Source-linked AI summary
Broadcasting with an Energy Harvesting Rechargeable Transmitter
Jing Yang, Omur Ozel, Sennur Ulukus
TL;DR
The paper addresses how to minimize completion time for two users’ data in an energy-harvesting AWGN broadcast channel with rechargeable storage. It characterizes optimal power allocation and uses that structure to develop a globally optimal offline scheduling algorithm. The key structure matches the single-user total-power policy and uses a cut-off level to divide power between stronger and weaker users.
Problem
The paper studies transmission completion-time minimization for fixed data amounts destined for two receivers in an energy-harvesting rechargeable-transmitter broadcast channel.
Method
The paper first characterizes the fixed-deadline maximum departure region, then uses duality, optimal-policy structure, and recursive single-user reductions to construct an offline schedule.
Results
The optimal total transmit power has the single-user structure, while a cut-off power allocates power below the cut-off to the stronger user and excess power to the weaker user.
Takeaways & Limitations
These structural properties yield an algorithm that finds the globally optimal offline scheduling policy by reducing the two-user broadcast problem to single-user problems as much as possible.
Abstract
from arXiv · showhide
In this paper, we investigate the transmission completion time minimization problem in a two-user additive white Gaussian noise (AWGN) broadcast channel, where the transmitter is able to harvest energy from the nature, using a rechargeable battery. The harvested energy is modeled to arrive at the transmitter randomly during the course of transmissions. The transmitter has a fixed number of packets to be delivered to each receiver. Our goal is to minimize the time by which all of the packets for both users are delivered to their respective destinations. To this end, we optimize the transmit powers and transmission rates intended for both users. We first analyze the structural properties of the optimal transmission policy. We prove that the optimal total transmit power has the same structure as the optimal single-user transmit power. We also prove that there exists a cut-off power level for the stronger user. If the optimal total transmit power is lower than this cut-off level, all transmit power is allocated to the stronger user, and when the optimal total transmit power is larger than this cut-off level, all transmit power above this level is allocated to the weaker user. Based on these structural properties of the optimal policy, we propose an algorithm that yields the globally optimal off-line scheduling policy. Our algorithm is based on the idea of reducing the two-user broadcast channel problem into a single-user problem as much as possible.
I. INTRODUCTION
The paper studies completion-time minimization for a two-user energy-harvesting AWGN broadcast channel with rechargeable storage and known offline energy arrivals. It derives optimal-policy structure and uses it to reduce the broadcast problem to recursive single-user scheduling problems.
- Motivation and problem: The transmitter must deliver B1 and B2 bits to two receivers while adapting rates and power to energy availability to minimize completion time.Energy arrivals and corresponding amounts are known at t = 0 for the offline schedule.
- Approach: The paper first solves a dual fixed-deadline problem that maximizes jointly delivered bits, then uses its structural properties for completion-time minimization.Varying weighted objectives characterizes the maximum departure-region boundary.
- Structural results: The optimal total transmit power is independent of the user weights and has the same majorization structure as the single-user solution.Power remains constant between relevant energy-harvesting epochs and changes when energy constraints become tight.
- Structural results: A cut-off power level determines the split: power below it serves the stronger user, while excess power serves the weaker user.The cut-off depends on the weighting ratio, so different cut-offs trace different boundary points.
- Algorithm: The resulting iterative algorithm recursively reduces each broadcast scheduling case to single-user problems and finds the globally optimal offline schedule.When the cut-off is below the initial total power, the stronger user receives a fixed portion and a fixed-point equation equalizes completion times; otherwise, the problem restarts with updated bits.
II. SYSTEM MODEL AND PROBLEM FORMULATION
The system is a two-user AWGN broadcast channel with one transmitter, two data queues, and an energy queue supplied by harvested energy. The transmitter chooses power and rate allocations subject to energy causality to finish both users’ data as early as possible.
- System model: The transmitter stores data for both receivers in separate data queues and harvested energy in an energy queue.The model includes one transmitter and two receivers.
- Channel model: The physical layer is a two-user AWGN broadcast channel with receiver 2 degraded because its noise variance is σ2 > 1.Receiver 1 has unit-variance Gaussian noise, while receiver 2 has variance σ2.
- Inputs: Initially available data consists of B1 bits for receiver 1 and B2 bits for receiver 2, while energy arrives at times sk in amounts Ek.The transmission policy adapts to available energy and remaining data.
- Optimization objective: The objective is to select transmit power and user-specific power portions that minimize the time T required to deliver all bits.The capacity region is parameterized by total power P and the fraction α assigned to the first user.
- Constraints: Energy causality requires cumulative consumed energy at every time t to be no greater than cumulative harvested energy.The constraint applies throughout transmission, not only at the final completion time.
III. CHARACTERIZING D(T): LARGEST (B1, B2) REGION FOR A GIVEN T
For a fixed deadline, the paper characterizes the achievable departure region and proves structural properties of its optimal policies. These properties then support completion-time minimization through a cut-off-power construction and an iterative scheduling algorithm.
- Fixed-deadline region: Optimal rates remain constant between consecutive energy arrivals, because equalizing changing rates lowers energy consumption under the convex cost.The saved energy can increase departures, making rate changes between harvests suboptimal.
- Power structure: The optimal total transmit power is independent of μ1 and μ2 and matches the single-user optimal transmit power.The power is constant between epochs where energy constraints are tight and changes only at such structural points.
- User allocation: There is an integer transition epoch after which the weaker user receives positive rate; before it, the weaker-user rate is zero.The stronger-user rate is constant or increasing, while the weaker-user share increases after the transition.
- Completion-time solution: The paper uses these properties to construct an efficient iterative algorithm for the globally optimal offline schedule.The conclusion states that the broadcast problem is reduced to single-user scheduling as much as possible.
- Completion-time solution: A constant cut-off power Pc allocates all power below Pc to the stronger user and all power above Pc to the weaker user.This structure is used to reduce completion-time optimization to single-user cases and equalize the users’ completion times.
IV. MINIMIZING THE TRANSMISSION COMPLETION TIME T FOR A GIVEN (B1, B2)
The section reduces the two-user completion-time problem using structural properties of optimal power allocation and a cut-off power for the stronger user. It then develops an algorithm that is feasible and globally optimal by reducing cases to single-user problems.
- The optimal policy finishes transmitting to both users at the same time when both have nonzero bit demands.
- For a fixed cut-off power Pc, the first user’s completion time decreases while the second user’s completion time increases with Pc.
- The cut-off power is selected so the two users’ completion times are equal, with bisection applicable because the relevant function is continuous and strictly decreasing.
- The resulting algorithm is feasible and optimal, using single-user power allocation and reducing the broadcast problem to simpler subproblems.
- When Pc exceeds the initial total power P1, P1 is allocated entirely to the stronger user and the residual problem is reduced recursively.
V. NUMERICAL EXAMPLES
The numerical examples evaluate optimal scheduling under specified AWGN broadcast-channel and energy-arrival settings. The plotted departure regions expand with completion time, and the reported policies finish before the final energy harvest.
- The examples use a band-limited AWGN broadcast channel with bandwidth W = 1 MHz and specified path losses and noise power spectral density.
- The energy arrivals occur at t = [0, 2, 5, 6, 8, 9, 11] s with harvested amounts E = [10, 5, 10, 5, 10, 10, 10] mJ.
- For T = 6, 8, 9, 10 s, the maximum departure region is convex and expands monotonically as T increases.
- With (B1, B2) = (15, 6) Mbits, the optimal policy finishes both transmissions at T = 9.66 s without using the harvest at t = 11 s.
- In the second example, both users’ rates monotonically increase and transmission finishes at T = 9.25 s without using the last energy harvest.
VI. CONCLUSIONS
The paper characterizes the optimal transmission policy for an energy-harvesting broadcast channel and uses those properties to develop a globally optimal off-line policy.
- The optimal total transmit power has the same structure as in the single-user channel.
- A cut-off power level exists for the stronger user.
- When total transmit power is below the cut-off, all power is allocated to the stronger user.
- When total transmit power exceeds the cut-off, all power above it is allocated to the weaker user.
- These structural properties yield an iterative algorithm for the globally optimal off-line transmission policy.