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Joint Power and Blocklength Optimization for URLLC in a Factory Automation Scenario

Hong Ren, Cunhua Pan, Yansha Deng, Maged Elkashlan, Arumugam Nallanathan

arXiv:1911.13050v1eess.SPcs.IT

TL;DR

The paper studies how to allocate blocklength and power for reliable, low-latency factory-automation URLLC when finite-blocklength effects make Shannon capacity inapplicable. It formulates and solves this problem across four downlink schemes, finding that relay-assisted transmission generally performs best while NOMA is effective under very stringent delay.

  • Problem

    URLLC short packets make Shannon’s capacity formula inapplicable, motivating resource allocation that explicitly handles finite blocklength, decoding error probability, latency, and energy constraints.

  • Method

    The paper jointly optimizes blocklength and transmit power for OMA, NOMA, relay-assisted, and cooperative NOMA transmission in a two-device factory-automation scenario, using low-complexity algorithms.

  • Results

    Relay-assisted transmission significantly outperforms the other schemes in most cases, while NOMA performs well when the delay requirement is very stringent.

  • Takeaways & Limitations

    Relay assistance is the strongest overall option among the compared schemes, whereas NOMA is competitive when available blocklength is very limited.

Abstract

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In URLLC, short packet transmission is adopted to reduce latency, such that conventional Shannon's capacity formula is no longer applicable, and the achievable data rate in finite blocklength becomes a complex expression with respect to the decoding error probability and the blocklength. To provide URLLC service in a factory automation scenario, we consider that the central controller transmits different packets to a robot and an actuator, where the actuator is located far from the controller, and the robot can move between the controller and the actuator. In this scenario, we consider four fundamental downlink transmission schemes, including orthogonal multiple access (OMA), non-orthogonal multiple access (NOMA), relay-assisted, and cooperative NOMA (C-NOMA) schemes. For all these transmission schemes, we aim for jointly optimizing the blocklength and power allocation to minimize the decoding error probability of the actuator subject to the reliability requirement of the robot, the total energy constraints, as well as the latency constraints. We further develop low-complexity algorithms to address the optimization problems for each transmission scheme. { For the general case with more than two devices, we also develop a low-complexity efficient algorithm for the OMA scheme.} Our results show that the relay-assisted transmission significantly outperforms the OMA scheme, while NOMA scheme performs well when the blocklength is very limited. We further show that the relay-assisted transmission has superior performance over the C-NOMA scheme due to larger feasible region of the former scheme.

I. INTRODUCTION

The paper addresses URLLC resource allocation for factory automation, where short packets make Shannon capacity inapplicable and stringent latency and reliability requirements apply. It jointly optimizes blocklength and power across four downlink schemes and develops low-complexity solution methods.

  • Motivation: URLLC short packets invalidate Shannon’s capacity formula, requiring achievable-rate modeling that accounts for finite blocklength and decoding error probability.The finite-blocklength rate depends on SNR, blocklength, and decoding error probability.
  • Motivation: Existing work largely analyzes finite-blocklength performance, while practical URLLC requires joint blocklength and power optimization under error-probability and latency constraints.The resulting optimization is difficult because the achievable-rate expression is neither convex nor concave in blocklength and transmit power.
  • System scenario: The considered factory-automation scenario has a distant actuator and a mobile robot receiving different packets from a central controller within strict transmission-time and error-probability limits.The robot can move between the controller and actuator, creating a two-device downlink setting for comparing transmission schemes.
  • Contributions: The study compares OMA, NOMA, relay-assisted transmission, and cooperative NOMA while minimizing actuator decoding error subject to robot reliability, energy, and blocklength constraints.The optimization jointly selects blocklength and transmit power for the two devices.
  • Contributions: Low-complexity algorithms are developed for the four schemes, including specialized blocklength searches, iterative procedures, and a bisection method under a convexity condition.For OMA, the paper also addresses the case of more than two devices with a low-complexity algorithm.
  • Results: Relay-assisted transmission significantly outperforms the other schemes in most cases, while NOMA performs well when the delay requirement is very stringent.The reported comparison concerns decoding or packet error probability and network availability performance.

II. SYSTEM MODEL

The system models URLLC downlink transmission from a central controller to a nearby robot and distant actuator under short-packet, finite-blocklength conditions. The design uses joint blocklength and power optimization for four transmission schemes.

  • The controller sends equal-size packets of D bits to a robot and an actuator, with the actuator located far away and the robot near the controller.
  • Transmission must finish within M symbols, corresponding to a latency of tmax = MTs seconds.
  • URLLC uses limited blocklengths, so decoding error probability cannot be ignored and Shannon capacity does not directly characterize transmission.
  • Finite-blocklength coding rate depends on blocklength, decoding error probability, and received SNR through a normal approximation.
  • The paper jointly optimizes transmission blocklength and power to minimize the actuator’s decoding error probability across four transmission schemes.

III. TRANSMISSION SCHEMES

The paper formulates resource allocation for OMA under actuator-error minimization, robot-reliability, energy, and latency constraints. OMA separates the robot and actuator transmissions into orthogonal channel uses and supports integer blocklength optimization.

  • The studied schemes minimize actuator decoding error while satisfying latency, robot reliability, and total-energy constraints.
  • A. OMA transmission: OMA assigns separate orthogonal channel uses or blocklengths to the robot and actuator.
  • A. OMA transmission: The OMA formulation allocates powers and blocklengths to meet the robot’s error requirement while minimizing the actuator’s error probability.
  • A. OMA transmission: The normalized channel gains are assumed to satisfy h1 > h2, giving the robot a stronger channel than the actuator.
  • A. OMA transmission: In OMA, the robot and actuator use integer blocklengths whose sum is constrained by the latency budget M.

2) Algorithm to solve Problem (7):

The OMA optimization is solved by narrowing feasible blocklength ranges, searching over robot blocklengths, and optimizing the remaining allocation variables. The procedure exploits equality constraints, monotonicity, and continuous relaxation where applicable.

  • The algorithm searches feasible integer robot blocklengths and, for each one, searches feasible actuator blocklengths.
  • For each robot blocklength, bisection finds the robot power satisfying its decoding-error constraint, and remaining energy determines actuator power.
  • 3) Special case of Problem (7):: For high SNR, V ≈ 1 enables an efficient special-case solution; the approximation is stated for γ > 20 dB.
  • When E2h2/(M−m1) ≥ e^-1, the relaxed actuator-blocklength objective is concave and the resulting problem is convex.

B. NOMA transmission

NOMA transmits both packets simultaneously using superposition coding and relies on successive interference cancellation at the robot. The optimization accounts for finite-blocklength decoding errors, SIC success or failure, and power allocation under reliability and energy constraints.

  • B. NOMA transmission: NOMA uses superposition coding so the controller transmits to both devices simultaneously over blocklength M.
  • B. NOMA transmission: The robot first decodes the actuator signal and then removes it before decoding its own signal when SIC succeeds.
  • B. NOMA transmission: If SIC fails, the robot decodes its own signal while treating the actuator’s information as interference.
  • B. NOMA transmission: The robot’s average decoding error probability combines the outcomes of successful and failed SIC.
  • B. NOMA transmission: The actuator directly decodes its own signal while treating the robot’s signal as interference.
  • B. NOMA transmission: The NOMA optimization allocates powers under energy and channel-ordering constraints, with the robot reliability constraint active at the optimum.

C. Relay-assisted transmission

The relay-assisted scheme uses the robot as a decode-and-forward relay, splitting transmission into broadcast and relay phases. Its optimization jointly searches blocklength and power under reliability, energy, and latency constraints, using a one-dimensional reduction and Algorithm 2.

  • Transmission protocol: The robot decodes the combined packet and, under decode-and-forward operation, forwards the actuator packet during a second phase of blocklength m2.The relay phase uses coding rate D/m2 and transmit power pr.
  • Transmission protocol: The controller broadcasts a combined 2D-bit packet to both devices during the first phase of blocklength m1.The combined packet uses power ps, and the received signals are specified for the robot and actuator.
  • Error analysis: The actuator error probability accounts for robot decoding failure, relay-to-actuator failure, and fallback decoding from the first-phase signal.The component error probabilities are ε1, ε2, and ˆε2, respectively.
  • Optimization problem: The resource allocation problem jointly selects powers and integer blocklengths subject to the robot reliability constraint, total energy, and total blocklength limits.The total energy and latency constraints are represented by m1ps + m2pr ≤ Etot and m1 + m2 ≤ M.
  • Optimization problem: Unlike the OMA and NOMA cases, the decoding-error constraint may be inactive at the optimum because the objective can also decrease with ε1.Consequently, the earlier OMA and NOMA algorithms cannot be directly applied.
  • Solution method: For fixed blocklengths, the power problem reduces to one-dimensional search by deriving feasible bounds for pr and ps and using pr = (Etot − m1ps)/m2.The overall procedure searches feasible blocklengths and solves the reduced power problem.
  • Solution method: Algorithm 2 solves the relay-assisted resource allocation problem through iterative blocklength-bound calculations and objective minimization.The algorithm evaluates feasible m1 and m2 pairs and selects the blocklength combination with minimum actuator error probability.

D. C-NOMA transmission

C-NOMA combines NOMA transmission with relay assistance: the controller sends both packets in the first phase, while the robot can forward the actuator packet in the second. The paper reduces its coupled optimization to bounded searches, while noting that its feasible region is smaller than relay-assisted transmission.

  • Transmission protocol: C-NOMA transmits two signals in the first phase using NOMA, then has the robot act as a relay and forward the actuator packet.The first phase uses signals x1 and x2, while the second phase uses relay transmission.
  • Error analysis: The robot uses successive interference cancellation to decode the actuator signal and has different error probabilities depending on whether SIC succeeds.The analysis includes separate expressions for perfect-SIC and SIC-failure cases.
  • Error analysis: The actuator error probability combines relay-phase decoding with the possibility that it must decode its packet from the first-phase signal.The corresponding probabilities include ε2 and ˆε2.
  • Optimization problem: The C-NOMA optimization allocates p1, p2, pr, m1, and m2 under reliability, energy, latency, integer-blocklength, and p1 ≤ p2 constraints.The energy and latency constraints are m1(p1 + p2) + m2pr ≤ Etot and m1 + m2 ≤ M.
  • Solution method: Because three power variables complicate bound derivation, the method treats p1 + p2 as a single quantity and searches over that sum.For fixed m1 and m2, p1 is obtained from a one-dimensional equation, p2 follows from the sum, and pr follows from the energy equality.
  • Solution method: Algorithm 3 uses iterative bounds and one-dimensional searches to solve the C-NOMA optimization problem.The procedure searches m1, m2, the summed first-phase power, and the resulting relay power.
  • Feasible-region comparison: The relay-assisted scheme has a larger feasible region because a feasible summed power in that scheme can correspond to infeasible separate C-NOMA powers.The paper gives this feasible-region relation as a reason for relay-assisted transmission's advantage.

IV. EXTENSION TO MORE DEVICES FOR THE OMA SCHEME

For more than two devices, the paper extends the OMA formulation and provides a low-complexity search procedure, while leaving extensions to other transmission schemes for future work.

  • Extension scope: The multi-device extension considers the OMA scheme and develops Algorithm 3 for its optimization procedure.The supplied text identifies this extension as applying to OMA rather than the other schemes.
  • Extension scope: The extension to transmission schemes other than OMA is left for future work.

A. Sytem Model and Problem Formulation

For K devices, the paper formulates OMA resource allocation to minimize the Kth device's decoding error while meeting the other devices' reliability requirements. It reformulates the problem, derives blocklength bounds iteratively, and uses a suboptimal search because global optimality is unavailable.

  • A. Sytem Model and Problem Formulation: The multi-device OMA model orders K devices by decreasing normalized channel gain and jointly allocates power and blocklength.The ordering is h1 > h2 > · · · > hK.
  • A. Sytem Model and Problem Formulation: The objective minimizes the decoding error probability of device K while guaranteeing the error requirements of devices 1 through K − 1.
  • A. Sytem Model and Problem Formulation: Unlike the two-device case, a globally optimal solution is unavailable, so the paper seeks a suboptimal solution for the general problem.
  • B. Problem Reformulation: The reformulation approximates V as one under high SNR and uses monotonic error behavior to show that key energy, latency, and reliability constraints are tight at optimum.The approximation is stated to be accurate when γ ≫ 1.
  • B. Problem Reformulation: Substitution reduces the number of optimization variables, but the reformulated problem remains difficult and begins with exhaustive search over mK followed by power optimization.
  • C. Bounds of mK: For short-packet OMA, each device generally uses fewer than 100 channel uses, enabling g(mk) to be treated as monotonically decreasing and convex.The supplied typical URLLC setting uses around 100 transmission bits and an error requirement around 10^-9.
  • C. Bounds of mK: The method derives tight bounds for mK by iteratively updating device blocklength bounds using monotonicity and bisection search.The procedure continues until the bounds converge.

D. Optimization of pK with Given mK

The section addresses the integer blocklength constraint by relaxing the optimization to continuous variables, solving the resulting convex problem, and converting the solution back to integers with a greedy search.

  • The integer constraint makes the blocklength optimization difficult to solve directly.
  • Relaxing blocklengths to continuous values yields a convex optimization problem solvable by Lagrangian dual decomposition.
  • For a fixed multiplier, optimal blocklengths are obtained using the convex Lagrangian formulation and bisection search.
  • The multiplier is found by solving F(λ) = M − mK because F(λ) is monotonically decreasing.
  • The continuous solution is converted to integer blocklengths with a greedy search that allocates each block to the device yielding the largest decrement in g(mk).

V. SIMULATIONS RESULTS

Simulations evaluate the four transmission schemes under path-loss-only and Rayleigh-fading settings. Relay-assisted transmission generally provides the strongest reliability, while NOMA is advantageous when blocklength is very limited.

  • Simulation setup: The simulations use a 1 MHz bandwidth, 100 us transmission delay, and robot error-probability requirement of 10^-9.The controller, robot, and actuator are positioned on a line, with the robot moving toward the actuator.
  • Blocklength: Larger blocklength improves actuator reliability for all schemes, and relay-assisted transmission outperforms the other schemes as M increases.
  • Blocklength: When M increases from 50 to 100, relay-assisted error probability decreases from 1 to 10^-22.
  • Blocklength: NOMA performs best at M = 50 and 60 because it uses the whole blocklength, whereas other schemes divide it between two transmission stages.With larger M, relay-assisted and C-NOMA transmission outperform NOMA.
  • Packet size: Larger packet size increases error probability, while relay-assisted advantage over OMA and NOMA shrinks as D increases.NOMA generally outperforms OMA, but C-NOMA becomes worse than OMA at D = 125 bits.
  • General OMA: For general OMA, the decoding error probability is almost identical to OMA when K = 2, while complexity is lower and actuator reliability worsens as K increases.Increasing the number of devices leaves less energy and blocklength for the Kth device.

B. Network Availability Performance (Channel Generation Times=1000)

Network availability generally improves with blocklength and energy, but relay-assisted transmission provides the strongest performance across most tested conditions. NOMA is competitive under very stringent latency, while relay-assisted and C-NOMA converge in higher-resource regimes.

  • Relay-assisted transmission achieves the best network availability over the packet-size region and generally outperforms the other schemes.
  • When D = 100 bits, relay-assisted and C-NOMA reach nearly 98% availability, whereas NOMA is as low as 87%.
  • Availability increases with M for all schemes; NOMA slightly exceeds relay-assisted transmission at M = 50, while C-NOMA reaches 97% at M = 100.
  • At M = 100, OMA and NOMA converge to low availability, with NOMA reported at 86%, while all schemes saturate at high M.
  • Increasing the energy limit improves every scheme; relay-assisted transmission leads, and its gap over C-NOMA disappears at high energy where both reach 98%.
  • At ˜Etot = 5 × 10^-4 Joule, OMA and NOMA are near 86%, and relay-assisted transmission exceeds NOMA by up to 30%.

APPENDIX A PROOF OF LEMMA 1

The appendix proves optimization properties using contradiction arguments, monotonicity, convexity, and derivative analysis. These results establish feasibility and structural behavior needed for the proposed resource-allocation solutions.

  • A contradiction argument shows that reallocating power while preserving total energy can improve the objective, ruling out a nonoptimal interior solution.
  • The proof uses monotonicity of Q(f(γ1, m1, D)) with respect to γ1 to construct a feasible solution with lower decoding error probability.
  • Under the stated condition, ˜J(mk) and J(mk) are convex, which implies convexity of g(mk).
  • The derivative analysis further establishes that g(mk) decreases monotonically when inequality (60) holds.
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