Minimum Length Scheduling for Full Duplex Time-Critical Wireless Powered Communication Networks

02/03/2020
by   Muhammad Shahid Iqbal, et al.
Kadir Has University
0

Radio frequency (RF) energy harvesting is key in attaining perpetual lifetime for time-critical wireless powered communication networks due to full control on energy transfer, far field region, small and low-cost circuitry. In this paper, we propose a novel minimum length scheduling problem to determine the optimal power control, time allocation and transmission schedule subject to data, energy causality and maximum transmit power constraints in a full-duplex wireless powered communication network. We first formulate the problem as a mixed integer non-linear programming problem and conjecture that the problem is NP-hard. As a solution strategy, we demonstrate that the power control and time allocation, and scheduling problems can be solved separately in the optimal solution. For the power control and time allocation problem, we derive the optimal solution by using Karush-Kuhn-Tucker conditions. For the scheduling, we introduce a penalty function allowing reformulation as a sum penalty minimization problem. Upon derivation of the optimality conditions based on the characteristics of the penalty function, we propose two polynomial-time heuristic algorithms and a reduced-complexity exact algorithm employing smart pruning techniques. Via extensive simulations, we illustrate that the proposed heuristic schemes outperform the previously proposed schemes for predetermined transmission order of users and achieve close-to-optimal solutions.

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I Introduction

Time critical wireless sensor networks have been widely used in emergency alert systems and cyber-physical systems due to many advantages, including easy installation and maintenance, low complexity and cost, and flexibility [wncs_sinem, intravehicle_sinem]. Several studies have been conducted on minimizing the schedule length given the traffic demand and limited battery lifetime of the users in these networks [wncs_ref83, wncs_ref194]. However, recent developments in energy harvesting technologies have the potential to provide perpetual energy, eliminating the need to replace batteries. Extending the battery lifetime of sensors with energy harvesting technologies have initially focused on generating energy from natural sources such as sun, vibration and pressure [natural_1, natural_2]. However, the dependency of the harvested energy on the environmental conditions so randomness in the generation of energy is a bottleneck for its usage in time-critical applications. On the other hand, energy harvesting technologies based on inductive or magnetic resonant coupling are practically infeasible for such type of networks due to their short energy transfer distance, large size and requirement of accurate calibration and alignment of coils. Considering the advantages of having full control on energy transfer, high range and small form factor, radio frequency (RF) energy harvesting is the most suitable technology [harvest_10]. The recent advances in the design of highly efficient RF energy harvesting hardware is expected to even further extend its usage [RFEH_efficientDesign1, RFEH_TFET_design, RFEH_efficientDesign2].

RF energy harvesting networks have been previously studied in the context of simultaneous wireless information and power transfer (SWIPT) and wireless powered communication network (WPCN). In SWIPT, the base station transmits energy and data simultaneously to multiple receivers in the downlink. The trade-off between wireless information transmission capacity and wireless energy transmission efficiency of a single user has been analyzed for point-to-point transmissions considering additive white gaussian noise (AWGN) channels [harvest_06], flat-fading channels [harvest_61], co-located and separated energy harvester and information decoder setup [harvest_08], and a non-linear energy harvesting model [harvest_new63]. SWIPT based multi-user systems mostly aim to maximize the weighted sum energy transfer or energy efficiency [harvest_19, harvest_21, harvest_42, harvest_60, harvest_new62] or minimize the total transmit power at the base station [harvest_59] while restricting minimum data rate provided to a subset of receivers as a quality of service (QoS) constraint. Minimum rate constraint does not provide any delay guarantee, required for time-critical wireless networks. Moreover, these studies assume the simultaneous transmission of energy and information without considering any scheduling. The scheduling of SWIPT based networks has been considered in a limited context in [harvest_01_ref146] and [harvest_01_ref169]. The time is divided into multiple slots. In each time slot, a single user is selected for information reception while energy is transferred to the remaining users. The scheduling algorithms are proposed for the selection of this single user in each time slot.

In WPCN, the wireless users harvest energy from the base station in the downlink and then transmit data to the base station in the uplink. The first protocol proposed for WPCN, called ”harvest-then-transmit”, is based on dynamic time-division multiple access (TDMA) in a half-duplex framework. Each TDMA frame is divided into non-overlapping intervals of variable durations for the wireless energy transmission from the access point to the users in the downlink and information transmission of the users in the uplink by using their harvested energy stored in a rechargeable battery [harvest_07]. The objective of the formulation is to maximize the total throughput within a total time constraint by the optimization of the time allocation for the downlink wireless power transfer of the single antenna access point and uplink information transmissions of the single antenna users. Since the objective of throughput maximization results in unfair achievable rates among different users, with the corresponding allocation substantially favoring near users with mostly better channel conditions, some of the later works have focused on alternative objective functions, such as maximization of minimum throughput [harvest_04], maximization of weighted sum rate of uplink information transmission [harvest_50, harvest_44, harvest_55], maximization of energy efficiency [harvest_51] and minimization of schedule length [harvest_sinem, harvest_elif]. Other studies, on the other hand, have included the usage of near users as relays by using some of their energy and time to relay information of the farther users [harvest_39_ref17, harvest_39, harvest_56]. The order of information transmission in the uplink does not matter due to the non-overlapping characteristic of the wireless energy and information transmission, thus, no scheduling algorithm is required in these half-duplex systems. Although these studies impose a transmit power constraint on the access point for wireless energy transmission in the form of either the assignment of a constant value [harvest_07, harvest_19, harvest_04, harvest_39_ref17, harvest_39, harvest_51, harvest_56] or constraint on its average and maximum value [harvest_50, harvest_44, harvest_55], no upper bound has been imposed on the transmit power of the users in their information transmission. Also, the initial battery level of the users are not considered in these works, except [harvest_51].

WPCNs recently started to incorporate full duplex technology with the goal of improving the amount of transferred energy by allowing the access point to simultaneously transfer wireless energy and receive information, and in some cases also enabling the concurrent reception of wireless energy and transmission of information at the users. Full duplex technology is implemented as out-of-band or in-band systems. In an in-band full duplex system, the energy and information are transmitted within the same frequency band. The main challenge in these systems is to mitigate the self-interference, where part of the transmitted signal is received by itself, thus interfering with the desired received signal. The recent advances in self-interference cancellation (SIC) techniques [harvest_41_ref26, harvest_41_ref27] and their practical implementations [harvest_30_ref19, harvest_30_ref26] placed full-duplex as one of the key transceiving techniques for 5G networks [harvest_new66]. Full-duplex WPCN systems have been mostly formulated with the goal of maximizing the sum throughput by assuming either only access point operating in full-duplex mode [harvest_30, harvest_40, harvest_50] or both access point and users operating in full-duplex mode [harvest_50, harvest_41, harvest_new68]. [harvest_30] assumes perfect self-interference cancellation, whereas [harvest_40, harvest_41, harvest_50] include residual self-interference, proportional to the transmit power of the access point. The QoS requirement is considered by placing a constraint of upper bound on the user transmission rates [harvest_40]. Only, [harvest_30] additionally considers the minimization of schedule length given the traffic demand of the links. In full-duplex systems, since the users can harvest energy during the transmission of other users, the order of transmission so scheduling of user transmissions is important. However, previous studies assume predetermined transmission order without considering the incorporation of any scheduling algorithm. Moreover, none of these studies consider any limitation on the transmit power of the users, while assuming either constant transmission power [harvest_30, harvest_40] or a maximum power constraint [harvest_50, harvest_41] for the access point. Furthermore, these studies assume that the energy required for the data transmission needs to be supplied by the wireless transfer, without considering the initial battery level of the users.

The goal of this paper is to determine the optimal time allocation, power control, and scheduling with the objective of minimizing the schedule length subject to the traffic requirement, the maximum transmit power constraint, and the energy causality constraint of the users, for a time-critical WPCN. The original contributions of this paper are listed as follows:

  • We propose a new optimization framework for an in-band full-duplex WPCN, employing the maximum transmit power and energy causality constraints and considering the initial battery levels and full duplex energy harvesting capability of the users.

  • We characterize Minimum Length Scheduling Problem () aiming at determining the optimal power control, time allocation and scheduling with the objective of minimizing the completion time of the schedule subject to data, energy causality and maximum transmit power constraints of the users. We formulate the problem mathematically as a mixed integer nonlinear programming (MINLP) problem, which is non-convex and thus generally difficult to solve for a global optimum. We further conjecture that is NP-hard based on the reduction of Single Machine Scheduling Problem () which is proven to be NP-hard [jiang2013single] to . Then, we propose a solution framework based on the decomposition of to optimal power control and time allocation, and optimal scheduling problems.

  • We formulate the power control and time allocation problem as a convex problem and derive the optimal solution in closed-form by evaluating the Karush-Kuhn-Tucker (KKT) conditions.

  • For the scheduling problem, we introduce a penalty function, defined as the difference between the actual and minimum possible transmission time of a user. This allows consideration of the schedule length minimization objective as the minimization of the sum of the penalties of the users. By exploiting the characteristics of the penalty function, we analyse optimality conditions for the optimal schedule. Based on the derived optimality conditions, we propose two polynomial-time heuristic algorithms and one exact exponential-time algorithm with significantly reduced complexity based on smart enumeration techniques.

  • We evaluate the performance of the proposed scheduling algorithms for various parameters, including different transmit power of HAP, different maximum transmit power of the users, and different network sizes, in comparison to the optimal solution and conventional schemes proposed for the minimum length scheduling of users with a predetermined transmission order. We illustrate that the proposed polynomial time heuristic algorithms perform very close to optimal while outperforming previously proposed algorithms significantly.

The rest of the paper is organized as follows. Section II describes the WPCN model and assumptions used in the paper. Section III presents the mathematical formulation of the minimum length scheduling problem, investigates its complexity and introduces our solution strategy based on the decomposition of the problem. Section IV presents the optimal power control and time allocation problem and derives its optimal solution. Section V presents the optimal scheduling problem, analyzes its optimality conditions, proposes one exact reduced complexity exponential-time algorithm and two polynomial-time heuristic algorithms. Section VI evaluates the performance of the proposed scheduling schemes. Conclusions are presented in Section VII.

Ii System Model and Assumptions

The system model and assumptions are described as follows:

Figure 1: System Model for Wireless Powered Communication Network
  1. The WPCN architecture, as depicted in Fig. 1, consists of a HAP and N users; i.e., sensors and machine type communication (MTC) devices. Both the HAP and the users are equipped with one full-duplex antenna. Full duplex antennas are used for simultaneous wireless energy transfer on the downlink from the HAP to the users and data transmission on the uplink from the users to the HAP.

  2. We consider Time Division Multiple Access (TDMA) as medium access control protocol for the uplink data transmission from the users to the HAP. The time is partitioned into scheduling frames, which are further divided into variable-length slots each allocated to a particular user. The energy transfer from the HAP to the users continues throughout the frame. Each user can use the energy it harvests from the beginning of the frame till the end of its transmission, including both its own dedicated time slot and the time slots allocated for the previously scheduled users. The energy harvested by a user after its dedicated slot can be stored in the battery for possible usage in the subsequent scheduling frames.

  3. The HAP is equipped with a stable energy supply and continuously transfers wireless energy with a constant power . Each user harvests energy from the HAP and stores it in a rechargeable battery with an initial energy at the beginning of the scheduling frame. The initial energy includes the harvested and unused energy in the previous frames.

  4. The channel gains for the downlink and uplink channels are assumed to be different. The downlink channel gain from the HAP to user is denoted by . The uplink channel gain from user to the HAP is denoted by . Both channels are assumed to be block-fading, i.e., the channel gains remain constant over the scheduling frame. This quasi-static channel assumption is commonly used in previous WPCN formulations [harvest_19, harvest_04, harvest_39_ref17, harvest_39, harvest_51, harvest_56]. We assume that the HAP has perfect channel state information; i.e., the channel gains are perfectly known at the HAP [harvest_07, harvest_04, harvest_50, harvest_44, harvest_55, harvest_51, harvest_30, harvest_40].

  5. The energy harvesting rate of user , denoted by , depends on the downlink channel gain , transmit power of HAP and antenna efficiency as .

  6. We assume user has a traffic demand bits to be transmitted over the scheduling frame.

  7. We use continuous rate transmission model, in which Shannon’s channel capacity formulation for an AWGN wireless channel is used in the calculation of the maximum achievable rate as a function of Signal-to-Interference-plus-Noise Ratio (SINR) as , where is the transmission rate of user , is the transmission power of user , is the channel bandwidth, and is defined as , in which the term is the power of self interference at the HAP and is the noise power. Although the networks are generally restricted to support discrete rates, the continuous rate assumption is conventionally used in most of the studies in the literature [harvest_new62, harvest_07, harvest_30, harvest_50].

  8. We use continuous power model in which the transmission power of a user can take any value below a maximum level imposed to avoid the interference to nearby systems.

Iii Minimum Length Scheduling Problem

In this section, we introduce the minimum length scheduling problem referred as . We first present the mathematical formulation of as an optimization problem and investigate its complexity. Then, we provide the solution strategy followed in the subsequent sections.

Iii-a Mathematical Formulation

The joint optimization of the time allocation, power control and scheduling with the objective of minimizing the schedule length given the traffic demands of the users while considering realistic transmission model for a full-duplex system is formulated as follows:

:

minimize (1a)
subject to (1b)
(1c)
(1d)
(1e)
variables (1f)

The variables of the problem are , the transmit power of user ; , the transmission time of user , and

, binary variable that takes value

if user is scheduled before user and otherwise. In addition, denotes an initial waiting time duration during which all the users only harvest energy without transmitting any information [harvest_50].

The objective of the optimization problem is to minimize the schedule length as given by Equation (1a). Equation (1b) represents the constraint on satisfying the traffic demand of the users. Equation (1c) gives the energy causality constraint: The total amount of available energy, including both the initial energy and the energy harvested until and during the transmission of a user, should be greater than or equal to the energy consumed during its transmission. Equation (1d) represents the scheduling constraint, stating that if user transmits before user , user cannot transmit before user . Equation (1e) represents the maximum transmit power constraint.

This optimization problem is a MINLP thus difficult to solve for a global optimum [opt_book].

Iii-B Complexity Analysis

We conjecture on the complexity of by illustrating the analogy to the single-machine scheduling problem studied in [jiang2013single], denoted by .

For completeness, is described as follows: Given a set of jobs, a normal processing time , a learning coefficient , an actual processing time function for each job , and a positive integer , is there a schedule with makespan , where makespan is defined as the completion time of the last scheduled job?

The actual processing time of a job scheduled in the position is defined as

(2)

where is the actual processing time of the job scheduled in the position with , where is the normal processing time of the job scheduled in the position. Note that normal processing time is the processing time of a job if it is scheduled first; i.e., .

The processing time given in Eq. (2) is based on a time and job-dependent learning model such that as the sum of actual processing time of the previously completed jobs increases, a particular job is processed in shorter time with respect to its normal processing time. We can assume that the schedule starts at time without loss of generality. Then, based on Eq. (2), the actual processing time of a job scheduled at time can be given as

(3)

Now consider . Jobs, processing times and makespan in correspond to users, transmission times and schedule length in , respectively. Moreover, the effect of learning on the processing time of a job in corresponds to the effect of energy harvesting on the transmission time of a user in . Due to energy harvesting, user can complete its transmission in less amount of time as it is scheduled later. In other words, transmission time decreases as a function of scheduling time as user harvests more energy to be able to transmit with larger transmit power. Then, for a sufficiently large value such that users cannot afford to transmit within a finite time horizon, the transmission time of user can be formulated as a function of scheduling time as

(4)

where is a time dependent function representing the dependency of the transmission time on the energy harvesting rate of user and is the transmission time of user if it is scheduled first; i.e., at . Then, the learning effect based processing time model given by Eq. (3) can be considered as a special instance of the transmission time model in Eq. (4) for ; i.e., is independent of time. Hence, considering the additional complexity introduced by the time-dependence to the problem, can be considered at least as hard as . Furthermore, is proven to be NP-complete [jiang2013single] which is an indicator of NP-completeness of the decision version of . To this end, it can be conjectured that is NP-hard.

Iii-C Solution Strategy

As the mathematical formulation and the complexity analysis presented in previous sections suggest, it is difficult to solve for a global optimum. In other words, finding a global optimum requires algorithms with exponential complexity. Such optimal algorithms, on the other hand, are intractable even for moderate problem sizes. In this paper, we present a solution framework to overcome this intractability based on the decomposition of the optimal power and time allocation and the optimal scheduling problems as described below:

  • For a given scheduling order of the users, requires determining the optimal power and time allocation of the users with minimum schedule length while considering their data, maximum transmit power and energy causality constraints. We first show that this problem is a convex optimization problem suggesting that it is polynomial-time solvable. Then, we provide the optimal solution based on the analysis of the KKT conditions.

  • Determining the optimal power and time allocation for a given scheduling order reduces to determining the optimal scheduling order. We first introduce penalty function characterized as the difference between the actual and minimum possible transmission time of a user and demonstrate the equivalence between schedule length minimization objective and the minimization of the sum of the penalties of the users. Then, based on the optimality conditions derived using the penalty function, we propose two polynomial-time heuristic algorithms that perform very close to optimal. Furthermore, we propose an exact exponential-time algorithm with reduced complexity based on smart pruning techniques.

Iv Optimal Power Control

In this section, we are interested in determining the optimal power control and time allocation to minimize the schedule length for a given transmission order of a set of users; i.e. ’s are given in .

We first illustrate that inclusion of , the initial waiting time, is not actually needed.

Lemma 1.

In the optimal solution of , .

Proof.

Suppose that is the optimal energy harvesting time, and are the sets of optimal transmission times and transmit powers, respectively. Then, the energy consumed by user is and due to the energy causality constraint. Now consider that, instead of waiting for a duration , user transmits the same amount of data in a time slot with length with transmit power . Since the energy required for the transmission of a fixed amount of data is a monotonically increasing function of transmit power ; i.e., , . Then, the energy causality constraint is not violated since . This is a contradiction. ∎

Note that can be interpreted as a delay in the transmission of a user which can complete its transmission without this delay in the same amount of time using less energy. Then, considering the transmission of an arbitrary user in a schedule, we have the following corollary.

Corollary 1.

Delaying the transmission of a user to harvest more energy also delays the completion time of the transmission of the user.

Next, we mathematically formulate the power control and time allocation problem, denoted by . Without loss of generality, we assume that user transmits in time slot . For brevity, we present the formulation with a variable transformation to illustrate its convexity as follows:

:

minimize (5a)
subject to (5b)
(5c)
(5d)
variables (5e)

where denotes the energy consumption of user during its transmission with transmit power in the time slot with length .

Lemma 2.

is a convex optimization problem.

Proof.

The objective of is a linear function of . The function in constraint (5b) is a convex function of and since its Hessian is positive semidefinite. Besides, constraints (5c) and (5d) are affine. Then, is a convex optimization problem. ∎

Since is a convex optimization problem, it can be solved by evaluating its KKT conditions which specify necessary and sufficient conditions for an optimal solution of a convex optimization problem; i.e., any feasible point satisfying the KKT conditions is a global optimum point [ex24].

KKT conditions of are as follows:

(6a)
(6b)
(7a)
(7b)
(7c)

where Eqs. (6a)-(6b) and Eqs. (7a)-(7c) represent the gradient and complementary slackness conditions, respectively, and is the KKT multiplier associated with the constraint of the user, where corresponding to constraints (5b)-(5d), respectively.

Lemma 3.

In an optimal solution of , the constraint must be satisfied with equality.

Proof.

Suppose that in an optimal solution of , the constraint is not satisfied with equality. For an optimal solution, KKT conditions given in Eqs. (6)-(7) should be satisfied. Since constraint is not binding, by Eq. (7a). Then, by Eq. (6b), and since . However, for , Eq. (6a) is violated. This is a contradiction. ∎

Lemma 4.

In an optimal solution of , either constraint or must be satisfied with equality.

Proof.

Suppose that none of the constraints and are binding. Then, and by Eqs. (7b) and (7c). Then, either Eq. (6a) or Eq. (6b) is violated for any , which violates KKT conditions. On the other hand, suppose that both constraints and are binding. Then, due to Lemma 3, these two constraints together with constraint specify an overdetermined system of equations for which a solution does not exist. This is a contradiction. ∎

Theorem 1.

In the optimal solution of , transmit power of a user is given by

(8)

where is the Lambert function [ex25],

(9)
(10)

Then, the optimal time allocation of a user is given by

(11)
Proof.

By Lemmas 3 and 4, either constraints and or and are satisfied with equality in an optimal solution. Let be the transmit power satisfying constraints and with equality. Note that may satisfy or violate constraint . Then,

(12)
(13)

where . Define . Then, Eq. (13) can be rearranged as

(14)

Inserting Eq. (14) into Eq. (12), Eq. (12) can be rearranged as

(15)

We can represent Eq. (15) in the form as

(16)

where

(17)

Solution to is . Then, omitting some steps for brevity, is obtained as

(18)

and since from Eq. (14), is

(19)

Since the energy consumed during transmission of a fixed amount of data is a monotonically increasing function of the transmit power, as given in the proof of Lemma 1, any power allocation violates the energy causality constraint ; whereas, any power allocation satisfies it; i.e., is the maximum transmit power satisfying the energy causality constraint. Then, if is feasible considering constraint ; i.e., , is optimal. Otherwise, if violates constraint ; i.e., , since an optimal solution should satisfy either constraints and or and with equality and the latter case does not yield a feasible power allocation, constraint is satisfied with equality in the optimal solution. Hence, is optimal. Therefore, optimal power allocation is given by the minimum of and ; i.e., . Besides, by Lemma 3, the optimal time allocation is

(20)

V Optimal Scheduling

The goal of this section is to determine the optimal schedule; i.e., the transmission order of the users, in order to minimize the length of the schedule. In Section IV, the optimal time allocation and power control have been determined for a given schedule. On the other hand, optimizing the schedule can further decrease the schedule length. For instance, prioritizing the users with greater amount of energy and delaying the users with less energy such that they can harvest more energy and thus transmit with larger power within a shorter time slot may result in a significant decrease in the completion time of the transmissions. A straightforward solution to find the optimal schedule would be a brute-force search algorithm that enumerates all possible orderings of the users and then determines the one with the minimum length. However, such an algorithm has exponential complexity which makes it computationally intractable for even a medium size network. Hence, fast and scalable solutions are required.

In this section, we propose two polynomial-time heuristic algorithms and an exponential-time optimal algorithm with significantly reduced complexity. Next, we first investigate the optimality conditions for a minimum length schedule and then present the algorithms.

Let and be the starting and ending time of the transmission of user in the schedule, respectively.

Definition 1.

The penalty function of user , , is defined as the difference between the actual transmission time and the minimum possible transmission time, and formulated as

(21)

where is the minimum possible transmission time of user corresponding to maximum transmit power and is the maximum penalty for user when it is scheduled first.

Figure 2: Illustration of penalty function . is the lower bound for while becomes equal to for the first time when user can afford to transmit with .

Penalty function of user is illustrated in Fig. 2. decreases as a function of until it becomes when user can afford to transmit with for the first time. This monotonic characteristic of the penalty function is stated in the following lemma.

Lemma 5.

The penalty function is a non-increasing function of .

Proof.

By Lemma 4, any user should either transmit with or consume all its energy during transmission. Let be the earliest time instant at which user transmits with . For , user consumes all its energy during transmission. As increases up to , since the harvested energy increases, the energy consumed by the user increases so the transmit power increases up to . Therefore, the transmission time decreases, resulting in a decrease in the penalty function. On the other hand, for any starting time , the user transmits with . Since the transmission time is equal to the minimum possible transmission time at transmit power , the penalty is equal to for . ∎

Lemma 6.

For , the objective of minimizing the schedule length is equivalent to minimizing the sum of the penalties .

Proof.

By definition of the penalty function,

(22)

Since is constant for any user , can be removed from the objective function. Then, since , the objective function in Eq. (22) is reduced to . ∎

Theorem 2.

If for user , then there exists an optimal solution to in which user is scheduled first with transmission time .

Proof.

Suppose that there exists exactly one optimal schedule with length in which user with is not scheduled first. Denote the set of users scheduled before and after user by and , respectively, and the penalty of each user by . Note that based on Lemma 5. Now, consider that schedule is updated such that user is scheduled first and the scheduling order of the other users remain the same. Denote the resulting schedule by with length and the penalty of each user in the schedule by . The reallocation of user will delay the starting time of first scheduled user in by . Then, by Corollary 1, the starting time and the corresponding ending time of each user will be delayed consecutively. Therefore, for any user , since the penalty is a nonincreasing function of starting time as given by Lemma 5. Hence, . Then, the transmissions of user and users in in schedule will be completed either earlier or at the same time compared to schedule . Consequently, the starting time and the corresponding ending time of the transmission of a user will either decrease or remain the same, by Corollary 1. Hence, the schedule length which is equal to the ending time of the last scheduled user in will not increase; i.e., . This is a contradiction. ∎

Theorem 2 can be interpreted that at time , it is optimal to schedule a user with zero penalty if such a user exists initially. Consequently, at any time , after the completion of the ongoing transmission, it is still optimal to schedule a user with zero penalty among the remaining unallocated users since making a scheduling decision on minimizing the schedule length at time requires minimizing the sum of the penalties of the remaining unallocated users. Then, we have the following corollary of Theorem 2 presenting an optimal online scheduling policy.

Corollary 2.

For any scheduling policy, at any time instant ,

  1. It is optimal to schedule a user with next after the currently scheduled user completes its transmission.

  2. It is optimal to schedule a user that can feasibly afford to start its transmission at time and complete it using maximum transmit power ; i.e., without violating the energy causality constraint, next after the currently scheduled user completes its transmission.

Then, considering the possibility that at any time instant, at least one user with zero penalty; i.e., that can feasibly afford maximum transmit power, can be found, next corollary can be stated.

Corollary 3.

A schedule given by the optimal set of transmission times and transmit powers is optimal if the penalty of each user is zero; i.e., .

Next, we introduce the scheduling algorithms based on the foregoing optimality analysis.

1:Input:
2:Output: ,
3: , 0,
4:while  do
5:    arg,
6:    + ,
7:    - ,
8:    + + ,
9:end while
Algorithm 1 Minimum Penalty Algorithm (MPA)

V-a Minimum Penalty Algorithm

Minimum Penalty Algorithm (MPA) aims at minimizing the sum of the penalties of the users in a greedy manner based on Lemma 6, as given in Algorithm 1. Denote the schedule by , where the element of is the user scheduled in the time slot. Let the schedule length be . Input of MPA algorithm is a set of energy harvesting users, denoted by . The algorithm starts by initializing the schedule to an empty set and the schedule length to (Line ). At each step of the algorithm, MPA picks the user with the minimum penalty among the unscheduled users (Line ). The current time slot is allocated to this minimum penalty user (Line ). Then, the scheduled user is discarded from set (Line ) and the schedule length is updated by adding the transmission time of the scheduled user (Line ). Algorithm terminates when all users in are scheduled (Line ) and outputs the schedule and the corresponding schedule length . Fig. 3 illustrates MPA graphically for users.

Figure 3: Graphical illustration of MPA algorithm for users. Steps (dotted black line) depict the process of MPA. MPA first allocates user at since . Then, after user completes its transmission at , user starts its transmission; i.e. . The length of the resulting schedule is . The alternative schedule, i.e., scheduling first user with a greater initial penalty and then user (dotted red line), would yield a greater schedule length .
1:Input:
2:Output: ,
3: , 0,
4:while  do
5:   determine optimal power for all at time ,
6:    arg,
7:    + ,
8:    - ,
9:    + ,
10:end while
Algorithm 2 Maximum Transmit Power Algorithm

V-B Maximum Transmit Power Algorithm

Maximum Transmit Power Algorithm (MTPA) picks the user that can afford maximum feasible transmit power among all users, based on Corollaries 2-3, where we show that allocating a user that can feasibly afford at any time instant is optimal. MTPA is given in Algorithm 2. Input of MTPA algorithm is a set of users denoted by while the output is the schedule and the corresponding schedule length . The algorithm starts by initializing to an empty set and to (Line ). At each step of the algorithm, optimal transmit power of each user is determined by Eq. (8) (Line ) and the user with maximum power is allocated to the current time slot with duration given by Eq. (11) (Lines ). Then, the scheduled user is discarded from set (Line ) and the schedule length is updated by adding the transmission time of the scheduled user (Line ). Algorithm terminates when all users in are scheduled (Line ).

V-C Fast Pruning Algorithm

In this section, we propose an exact and efficient algorithm based on pruning mechanisms based on the analysis presented in Theorem 2 and the following corollaries, as given in Algorithm 3.

Consider a tree model with levels where is the set of users to be scheduled. Each level of the tree specifies the user allocated in the time slot of the schedule. A branch of the tree, consisting of one user from each level of the tree, corresponds to one feasible schedule for . A node of the tree at the level specifies the set of users allocated in first slots of the schedule. Size of a node is defined as the number of scheduled users specified by ; i.e., a node at the level of the tree has size . For instance, a node of the tree at the level specifies that users and are allocated in first two slots of the schedule, respectively, and has size . Note that maximum size for a node is corresponding to a branch; i.e., a feasible schedule consisting of all users. We further define the penalty and the transmission length of a node as the penalty of the last scheduled user by and the sum of the transmission times of the users scheduled by , respectively.

The aim of FPA algorithm is to determine the optimal schedule without generating all possible branches of the tree; i.e. without generating all feasible schedules. Two pruning mechanisms are employed to decrease the search space for the feasible schedules in this regard. Unless a node of the tree is pruned out by the algorithm via these mechanisms, it is branched into children nodes, each corresponding to the allocation of a new user to the already scheduled users by the ascendant node. First pruning mechanism is based on Corollary 2. If the penalty of a node is , all children nodes of its ascendant are pruned out since the other nodes originating from the same ascendant cannot yield a better schedule than that particular node. Second one is pruning out the nodes which cannot end up in a minimum length schedule. If the transmission length of a node is greater than or equal to the current minimum schedule length, that particular node is pruned out since adding new users to a node will only increase its transmission length.

1:Input:
2:Output: ,
3: ,
4:while   do
5:   determine of ,
6:    the set of nodes in with degree ,
7:   if  then
8:       the node with size ,
9:      if  then
10:         ,
11:          ,
12:      end if
13:      discard node from ,
14:      continue,
15:   end if
16:    arg,
17:   if  then
18:      prune all nodes in with size and same ascendant node with ,
19:   end if
20:   if  then
21:      prune node ,
22:   else
23:      generate set of children nodes of node ,
24:       + ,
25:   end if
26:   discard node from ,
27:end while
Algorithm 3 Fast Pruning Algorithm

Input of FPA algorithm is a set of users denoted by while the output is the schedule and the corresponding schedule length . FPA algorithm keeps track of the set of nodes of the tree which are not evaluated so far. The algorithm starts by initializing to set containing all nodes in level of the tree each corresponding to the allocation of one user in to time slot and to (Line ). Let be the largest size of a node in and be the set of nodes in with size . FPA determines and at each iteration (Lines ). Unless includes a node with size ; i.e., a branch, FPA picks the node with the minimum penalty in , denoted by (Line ). If the penalty of node is (Line ), then all nodes with the same size ; i.e., having same ascendant node with , are pruned out (Line ). If the transmission length of node is greater than or equal to the current minimum schedule length (Line ), then is also pruned out (Line ). Otherwise, is branched into its children nodes, denoted by set (Line ), which is added to set (Line ). Finally, is discarded from since its evaluation is completed (Line ). If, at a particular iteration, includes a node with maximum size (Line ), then the node with degree (Line ) is a branch and specifies a feasible schedule. Therefore, algorithm evaluates whether it outperforms the current best feasible schedule and updates and the corresponding current minimum schedule length if the transmission length of this node is less than (Lines ). Then, is discarded from (Line ) and the algorithm continues with the next iteration (Line ). FPA terminates when all nodes in are evaluated; i.e., (Line ). Fig. 4 illustrates FPA algorithm for users, through an example.

Figure 4: Graphical illustration of FPA algorithm for users. Algorithm starts by generating the level nodes , , , and . The green circles represent the nodes evaluated and not pruned by FPA. The yellow circles represent the pruned nodes having the same ascendant with a zero penalty node while the red circles represent the pruned nodes that cannot yield a better schedule compared to the current best feasible schedule determined by FPA. Blue circle denotes the optimal schedule yielded by FPA after all nodes are either evaluated or pruned out. The nodes are evaluated by FPA in the following order: .
Theorem 3.

FPA algorithm determines an optimal solution for .

Proof.

We will first show that FPA generates all possible schedules for the set of users unless a node of the tree is pruned out. Consider any node with size corresponding to a feasible schedule for the set of users where denotes the user allocated in the time slot of the schedule represented by node . The ascendant of node , say , is the node with size and corresponds to the allocation of first users scheduled by node . Then, by the same logic, one can iteratively determine the ascendants of up to the node with size corresponding to the allocation of only the first user scheduled by node . Node is one of the nodes in the initially specified set by FPA algorithm (Line ). This guarantees the generation of the schedule represented by node by FPA unless one of the ascendants of node is pruned out. Furthermore, since the branch of the tree originating from node and consisting of the successive ascendants of node up to is uniquely determined, the schedule represented by node is generated only once. On the other hand, any node pruned out by FPA cannot yield an optimal solution since it is pruned either due to the existence of a better feasible schedule determined by FPA so far (Lines ) or because one node having the same ascendant with has penalty and hence yields a better schedule (Lines ). Thus, the optimal schedule will be one of the nodes generated by FPA. This completes the proof. ∎

Vi Performance Evaluation

The goal of this section is to evaluate the performance of the proposed algorithms in comparison to the optimal solution and previously proposed algorithms. The previously proposed algorithm in [harvest_30], denoted by PCA, aims at minimizing the schedule length in a full-duplex system for a given transmission order of the users, without considering scheduling and maximum transmit power constraint. The algorithm determining the optimal time and power allocation for a predetermined transmission order of users based on Theorem 1 is included for a fair comparison to PCA, denoted by OTPA. The brute-force optimal algorithm which evaluates the length of all possible schedules and selects the minimum length schedule is denoted by BFA.

Simulation results are obtained by averaging

independent random network realizations. The users are uniformly distributed in a circle with radius of

m. The attenuation of the links considering large-scale statistics are determined using the path loss model given by where is the path loss at distance , is the reference distance, is the path loss exponent, and

is a zero-mean Gaussian random variable with standard deviation

. The small-scale fading has been modeled by using Rayleigh fading with scale parameter set to mean power level obtained from the large-scale path loss model. The parameters used in the simulations are for ; bits for ; MHz; m; dB; , [harvest_50]. The self interference coefficient is taken as dBm.

Figure 5: Schedule length vs. maximum transmit power

Fig. 5 illustrates the schedule length for different values in a network of users. The initial battery level of the users are assumed for a fair comparison between PCA and the proposed algorithms. The algorithms incorporating scheduling, i.e. MPA, MTPA and FPA, perform significantly better than the algorithms developed for a given scheduling order, i.e. PCA and OTPA. The performance improvement due to scheduling becomes more evident as increases. This is due to the fact that as value increases, the first time instant at which a user has zero penalty increases. The slight improvement of OTPA over PCA is a result of energy harvesting capability of a user during transmission for OTPA. Moreover, MPA and MTPA perform very close to optimal and show robustness against increasing . In addition, MPA slightly outperforms MTPA since while the former directly considers the transmission time of users through penalty minimization, the latter considers maximizing the transmit powers, which does not necessarily correspond to picking users with minimum penalty. Note that, penalty function takes data requirements and channel conditions into account while the transmit power of a user is just a function of the energy available for a user at a time.

Figure 6: Schedule length vs. HAP transmit power

Fig. 6 illustrates the schedule length for different HAP transmit power values in a network of users, assuming zero initial battery levels and mW. The schedule length decreases with the increasing transmit power since higher HAP power allows users to harvest more energy and complete their transmission in shorter time as long as they can transmit with higher transmit powers. Moreover, the algorithms MTPA, MPA and FPA employing scheduling significantly outperform the algorithms OTPA and PCA without scheduling. However, as increases to relatively large values around dB, performance of the algorithms converge to each other since users can reach to maximum transmit power values faster and this removes the necessity for scheduling: each user transmits at high data rates and can complete its transmission in a time slot with length close to . Note that the lower bound on the minimum schedule length is . Similar to Fig. 5, MTPA, MPA perform very close to optimal, indicating that the algorithms can adapt their schedules based on the energy harvested by each user efficiently.

Figure 7: Schedule length vs. number of users

Fig. 7 illustrates the effect of the network size on the performance of the proposed algorithms. For OTPA and PCA in which no scheduling is performed, the addition of each user increases the schedule length by almost a constant amount since increase in the schedule length is just caused by the time slot length of that particular user. On the other hand, for MTPA, MPA and FPA, each new user yields a diminishing increase in the schedule length since the transmission order may change by the addition of a user. For instance, a new user with good channel conditions such that it can initially transmit with will be scheduled first and this will allow the other users to harvest more energy to be able to complete their transmissions in shorter time durations. Note that, as discussed in the previous sections, delaying the transmission of a user decreases its transmission time if its transmission power increases. Then, the increase in the schedule length will be less than the time slot length of the new user by the sum of penalty reductions of the other users. MTPA and MPA perform very close to optimal as the number of users increases with a small gap for users caused by the exponential nature of the problem complexity. Note that for users, there are

possible schedules among which MTPA and MPA determine only one in polynomial-time. However, their performance is still very robust to the network size since having larger number of users will increase the probability of having at least one user with zero penalty or

transmit power to be picked by MPA and MTPA at a time, respectively.

Figure 8: Runtime vs. number of users

Fig. 8 shows the average runtime of the proposed algorithms for increasing number of users. The runtime of BFA increases exponentially with increasing number of users, therefore, has a large computational burden. On the other hand, the proposed optimal algorithm FPA decreases the runtime significantly by reducing the search space for the optimum schedule via smart pruning mechanisms discussed in Section V-C. For a network of users, FPA achieves a runtime around one thousandth of the runtime of BFA. Furthermore, the runtime of the proposed polynomial-time algorithms MTPA and MPA increases almost linearly, as expected. Evaluating Fig. 7 and Fig. 8 together, we can observe that MTPA and MPA are scalable algorithms that can achieve close to optimal solutions in reasonable runtimes even for large network sizes for which use of an exponential-time algorithm would be intractable.

Vii Conclusion

In this paper, we have considered a WPCN where multiple users can harvest energy from and communicate data to a hybrid access point that can supply RF energy in full duplex manner. We have investigated the minimum length scheduling problem to determine the optimal power control, time allocation and transmission schedule subject to data, energy causality and maximum transmit power constraints. We have formulated the problem as a mixed integer non-linear programming problem which is generally difficult to solve for a global optimum and conjectured that the problem is NP-hard. To solve the problem for a global optimum, we have provided a solution strategy in which the power control and time allocation, and the scheduling problems are decomposed. For the power control and time allocation problem, we have proposed optimal policies using convex optimization techniques. For the scheduling, we have proposed a novel idea by introducing the penalty function definition through which we have analyzed the characteristics of the optimal solution. We have proposed two polynomial time heuristic algorithms based on the optimality conditions. Through extensive simulations, we have illustrated that these heuristic algorithms perform very close-to-optimal while outperforming the conventional schemes significantly. We have also provided an exact algorithm that converges to the optimal solution fast by using smart pruning techniques. Simulations show that the exact algorithm yields much lower average runtime with respect to an exhaustive enumeration algorithm.

References