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Bound Predicts 5G Delay Violations Under Real HARQ Timing

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2026-07-08 14:23 UTC pith:7MVMTGDP

load-bearing objection Tractable DVP bound for 5G HARQ with practical timing — mostly solid, with one load-bearing approximation worth probing the 3 major comments →

arxiv 2607.06169 v1 pith:7MVMTGDP submitted 2026-07-07 cs.IT cs.NIcs.SYeess.SYmath.IT

Delay Violation Probability Modeling for 5G Systems with HARQ Operation

Sangwon Seo , Vishnu N Moothedath , Niloofar Mehrnia , Neda Petreska , Bernhard Kloiber , James Gross This is my paper
classification cs.IT cs.NIcs.SYeess.SYmath.IT
keywords delay violation probabilityHARQ5G NRqueueing theoryMarkov chainultra-reliable low-latency communicationcontrol signalingquality of service
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper derives a tractable upper bound on the delay violation probability (DVP) for 5G systems using HARQ retransmissions, where DVP is the probability that a packet's end-to-end delay exceeds a target threshold. The central contribution is a decomposition of total delay into a waiting component (from queueing and periodic control-signal blockage) and a service component (from transmissions, decoding delay, feedback delay, and retransmissions), combined into a single bound via a discrete-time Markov chain that captures the stationary queue backlog. The key mechanism is a recursive algorithm that computes the waiting-delay distribution conditioned on the number of packets ahead, weighted by the Markov chain's stationary distribution, and then convolved with a service-delay violation probability that explicitly accounts for the round-trip time between transmission attempts (one transmission slot plus decoding and feedback delays) and the periodic loss of data slots to control signaling. The resulting bound in Theorem 2 is parameterized by arrival rate, queue size, control-signal periodicity, decoding and feedback delays, maximum retransmission attempts, and per-attempt packet error rates, and is validated against both MATLAB queueing simulations and ns-3 5G-LENA packet-level simulations across varying delay targets, SNRs, arrival rates, and control-signal periodicities.

Core claim

The paper's central result is Theorem 2 (Eq. 21), which provides a closed-form upper bound on the total DVP by summing over all possible waiting-delay values k, weighting each by the waiting-delay PMF f_{τw}(k) computed from the Markov chain stationary distribution and Algorithm 1, and multiplying by a service-delay violation probability that splits into a no-blockage case and a control-signal-blockage case. The bound reveals that under stringent latency targets, DVP exhibits a floor dominated by protocol timing (decoding, feedback, control-signal blockage) rather than physical-layer reliability, meaning that increasing signal power or radio resources beyond a certain point yields no further

What carries the argument

Discrete-time Markov chain over (queue backlog, HARQ attempt index, control-signal phase); recursive waiting-delay PMF computation (Algorithm 1); decomposition of DVP into waiting and service components; control-signal blockage approximated by union bound β_k = min(1, k/ξ) for waiting and α = min(1, M/ξ) for service

Load-bearing premise

The bound assumes that at most one data slot is lost to control signaling during a packet's service period, reducing the effective delay budget by one slot. When the service window spans multiple control-signal cycles (which can happen with several retransmissions or a large delay target), more than one blocked slot may occur, and the single-slot reduction may underestimate the true delay impact.

What would settle it

If ns-3 5G-LENA simulations under configurations with small control-signal periodicity ξ relative to the delay target show DVP values consistently exceeding the proposed upper bound, the bound would be violated and the single-slot blockage approximation invalidated.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Network operators can use the bound to determine minimum radio resource allocations (resource blocks) needed to meet a target DVP without running full system simulations, enabling faster QoS-aware provisioning.
  • The delay floor phenomenon identified by the model implies that for ultra-low-latency URLLC services, reducing decoding or feedback delays (e.g., via faster processing or shorter feedback timing) may be more effective than improving channel coding or SNR.
  • The framework can inform control-signal periodicity configuration (ξ): the model quantifies exactly how much DVP degrades when control signals are more frequent, supporting a trade-off between signaling overhead and latency guarantees.
  • The Markov chain structure is extensible to other periodic blockage patterns (e.g., TDD guard periods or measurement gaps) by modifying the phase variable, broadening applicability beyond control signaling.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The single-slot reduction for control-signal blockage during service (Eq. 15) could be tightened by allowing multiple blocked slots, which would improve the bound's accuracy when the delay target is large relative to ξ or when multiple HARQ retransmissions extend the service window across several control-signal cycles.
  • The framework could be extended to non-Bernoulli arrivals (e.g., periodic or bursty traffic typical of industrial control loops) by modifying the Markov chain transition probabilities, potentially revealing different DVP floors for different traffic patterns.
  • The delay floor suggests a natural design principle: there exists a minimum achievable DVP for a given protocol timing configuration, and below that floor, no amount of physical-layer improvement helps — a regime boundary that could be computed in closed form from the model parameters.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. This paper proposes a tractable upper bound on the delay violation probability (DVP) for slot-based 5G systems with HARQ. The framework decomposes total delay into waiting delay (from arrival to first transmission) and service delay (from first transmission to success or discard). The waiting-delay PMF is derived from a discrete-time Markov chain (DTMC) over queue length, HARQ attempt index, and control signal periodicity (CSP) phase, combined with a recursive algorithm (Algorithm 1). The service-delay bound accounts for decoding delay, feedback delay, retransmission attempts, and periodic CS blockage. The final DVP bound (Theorem 2) combines both components and is validated against a MATLAB queueing simulation and ns-3 5G-LENA packet-level simulations across arrival rates, SNRs, CSPs, and delay targets.

Significance. The paper addresses a practically important problem for URLLC DVP prediction in 5G HARQ systems. The derivation from the DTMC steady-state distribution through the waiting-delay PMF to the service-DVP bound is logically coherent, with clean proofs in Appendices A and B. The bound is parameter-free given system inputs (f, Qmax, xi, tau_d, tau_f, M, p). The ns-3 5G-LENA validation is comprehensive (Figs. 4-6, Appendix C) and the bound is consistently tight across tested scenarios. Reproducible code is provided via GitHub. The inclusion of decoding/feedback timing and CS blockage in a unified DVP framework is a useful contribution over prior work that assumes instantaneous feedback or omits these components.

major comments (3)
  1. Section III.A, Table II, Contribution 1: The paper claims to 'model the parallel HARQ processes using a multi-server queue,' but the DTMC state (q, m, k_xi) tracks a single packet's attempt index m, not the states of multiple in-flight HARQ processes. The transitions (e.g., (q,m) -> (q, m+1) with probability I*f'*p_m) assume that a failed packet is immediately ready for retransmission in the next slot, which corresponds to same-slot (instantaneous) feedback. This contradicts the non-zero RTT = 1 + tau_d + tau_f (Eq. 12) that is explicitly modeled in the service delay (Section III.C). The parallel HARQ behavior is captured only in the service-delay timing analysis, not in the queue dynamics that determine the waiting-delay distribution. The paper should clarify what the 'multi-server queue' refers to, explicitly state this simplification, and discuss its implications for the DVP bound's紧度
  2. Section IV: The ns-3 validation covers arrival rates f^{-1} in {3, 6, 8, 10}, all relatively low. With RTT = 2 slots (tau_d = tau_f = 0.5 ms), the expected number of in-flight packets at f = 1/3 is approximately f * RTT = 2/3, which is insufficient to test the regime where in-flight packets saturate available HARQ processes and constrain new transmissions. Since the DTMC does not track in-flight packets, the bound's validity in higher-load regimes (e.g., f = 1/2 or f = 1) is untested. Additional validation at higher arrival rates would strengthen the generality claim.
  3. Section III.A, Table II: The DTMC's attempt index m tracks only the packet currently being transmitted, not the states of other packets awaiting feedback in parallel HARQ processes. The scheduler's next action depends on which processes have pending retransmissions (the paper states retransmissions have priority), but this scheduling dependency on multiple in-flight process states is not captured. The paper should discuss whether and how this affects the stationary queue-length distribution pi_q (Eq. 8) and the resulting waiting-delay PMF.
minor comments (6)
  1. Eq. (9): The text states the bound is 'accurate when q << xi,' but beta_k depends on k (the waiting interval length), not q. This should read 'accurate when k << xi.'
  2. The paper should specify the number of HARQ processes assumed in the ns-3 5G-LENA simulations, as this directly affects the parallel HARQ behavior being modeled.
  3. Figure 2 is difficult to read; consider enlarging or simplifying the timing diagram.
  4. References [40], [44], [45] are to very recent or forthcoming work (2025); ensure final versions are properly cited.
  5. In Table II, the notation f' and p'_m are defined but the prime notation for p'_m = 1 - p_m could be confused with the attempt index; a brief reminder in the table caption would help.
  6. Section IV.A states the MATLAB validation 'focuses on the modeled queueing dynamics rather than the detailed 5G protocol stack.' It would be helpful to explicitly state that the MATLAB simulation implements the same timing assumptions as the DTMC (including instantaneous feedback), so readers understand that the ns-3 validation is the independent test.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful and substantive review. The referee's three major comments center on a common and valid observation: the DTMC in Section III.A does not track the states of multiple in-flight HARQ processes, and the transition structure implicitly assumes same-slot feedback, which is inconsistent with the non-zero RTT modeled in the service-delay analysis. We agree that the terminology 'multi-server queue' is imprecise and that this simplification must be explicitly stated and discussed. We also agree that the validation is limited to relatively low arrival rates. Below we respond to each comment in detail.

read point-by-point responses

  1. Referee: Section III.A, Table II, Contribution 1: The paper claims to 'model the parallel HARQ processes using a multi-server queue,' but the DTMC state (q, m, k_xi) tracks a single packet's attempt index m, not the states of multiple in-flight HARQ processes. The transitions assume that a failed packet is immediately ready for retransmission in the next slot, which corresponds to same-slot (instantaneous) feedback. This contradicts the non-zero RTT = 1 + tau_d + tau_f (Eq. 12) that is explicitly modeled in the service delay (Section III.C). The parallel HARQ behavior is captured only in the service-delay timing analysis, not in the queue dynamics that determine the waiting-delay distribution. The paper should clarify what the 'multi-server queue' refers to, explicitly state this simplification, and discuss its implications for the DVP bound's tightness.

    Authors: The referee is correct on all counts. We acknowledge the following: (1) The DTMC state (q, m, k_ξ) tracks the attempt index of the single packet currently being served by the scheduler, not the states of all in-flight HARQ processes. (2) The transitions in Table II, e.g., (q, m) → (q, m+1) with probability I(k_ξ)·f'·p_m, implicitly assume that the outcome of the m-th attempt is known by the next slot, i.e., same-slot feedback. This is inconsistent with the non-zero RTT = 1 + τ_d + τ_f modeled in Section III.C. (3) The term 'multi-server queue' in Contribution 1 is therefore misleading. The parallel HARQ behavior—specifically, the fact that new packets can be transmitted while earlier ones await feedback—is captured only in the service-delay timing analysis (Section III.C), not in the queue dynamics that determine the waiting-delay distribution π_q and the PMF f_τw(k). We will revise the manuscript as follows: (a) Replace the 'multi-server queue' terminology with an accurate description of what the DTMC models—a single-server queue with an attempt-index-augmented state that approximates pipelined service by allowing the attempt index to advance without blocking on feedback. (b) Add an explicit paragraph in Section III.A stating the simplification: the DTMC assumes same-slot feedback for queue dynamics, while the full RTT is accounted for in the service-delay bound. (c) Discuss the implications: at low arrival rates (f ≪ 1/RTT), the expected number of in-flight packets is small, so the queue dynamics are dominated by arrivals and occasional retransmissions rather than by pipeline saturation. In this regime, the simplification has minimal impact on π_q and hence on f_τw(k), and the service-delay analysis (which does account for the full RTT) dominates the DVP bound. At更高载s revision: yes

  2. Referee: Section IV: The ns-3 validation covers arrival rates f^{-1} in {3, 6, 8, 10}, all relatively low. With RTT = 2 slots (tau_d = tau_f = 0.5 ms), the expected number of in-flight packets at f = 1/3 is approximately f * RTT = 2/3, which is insufficient to test the regime where in-flight packets saturate available HARQ processes and constrain new transmissions. Since the DTMC does not track in-flight packets, the bound's validity in higher-load regimes (e.g., f = 1/2 or f = 1) is untested. Additional validation at higher arrival rates would strengthen the generality claim.

    Authors: The referee is correct that the tested arrival rates correspond to low in-flight packet counts and that higher-load validation would strengthen the paper. We will add ns-3 5G-LENA simulation results for f^{-1} = 2 (i.e., f = 1/2) and, if feasible within the revision timeline, f^{-1} = 1 (i.e., f = 1, every slot has an arrival). At f = 1/2 with RTT = 2, the expected in-flight count is approximately 1, which begins to exercise the pipeline. At f = 1, the pipeline is fully saturated and the simplification in the DTMC is most stressed. We expect the bound to remain valid (as an upper bound) because the service-delay analysis conservatively accounts for RTT and CS blockage, but we will report the actual gap between the bound and simulation at these higher loads. If the bound becomes loose at high load, we will explicitly state the load regime where the model is recommended. We will also add a remark in Section IV acknowledging that the model's tightness is expected to degrade as the arrival rate approaches 1/RTT, since the DTMC simplification becomes more impactful when multiple packets are simultaneously in flight. revision: partial

  3. Referee: Section III.A, Table II: The DTMC's attempt index m tracks only the packet currently being transmitted, not the states of other packets awaiting feedback in parallel HARQ processes. The scheduler's next action depends on which processes have pending retransmissions (the paper states retransmissions have priority), but this scheduling dependency on multiple in-flight process states is not captured. The paper should discuss whether and how this affects the stationary queue-length distribution pi_q (Eq. 8) and the resulting waiting-delay PMF.

    Authors: This comment is closely related to Comment 1 and identifies the same underlying issue from the scheduling perspective. The referee is correct: the scheduler's retransmission-priority policy depends on the states of all in-flight HARQ processes, but the DTMC collapses this into a single attempt index m for the currently served packet. This means the DTMC cannot distinguish between, for example, a slot where a retransmission is prioritized because a NACK arrived for an earlier packet versus a slot where a new packet is transmitted because no retransmission is pending. The practical effect on π_q is as follows: when the system has few in-flight packets (low f), retransmissions are infrequent and the priority policy rarely changes the scheduling decision, so the impact on π_q is negligible. As f increases and multiple packets are in flight, the retransmission-priority policy can delay new transmissions, increasing the queue backlog beyond what the DTMC predicts. This would cause the DTMC to underestimate π_q at larger q values, which in turn underestimates the waiting delay and could make the bound tighter (i.e., lower) than it should be—potentially compromising the upper-bound property. We will add a discussion of this effect in Section III.A, explicitly stating that the bound's validity as an upper bound is expected to hold in the low-to-moderate load regime (f · RTT ≪ number of HARQ processes) and that the simplification becomes increasingly conservative at higher loads. We will also note that the ns-3 validation (which uses the full retransmission-priority scheduler) serves as an empirical check: if the bound remains above the simulation DVP at higher arrival rates (per Comment 2), this provides evidence that the simplification does not violate the upper-bound property. revision: yes

Circularity Check

0 steps flagged

No significant circularity; the DVP bound is derived from system parameters and externally-obtained PER values, not from the DVP data being predicted.

full rationale

The paper's derivation chain is self-contained against external benchmarks. Theorem 2 (Eq. 21) is derived from: (1) a DTMC (Table II) constructed from system parameters (f, Qmax, ξ, M) and the PER vector p, yielding stationary distribution π_q (Eq. 8); (2) Algorithm 1, which computes the conditional waiting-delay PMF from p and queue state q; (3) service-delay bounds (Eqs. 13–18) computed from τd, τf, ξ, and p; and (4) standard probability conditioning (Appendix B). The PER vector p (Eq. 4) is obtained from ns-3 5G-LENA link-level simulation — a different quantity from the DVP being predicted. The DVP bound is then validated against independent ns-3 5G-LENA end-to-end simulations (Figs. 3–8). The only self-citation is [45] (overlapping authors: Moothedath, Seo, Petreska, Kloiber, Gross), which provides the PER abstraction Φ(·). However, this is an input to the model, not the target result, and the paper explicitly notes PER can also come from [34], [35] (Polyanskiy et al., non-self-citations) or measurements. The self-citation is not load-bearing for the central DVP bound. No step in the derivation reduces to its own inputs by construction. The skeptic's concern about the DTMC not tracking in-flight packets is a modeling correctness issue, not a circularity issue.

Axiom & Free-Parameter Ledger

7 free parameters · 6 axioms · 0 invented entities

No new physical or mathematical entities are invented. The model uses standard queueing-theoretic constructs (DTMC, multi-server queue abstraction) and standard 5G protocol elements (HARQ processes, RBs, CSP).

free parameters (7)
  • PER vector p = [p1, ..., pM] = Obtained from link-level simulation (5G-LENA) or analytical FBL approximations
    Input to the DVP bound; not fitted to DVP data but obtained externally from link-level simulation per Eq. 4
  • f (arrival rate) = Default 1/3
    System parameter, not fitted to the DVP result
  • ξ (CSP) = Default 80
    System parameter from 3GPP configuration
  • τd (decoding delay) = 0.5 ms
    System parameter
  • τf (feedback delay) = 0.5 ms
    System parameter
  • M (max transmission attempts) = Not specified numerically in Table III
    System parameter
  • Qmax (queue size) = 16
    System parameter
axioms (6)
  • domain assumption Packet arrivals are i.i.d. Bernoulli with probability f per slot
    Section II: 'Packet arrivals are modeled as an independent and identically distributed (i.i.d.) Bernoulli process.' This is a standard simplification for periodic traffic.
  • domain assumption NACKs for failed packets are always delivered successfully in the downlink
    Section II: 'We assume that NACKs for failed packets are always delivered successfully in the downlink [43].' This eliminates feedback error from the model.
  • domain assumption At most one transmission per data slot
    Section II: 'We assume at most one transmission per (data) slot.' This is consistent with single-UE scheduling.
  • ad hoc to paper CS blockage bound β_k = min(1, k/ξ) is accurate when q ≪ ξ
    Eq. (9) and surrounding text: 'which follows from a union bound and is accurate when q ≪ ξ.' This is a simplification specific to this paper's derivation.
  • ad hoc to paper At most one blocked slot occurs during the service period, reducing the delay target by one slot
    Eq. (15): M^{CS}_{d̄} uses d̄ + τf − 1 in the numerator, assuming a single blocked slot. The bound α_{d̄−k} = min(1, M^{noCS}_{d̄−k}/ξ) in Eq. (19) upper-bounds the probability of this event.
  • domain assumption Retransmission has priority over new transmissions when pending
    Section II: 'We additionally give priority to schedule the retransmission if one or more HARQ processes have a pending retransmission.' This scheduling policy is assumed throughout.

pith-pipeline@v1.1.0-glm · 20306 in / 4260 out tokens · 288120 ms · 2026-07-08T14:23:45.506527+00:00 · methodology

0 comments
read the original abstract

Meeting the growing demand for quality-of-service (QoS) guarantees in 5G networks requires an accurate characterization of delay performance, commonly captured by the delay violation probability (DVP) at a specified delay target. Although hybrid automatic repeat request (HARQ) is a fundamental reliability mechanism in wireless systems and is central to supporting QoS, many existing approaches to DVP prediction for HARQ remain overly simplified. In particular, they omit important delay components and adopt assumptions that do not reflect the operation of HARQ in slot-based systems such as 5G. Consequently, these models can substantially underestimate the DVP, especially under stringent latency requirements, where the contribution of the neglected components becomes critical. To address this gap, we develop a tractable DVP characterization for 5G HARQ that accounts for queueing, transmission, decoding, and feedback delay, as well as the contribution of Control Signaling (CS) transmissions to the overall delay, under practical timing assumptions consistent with 3GPP operation. Moreover, we incorporate parallel packet transmissions that proceed without waiting for earlier packets to succeed, an essential HARQ behavior frequently overlooked in prior work. Using tools from queueing theory and Markov analysis, we then derive upper bounds on the DVP and validate them against ns-3 5G-LENA simulations.

Figures

Figures reproduced from arXiv: 2607.06169 by Bernhard Kloiber, James Gross, Neda Petreska, Niloofar Mehrnia, Sangwon Seo, Vishnu N Moothedath.

Figure 1
Figure 1. Figure 1: System model, showing the retransmission process. Different delay components are shown where the packets experience them [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 4
Figure 4. Figure 4: DVP vs. allocated NRB per slot for different arrival rates f. B. Performance Comparison and Impact Analysis 1) Impact of the Offered Load (Arrival Rate f) [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 3
Figure 3. Figure 3: Validation of the Proposed model against queueing simula￾tion. constraints. In both Figures 3a and 3b, the Proposed model remains identical to or slightly above the simulation results over the NRB range. For the more stringent delay target d = 2, the two results almost overlap, indicating that the Proposed model provides a very tight upper bound while characterizing both the sharp DVP decrease in the low-N… view at source ↗
Figure 5
Figure 5. Figure 5: DVP vs. allocated NRB per slot for different SNR γ. 2) Impact of Link Quality (SNR γ) [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: DVP vs. allocated NRB per slot for different arrival rates f. 5 10 15 20 25 PRB per slot 1 10−1 10−2 10−3 10−4 10−5 DVP γ=10 γ=20 5G Simulation Proposed Max Throughput Fixed Tx Rate (a) d = 3 ms. 5 10 15 20 25 PRB per slot 1 10−1 10−2 10−3 10−4 10−5 DVP γ=10 γ=20 5G Simulation Proposed Max Throughput Fixed Tx Rate (b) d = 10 ms. 5 10 15 20 25 PRB per slot 1 10−1 10−2 10−3 10−4 10−5 DVP ξ=80 ξ=320 No ξ 5G S… view at source ↗
Figure 8
Figure 8. Figure 8: DVP vs. allocated NRB per slot for different parameters: (a) and (b) SNR γ, (c) and (d) different CSP ξ. Appendix B Proof of Theorem 2 Since τΣ = τw + τs, conditioning on the waiting delay yields P(τΣ > ¯d) = ∑ ⌊d¯⌋ k=0 P(τw = k) P(τs > ¯d − k | τw = k). Applying Lemma 1 and using P(τw = k) = fτw (k) given in (11) gives P(τΣ > ¯d)≤ ∑ ⌊d¯⌋ k=0 fτw (k)   ( 1−αd¯−k ) MnoCS ∏d¯−k i=1 pi+αd¯−k MCS ∏d¯−k i=1 p… view at source ↗

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