Energy Harvesting Broadband Communication Systems with Processing Energy Cost
Abstract
Communication over a broadband fading channel powered by an energy harvesting transmitter is studied. Assuming noncausal knowledge of energy/data arrivals and channel gains, optimal transmission schemes are identified by taking into account the energy cost of the processing circuitry as well as the transmission energy. A constant processing cost for each active subchannel is assumed. Three different system objectives are considered: 1) throughput maximization, in which the total amount of transmitted data by a deadline is maximized for a backlogged transmitter with a finite capacity battery; 2) energy maximization, in which the remaining energy in an infinite capacity battery by a deadline is maximized such that all the arriving data packets are delivered; 3) transmission completion time minimization, in which the delivery time of all the arriving data packets is minimized assuming infinite size battery. For each objective, a convex optimization problem is formulated, the properties of the optimal transmission policies are identified, and an algorithm which computes an optimal transmission policy is proposed. Finally, based on the insights gained from the offline optimizations, lowcomplexity online algorithms performing close to the optimal dynamic programming solution for the throughput and energy maximization problems are developed under the assumption that the energy/data arrivals and channel states are known causally at the transmitter.
ptptptptptptptptptptptptptptptptptptptptptptptptptptptptptptptptptptptptptptptpt
Energy Harvesting Broadband Communication Systems with Processing Energy Cost
Oner Orhan, Deniz Gündüz, Senior Member, IEEE, and Elza Erkip, Fellow, IEEE
Index Terms
Offline power optimization, throughput maximization, remaining energy maximization, transmission completion time minimization, online algorithms.
I Introduction
Wireless sensor nodes are typically designed to have low cost and small size. These design objectives impose restrictions on the capacity and efficiency of the energy storage units that can be used. As a result, continuous operation of the sensor network requires frequent battery replacements, which increases the maintenance cost. Energy harvesting (EH) devices are able to overcome these challenges by collecting energy from the environment. However, due to the nature of the ambient energy sources, the amount of useful energy that can be harvested is limited and unreliable. Consequently, optimal management of the harvested energy becomes a new challenge for EH wireless nodes.
In most communications literature the energy cost of operating transmitter circuitry, such as digitaltoanalog converters, mixers, filters, etc. is ignored. In short range communications, as in most wireless sensor networks, where internode distances are less than 10m, processing energy consumption can be comparable to the transmission energy [1]. When the processing cost is negligible, increasing the transmission time and lowering the transmission power increases the energy efficiency (natsperjoule), provided the ratepower function is monotonically increasing and concave, properties satisfied by most common transmission schemes as well as Shannon’s capacity function. However, as shown in [2], when processing cost is taken into account, bursty transmissions separated by “sleep” periods become optimal. In EH communication systems, this affects the optimal power allocation scheme considerably since both the power allocation and the sleep intervals will depend on the energy arrival profile.
In this paper, we consider an EH transmitter with processing cost communicating over a broadband fading channel, modelled as parallel subchannels with each subchannel having independent fading. Following the power consumption model in [2] and [3], processing energy cost is modelled as a function of the transmission bandwidth and time and is assumed to be equal to a constant value for each subchannel. We characterize optimal transmission policies for three different system objectives under the offline optimization framework which assumes that all channel gains and the sizes of arriving energy and data packets are known noncausally before transmission starts. First, we only consider energy packet arrivals over time for a backlogged transmitter^{1}^{1}1A backlogged transmitter is the one that always has data available for transmission. with a finite capacity battery, and we maximize the amount of total data delivered by a deadline . We call this the throughput maximization problem [4]. Throughput maximization is an important objective for high data rate applications. Then, we consider both data and energy arrivals over time and an unlimited battery, and maximize the remaining energy in the battery by the deadline. This is the energy maximization problem [5] most suitable for energy efficient, green applications. Finally, for the joint energy and data arrival scenario we also find the minimum delivery time of all the data packets. This is called the transmission completion time (TCT) minimization problem [6], is important for delay limited applications. For each of these problems we identify the structure of the optimal transmission policy by solving a convex optimization problem, and based on this structure we provide an algorithm which finds the optimal transmission policy.
We next consider a more realistic model assuming only the causal knowledge of energy/data arrivals and channel gains, and study the online optimization problem. Since the optimal solution of the online optimization problem based on dynamic programming is prohibitively complex, we propose simple algorithms for the throughput and energy maximization problems based on the insights gained from the optimal solutions of the corresponding offline optimization problems.
In recent years, optimal transmission policies for EH communication systems have been studied extensively under various assumptions regarding the knowledge at the transmitter about the energy harvesting process. Within the offline optimization framework optimal transmission policies have been investigated for pointtopoint [6][7] and various multiuser communication scenarios, including broadcast channel [8], [9], [13], interference channel [10] and twohop networks [11], [12]. In addition, battery imperfections in terms of leakage, finite energy storage capacity, and energy storing and retrieving losses are investigated in [13], [4], and [14], respectively.
Online optimization of EH communication systems has also received considerable interest. Optimal transmission policies for EH nodes based on Markov decision processes are studied [15][16]. In [7, 14, 19], heuristic online policies are presented. A more practically oriented learningtheoretic approach to EH system optimization is studied in [17]. See [18] for a general overview of EH communication systems under offline, online and learningtheoretic frameworks.
The effect of processing cost on EH communication systems have been investigated in [19][22]. Optimal transmission policies that maximize the average throughput are studied for a constant singlelink in [19][20], and parallel channels in [19]. In [21], the throughput maximization problem is studied for a timeslotted system using suboptimal slot selection and power allocation. Our previous work [22] and [5] consider a narrowband fading channel with processing cost and study the throughput maximization and energy maximization problems. The current paper extends all the prior literature by considering a broadband fading EH communication system with processing cost.
In the next section, we describe the system model. In Section III, we summarize the gluepouring algorithm which provides the optimal power allocation strategy in a battery operated communication system when the processing energy cost is taken into account [2]. We investigate the structure of the optimal offline transmission policies and provide directional gluepouring interpretations for the throughput maximization, energy maximization and the TCT minimization problems in Section IV, V, and VI, respectively. In Section VII, we propose online algorithms for the throughput and energy maximization problems. In Section VIII, numerical results are presented. Finally, we conclude our paper in Section IX.
Ii System Model
We consider an EH transmitter communicating over a broadband fading channel modelled as parallel independently fading subchannels. Each subchannel has additive white Gaussian noise (AWGN) with unit variance. The real valued channel gain for subchannel at time is denoted as , . Without loss of generality, Shannon capacity, defined as (nats/s/Hz), , is considered as the transmission ratepower function, where is the transmission power of subchannel at time .
We assume that finite number of energy and data packets arrive at the transmitter in time interval each carrying finite amount of energy and data, respectively. We assume that the energy and data packet arrival times are denoted as and , respectively. A rechargeable battery with a finite capacity of is available at the transmitter. We assume that the harvested energy is first stored in the battery before being used by the transmitter. Accordingly, the size of an harvested energy packet is less than without loss of generality. In addition, we assume that the battery is able to store and preserve the harvested energy without any loss. We also assume that changes at the time instances , and remains constant in between. In order to simplify the problem formulation, all channel changes and energy/data arrival events are combined in a single time series as by allowing zero energy/data arrivals when the channel gain of any subchannel changes, or the channel gains to remain constant when an energy/data packet arrives. We define an epoch as the time interval between two consecutive events. We denote the duration of the ’th epoch as . The size of the energy and data packet arriving at time is referred to as and , respectively, and indicates the channel gain of subchannel in epoch .
In addition to the energy used for transmission, we consider the processing energy cost of the transmitter circuitry which models the energy dissipated by the microprocessors, mixers, filters, and converters. Using the system level power consumption model of a wireless transmitter in [3], we take into account the dependence of the processing cost on the transmission bandwidth. We assume a processing cost of joules per second for a subchannel simplicity. This constant processing energy per subchannel, independent of the transmission power, is consumed only during the time the corresponding subchannel is used.
Using optimality of constant power transmission within each epoch [6], we denote the nonnegative transmission power within epoch of subchannel as . As argued in [2], due to the processing cost it may not be optimal to transmit continuously, i.e., bursty transmission can be optimal. Therefore, we denote transmission duration of as , . Accordingly, a transmission policy refers to power levels with durations , , that determine the energy allocated to each subchannel at each epoch . Any feasible transmission policy should satisfy the energy causality constraint:
(1) 
Moreover, since increasing the transmission power or duration strictly increases the amount of transmitted data, an optimal transmission policy must avoid battery overflows by utilizing all the harvested energy. Therefore, an optimal transmission policy must also satisfy the following battery overflow constraint:
(2) 
Data arrivals over time also impose data causality constraints on the feasible transmission policy as follows:
(3) 
In Sections IVVI, we identify the optimal offline transmission policies, in which all energy/data arrivals and channel gains are known before transmission starts, for three different system objectives stated below. Mathematical formulations are deferred to Sections IV, V, and VI.

Throughput maximization: Assuming that the transmitter has sufficient data in its data buffer before transmission starts, i.e., backlogged system with , , , we maximize the total amount of data delivered by the deadline .

Energy maximization: Relaxing the battery size constraint, i.e., , we maximize the remaining energy in the battery by the deadline while guaranteeing that all the arriving data is delivered to the destination.

TCT minimization: We minimize the delivery time of all data packets arriving at the transmitter while assuming an infinite size battery, i.e., .
In addition, we consider online transmission policies in which we assume that all energy/data arrivals and channel gains are known causally for throughput and energy maximization problems in Section VII.
Iii Preliminaries
For ease of exposure, we first illustrate the optimal transmission policy for throughput maximization for , . This models a battery operated system. For a single energy arrival at time , a channel state and processing cost , for , maximum throughput is given by the solution of the following optimization problem:
(4) 
where is the total transmission duration and is the transmission power. The corresponding optimal transmission power [2] satisfies
(5) 
The above equation has only one solution for the optimal power level which is given by (11) in [2]. Note that increases as the channel gain decreases^{2}^{2}2This follows from (III) by taking the derivative of with respect to .. Moreover, does not depend on the available energy . For finite transmission deadline , if , then the above solution is still optimal. On the other hand, if , transmitting at power cannot be optimal because some energy would remain in the battery at time . In this case, we can increase the throughput by increasing the transmission power so that all the available energy is consumed by time , and the optimal transmission power is given by .^{3}^{3}3A similar observation is made in [13] where constant rate battery leakage is considered instead of processing cost. This correspondence does not extend to multiple energy packets or fading channels as will be seen later in the paper.
In the case of multiple fading levels, again for single subchannel , single energy arrival and no transmission deadline (), the optimal transmission policy is given by the gluepouring algorithm [2]. For two fading levels with durations , , respectively, the gluepouring solution is summarized below. In the following, and denote the transmission durations for epochs with fading levels and , and and denote the solutions of (III) for channel gains and , respectively.

If , then the optimal transmission policy is and with power levels and , respectively.

If , then the optimal transmission policy is and with power levels and , respectively.

If , then the optimal transmission policy is and with power levels and , respectively.

If , then the optimal transmission policy is obtained through the classical waterfilling algorithm.
Based on the above solution, for the general system model with subchannels glue level in epoch of subchannel is defined as the sum of the transmission power and the inverse channel gain in that epoch, i.e., .
Iv Throughput Maximization
In this section, we consider the throughput maximization problem introduced in Section II, that is, we maximize the total delivered data until the deadline . We assume that and , , and the last event corresponds to the transmission deadline, i.e., . Mathematically, the problem can be formulated as follows.
(6a)  
s.t.  (6d)  
where we have defined , for and . Notice that is equivalent to the total allocated transmission energy to epoch of subchannel . In the above optimization problem, the constraints in (6d) and (6d) are due to the energy causality and battery overflow constraints in (II) and (II), respectively. The term is the perspective function of the concave function . Here, we take when . Since perspective operation preserves concavity [23], the objective function in (6a) is concave. In addition, the constraints in (6d)(6d) are linear. Therefore, the optimization problem in (IV) is convex, and efficient numerical solutions exists [23].
The optimal allocated transmission energy to epoch of subchannel , and the corresponding optimal transmission duration , for and , must satisfy the following KKT conditions:
(7)  
(8) 
for and . Here is the Lagrangian of (IV) with , , , , and as Lagrange multipliers for constraints in (6d)(6d), respectively. The complementary slackness conditions are
(9)  
(10)  
(11) 
We next identify some properties of an optimal transmission policy for the throughput maximization problem based on the KKT conditions in (IV)(IV) which are both necessary and sufficient due to the convexity of the optimization problem in (IV):

If or , then the optimal transmission power must be zero.

If and , then due to the complementary slackness conditions in (IV). Therefore we can compute the optimal transmission power in terms of and as follows:
(12) which is obtained by substituting into (IV). By combining (IV) and (IV) we can obtain
(13) When we replace in (IV) with , we obtain
(14) Note that when , i.e., , (IV) is equivalent to (III). Therefore, it has a unique solution for given and . We denote the solution of (IV) when as . Since (IV) depends only on and , we can compute the optimal transmission power directly without solving the optimization problem in (IV). When , i.e., , it can be argued from (IV) that the optimal transmission power must satisfy .
Remark IV.1
When there is no processing cost, i.e., and , , and when , . To see this suppose and . In this case we can argue that (IV) leads to when . However this contradicts with the assumption on , since . Therefore, when , there is no bursty transmission. Consequently the optimal transmission policy for leads to the classical waterfilling over subchannels [25].
Lemma 1
In the optimal transmission policy, whenever the glue level in subchannel , i.e., , decreases (increases) from one epoch to the next, the battery must be full (empty).
Proof: The optimal transmission power satisfies (IV) whenever a nonzero transmission energy is allocated to epoch of subchannel , and . In addition, from the complementary slackness conditions (IV)(IV), we can argue that the battery is empty whenever and , and the battery is full whenever and . This is because whenever the constraint in (6d) is satisfied with equality, i.e., , the constraint in (6d) cannot be satisfied with equality, i.e., , and vice versa. From (IV) we see that implies and , since and leads to an increase in the denominator of RHS of (IV). Similarly, implies and . Therefore, we can conclude that whenever the glue level in subchannel , , decreases (increases) from one epoch to the next, the battery must be full (empty).
Lemma 2
In the optimal transmission policy, the glue levels in an epoch are the same for all subchannels to which nonzero transmission energy is allocated.
Proof: Rearranging (IV) we obtain
(15) 
for . Note that right hand side of (IV) must be the same for all subchannels in epoch to which nonzero transmission energy is allocated. Therefore, we can conclude that the glue level in an epoch is the same for all subchannels with nonzero transmission energy.
Remark IV.2
It is possible to show that , the solution of (IV) when , is a decreasing function of . Since the optimal transmission power in an epoch of subchannel must satisfy , the optimal transmission policy utilizes epochs with the highest channel gain under the energy causality and battery size constraints.
Remark IV.3
The optimization problem in (IV) may have multiple solutions. Consider a subchannel with multiple epochs having the same channel gain. In an optimal transmission policy, if these epochs are partially utilized, i.e., , then the corresponding optimal transmission power must be equal to . Then, the corresponding optimal values for in (6a) must also be the same, therefore, we can obtain another transmission policy by transferring some of the energy between these epochs under the energy causality and battery size constraints. Similarly, if an epoch has multiple partially utilized subchannels having the same channel gain, we can find another optimal transmission policy by transferring energy between these subchannels.
Iva Directional Backward GluePouring Algorithm
The directional backward gluepouring algorithm, introduced in [22] for the throughput maximization problem with a single fading channel (), is an adaptation of the gluepouring algorithm in Section III to the EH model, where the energy becomes available over time. Due to the energy causality constraint, harvested energy can only be allocated to epochs . When energy of amount is transferred to future epochs , the constraint (6d) is satisfied with strict inequality, i.e., . Then the glue level cannot increase as argued in Lemma 1. Conversely, if there is a glue level increase, that is, if , then the constraint (6d) is satisfied with equality, and no energy is transferred to future epochs. In addition, due to battery size constraint, the amount of energy that can be transferred to epoch is limited by . When the transferred energy is less than , the battery size constraint in (6d) must be satisfied with strict inequality, i.e., , and the glue level does not change as argued in Lemma 1. Conversely, when there is a glue level decrease, that is, if , the amount of transferred energy to the ’th epoch is . Therefore, we can allocate the harvested energy to epochs, starting from the last nonzero energy packet to the first, under the energy causality and battery size constraints. Moreover, the optimal transmission power for different subchannels of an epoch must have the same glue level while satisfying the condition . These suggest that, the optimal transmission policy can be obtained through the directional backward gluepouring algorithm over the epochs of subchannels. Accordingly, the optimal transmission policy can be computed as in Table I.
To illustrate the directional backward gluepouring algorithm, consider the example in Fig. 1. There are two subchannels () and two fading levels () in each subchannel. The inverse channel gains are shown as heights of the solid blocks. The dashed lines above the block are used to express optimal power levels for , such that corresponds to the difference in height between the solid block and the dashed one. We consider two energy arrivals at the beginning of each epoch which are indicated by the downward arrows in the figure. As argued above, the algorithm first allocates power to the second epoch using the last harvested energy , as shown in Fig. 1, then considers the first energy packet for the first and second epochs together. The glue levels are the same among the subchannels for which the condition holds, as shown in Fig. 1. Note that due to the limited battery capacity, the transferable energy from the first epoch to the second is limited by , which explains the glue level difference between the first and second epochs in Fig. 1.
V Energy Maximization
In this section, we study the energy maximization problem introduced in Section II, that is, we maximize the remaining energy in the battery by the deadline such that all the data packets , , are delivered. We assume that the last event corresponds to the transmission deadline, i.e., , and relax the finite battery size constraint, i.e., . The optimization problem for the energy maximization can be formulated as follows:
(16a)  
s.t.  (16e)  
where we have defined for and . Here, can be considered as the total amount of data transmitted within epoch of subchannel . In the above optimization problem, (16e) and (16e) are due to the energy and data causality constraints in (II) and (II), respectively. Constraint (16e) arises as a result of the delivery requirement of all data packets by the deadline. Note that, the term is the perspective function of a strictly convex function . Here, we take when . Since the perspective operation preserves convexity [23], the objective function in (16a) is concave, and the constraint set defined by (16e)(16e) is convex. Therefore, the optimization problem in (V) is convex. The constraint set of (V) can be empty due to insufficient harvested energy to deliver all the data packets. Feasibility of (V) can be checked by solving the optimization problem in (V) with a new objective function , and a new constraint replaced with (16e). Note that corresponds to the additional amount of data that can be delivered in the last epoch for the given energy profile. If the optimal value of this optimization problem is nonnegative, i.e., , then the constraint set defined by (16e)(16e) has a feasible solution.
The optimal value of the total transmitted data and the corresponding transmission duration for epoch of subchannel , and , must satisfy the following KKT conditions:
(17)  
(18) 
for and . Here, is the Lagrangian of (V) with , , , , and as Lagrange multipliers corresponding to constraints (16e)(16e), respectively. The complementary slackness conditions are given as:
(19)  
(20)  
(21)  
(22) 
Similar to Section IV, we characterize the properties of the optimal transmission policy for the energy maximization problem using the KKT conditions in (V)(V).
We observe that the optimal power and transmission duration for epoch of subchannel for and , satisfy the following:

If , must be zero as no data is transmitted in that epoch.

If , then due to the complementary slackness conditions in (V). In this case, the optimal transmission power can be computed in terms of and , , as follows
(23) This is obtained by using (V) and replacing with . In addition, we can obtain the following from (V):
(24) When we replace with , we get
(25)
Remark V.1
Similar to the throughput maximization problem in Section IV, as argued in Remark IV.1, the optimal transmission policy over subchannels becomes the classical waterfilling solution when there is no processing cost, i.e., . This follows from the fact that (V) leads to , when and , as argued in Remark IV.1.
Lemma 3
In the optimal transmission policy, whenever the glue level in subchannel , , increases from one epoch to the next, either the battery depletes and a new energy packet is harvested, or the data buffer empties and a new data packet arrives.
Proof: The optimal transmission power satisfies (V) when there is nonzero data transmission in epoch of subchannel for and . We can also conclude from the complementary slackness conditions in (V)(V) that whenever , the battery depletes, and whenever , the data buffer empties. Therefore, the glue level increases from one epoch to the next, when either the battery depletes and a new energy packet is harvested, or the data buffer empties and a new data packet arrives.
Similar to Lemma 2, in the optimal transmission policy, the glue levels in an epoch are the same for all the subchannels .
Note as in Section IV, the optimal transmission policy must satisfy . Therefore, Remark IV.2 is valid for the energy maximization problem as well.
Remark V.2
Similar to the throughput maximization problem in (IV), the energy maximization problem in (V) may have multiple solutions. The optimal transmission power is equal to if the optimal transmission duration of an epoch of a subchannel satisfies . If multiple epochs have the same channel gain, the optimal values satisfying are the same. Therefore, as can be argued from the objective function of (V), we can find another optimal transmission policy satisfying the energy and data causality constraints by transmitting some of the data in a different epoch with the same optimal transmission power.
Va Directional Backward GluePouring Algorithm with Data Arrivals
The directional gluepouring algorithm of Section IVA can be modified to solve the energy maximization problem by taking into account data arrivals. A data packet can only be transmitted after it has arrived due to the data causality constraint. When part of the data is transferred to future epochs , the constraint (16e) is satisfied with inequality, i.e., . Then the glue level remains the same as argued in Lemma 3. Conversely, if there is a glue level increase, i.e., if , then the constraint (16e) is satisfied with equality, and no data is transferred to future epochs. By Lemma 1 the optimal transmission policy must satisfy the condition , and the glue levels are the same for all subchannels in an epoch. Accordingly, we can schedule transmission of the data starting from the last nonzero data packet to the first, such that the required energy to transmit the data satisfies the energy causality constraint. Therefore, the optimal transmission policy can be computed using a directional backward gluepouring algorithm with data arrivals in which the data packet is transmitted over subsequent epochs, and the energy allocation for each data packet is done using the gluepouring algorithm in Section IVA. Accordingly, the optimal transmission policy can be computed as in Table II.
To illustrate the directional backward gluepouring algorithm with data arrivals, we consider the algorithm for two subchannels with two fading states in each subchannel as shown in Fig. 2. The inverse channel gains are indicated by solid blocks in the figure. The optimal power levels are indicated with dashed lines which are above the solid blocks. In addition, the energy and data arrivals are showed with downward arrows, respectively. The algorithm first allocates power to the second epoch such that bits are transmitted in this epoch and the glue levels are the same in the subchannels in which the condition , , is satisfied (see Fig. 2). Note that, although both energies and are available for the transmission of bits, is used first, as can also be used to transmit the bits in the first data packet. If was not enough to transmit bits, some of the energy from the first arrival would also be used. Then, the algorithm considers the first data packet and allocates power according to the gluepouring algorithm in Section IVA as shown in Fig. 2.
Vi Transmission Completion Time (TCT) Minimization
In this section, we consider the TCT minimization problem introduced in Section II. Our goal is to identify an optimal transmission policy which minimizes the delivery time of all the data packets , . We again assume that the battery has infinite size . We first discuss the relation between the TCT minimization and energy maximization problems, and then we propose an algorithm which finds the optimal transmission policy for TCT minimization.
As argued in Remark V.2, the optimal transmission scheme for the energy maximization problem may have multiple solutions which can lead to different TCTs. Since we are seeking the minimum TCT, without loss of optimality, we put some restrictions on the optimal transmission policy obtained by the energy maximization problem before we relate the two problems: i) nonzero power is always allocated at the beginning of an epoch, i.e., during the time interval ; ii) if there are multiple epochs with the same channel gain in a subchannel , , the transmission power is allocated starting from the earliest epoch satisfying the energy and data causality constraints; and iii) if all the utilized subchannels in the last epoch have the same channel gain, then the transmission power is allocated to those subchannels for which the transmission duration is the same.
Denoting the minimum TCT time as we note that the battery must be depleted by the time , otherwise we could increase the transmission power and deliver the arrived data in a shorter time. Therefore, we can conclude that the remaining energy in the battery obtained by the energy maximization problem must be zero when the deadline is equal to . Any delay constraint , for which the energy maximization problem leads to zero remaining energy in the battery, satisfies , as the transmission power in the time interval can be zero.
Following the arguments above, the smallest transmission deadline , for which the energy maximization problem has a feasible solution, is an upper bound on . This suggests that for , the harvested energy is insufficient to transmit all the arrived data packets, and is a lower bound on . Note that due to the requirement of transmitting all the arriving data packets, we also need to ensure that the last nonzero data packet arrival instant is upper bounded by . After identifying the time interval , we formulate a convex optimization problem which minimizes the maximum of the transmission durations of subchannels in epoch , i.e., to find . The TCT minimization algorithm is outlined next.
Via TCT minimization
In order to compute , we first find the smallest , such that is greater than the last nonzero data packet arrival time , and the directional backward gluepouring algorithm in Section VA with has a feasible s