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Joint Scheduling of URLLC and eMBB Traffic in 5G Wireless Networks

Arjun Anand, Gustavo de Veciana, Sanjay Shakkottai

arXiv:1712.05344v2cs.NI

TL;DR

5G scheduling must simultaneously support high-rate eMBB and immediate, reliable URLLC delivery despite URLLC puncturing or superposition over ongoing eMBB transmissions. The paper formalizes joint scheduling under linear, convex, and threshold loss models and derives online policies with theoretical guarantees. Its key result is that linear loss permits a decomposition, while convex and threshold losses generally require joint optimization.

  • Problem

    The paper asks how to jointly schedule slot-scale eMBB allocations and minislot-scale URLLC overlaps while maximizing eMBB utility and immediately satisfying URLLC demands.

  • Method

    The paper characterizes feasible throughput regions and develops online joint schedulers for linear, convex, and threshold eMBB rate-loss models.

  • Results

    Linear loss permits an iterative gradient eMBB scheduler using expected URLLC loss and an URLLC scheduler oblivious to eMBB states and allocations; convex and threshold losses do not generally de-couple.

  • Takeaways & Limitations

    The dual scheduling objectives can be met through clean decomposition for linear loss, whereas more general loss models require joint optimization within the studied policy classes.

Abstract

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Emerging 5G systems will need to efficiently support both enhanced mobile broadband traffic (eMBB) and ultra-low-latency communications (URLLC) traffic. In these systems, time is divided into slots which are further sub-divided into minislots. From a scheduling perspective, eMBB resource allocations occur at slot boundaries, whereas to reduce latency URLLC traffic is pre-emptively overlapped at the minislot timescale, resulting in selective superposition/puncturing of eMBB allocations. This approach enables minimal URLLC latency at a potential rate loss to eMBB traffic. We study joint eMBB and URLLC schedulers for such systems, with the dual objectives of maximizing utility for eMBB traffic while immediately satisfying URLLC demands. For a linear rate loss model (loss to eMBB is linear in the amount of URLLC superposition/puncturing), we derive an optimal joint scheduler. Somewhat counter-intuitively, our results show that our dual objectives can be met by an iterative gradient scheduler for eMBB traffic that anticipates the expected loss from URLLC traffic, along with an URLLC demand scheduler that is oblivious to eMBB channel states, utility functions and allocation decisions of the eMBB scheduler. Next we consider a more general class of (convex/threshold) loss models and study optimal online joint eMBB/URLLC schedulers within the broad class of channel state dependent but minislot-homogeneous policies. A key observation is that unlike the linear rate loss model, for the convex and threshold rate loss models, optimal eMBB and URLLC scheduling decisions do not de-couple and joint optimization is necessary to satisfy the dual objectives. We validate the characteristics and benefits of our schedulers via simulation.

I. INTRODUCTION

5G systems must support high-rate eMBB and highly reliable, ultra-low-latency URLLC traffic through slot- and minislot-scale multiplexing. The paper formalizes joint scheduling under multiple eMBB rate-loss models and develops corresponding online algorithms and structural results.

  • eMBB requires gigabit-per-second rates with moderate latency, while URLLC targets 0.25–0.3 msec packet delays and 99.999% reliability.
  • eMBB allocations are fixed at slot beginnings, whereas URLLC packets are scheduled in the next minislot by superposition or puncturing.Slots are proposed to last one millisecond and minislots 0.125 msec; puncturing removes eMBB transmission power during overlap.
  • The central problem is jointly allocating eMBB resources at slot boundaries and placing stochastic URLLC overlaps at minislot boundaries while accounting for resulting eMBB rates.Overlap placement can affect the rates received by scheduled eMBB users during the slot.
  • The paper characterizes joint scheduling under linear, convex, and threshold eMBB rate-loss models and derives online scheduling policies for these settings.The framework includes capacity-region characterizations and algorithms with theoretical guarantees.
  • For linear loss, URLLC placement can be randomized uniformly while eMBB scheduling uses an iterative gradient algorithm that accounts for expected loss.For convex and threshold models, the paper studies restricted minislot-homogeneous policies and derives corresponding optimization methods.
  • The work addresses a gap in prior studies, which considered related resource allocation or rate expressions but did not design joint eMBB–URLLC scheduling algorithms using puncturing or superposition.

III. LINEAR MODEL FOR SUPERPOSITION/PUNCTURING

Under linear superposition/puncturing loss, joint scheduling may de-couple: an eMBB scheduler can account for expected degradation while URLLC placement ignores eMBB state and allocation decisions.

  • Optimal joint scheduling may protect the lower-rate user for fairness or place URLLC opportunistically on the better-channel user to improve total throughput.
  • With linear loss, the URLLC scheduler can be oblivious to channel states, utility functions, and actual eMBB rate allocations when the eMBB scheduler anticipates puncturing degradation.
  • The general dependence between eMBB scheduling and URLLC puncturing therefore admits a simpler decomposition under the linear loss model.

A. Characterization of capacity region

Under the linear loss model, uniform random URLLC placement characterizes the full feasible throughput region and supports a decomposed online utility scheduler. The eMBB scheduler accounts for expected puncturing loss, while URLLC placement can remain channel- and utility-oblivious.

  • Capacity-region characterization: The resulting capacity region is convex, closed, and bounded for any fixed URLLC load ρ ∈ (0, 1).Uniform placement scales the eMBB constraints by the factor (1 −ρ), preserving convexity.
  • Capacity-region characterization: Uniform random URLLC placement achieves the throughput of any feasible joint policy under the linear superposition/puncturing loss model.Theorem 1 states that the full capacity region equals the region achieved with uniform random placement.
  • Utility maximization: The utility problem can therefore be optimized over policies using uniform random URLLC placement rather than arbitrary joint placements.This reduction follows directly from the equality of the unrestricted and randomized capacity regions.
  • Utility maximization: The eMBB scheduler uses an iterative gradient method with the expected puncturing correction (1 −ρ), while URLLC traffic is placed uniformly at random in each minislot.Actual eMBB rates are fed back after each slot to update the scheduler’s rate estimates.
  • Utility maximization: An optimal solution can use URLLC placement that is oblivious to eMBB channel states, utilities, and allocation decisions.This provides a simple URLLC scheduling rule despite the apparent possibility of targeting lower-rate or lower-marginal-utility eMBB users.
  • Utility maximization: Random puncturing is not optimal with a static eMBB scheduler: at 50% URLLC load, opportunistic puncturing yields 0.875 packets/slot per user versus 0.75.The improvement depends critically on opportunistic eMBB scheduling, which operates on the Pareto frontier of achievable rates.

IV. CONVEX MODEL – MINISLOT-HOMOGENOUS POLICIES

For convex loss models, the paper studies channel-state-dependent policies whose eMBB allocations and URLLC placements are homogeneous across minislots. Feasibility couples their allocation parameters through a sharing constraint.

  • Policy class: The convex-loss analysis restricts attention to minislot-homogeneous eMBB/URLLC schedulers and uses a concavity condition for stochastic-approximation utility maximization.The policy class remains channel-state dependent while keeping allocations homogeneous across minislots.
  • eMBB allocations: Minislot-homogeneous eMBB allocations assign the same per-minislot resource share throughout an eMBB slot.The overall slot allocation is represented by the common allocation associated with the first minislot.
  • URLLC placements: URLLC placements are proportional to pre-specified weights that remain time-homogeneous across minislots.The weight matrix γ determines the induced URLLC load for each eMBB user and channel state.
  • Coupled feasibility: The eMBB and URLLC decisions are coupled because induced URLLC load cannot exceed the eMBB resources allocated to a user.This constraint applies almost surely across minislots.
  • Coupled feasibility: With a (1 −δ) URLLC sharing factor, feasible policies satisfy (1 −δ)γ ≤ φ, and the policy set ΠH,δ is convex.The sharing factor bounds peak URLLC demand below the full resource capacity, leaving resources available for eMBB traffic.

B. Characterization of the throughput region

The section characterizes throughput under minislot-homogeneous policies and formulates utility maximization as a convex optimization problem. An online stochastic-approximation scheduler asymptotically maximizes eMBB utility under the stated assumptions.

  • Throughput characterization: Minislot-homogeneous policies define feasible throughput regions for eMBB/URLLC scheduling, including their convex hull through time-sharing or randomization.The convex-hull rates are achievable by time-sharing among minislot-homogeneous policies.
  • Throughput characterization: Under Assumption 2, the achievable throughput region equals its convex hull, so minislot-homogeneous policy randomization is unnecessary.The result is stated as CH,δ = ˆCH,δ.
  • Optimization formulation: Concavity of the induced rate functions makes utility maximization over (φ, γ) a convex optimization problem with a concave objective and convex constraints.This formulation supports iterative updates of eMBB allocations and URLLC placement factors.
  • Online scheduler: The scheduler fluidizes bandwidth, allocates fraction ˜φu(t) to each eMBB user, and assigns URLLC traffic through the placement vector ˜γ(t).Received eMBB rates are fed back at the end of each slot for online updating.
  • Online scheduler: The stochastic-approximation online algorithm asymptotically maximizes eMBB utility and is optimal under Assumptions 3 and 2.Theorem 4 compares its average rate vector with the offline optimum r∗.

D. Optimality of Minislot-Homogeneous Policies

This section examines whether allowing URLLC placement to depend on minislot history improves scheduling. Under finite discrete demands and homogeneous convex losses, an optimal minislot-homogeneous placement policy exists.

  • Policy classes: Causal minislot-dependent policies can use the minislot index and previous URLLC demands, whereas minislot-homogeneous policies cannot.The broader policy class is potentially more capable but may require prohibitively large MDP state spaces.
  • Policy classes: A causal joint policy first selects eMBB allocation φπ,s, then determines minislot puncturing through history-dependent placement functions γπ,su,m(·).These functions must satisfy the policy constraints for every minislot and demand history.
  • Homogeneous losses: Homogeneous loss functions retain useful user- and state-dependent models while imposing a structural restriction on loss scaling.The paper introduces homogeneity as the key restriction used in the optimality result.
  • Optimality result: For finite discrete URLLC demand support and homogeneous convex eMBB losses, an optimal solution has minislot-homogeneous URLLC placement.Thus, history-dependent minislot placement is not required for optimality under these conditions.

E. Optimal eMBB Slot Slicing

The section compares time and frequency slicing of eMBB resources under convex puncturing losses. With equal total allocated resources and equal mean puncturing, frequency slicing yields lower expected eMBB loss.

  • Configurations: Time slicing assigns each eMBB user the full frequency band during its own subset of minislots, while frequency slicing shares bandwidth throughout the slot.Both configurations allocate the same total area in the time-frequency plane.
  • Puncturing variability: Both slicing configurations have the same mean total puncturing, but frequency slicing produces smaller variability in each user’s total puncturing.The puncturing totals differ because time slicing concentrates exposure in fewer minislots, whereas frequency slicing spreads it.
  • Optimal slicing: Under convex loss functions, the expected loss from puncturing is higher for time slicing than for frequency slicing.The result follows from the lower puncturing variability of frequency slicing under equal mean exposure.
  • Optimal slicing: The main design implication is to spread each eMBB user’s allocation over time while sharing frequency to mitigate convex puncturing losses.This recommendation concerns eMBB resource slicing under the section’s convex-loss setting.

V. THRESHOLD MODEL AND PLACEMENT POLICIES

The threshold-loss section studies structured URLLC placement policies that simplify joint scheduling. Resource-proportional and threshold-proportional placement yield tractable throughput characterizations and online algorithms, while threshold-proportional placement minimizes the probability of any eMBB loss for a fixed eMBB allocation.

  • Computational scope: Joint optimization can become computationally challenging as the number of users increases because the stochastic-approximation problem jointly optimizes φ and γ.This is identified as a computational limitation of the preceding algorithm.
  • Threshold model: The threshold model treats eMBB traffic as unaffected until a puncturing threshold is reached, after which throughput loss is complete.The section focuses on policies imposing structural conditions on the puncturing matrix γ.
  • Online algorithms: The added RP and TP structure enables simpler online algorithms, including a rate-based iterative gradient scheduler and, for some cases, one-dimensional searches.The rate-based scheduler replaces the linear-model factor with a user-dependent FD(αs) term.
  • Placement policies: Resource Proportional placement assigns URLLC demands in proportion to eMBB slot allocations, while Threshold Proportional placement uses users’ loss thresholds.RP is motivated as a deterministic analogue of random placement; TP aims to avoid losses.
  • Throughput regions: For RP and TP policies, the paper characterizes induced loss probabilities, user throughputs, and achievable throughput regions under time-homogeneous scheduling.Convexity or joint concavity conditions establish equality between each policy’s region and its convex hull.
  • Threshold Proportional placement: Threshold Proportional placement achieves the minimum probability of any eMBB loss among policies using the same eMBB resource allocation.The paper distinguishes minimizing loss probability from minimizing eMBB rate loss.

VI. SIMULATIONS

Simulations compare optimal joint scheduling with simpler placement policies under convex and threshold loss models, and examine the trade-off between eMBB utility and URLLC delay.

  • Convex loss model: RP performs poorly at increasing URLLC loads because its placement ignores eMBB sensitivity, causing repeated resource shifts toward already-punctured sensitive users.Sensitive users receive more bandwidth because of higher marginal utility, then receive more puncturing under RP.
  • Convex loss model: At ρ = 0.4, RP yields 15% lower robust-user throughput and nearly similar sensitive-user performance than the optimal algorithm.Under convex loss, RP degrades further as URLLC load increases.
  • Convex loss model: At ρ = 0.6, RP throughput is 35% lower for robust users and 26% lower for sensitive users than under the optimal algorithm.
  • Threshold loss model: Under the threshold model, TP Placement tracks the optimal policy very well.The comparison uses αs = 0.3 for 50% of eMBB states and αs = 0.7 for the remainder.
  • Utility-delay trade-off: Larger δ limits URLLC service per minislot but enlarges the eMBB feasible constraint set, producing higher eMBB utility while increasing the URLLC delay trade-off.The delay study measures the probability that URLLC traffic waits more than two minislots, or 0.25 msec.
  • Overall findings: The simulations support a joint scheduling framework with theoretical guarantees for multiplexed URLLC and eMBB traffic.

APPENDIX

The appendix establishes capacity-region and convexity properties used to restrict policies and analyze joint eMBB/URLLC scheduling.

  • Theorem 1: Any feasible policy can be transformed into one using randomized URLLC placement across minislots while achieving the same long-term throughputs.
  • Convex loss model: For convex loss functions, the perspective construction l(φ, γ) = φh(γ/φ) is jointly convex in its arguments.This convexity supports optimization over eMBB allocation and URLLC placement variables.
  • Threshold loss model: For threshold-based loss functions, the appendix applies the corresponding perspective-function argument to derive the stated result.

E. Proof of Theorem 4

The proof analyzes the online stochastic approximation algorithm through an associated differential equation and a Lyapunov function.

  • Proof strategy: The stochastic approximation algorithm is analyzed through a continuous-time differential equation approximating its trajectories for large t.
  • Stability: The differential equation is globally asymptotically stable, and trajectories initialized in CH,δ converge to the unique optimum r∗.
  • Lyapunov argument: The Lyapunov function L(x) = U(r∗) − U(x) is positive away from r∗ and has nonpositive drift along trajectories.Strict concavity of the utility functions gives uniqueness of the optimal point.
  • Conclusion: Applying the stochastic-approximation convergence theorem yields R(t) → r∗ almost surely.

F. Proof of Theorem 5

The proof establishes optimality by comparing a hypothetical non-causal problem with the causal problem, then removing dependence on individual minislot demands under Assumption 4. This yields an optimal minislot-homogeneous joint scheduler.

  • Hypothetical non-causal scenario: The proof first considers a non-causal setting where all minislot URLLC demands are revealed after eMBB allocation.The resulting URLLC placement may initially depend on the minislot index and the full demand realization.
  • Demand-independent placement: The optimal policy can therefore use the same minislot-homogeneous URLLC placement across minislots while preserving optimality.The construction begins from the total puncturing experienced by each eMBB user and applies the corresponding placement factor to every minislot.
  • Hypothetical non-causal scenario: Any feasible causal policy can be transformed into a minislot-homogeneous policy depending only on total URLLC demand, so the non-causal optimum upper-bounds the causal optimum.The transformation preserves feasibility under the joint scheduling constraints.
  • Conclusion: Combining the upper bound with the demand-independent construction proves that an optimal minislot-homogeneous policy exists for the original causal problem.The constructed policy attains the upper bound established through the non-causal formulation.
  • Demand-independent placement: Under Assumption 4, an optimal non-causal solution exists whose URLLC placement policy is independent of total URLLC demand.The proof constructs this independence by showing that the same placement satisfies the relevant K.K.T. conditions across demand values.

G. Proof of Theorem 6

The proof uses Jensen’s inequality for the convex-loss analysis and then shows that pooled threshold resources provide a lower bound attained by the threshold proportional strategy.

  • Convex loss model: Jensen’s inequality is applied to the rewritten expression for the convex-loss analysis.Because minislot demands are i.i.d., the resulting right-hand side equals the left-hand side, completing the argument.
  • Threshold loss model: For threshold losses, the probability of loss depends on minislot demands and users’ puncturing thresholds.Relaxing the sequential allocation constraint allows demands and thresholds to be pooled into a single superposition/puncturing resource pool.
  • Threshold loss model: The pooled-resource loss probability is a lower bound for every placement policy, and the threshold proportional strategy attains this bound.Thus, the threshold proportional strategy minimizes the probability of loss on a given eMBB slot.
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