Cloud radio access network (C-RAN) , which centralizes the baseband functions at the baseband units (BBUs), can efficiently reduce the complexity of the remote radio units (RRUs), and thus the operation and deployment costs. Centralized baseband processing also enables efficient cooperative signal processing to increase the network capacity. In C-RAN, the fronthaul network transports the baseband signals between the BBUs and the RRUs. However, for fully centralized C-RAN, i.e., all baseband functions are centralized at the BBUs, the fronthaul rate requirement is high, which poses a major design challenge on C-RAN. For example, in a single 20MHz LTE antenna-carrier system, 1Gbps fronthaul rate is required with the standard CPRI interface . To support massive MIMO and other emerging technologies, the required fronthaul rate will be too high to bear.
Different from fully centralized C-RAN, by placing some baseband and network functions at RRUs, functional split is a promising technique to reduce the fronthaul rate requirement [8, 2]. There are multiple candidate functional split modes corresponding to different split points on the chain of baseband functions, as illustrated in Fig. 1. For each mode, the functions placed at the right side of the corresponding vertical dashed line are placed at the RRU, while the others are centralized at the BBU. The fronthaul rate requirement and processing complexity requirement at the RRUs vary, under different functional split modes. In general, with more baseband functions at the RRUs, the required fronthaul rate is lower, but the processing complexity is higher [6, 28], which also means more energy consumption at the RRUs. With certain functional split modes, for example, split between the physical layer and the MAC layer, the required fronthaul rate depends on the traffic load, and thus exploiting the fronthaul statistical multiplexing gain can further reduce the fronthaul rate requirement [19, 26]. With the development of software defined network (SDN) and network function virtualization (NFV), baseband functions can be virtualized and implemented on the general purpose computation platforms [4, 20]. As a result, the functions placed at the RRUs and the BBUs can be reconfigured according to the network state [13, 17].
By harvesting renewable energy from the environment, the RRUs are able to consume less or no energy from the power grid [25, 11, 30]. Another benefit is that the RRUs can be flexibly deployed at the places where the grid does not reach. However, reliable communication is challenging due to the randomness of renewable energy arrivals and the limitation of batteries, and thus the operation of RRUs should be well managed . In terms of power control, different from conventional “water-filling”, the throughput-optimal “directional water-filling” power control policy is found in a fading energy harvesting channel, where the “water”, i.e., the energy, can only flow from the past to the future. If the processing energy consumption is considered, the throughput-optimal transmission policy should become bursts, a “glue pouring” power control policy is proved to be optimal when there is only one energy arrival and no transmission deadline . The burst transmission is due to the fact that more processing energy is consumed with longer transmission time. For energy harvesting system with processing cost and multiple energy arrivals, a “directional backward glue-pouring” algorithm is proposed in .
There are some recent works on the flexible functional split mode selection in energy harvesting C-RAN systems. The grid power consumption and system outage rate are jointly studied by optimizing the offline placement of baseband functions, where the small base station is powered by renewable energy and the macro base station is powered by the grid 
. Reinforcement learning based online placement of functional split options is studied in for efficient utilization of the harvested energy, where the small cell is powered by renewable power with flexible functional split modes. To improve energy efficiency and throughput, RRU active/sleep mode and functional split mode selection in the energy harvesting C-RAN are determined according to the renewable energy levels and the number of users in the covering area of the RRU . However, to the best of our knowledge, the joint optimization of power control and flexible functional split mode selection has not been considered yet.
If the functional split mode is fixed in the energy harvesting communication system, the processing power is a constant, and thus “directional backward glue-pouring” algorithm  can be used to find the optimal power control policy. However, it is expensive and sometimes difficult to deploy fibers between the RRUs and the BBUs, and thus wireless fronthaul may be used as a low cost solution . Especially for RRUs powered by energy harvesting, they in general have no wired connection neither for power supply nor for fronthaul. In this case, flexible functional split is necessary, due to not only the fronthauling overhead brought by the wireless fronthaul, but also the unstable renewable energy supply. To this end, there are more than one candidate functional split modes, with different processing costs, and thus existing schemes like “directional backward glue-pouring” no longer apply. Functional split can tradeoff between the baseband processing complexity of RRUs and the fronthaul data rate requirement. In general, with more baseband functions at the RRU, the baseband processing complexity is higher, but the required fronthaul data rate is lower. Conversely, with more baseband functions at the RRU, the baseband processing power is lower, but the required fronthaul data rate is higher. This calls for new mechanisms that can determine the optimal functional split with the joint consideration of fronthaul properties and renewable energy arrivals.
In this paper, we study the selection of the functional split modes and power control policy for an energy harvesting RRU in C-RAN. We first consider the offline case, where the energy arrivals and the channel fading are non-causally known in advance. The functional split is jointly determined with the corresponding user data transmission duration and transmission power, and the objective is to maximize the throughput, while satisfying the energy and the average fronthaul rate constraints. For the optimal offline policy, we find that in each interval between successive energy arrivals, at most two modes are selected, the transmission power of the modes are the same for each channel fading block. We further analyze the scenarios with only one instance of energy arrival and two alternative functional split modes, and get the closed-from expression of the transmission power and transmission duration for each split mode, given the average fronthaul rate constraint. Based on the analysis, we propose a heuristic online policy, where the future energy arrivals and the channel fading are unknown in advance. Numerical results show that the heuristic online policy has similar performance with the optimal online policy developed by solving the Markov decision process (MDP) formulation.
The main contributions of this paper are summarized as follows.
We jointly optimize the functional split mode selection and power control for an RRU powered with renewable energy, to maximize the throughput under the average fronthaul rate constraint and random energy arrival.
For the offline problem where the energy arrivals and the channel fading are non-causally known, the throughput maximization problem is formulated and analyzed. We find the structure of the optimal solution that at most two functional split modes are selected between two successive energy arrivals. The online problem where the channel fading are causally known, is solved by its corresponding MDP formulation through value iteration.
To deal with the curse of dimensionality in solving an MDP, the closed-form expression of the transmission power and transmission duration in the special case with one energy arrival is derived, based on which a low-complexity heuristic online policy is proposed, and is shown to have near-optimal performance via extensive simulations.
The paper is organized as follows. The system model is described in Section II. The offline optimization problem is formulated and analyzed in Section III, and the online problem is introduced and solved by an MDP formulation in Section IV. The expression of optimal power control policy with one energy arrival, two functional split modes is derived in Section V. A heuristic online policy is proposed in Section VI. The numerical results are presented in Section VII. The paper is concluded in Section VIII.
Ii System Model
Consider a two-tier network, where a macro base station (MBS) covers a large area, while an RRU has small coverage areas within the coverage area of the MBS. The MBS has stable power supply, while the RRU is powered by renewable energy. The RRU transmits as much data as possible to the users with the harvested energy, while the remaining data is transmitted via the MBS. We thus aim to maximize the throughput of the RRU to reduce the traffic load of the MBS. We consider the downlink transmission from a particular RRU to its users, as described in Fig. 2(a). Assume that the BBU has sufficient data to transmit to users.
The system is slotted with normalized slot length. Assume that the wireless channel of the users is block fading, where the channel gain varies every block but remains constant within one block. Each block has slots. For each slot, the RRU serves the user with the best channel state, i.e., the user with the largest channel gain.
blocks, which is denoted by an epoch. We assume that the energy only arrives at the beginning of each epoch. The approximation is adopted to analyze the effect of different time scales of energy arrival and channel fading on the power control policy. As illustrated in Fig.2(b), units of energy arrives at the beginning of the -th epoch. The arrived energy is stored in a battery with capacity before it is used. Without loss of generality, we assume that , i.e., the amount of arrived energy is at most . There is no initial energy in the battery, i.e., the battery is empty before the first epoch. For the -th block of epoch , the maximum channel gain of the users is denoted as , which corresponds to the modulation and coding scheme (MCS) with the highest transmission rate, that the channel gain can support. Note that is measured when the reference transmit power is 1W.
For the scenario with multiple carriers, we assume that all carriers are used for transmission at the same time and have the same transmission power. If there are carriers, the channel gain of carrier in the -th block of epoch is denoted by . The spectrum efficiency is
where is the transmission power. In the optimal power control policy, blocks with good channel states are used for transmission, and the transmission power should be large enough due to the baseband processing power. The values of should be large, and thus the approximation is accurate. We now get the approximated channel gain in each block, i.e., , and the problem with multiple carries can be approximated as scenarios with single carrier. We thus explore the scenario with single carrier in the remaining part of this paper.
The RRU can be configured with candidate functional split modes. In each block, one or more functional split modes can be selected, but at most one functional split mode can be selected at any slot. In the -th block of epoch , the number of slots that functional split mode is selected is denoted by . Note that means that mode is not selected in the -th block of epoch . During one block, the total number of slots used for transmission of the modes should satisfy , where is number of slots in each block. The transmission power with mode in each block should be constant, denoted by . The maximum transmission power is , i.e., . The processing power of mode is denoted by , and the fronthaul rate requirement is denoted by . The processing power and fronthaul rate requirement are related to the MCS , and thus related to the transmission power, which makes the problem difficult to analyze. To simplify the problem, we assume that for each functional split mode , the processing power and fronthaul rate requirement are constant, which correspond to the MCS with the maximum transmission power . Also for the overhead of fronthaul, the average fronthaul rate is constrained to be no more than a given threshold . As the downlink scenario is considered, the energy consumption of the fronthaul happens at the BBU. The RRU only consumes energy when it is transmitting data to the users. In this case, bits of data are transmitted to the users with energy consumption in the -th block of epoch with mode .
For scenarios with multiple RRUs, if RRUs are self-powered and there is no cooperative transmission, the functional split selection and power control can be done separately at each RRU while treating the signals of other RRUs on the same frequency as noise. As for the scenario with cooperative transmission, we need to further optimize the precoding, and each RRU has its own energy constraints. However, due to the wireless fronthaul implementation and much more complex fronthaul topology, fronthaul sharing and and multiplexing gain should be further considered. Scenarios with cooperative transmission and fronthaul resource management are left as future work.
Iii Maximizing the Throughput
We consider the offline throughput maximization problem over a finite time of epochs. Due to the causality constraints, the energy that has not arrived can not be used, we have
note that is the energy consumed in epoch . There may be energy waste due to the limited battery size when the maximum transmit power is limited, which makes the energy constraints difficult to express. We thus ignore the maximum transmit power constraint when establishing the offline throughput maximization problem, and then approximate the transmit power that is larger than as in the optimal solution of the problem. As the energy in the battery at any time can not exceed the battery capacity, at the beginning of epoch , at which time the battery has the most energy in epoch , there should be
Denoted by , which is the energy consumed by the radio transmission in the -th block of epoch with mode , the optimization problem can be formulated as
where Eq. (6) is the constraint of the average fronthaul rate, and Eq. (9) is the constraint of the block length. Note that the functional split mode is included in the optimization of , i.e., means that mode is selected in the -th block of epoch , otherwise mode is not selected. Note that we can treat the number of slots as a continuous variable in the first place, in which case the complexity of solving the optimization problem can be greatly reduced, and some intuitive results can be given, while at the same time the effect on the throughput is small after approximating into an integer when is large. As the optimization objective in Eq. (5) is convex, and the constraints are linear, this is a convex problem. With Lagrangian multiplier method, we are able to get the following structure of the optimal solution.
In the -th block of epoch , during which the channel gain stays constant, the optimal transmission power of the selected modes are the same for any mode in the optimal solution.
The Lagrangian with , , , , and can be written as
Taking derivatives with respect to and , there should be
If mode is selected in the -th block of epoch , we have , with the complementary slackness condition , we have . According to (13), let ,
i.e., for and , the transmit power can be expressed as
The values of are the same for any selected mode in the -th block of epoch . ∎
Proposition 1 reveals that in one block, the transmission power with different functional modes are the same. Further more, we can find that the sum of the transmit power and the reciprocal of the channel gain are the same for any selected mode and block in epoch according to Eq. (16).
In each epoch, i.e., the duration between successive energy arrivals, the optimal functional split mode selection policy satisfies that at most two functional split modes are selected.
Denoted by the optimal transmission power in the -th block of epoch , and the corresponding transmission duration with functional split mode is . The baseband data amount transmitted via fronthaul in epoch is defined as , which can be expressed as . The number of slots used for transmission in block is defined as , i.e., . Given and , the throughput and the energy consumed by radio transmissions are fixed. The transmission duration should be an optimal solution of the following subproblem:
where the optimization objective is the energy consumed by baseband processing, which means that with transmission duration , the least energy is consumed by baseband processing, i.e., we aim to minimize the energy consumption while guaranteeing the transmission time and average fronthaul rate constraint. The number of slots used to transmit in epoch is defined as , where . We consider the constraints of the total transmission duration in each epoch, instead of the constraint of the total transmission duration in each block, the subproblem can be relaxed as:
In epoch , the energy consumed by baseband processing and the amount of data transmitted via fronthaul are only related to the total transmission duration of each mode, i.e., . For any optimal solution of the relaxed subproblem, we can find an equivalent solution of the subproblem, and thus the optimal solution of the subproblem is also the optimal solution of the relaxed subproblem. The Lagrangian of the relaxed subproblem is
Taking derivatives with respect to , we have . If mode is selected in the -th block of epoch , we have according to the complementary slackness condition that , we have . Let , there should be . If more than two functional split modes are selected, assume that the number of selected functional split modes is , and the selected modes are for . The following equations should have solution
Note that the formulation (20) has solution only when , or and satisfies that
for any 3 selected modes, which is a trivial scenario that can be ignored, and thus at most two functional split modes can be selected at each epoch. ∎
The solution obtained with continuous transmission duration is denoted by ‘upper bound’. We now introduce how to round the ‘upper bound’ into integer transmission duration. Slots with good channel states are used for transmission. The number of slots used for transmission with functional split mode in block is denoted by , with the corresponding transmission power , the energy used for transmission is . The energy used for baseband processing is , where is the baseband processing power. Number of slots used for transmission of each selected functional split mode is rounded into integer, denoted by . Besides the baseband processing energy, i.e., , the energy used for transmission is . The transmission power of each slot after rounding is then calculated according to Proposition 1, with the constraints of the total transmission energy .
According to Proposition 2, we conclude that at most two functional split modes are selected in one epoch, which means that the functional split mode selection can be determined at the time scale of energy arrival, rather than at the time scale of channel fading. In this sense, the switching of functional split mode can be done in a large time scale. The switching of functional split mode can be implemented by activating and deactivating functions in RRUs and BBU when RRUs and BBU are constructed by using container technologies, the introduced delay (less than millisecond ) and energy can be neglected.
Iv Optimal Online Policy
For the online policy, only the causal (past and present ) energy states and channel states are known at the RRU. To find the optimal online policy, we formulate the online problem as an MDP. The channel gain varies at the beginning of each block, and each block has slots. The beginning of the -th block is the
-th slot. The channel gain is modeled as a Markov chain withstates, and the channel gain of state is
. The transition probability from stateto state at the beginning of the -th block is denoted as .
The energy arrives once an epoch. We assume that the energy arrives at the beginning of each epoch. An epoch has blocks, i.e., slots. The energy arrival is modeled as a finite state Markov chain with states, and the arrived energy amount with state is . The transition probability from state to state at the beginning of the -th epoch is . The arrived energy is stored in a battery with capacity before it is used. The transmission power in slot is denoted as . Denoted by the functional split mode selected in slot , the baseband processing power is .
The energy is consumed only when the RRU transmits data to the users, i.e., when , the state of energy in the battery is updated as
To simplify the expression, we introduce a new variable, defined as
then the battery state is updated as
The system state is
where is the state of energy available in the battery, is the energy arrived in stage , records the energy arrival rate of the current epoch, is the channel gain, indicates how many blocks the current epoch has lasted, indicates how many slots the current block has lasted. The state transition probability is
The value of varies at the beginning of each block, and the state transition is described in Fig. 3(a). The transition probability of is expressed as
where is modulus operation which returns the remainder after division of by . The value of varies at the beginning of each slot. The transition probability of is expressed as
The state transition of channel state is described in Fig. 3(b), and the transmission probability of is
The transition probability of battery state is
As energy arrives every blocks, the energy only arrives at the beginning of each epoch. The state transition of energy arrival state is described in Fig. 3(c), and the transmission probability of is expresses as
The value of only changes after a new instance of energy arrival. The transmission probability of is expressed as
Due to the constraints of the energy in the battery and the maximum transmit power, the transmit power should be constrained as
where is the maximum allowed transmission power. According to Shannon’s equation, denoted by
The objective function is set as
where is the transmission rate of stage given the transmission power and the channel gain , which corresponds to the throughput in stage , the expectation is taken over the channel gain and the energy arrival rate; is the amount of baseband signals transmitted via fronthaul in slot , i.e., the fronthaul overhead, which corresponds to the average fronthaul rate. The optimization variable is the transmission power and the functional split mode selection, and is a weighting factor. We can tradeoff between the throughput and the fronthaul overhead by adjusting . With large , we have stringent constraint on the average fronthaul rate. To satisfy a given constraint of average fronthaul rate, we can iterate the weighting factor with algorithms such as the gradient descent method .
The average throughput maximization problem is formulated as an MDP, and the value iteration algorithm can be used to find the optimal policy . Every slot is treated as a stage. Denoted by the action taken in stage . The reward function in stage is denoted by
The objective is to minimize the average per-stage reward of the infinite horizon problem, which is denoted by
where is the initial state, is the possible policy. Problem (37) can be solved with value iteration algorithm. Denoted by the average per-stage reward, the relative reward when starting at state , the Bellman equation is expressed as
where is the set of all possible states. Initialize . Given any state , for the -th iteration, we have
note that converges to . A more general iteration formulation is
where . Denote the gap between and as , i.e.,
The iteration is considered as convergence when
where is a threshold which determines the convergence speed. The detailed value iteration algorithm is described in Algorithm 1.
For the optimal online problem, the state number of the MDP model is , and the number of actions is . The state space can be very large if some of the elements is of large size. The value iteration algorithm may encounter curse of dimensionality. In this case, lower-complexity algorithm is in need. In the next section, we will first analyze the power control policy with one instance of energy arrival, based on which a heuristic online algorithm is proposed.
V Single Energy Arrival, Constant Channel Gain
According to Proposition 2, at most two functional split modes are selected in each epoch in the optimal offline problem. To gain some insights, we will give some intuitive results when there is only one instance of energy arrival, and the channel gain is constant, i.e., , . Note that if the channel gain is averaged over an epoch, one epoch can be approximated to have only one block, where the approximated channel gain is the average channel gain over the epoch. For brevity, we will use , , , instead of , , and in this section, i.e., and are the corresponding transmission durations with the 2 functional split modes, and are the transmission power, the amount of available energy in this epoch is denoted by , the epoch length is denoted by . If there are more than two candidate functional split modes, i.e., , we can first calculate the throughput when any two of the functional split modes are selected (there are totally possible combinations), and obtain the optimal power control policy by comparing the throughput of all the possible scenarios.
If only one mode is selected, denoted by mode , the optimal power control policy can be obtained with “glue pouring” . Given the processing power and channel gain , and without maximum transmission duration constraint, the throughput maximization problem can be simplified to
where is the transmission power, and denote as the optimal transmission power obtained by solving the optimization problem. The optimal transmission power satisfies:
Note that the expression on the left side of the equality is an increasing function of