Resource Allocation for Multi-User Downlink MISO OFDMA-URLLC Systems

I. INTRODUCTION

• Shannon-based designs do not capture the relation among achievable rate, decoding error, and delay for short packets, motivating URLLC-specific resource allocation.The paper notes that applying Shannon capacity can underestimate latency and overestimate reliability.Source: §I · pp. 2–3
• The optimization maximizes weighted sum throughput subject to minimum transmitted bits, maximum packet error probability, and maximum packet delay.The delay constraint concerns transmission delay, not the other components of end-to-end delay.Source: §I · pp. 3–4
• The non-convex mixed-integer problem is reformulated as monotonic optimization and solved globally using polyblock outer approximation.The paper also develops a local-optimal, low-complexity method using difference-of-convex programming and successive convex approximation.Source: §I · p. 4
• The proposed low-complexity algorithm closely approaches the optimal algorithm and both outperform two baselines, especially for heterogeneous delay requirements.The results also identify multiple antennas as instrumental for low latency and high reliability.Source: §I · p. 4
• This paper extends conference version from single-antenna transceivers to a multi-antenna base station and adds an optimal policy and extensive simulations.The conference version provided only a sub-optimal algorithm.Source: §I · p. 4

A. System Model

• The system model has a multi-antenna base station serving single-antenna URLLC users over OFDMA subcarriers and time slots.Users’ delay requirements are assumed known, and only users whose requirements may be met in the current resource block are admitted.Source: A. System Model · p. 5

B. Channel Model

• The channel remains constant over the resource frame, and the base station is assumed to have perfect channel state information; interference from other users is treated as noise.The frame duration is shorter than the assumed channel coherence time.Source: B. Channel Model · pp. 5, 7

III. RESOURCE ALLOCATION PROBLEM FORMULATION

• The formulation optimizes MISO OFDMA-URLLC resource allocation using short-packet achievable rates and user QoS requirements.The QoS requirements include transmitted bits, packet error probability, and maximum delay.Source: §III · p. 7

A. Achievable Rate for SPC

• Finite-blocklength coding relates achievable bits to packet length and decoding error probability, addressing the short-packet regime where Shannon capacity does not apply.The paper bases its resource allocation on the normal approximation for finite blocklength codes.Source: A. Achievable Rate for SPC · pp. 7–8

B. QoS and System Performance Metric

• Each URLLC user has a minimum received-bit target, packet-error target, and maximum transmission delay; its symbols are assigned within the first D_k slots.A user can begin decoding after receiving all its OFDMA symbols, after D_k slots.Source: B. QoS and System Performance Metric · p. 8
• Weighted sum throughput uses per-user weights μ_k, with larger weights assigning users higher priority and throughput.The weights may be specified in the MAC layer.Source: B. QoS and System Performance Metric · pp. 8–9

C. Optimization Problem Formulation

• The optimization maximizes weighted system throughput subject to each user’s bit, reliability, and latency requirements.Constraint C1 ensures the minimum B_k bits, C2 limits total base-station power, and C3 serves users within D_k slots.Source: C. Optimization Problem Formulation · p. 9
• The problem is non-convex because of the SINR form and the short-packet normal approximation; the paper proposes monotonic optimization and a low-complexity successive-convex-approximation approach.The low-complexity approach also uses difference-of-convex programming to obtain close-to-optimal performance.Source: C. Optimization Problem Formulation · p. 9

IV. OPTIMAL SOLUTION OF THE OPTIMIZATION PROBLEM

• The optimal method applies monotonic optimization through an iterative algorithm that solves a semi-definite relaxation problem at each iteration.The relaxation is introduced to facilitate the optimal solution procedure.Source: §IV · p. 10, A. Semi-Definite Programming Relaxation · p. 10

B. Problem Transformation

• The reformulation bounds SINR from below with auxiliary variables and uses upper bounds to express the increasing functions in monotonic form.For the objective, V(z) is replaced by V(z_max) − t; analogous auxiliary variables ζ_k are used for user constraints.Source: B. Problem Transformation · pp. 11–12
• After dropping the rank constraint, the authors introduce auxiliary SINR variables z and prove the transformed problem yields the same beamforming-matrix solution as the original.The equivalence is stated for problems (25) and (14).Source: B. Problem Transformation · pp. 11–12
• The objective and a constraint in the transformed problem are differences of monotonic concave functions, so the authors convert them into monotonic form in two steps.The objective and constraints are then monotonically increasing or convex, placing the reformulated problem in the monotonic-optimization class.Source: B. Problem Transformation · pp. 12–13

C. Polyblock Algorithm

• Because the optimum lies on the feasible-set boundary but that boundary is unknown, the algorithm encloses the feasible set with an initial polyblock from above.The method then constructs nested outer polyblocks around the feasible set.Source: C. Polyblock Algorithm · p. 13, Figure 2 · p. 14
• At each iteration, the algorithm finds the boundary intersection along a vertex-to-origin line and uses it to construct a smaller polyblock.Vertices that no longer affect the polyblock are removed as the enclosing approximation is refined.Source: C. Polyblock Algorithm · p. 14
• The procedure continues with nested polyblocks enclosing the feasible set, then selects the vertex maximizing the objective function.The figure depicts the optimal solution on the feasible-set boundary.Source: C. Polyblock Algorithm · p. 14, Figure 2 · p. 14

D. Calculation of Intersection Point

• Each iteration seeks λ > 0 such that Φ(v(j)) = λv(j), the intersection of the vertex ray with the feasible-set boundary.Bisection tests feasibility of a convex problem, including the condition λt(j) + V(λz(j)) ≤ V(z_max).Source: D. Calculation of Intersection Point · pp. 14–16
• The SDP solution has rank at most one, and Proposition 1 states that the transformed and original problems yield the same beamforming-matrix solution.The proof connects the rank property to equivalence of the solutions.Source: D. Calculation of Intersection Point · p. 16
• The optimal scheme’s computational complexity is exponential in the number of vertices D per iteration, while the proposed sub-optimal algorithm has polynomial complexity and close-to-optimal performance.The global optimum serves as a performance upper bound for sub-optimal resource-allocation algorithms.Source: D. Calculation of Intersection Point · p. 16

• Penalized successive convex approximation provides a low-complexity, locally optimal solution to optimization problem (13).Source: §V · p. 16

A. Difference of Convex Programming

• The reformulation expresses the objective and constraints as differences of convex functions, enabling application of difference-of-convex programming.Source: A. Difference of Convex Programming · pp. 16–18
• The bilinear term in constraint C6 is non-convex, so it is rewritten as a difference of convex functions.This enables the constraint to be handled within difference-of-convex programming.Source: A. Difference of Convex Programming · p. 17
• First-order Taylor approximations of non-convex terms produce a convex optimization problem that can be solved iteratively.The resulting sequence converges to a local optimum; the solution matrices have rank at most one.Source: A. Difference of Convex Programming · pp. 18–19

B. Penalized Successive Convex Approximation

• Slack variables relax QoS constraint C1, while an increasing penalty weight drives violations toward zero when a feasible point exists.If the slack variables do not converge to zero, the original problem is infeasible; the penalty weight is capped to avoid numerical issues.Source: B. Penalized Successive Convex… · p. 19

VI. PERFORMANCE EVALUATION

• Simulations use a single cell with 50 m and 250 m inner and outer radii, Rayleigh fading, equal user weights, and 1,000 realizations.The stated setup applies unless otherwise specified.Source: §VI · p. 20

A. Performance Metric

• Infeasible channel realizations receive zero sum throughput, and average system throughput is computed across the considered realizations.Source: A. Performance Metric · pp. 20–21

B. Performance Bound and Benchmark Schemes

• Benchmarks include an unachievable Shannon-capacity upper bound, a Shannon-based allocation evaluated with short-packet rates, and an MRT scheme solved by successive convex approximation.Source: B. Performance Bound and Benchmark… · p. 21

C. Simulation Results

• Algorithm 3 matches Algorithm 1's performance in Figure 3 while converging in fewer than 5 iterations; Algorithm 1 needs about 2,000–3,000 iterations, and Algorithm 3 needs at most 5 in larger tests.The larger-parameter convergence tests cover the combinations considered in Figure 4.Source: C. Simulation Results · p. 22, Figure 3 · p. 22, Figure 4 · p. 22
• The proposed low-complexity scheme virtually matches the optimal scheme across transmit powers, while both outperform the baselines; Shannon-based allocation can violate QoS constraint C1.At high transmit power, Shannon-based baseline 1 yields similar performance because the finite-blocklength correction becomes negligible.Source: C. Simulation Results · pp. 22–23, Figure 5 · p. 23
• At Pmax = 40 dBm in S1, increasing BS antennas from 4 to 8 raises average system sum throughput by 96.77%; stricter S1 delays reduce throughput relative to S0.At NT = 8 and Pmax = 40 dBm, S1 lowers the upper bound and proposed-scheme throughput versus S0 by 6.2 and 10 bits/s/Hz, respectively; Figure 7 also shows throughput rising with antenna count.Source: C. Simulation Results · pp. 23–24, Figure 6 · p. 23, Figure 7 · p. 25
• Average throughput increases with delay parameter D but falls as more users require D < N; baseline 2 also loses throughput under strict delays.The paper attributes baseline 2's decline from S0 to S1 to fixed MRT beamforming's inability to adequately support delay-sensitive users.Source: C. Simulation Results · pp. 24–25, Figure 8 · p. 25
• As user count rises, throughput gains from multi-user diversity compete with a widening gap to the upper bound from finite-blocklength effects; baseline 2 cannot support more than 6 users in S1.For small user counts, the low-complexity scheme is close to the upper bound because users can receive enough resource blocks to make finite-blocklength effects negligible.Source: C. Simulation Results · p. 25, Figure 9 · p. 26

VII. CONCLUSION

• The proposed low-complexity algorithm closely approaches optimal performance and significantly outperforms two heuristic baselines; Shannon-based allocation may yield infeasible solutions.The conclusion also reports that multiple BS antennas improve URLLC reliability and reduce latency, while stringent delay requirements lower throughput.Source: §VII · pp. 25–26

APPENDIX A

• The objective is treated as a difference of two monotonic, concave functions, with F(z) identified as a sum of logarithmic functions.The appendix uses these properties to establish the difference-of-concave structure.Source: Appendix A · p. 26
• For positive z_k[m,n], V_k[m,n] is increasing and concave: its first derivative is positive and its second derivative is negative.The appendix extends these properties to sums and weighted sums of the component functions.Source: Appendix A · p. 27
• The square root of a monotonic concave function remains monotonic and concave, supporting the appendix's concavity argument for V(z).The argument also invokes the non-decreasing property of the square-root function and closure of weighted sums.Source: Appendix A · p. 27

APPENDIX B

• The SDP is jointly convex and satisfies Slater’s constraint qualification, so strong duality makes its dual problem equivalent to the primal problem.Source: Appendix B · p. 27
• The appendix studies the optimal beamforming structure through the KKT conditions and formulates the SDP dual using multipliers for constraints C6, C2, C3, and C4.Source: Appendix B · p. 28
• The optimal beamforming matrix is rank one or zero; the zero case means no transmission to that user on that subcarrier and time slot.Source: Appendix B · p. 29