOs, the WPT efficiency of the cell-free IoT has been improved significantly.
In cell-free massive MIMO systems, random pilots are wildly adopted since it is impossible to allocate finite orthogonal pilots to massive amount of sensors [15], [16]. Intuitively, the non-orthogonality of random pilots will reduce the accuracy of channel estimation. Then, the degradation of channel estimation can further decrease the efficiency of WPT and data transmission. Therefore, user scheduling is critical to satisfy the massive connections in wirelessly powered cellfree IoT. The traditional scheduling methods usually focus on improving the instantaneous network performance [17], [18], which may cause some sensors with bad channel conditions never scheduled. Therefore, it is more appropriate to design a scheduling strategy to improve the long-term system performance. Lyapunov optimization method is an effective way to improve the infinite horizon average objective function without predicting the future state [19], [20], which has been used to design scheduling strategies in some works [20], [21]. In particular, with the help of Lyapunov optimization, Zhai et al. designed an energy-efficient user scheduling for NOMA-based IoT networks to minimize the time average power consumption while satisfying the time average rate requirements for all devices [21]. Through optimizing energy beamforming, Choi et al. minimized the time average power consumption of the AP while meeting the given time average rate requirements for all nodes in a WPCN [22]. It is challenging to jointly design the long-term scheduling and power control scheme for a wirelessly powered network, since we need to strike a balance between the WPT and data transmission and to guarantee the fairness between users.
Contributions: Our contributions in this work are three-fold: • We provide an asymptotic analysis to obtain closed-form expressions for the harvested energy and the achievable rates, that are independent of random pilot sequences.
• To serve massive amount of sensors, a long-term scheduling and power control optimization problem is formulated to maximize the minimum time average achievable rate, while maintaining the battery state of each sensor higher than a predefined level. By solving the problem, the transmission mode (energy harvesting or data transmission), the active sensor set, the power control coefficients for each time slot are jointly determined.
• Under the Lyapunov optimization framework, the longterm problem is transformed into a sequence of optimization problems of minimizing the Lyapunov drift plus penalty for each time slot, which can be solved efficiently using our proposed low-complexity methods.
Simulation results reveal that our scheduling and power control scheme can boost the minimum time average achievable rate significantly. The remainder of this paper is organized as follows. In Section II we describe system model and formulate the problem. The asymptotic analysis is provided in Section III. In Section IV, we transform the long-term scheduling and power control problem into Lyapunov optimization. The algorithm for solving the uplink and downlink sub-problems in each time slot are described in Section V. Simulation results are given in Section VI. Finally Section VII concludes the paper.

A. System Model
We consider a wirelessly powered cell-free IoT network [15] with L APs and a set of randomly distributed single-antenna sensors denoted as K with |K| = K. Each AP equipped with N antennas is connected to a central processing unit (CPU) via a perfect back-haul network. In each time slot t, only a subset K (t) a with |K (t) a | = K a are scheduled as active sensors, while the remaining sensors are inactive. As shown in Fig. 1, each time slot of duration ∆ seconds contains T c OFDM symbols, in which τ OFDM symbols are used for channel estimation, while the remaining T c − τ symbols are used for downlink WPT if δ (t) = 1 or uplink data transmission if δ (t) = 0, where indicates the transmission mode i.e., the set of active sensors is K The channel between the k-th sensor and the n-th antenna of the l-th AP in time slot t is where β l,k is the large-scale fading coefficient, which depends on the location and is assumed known to the APs. h (t) (l,n),k ∼ CN (0, 1) represents the small-scale fading coefficient, which remains invariant in each time slot, but varies independently from one slot to another. The channel between the l-th AP and the k-th sensor is denoted as while the channel between the n-th antenna of l-th AP and all sensors is denoted as

B. Channel Estimation Phase
At the channel estimation phase in each time slot, all K a active sensors simultaneously transmit their pilot sequences to APs. Let Ψ Ψ Ψ = [ψ ψ ψ 1 , · · · , ψ ψ ψ K ] ∈ C τ ×K , where ψ ψ ψ k ∼ CN 0 0 0, 1 τ I I I τ denotes the pilot sequence of the k-th sensor. At the t-th slot, the received pilots at the n-th antenna of the l-th AP is given by where w w w (t) (l,n) ∼ CN (0 0 0, σ 2 I I I τ ) is the additive noise, ρ p is the pilot transmit power, and Θ Θ Θ (t) = diag(θ θ θ (t) ). Using the received pilots y y y (t) (l,n) , the LMMSE channel estimateĝ g g which implies that the k-th entry ofĝ g g (l,n) iŝ where l , = diag (β l,1 , · · · , β l,K ) Θ Θ Θ (t) , and a a a 3 The mean squared value of the channel estimateĝ (t) (l,n),k is given by C. Tranmission Phase 1) Downlink WPT: If δ (t) = 1, during the remaining T c −τ symbols, the APs jointly perform energy beamforming to the active sensors based on channel estimates, i.e., the transmitted signal from the l-th AP within the ℓ-th OFDM symbol is where q (t,ℓ) j ∼ CN (0, 1/(T c − τ )) is the ℓ-th symbol to the j-th sensor, and denote the downlink power control coefficients. The transmit power of each AP is constrained by where N ρ d is the maximum transmit power of each AP. Substituting (6) and (7) into (9), we have Then, the received signal at the k-th sensor is where v (t) k ∼ CN (0, σ 2 /(T c − τ )) is the noise. The amount of harvested energy at the each active sensor during each time slot can be expressed as where α = τ /T c and ζ ∈ (0, 1) represents the energy conversion efficiency.
2) Uplink Data Transmission: If δ (t) = 0, during the remaining T c − τ symbols, K a active sensors simultaneously deliver their data to APs. The received signal at the l-th AP within the ℓ-th OFDM symbol is given by l,j s (t,ℓ) j + n n n The energy consumption of each active sensor for data transmission in each time slot is where is the battery state of the k-th sensor at the beginning of time slot t. Each AP individually performs beamforing and then sendsĝ g g H l,k r r r (t,ℓ) l to the CPU, which detects s (t,ℓ) k using matched filtering (MF) as followŝ where C 1 is the desired signal, and C 2 +C 3 +C 4 is the effective noise. Since C 1 , C 2 , C 3 , and C 4 are uncorrelated, the achievable rate of the k-th sensor is lower bounded by [13], [14] R (t) where the effective SINR Γ k is where the expectation is taken over small-scale fading.
3) Battery Model: Let b max denote the capacity of the battery. Since b (14), the battery state of the k-th sensor b From (14), in order to guarantee certain minimum data rate, we constrain the battery state of each sensor at the beginning of each time slot to be above a predefined threshold b 0 , i.e.,

D. Outline of Results
Note that the quantities γ k given in (6), (11), and (15) are functions of the random pilot sequences Ψ Ψ Ψ. In Section III, an asymptotic analysis is performed as τ → ∞ while keeping α, κ = τ /K a and E p fixed, which shows that γ l,k ,Ẽ k (t), and R k (t) become independent of Ψ Ψ Ψ in this regime, i.e., k (that are independent of Ψ Ψ Ψ ) can be used to predict the performance of the general case with large but finite length of pilot.
In Section IV, we consider a long-term scheduling and power control problem to maximize the minimum time averaged achievable rate min k∈K lim T →∞ k , while meeting all the constraints detailed in Section II.A. Specifically, given the large-scale fading coefficients {β l,k }, the overall policy in time slot t consists of the transmission mode δ (t) , sensor states θ θ θ (t) , downlink power control coefficients η η η (t) and uplink power control coefficients ξ ξ ξ (t) , i.e., and the long-term optimization problem is formulated as (10), (13), (14), (17), (18).
In general, P 1 is NP-hard due to the following reasons. Firstly, it is a max-min problem with an infinite horizon average objective function. Secondly, the constraint region is also over infinite horizon. In addition, it is a mixed integer program due to the 0-1 constraints w.r.t. the sensor states.
To overcome these difficulties, in Section IV, we first relax P 1 into a maximization problem P 2 with long-term constraints. Through establishing virtual queues, P 2 is then reformulated into a maximization problem P ′ 2 with the constraints on the rate stability of virtual queues. Using Lyapunov optimization, problem P ′ 2 is then decomposed into a sequence of optimization problems of minimizing the Lyapunov drift plus penalty for each time slot. To solve the problem for each time slot, in Section V, we propose two low-complexity optimization methods for the downlink WPT and the uplink data transmission, respectively.
a /{k}. Then, it is straightforward to obtain the following lemma according to Theorem 1 and Theorem 2 in [24].
l,k is given as a /{k} is the unique solution to the following set of fixed-point equations l,k is given as A. Asymptotic Analysis of γ (t) l,k Theorem 1: As τ → ∞, the mean squared value of channel estimate given in (6) almost surely converges to a deterministic value independent of the random pilots Ψ Ψ Ψ, i.e., Proof: See Appendix A.

B. Asymptotic Analysis of E (t) k
To analyze E (t) k in (12), we first introduce the following lemma from [15] on the LMMSE channel estimation with random pilots.
Lemma 2: For m, l ∈ {1, · · · , L} with m = l and k ∈ K m,k = 0 0 0, Lemma 2 reveals that the channel estimates between different APs and a sensor are uncorrelated. However, the channel estimates between an AP and different sensors are correlated, sinceĝ a , which can be seen from (3) and (4). Since the channel estimation errorg g g (11) can be rewritten as Since S k1 and S k2 are uncorrelated and zero-mean, we have where step (a) is obtained according to Lemma 2 and For given pilots Ψ Ψ Ψ, the SINR in (16) can be expressed as [15] where l,k , and Theorem 2: As τ → ∞, the rate in (15) almost surely converges to a deterministic value independent of the random pilots Ψ Ψ Ψ, i.e., , bits/s/Hz, (28) Proof: See Appendix B.

D. Impact of More Active Sensors
It is noted thatγ k are functions of the largescale fading coefficients of active sensors. We now examine the effect of adding more sensor to the current set K  l,k , k ∈ K (t) a , l = 1, · · · , L.
Proof: See Appendix C. Remark 1: The mean squared error of the channel estimate is Lemma 3 reveals that the accuracy of channel estimation is reduced by activating inactive sensors. The reducedγ

IV. LONG-TERM SCHEDULING AND POWER CONTROL
In P 1 , we aim to achieve the max-min fairness over infinite horizon, which is hard to tackle. Similarly as in [25], we first introduce a sequence of auxiliary variables r (t) bounded between 0 and r max , where r max = max is the achievable rate of the k-th sensor in (28) with full transmit power, perfect channel estimation and no interference. Then, we have Thus, the max-min problem P 1 is equivalent to maximizing lim T →∞ 1 T T −1 t=0 r (t) with the following two extra constraints [19], [25], Another challenge resides in constraint (18), which makes the control policy P (t) over different time slots coupled due to the dynamics of b (t) k in (17). We therefore relax (18) to the following long-term constraint, which enable us to design the policy using Lyapunov optimization. Hence, P 1 is relaxed to A. Problem Reformulation with Queue Stability Constraints P 2 is still hard to tackle due to the long-term constraints (31) and (33). Similarly as in [21], [23], we transform the long-term constraints into queue stability constraints.
Define {X k (t) : k ∈ K} as the virtual queues associated with constraint (33). In each time slot, the virtual queue X k (t) is updated according to where can be considered as the arrival rate and the departure rate of the virtual queue X k (t), respectively. We say X k (t) is rate stable if lim t→∞ X k (t) t = 0 [21], [23]. To maintain the rate stability of X k (t), the departure rate b (t+1) k must be no less than the arrival rate b 0 , which coincides with constraint (33).
Similarly, we also define {Y k (t) : k ∈ K} as the virtual queues associated with constraint (31), where Y k (t) is updated as To reveal the relation between the long-term constraints (31) and (33) in P 2 and the rate stability of {X k (t), Y k (t) : k ∈ K}, we have the following lemma.

Lemma 4:
If {X k (t), Y k (t) : k ∈ K} are rate stable with finite initial values, then the long-term constraints (31) and (33) are satisfied.
Proof: Without loss of generality, we take X k (t) for example. (34) can be rewritten as Sum over t = 0, · · · , T − 1, we have Dividing both sides of (36) by T and taking the limit, we obtain where (a) is due to 0 ≤ b (t) k ≤ b max . Hence, the rate stability of X k (t) and the finiteness of X k (0) imply lim inf 1 Based on Lemma 3, we then replace the long-term constraints (31) and (33) in P 2 by rate stability constraints, leading to the following formulation with stricter constraints.

B. Lyaopunov Drift Plus Penalty Method
To investigate the rate stability of all virtual queues {X k (t), Y k (t) : k ∈ K}, we define the Lyapunov function [19] The Lyaopunov drift is defined as Let [P (t) ] * denote the policy that minimizes the Lyaopunov drift in each time slot, i.e., k ∈ K} must be rate stable.
Proof: See Appendix D. Theorem 3 reveals that the minimizer of D(t) in each time slot is more likely to achieve the rate stability of {X k (t), Y k (t) : k ∈ K} compared with any other feasible policies. In P ′ 2 , we aim to maximize lim T →∞ 1 T T −1 t=0 r (t) while keeping all queues {X k (t), Y k (t) : k ∈ K} rate stable. To achieve this goal, we minimize the following drift plus penalty in each time slot [19], i.e., min where W is a positive weight to balance the two items in (41). However, minimizing Φ (t) is not easy to tackle due to the quadratic form of the Lyapunov drift D(t). Similarly as in [19], we will minimize the upper-boundΦ (t) given in the following theorem instead of minimizing Φ (t) directly. Theorem 4: Φ (t) is upper bounded by the following linear functionΦ From (34), we have Substituting (42) and (43) into (41), we can conclude the proof.
It is noted that minimizing the upper-boundΦ (t) is also helpful to strike a balance between the objective function and the rate stability of virtual queues. For example, if X k (t) tends to rate unstable, then we have X k (t) → ∞ and According to (39) and (42), we We can interpret the maximization ofΦ (t) as follows. The violation of the constraint b (t+1) k > b 0 leads to the growth of the backlog X k (t). When X k (t) is sufficiently large to dominateΦ (t), maximizingΦ (t) is equivalent to maximizing b (t+1) k , which tends to satisfy b (t+1) k > b 0 and further reduce X k (t+1). Similarly, a large backlog Y k (t) leads to maximizing R (t) k to reduce Y k (t + 1).
In P 3 , r (t) is independent of P (t) , hence the optimal r (t) depends only on the backlogs {Y k (t) : k ∈ K}, i.e., Next, given δ (t) , we set the value of θ θ θ (t) , or equivalently the set of active sensors K (t) a , in a greedy way. Specifically, we choose the K a active sensors with the largest X k (t) for δ (t) = 1 and choose the K a active sensors with the largest Y k (t) for δ (t) = 0. Then P 3 is decomposed into the following two subproblems. For δ (t) = 1, we aim to optimize the WPT through the downlink power control coefficients η η η (t) , i.e., (10), (17).
Denote Q ′ and Q ′′ as the optimal objective values of P ′ 3 and P ′′ 3 , respectively. Then, the transmission mode δ (t) is determined as A. Solution to P Given δ (t) = 1, the battery state is updated according to which is NP-hard due to the non-convex objective function.

C. Overall Algorithm
Finally, the proposed long-term scheduling and power control algorithm for solving P 1 in (20) is summarized in Algorithm 1.

VI. SIMULATION RESULTS
In this section, simulation results are provided to verify the accuracy of closed-form expressions and the performance of our proposed scheduling and power control approach.

A. Simulation Setup
We consider a large square hall of 50×50 m 2 with wrappedaround to avoid boundary effects. L = 100 APs are placed on the h AP = 7m high ceiling to form a uniform square array. K = 200 sensors with height h s = 1.65m are randomly distributed in this area. We model the large scale fading β l,k as β l,k = L l,k 10 σ sh z l,k 10 where 10 σ sh z l,k 10 denotes the shadow fading with σ sh = 8dB and z l,k ∼ CN (0, 1), and the path loss L l,k (dB) is given by Algorithm 1 Proposed long-term scheduling and power control algorithm.

B. Accuracy of Expressions
Through a realization with K a = 30 sensors randomly scheduled, the accuracy of the closed-form expressionsĒ (28) are verified in Fig. 2 and Fig. 3, respectively. In Fig 2, the closed-form expressionsĒ k independent of random pilots are compared with the simulation results obtained through 500 realizations of random pilot sequences with the uniform power control, i.e., η (t) l,k γ (t) l,k = 1/K a , ∀l, and k ∈ K (t) a . The figure shows that the closed-form expressions agree well with the mean of simulation results. In addition, the small variances of simulation results stemming from random pilots reveals that pilot optimization over random pilots is not necessary.
In Fig. 3, the closed-form expressionsR (t) k independent of random pilots are compared with the simulation results obtained through 500 realizations of random pilot sequences with uniform power control, i.e., ξ  as in Fig. 2, it can be seen that the difference between the closed-form expressions and the simulation results is small. To investigate the impact of enlarging the active set K (t) a , the averageĒ k obtained through 500 random schedule realizations versus K a is plotted in Fig. 4. As noted in Remark 1, the metricsĒ (t) k andR (t) k decrease as K a increases. In addition, it can be seen thatĒ (t) k is more sensitive to K a , which implies the importance of scheduling during WPT.

C. Performance Comparison
In this subsection, we evaluate the performance of our proposed long-term scheduling and power control approach. For comparison, we consider the following simple greedy scheme as a benchmark. In each time slot for transmission mode, K a sensors with the largest b (t) k are scheduled for data transmission with ξ a . The transmission mode continues until there exits some sensors whose battery   is depleted, i.e., b (t) k = 0, and then the harvesting mode is triggered. In each time slot for harvesting, the K a sensors with the lowest b (t) k are scheduled for WPT with the uniform power allocation, i.e., η k ≥ b 0 , k ∈ K, the transmission mode is triggered again. In our simulations, we consider K a = 30 and 100. Fig. 5 and Fig. 6 plot the dynamic of the minimum time average rate with K a = 30 and 100, respectively. It can be seen that the minimum time average rate becomes stable after about 1000 time slots. The shadowed error bar iŝ k }/K is the mean of time average rates over all sensors. The small shadowed error bar reflects the max-min fairness of our proposed approach. Compared with the greedy benchmark, our proposed approach can boost the minimum time average rate significantly. The improvement mainly results from two aspects: On one hand, the power consumption for data transmission is significantly reduced since (51) strikes a balance between the spectrum efficiency and the power consumption, instead of focusing only on the spectrum efficiency. On the other hand, the WPT efficiency is improved through optimizing the downlink power control coefficients and scheduling. Moreover, it is seen that a larger W leads to a higher minimum time average rate, and requires more time slots to achieve the rate stability. Comparing Fig. 5 and Fig. 6, the minimum time average rate becomes smaller as K a increased from 30 to 100 although more sensors are active in each time slot. This is because the accuracy of channel estimation is reduced by enlarging the active set K (t) a , which is noted in Remark 1. To investigate the rate stability of the virtual queues, Fig. 7 shows the sum of time average backlogs X(t) and Y (t) versus The non-increasing X(t) and Y (t) reveal that {X k (t), Y k (t) : k ∈ K} are rate stable, which implies the long-term constraints (31) and (33) of problem P 2 are satisfied. With the increase of W , the time average backlogs X(t) and Y (t) also increase. k is smaller than b 0 for some continuous time slots, the corresponding backlog X k (t) would increase continuously, and then trigger power transfer to increase b (t) k and avoid the depletion of the battery.

VII. CONCLUSIONS
In this paper, we have considered the long-term scheduling and power control in a wirelessly powered cell-free IoT network. We first derived closed-form expressions for the harvested energy and the achievable rates, and then formulated a long-term scheduling and power control problem to maximize the minimum time average achievable rate. Following the Lyapunov optimization approach, the transmission mode, the sensor state, the uplink and downlink power control coefficients are jointly determined for each time slot. Simulation results reveal that the proposed long-term scheduling and power control approach can boost the max-min time average achievable rate significantly.

A. Proof of Theorem 1
To prove Theorem 1, we first introduce the following two lemmas.
Lemma 5 ( [24]): Let A A A ∈ C τ ×τ is a Hermitian invertible matrix. Then, for any vector x x x ∈ C τ and any scalar a ∈ C such that A A A + ax x xx x x H is invertible, Lemma 6 ( [24]): Let A A A ∈ C τ ×τ , and x x x, y y y ∼ CN (0 0 0, 1 τ I I I τ ). Assume that A A A has uniformly bounded spectral norm (with respect to τ ) and that x x x and y y y are mutually independent and independent of A A A, we have x x x H A A Ay y y a.s.

B. Proof of Theorem 2
Since a a a