27-EFT.WP.Packets.Light v1.0 | Chapter 10 — Queueing, Delay & Conservation (Bursts / Guard Bits)

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Chapter 10 — Queueing, Delay & Conservation (Bursts / Guard Bits)

One-sentence goal: Establish a unified convention for queueing and delay on links and nodes (OCS/OPS) and a guard-bit conservation rule, with computable equations for configured vs measured dual forms, accompanying contracts, and manifest-ready interfaces.

I. Scope & Objects

1. Inputs
   • Framing & timebase: frame_spec = { T_epoch, T_f, T_slot, T_sym, T_guard } (see Chapter 3), unified clocks tau_mono / ts.
   • Traffic & burstiness: arrival parameters { λ_pkt, L_pkt } (packets) or { A, H } (circuit Erlang), token-bucket burst ( r, p, σ ).
   • Nodes & resources: service rate C (bit/s or sym/s), number of ports P, FDL set { d_i } (OPS), bufferless/buffered policy, guard-bit policy.
   • Path & switching: route and crossbar matrix (see Chapter 9), label label = ( lambda, k_sub, S_sel ) (see Chapter 5).
2. Outputs
   • Delay decomposition: lat_total = lat_prop + lat_ser + lat_sw + lat_q + lat_proc;
   • Conservation & budgets: T_guard_budget, utilization, collision/drop rates;
   • Dual-form gap: configured (model/simulation) vs measured (timestamps) discrepancy delta_form_queue;
   • Manifest: manifest.packet.queue.* plus a contract report.
3. Boundary
   Physical propagation/compensation: Chapters 2 / 6; switching/routing: Chapter 9. This chapter focuses on queueing, delay, and guard-bit conservation.

II. Terms & Variables

• Rates & arrivals: λ_pkt [1/s], E[L] [bit], utilization ρ = λ_pkt * E[L] / C; circuit offered load A (Erlangs).
• Service & buffering: μ = C / E[L] (mean service rate), buffer capacity K [pkt] or FDL set { d_i }.
• Delay components: propagation lat_prop; serialization lat_ser = L / C; switching & reconfiguration lat_sw; queueing lat_q; processing lat_proc.
• Guard bits: T_guard ≥ 0, conservation budget T_guard_budget.
• Token bucket / burst: ( r, p, σ ); network calculus ( σ, ρ ).
• Two forms: configured (analysis/simulation/network calculus) vs measured (timestamp differences), gap delta_form_queue.
• Dimensions: unit(lat_*) = "[T]", unit(ρ) = "1", unit(T_guard) = "[T]".

III. Postulates P610-*

• P610-1 (Two forms in parallel): Any published delay must include both configured and measured forms, and record delta_form_queue.
• P610-2 (Hierarchical conservation): T_slot = N_sym*T_sym + N_gap*T_guard must hold; across nodes, guard-bit consumption must not drive downstream T_guard below T_guard_min (see Chapter 3).
• P610-3 (Explicit measures): Time / event / set–domain integrals or counts are explicit: ( ∫_{t∈W} • dt ), ( ∑_{pkt∈W} • ).
• P610-4 (Dimensional compliance): All published quantities pass check_dim( y − f(x) ); any dB↔linear conversion is annotated in the manifest.
• P610-5 (Fail-Closed): If conservation/thresholds fail or the dual-form gap exceeds limits, enforce rate-limiting / guard-bit enlargement / bypass or rollback, and persist the strategy card.

IV. Minimal Equations S610-*

1. Mean queueing delay (stochastic models)
   • M/M/1:
     lat_q = ρ / ( μ (1 - ρ) ) = ( λ_pkt / μ ) / ( μ - λ_pkt );
     total delay lat_total = lat_prop + E[L]/C + lat_sw + lat_q + lat_proc.
   • M/D/1:
     lat_q = ( ρ^2 ) / ( 2 μ (1 - ρ) ).
   • Finite buffer M/M/1/K:
     blocking P_block = ( (1-ρ) ρ^K ) / ( 1 - ρ^{K+1} ); effective delays computed conditional on K.
2. OPS without buffer / with FDL (approximations)
   • No buffer, no conversion: single-output contention
     P_cont ≈ 1 - (1 - ρ / P_out)^{N_in}; drop rate ≈ P_cont.
   • With FDL { d_i }: schedulability P_sched = P(∃ d_i ≥ Δt_conflict), effective service μ_eff = μ • P_sched, ρ_eff = λ_pkt / μ_eff,
     lat_q ≈ ρ_eff / ( μ_eff (1 - ρ_eff) ), bound lat_FDL ≤ max(d_i).
3. Network calculus (deterministic bounds)
   • Arrival curve ( σ, ρ ) and service curve β(t) = R(t - T_0)^+:
     • Delay bound D_max ≤ σ / (R - ρ) + T_0 (with R > ρ);
     • Backlog bound B_max ≤ σ + ρ T_0.
   • Guard-bit budget: T_guard_budget ≥ D_max + jitter_margin.
4. Token-bucket shaping/policing
   Token bucket ( r, p ): instantaneous arrivals bounded by A(t) ≤ r t + σ; shaper output satisfies ( σ_out, r ), shaping delay bound W_shaper ≤ σ / ( C - r ).
5. End-to-end delay composition
   • lat_total = ∑_{links} ( lat_prop + E[L]/C + lat_sw + lat_q + lat_proc );
   • 95th-percentile composition: lat_total,p95 ≈ ⊕_i lat_i,p95 (conservative summation or convolution under independence; manifest records the convention).
6. Dual-form gap
   delta_form_queue = | lat_total^{config} - lat_total^{meas} |; publication gate: delta_form_queue ≤ tol_queue.

V. Metrology Pipeline M60-10 (Ready → Model → Measure → Verify → Persist)

• Ready: freeze frame_spec and RefCond; gather node/port/FDL/rate and queueing policies; configure tol_queue / T_guard_min.
• Model: choose analytic forms (M/M/1, M/D/1, M/M/1/K, OPS contention/FDL) and/or network calculus ( σ, ρ; R, T_0 ); derive lat_total^{config} and T_guard_budget.
• Measure: use timestamps/counters to collect lat_sw / lat_q / lat_total, P_cont / P_block, and the actual T_guard consumption.
• Verify: compute delta_form_queue; validate T_guard ≥ 2σ_J + ΔCD + ΔPMD (see Chapter 3), P_cont / P_block, ρ and resource usage; trigger strategy cards (rate-limit / enlarge guard / reschedule / bypass).
• Persist:
  manifest.packet.queue.* = { frame.hash, RefCond, model:{ type, params }, lat^{ config, meas }, delta_form_queue, ρ, P_cont / P_block, T_guard_budget, contracts.*, signature }.

VI. Contracts & Assertions C60-10x (Suggested Thresholds)

• C60-1001 (Dual-form gap): delta_form_queue_p95 ≤ tol_queue (typical tol_queue = 0.05 • lat_total^{config}).
• C60-1002 (Guard-bit conservation): T_guard ≥ 2σ_J + ΔCD + ΔPMD and T_guard ≥ T_guard_min.
• C60-1003 (Load & stability): ρ < 1 (or the calculus condition R > ρ); else rate-limit or scale out.
• C60-1004 (Contention / blocking): P_cont ≤ P_cont_max, P_block ≤ P_block_max.
• C60-1005 (Dimensional compliance): all lat_* / ρ / P_* / T_guard fields pass check_dim.
• C60-1006 (Freshness & coverage): measurement window and panel updates ≤ Δt_panel_max, coverage ≥ cov_min.

VII. Implementation Bindings I60-10* (interfaces, I/O, invariants)

• I60-101 queue_model(params:{ λ, E[L], C, K }) -> { lat_q, P_block }
• I60-102 ops_fdlsched(params:{ λ, μ, P_out, N_in, FDL, WC }) -> { P_cont, μ_eff, lat_q }
• I60-103 netcal_bound(arrival:{ σ, ρ }, service:{ R, T0 }) -> { D_max, B_max }
• I60-104 shaper_delay(token_bucket:{ r, σ }, C) -> { W_shaper }
• I60-105 compose_latency(links[]) -> { lat_total_{ mean, p95 } }
• I60-106 guard_budget(jitter, residuals:{ ΔCD, ΔPMD }) -> { T_guard_min }
• I60-107 measure_latency(ts_stream) -> { lat_prop, lat_ser, lat_sw, lat_q, lat_proc, lat_total }
• I60-108 compare_dual_queue(config, meas) -> delta_form_queue
• I60-109 emit_queue_manifest(results, policy) -> manifest.packet.queue
  Invariants: two_forms_present = true; check_dim(*) passes; frames and routes are traceable; rate-limit / guard-bit expansion actions persisted.

VIII. Cross-References

• Framing/alignment & guard bits: Chapter 3;
• Switching & routing: Chapter 9;
• Arrival-time harmonization: Chapter 8;
• Physical compensation & residuals: Chapter 6 (for ΔCD / ΔPMD in the conservation term);
• Metrology & dashboards: Chapter 11 (delay / load / contention panel fields).

IX. Quality & Risk Control

1. SLI / SLO: delta_form_queue_p95, lat_total_p95, ρ_p95, P_cont / P_block, T_guard_violation_rate, throughput_stability.
2. Fallback strategies:
   • ρ ≥ 1 or rising P_cont: rate-limit / scale out / reschedule;
   • Rising delta_form_queue: back-write model parameters or prefer measured form;
   • Guard shortfall: increase T_guard, reduce throughput, or defer scheduling;
   • SLA delay exceedance: priority / reserved slices or bypass.
3. Audit: models/parameters and measurement evidence, threshold changes, strategy-card execution logs, and the manifest signature chain with replay consistency.

Summary

• Centering on queueing/delay and guard-bit conservation, this chapter provides unified equations and publication rules across M/M/1, FDL, and network-calculus formulations.
• Through P610 / S610 / M60-10 / C60-10x / I60-10* and the manifest.packet.queue.* schema, queueing and delay become measurable, auditable, and rollback-ready.