25-EFT.WP.STG.Dynamics v1.0 | Chapter 14 — Runtime and Streaming (Windows / Caches / Panels)

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Chapter 14 — Runtime and Streaming (Windows / Caches / Panels)

One-Sentence Goal
Provide a low-latency, auditable streaming execution framework for dynamic estimation and forecasting on G = (V, E), unifying windowing, caching, backpressure, replay, and dashboarding, and enforce SLOs via contracts.

I. Scope and Objects

1. Objects
   • Streaming graph-dynamics pipeline: ingest → align → window → model eval → assimilation / uncertainty → publish / panel.
   • Time semantics: event time t_evt, processing time t_proc, monotone time base tau_mono (see …TimeBase v1.0).
   • Stateful operators: window aggregates, online filters (Ch. 12), and uncertainty estimators (Ch. 13).
2. Inputs
   Data stream { (TraceID, ts, y, meta) }, graph operators { L, A }, models and versions model.hash, window & watermark configs, cache & fallback policies, SLO targets.
3. Outputs
   Published x̂, covariance/uncertainty, window statistics, runtime panel panel.*, and manifest.stg.runtime.
4. Boundaries & constraints
   Resource ceilings (CPU/mem/bandwidth), max out-of-order lateness_max, replay window Δt_replay, and unit/dimension coherence.

II. Terms and Variables

• Windowing: Δt_win (width), Δt_slide (slide), W(t) (domain).
• Watermark: wm(t) with permitted out-of-order lateness_max.
• Cache: cache.cap, cache.ttl, cache.policy ∈ { LRU, LFU, ARC }, hit-rate h.
• Rates & latency: arrival λ, service μ, total latency L_total = L_ingest + L_window + L_model + L_emit.
• Backpressure indicators: queue depth q, drop_rate, replay_rate.
• Dual-form (stream): delta_form_stream = | metric_evt − metric_proc |.
• Panel KPIs: latency_p95, lag_evt, cpu_pct, mem_pct, ESS_p10 (PF), nis_p95 (KF/UKF).

III. Axioms P714-*

• P714-1 (Event-time priority) — Unless stated otherwise, aggregation and gating use event time t_evt; windows close when wm(t) crosses the boundary.
• P714-2 (Explicit windowing) — Every operator declares { type ∈ { tumbling, sliding, session }, Δt_win, Δt_slide } and an associative/commutative reducer.
• P714-3 (Cache contract) — Cache keys include TraceID | node | unit | RefCond | model.hash; cache.ttl and invalidation strategy are persisted.
• P714-4 (Backpressure safety) — If λ ≥ ρ • μ (with ρ < 1) holds for > T_bp, rate-limit/degrade or fallback to lightweight models.
• P714-5 (Parallel dual forms) — Key SLIs are computed in both event-time and processing-time; record delta_form_stream under contracts.
• P714-6 (Idempotency & dedup) — Use TraceID, seq, ts as the idempotent key; duplicate messages must be side-effect free.
• P714-7 (Version coherence) — L/A.hash, model.hash, H.hash must match the executing versions; otherwise, refuse publication.

IV. Minimal Equations S714-*

• S714-1 (Window reduction):
  R_W(t) = ( ∫_{τ ∈ W(t)} φ( y(τ), meta ) dτ ) or R_W(t) = ( Σ_{τ ∈ W(t)} φ(•) ), with W(t) = [ t − Δt_win, t ).
• S714-2 (Closing condition):
  Window W = [ T0, T1 ) closes when wm(t_proc) ≥ T1 − lateness_max; later arrivals go to a patch stream and are tagged.
• S714-3 (Stability criterion):
  Steady-state boundedness requires λ < μ; queue evolves q_{k+1} = max( 0, q_k + a_k − s_k ); E[q] bounded if E[a_k] < E[s_k].
• S714-4 (Token-bucket rate limit):
  Token rate r, bucket depth B; bursts ≤ B, mean throughput ≤ r; choose r = ρ • μ with ρ < 1.
• S714-5 (LRU hit approximation—Che’s):
  With independent request rates λ_i, characteristic time T_c solves Σ_i ( 1 − e^{−λ_i T_c} ) = cache.cap; then h_i ≈ 1 − e^{−λ_i T_c}, overall h = ( Σ λ_i h_i ) / ( Σ λ_i ).
• S714-6 (SLO decomposition):
  L_total = L_ingest + L_window + L_model + L_emit; budget B = { B_ing, B_win, B_mod, B_emit }; require p95(L_total) ≤ SLO.latency_p95.
• S714-7 (Dual-form gap):
  delta_form_stream = | R_W^{evt}(t) − R_W^{proc}(t) | ≤ tol_stream.
• S714-8 (Panel smoothing):
  EWMA_α(z)_k = α z_k + (1 − α) EWMA_α(z)_{k−1}, with default α = 2/(N+1) for a view-window of N.

V. Metrology Workflow M7-14 (Ready → Model/Estimate → Execute → Validate → Persist)

1. Ready
   • Fix time semantics (event/processing), wm(t) policy, lateness_max; define window families & reducers.
   • Bind L/A.hash, model.hash, RefCond/units; set SLO budget B and degrade/replay strategies.
2. Model / Estimate
   • Configure layered caches (RAM → local SSD → remote KV), size capacities and estimate hit-rate;
   • Choose rate-limiting & backpressure controls (token bucket / congestion window);
   • Define panel metrics and alert thresholds.
3. Execute
   • Ingest → align → window reduce → model calls (Ch. 12 KF/UKF/PF; Ch. 13 uncertainty) → publish;
   • Use wm(t) for closing and late-data patching; on cache miss, back-source or use degrade path.
4. Validate
   • Monitor latency_p95, drop_rate, delta_form_stream, ESS_p10 / nis_p95;
   • Enforce version & unit coherence; under backpressure, verify rate limiting and fallbacks.
5. Persist / Publish
   manifest.stg.runtime = { graph.hash, L/A.hash, model.hash, window: { type, Δt_win, Δt_slide, lateness_max }, watermark, cache: { cap, ttl, policy, h }, rate_limit: { r, B }, slo: { budget, BreachEvents }, deltas: { delta_form_stream_p95 }, panel.snapshot_id, RefCond, units }.

VI. Contracts & Assertions C70-14xx

• C70-1401 (Timely closing): Pr( window_close_time − T1 ≤ lateness_max ) ≥ 1 − α (suggest α = 0.01).
• C70-1402 (Dual-form stream gap): delta_form_stream_p95 ≤ tol_stream (default tol_stream = 0.05 • range(metric)).
• C70-1403 (SLO attainment): latency_p95 ≤ SLO.latency_p95 with consecutive-window success ≥ 0.99.
• C70-1404 (Backpressure safety): if q > q_high for > T_bp, must trigger rate_limit or degrade_mode, and drop_on_floor = false unless explicitly allowed.
• C70-1405 (Cache coherence): changes in model/operator hashes trigger cache.invalidate(selector); otherwise flag a consistency violation.
• C70-1406 (Idempotency & dedup): repeated TraceID produces stable result hash hash(out) = hash(ref).
• C70-1407 (Unit/dimension): check_dim( R_W ) = unit(y), check_dim( L_total ) = "[T]".

VII. Implementation Bindings I70-14*

• I70-141 build_window_op(type, Δt_win, Δt_slide, φ) -> WOP
• I70-142 update_watermark(stream_meta) -> wm(t)
• I70-143 reduce_window(WOP, batch) -> R_W, meta
• I70-144 cache_getput(key, val, policy, ttl) -> { val', hit }
• I70-145 rate_limit(token_bucket, now) -> permit
• I70-146 handle_backpressure(q, λ, μ, policy) -> action (action ∈ { shape, degrade, spool, reject })
• I70-147 panel_update(metrics) -> snapshot_id
• I70-148 hot_swap_model(model_new) -> ok (atomic swap + rollback point)
• I70-149 replay_gap_fill(range, source) -> patches
• I70-14A assert_runtime_contracts(state, rules) -> report
• I70-14B emit_runtime_manifest(results, policy) -> manifest.stg.runtime

Invariants: non_decreasing(tau_mono); monotone wm(t); cache.cap > 0; drop_rate ≤ policy.max_drop; versions/units traceable.

VIII. Cross-References

• Assimilation & filtering: Ch. 12 (real-time x̂, P and residual/NIS panels).
• Uncertainty & guardbands: Ch. 13 (online u_c and guardband alerting).
• Numerical integration & stability: Ch. 9 (runtime step switching & stiffness control).
• PathCorrection / TimeBase / Sync: time-base alignment and offset / skew / J logging & correction.

IX. Quality & Risk Control

1. Suggested SLI/SLOs
   • latency_p95 ≤ 200 ms, lag_evt_p95 ≤ 1 • Δt_win, delta_form_stream_p95 ≤ tol_stream;
   • drop_rate ≤ 0.1%, replay_gap_covered ≥ 99.5%, cache_hit ≥ 80% (hot-key workloads).
2. Fallbacks
   • Rate shaping: set r = 0.8 • μ and increase B.
   • Model degrade: KF ← UKF ← PF staged fallback; disable part of uncertainty branches to preserve core latency.
   • Storage degrade: switch to local ring buffer and batch windowing.
   • Replay repair: asynchronously patch windows for t ∈ [ now − Δt_replay, now ) and republish corrections.
3. Audit
   Persist watermark trajectories, lateness histograms, q(t), SLO-breach logs, hot-key distributions and cache-hit curves, panel snapshots, and manifest.stg.runtime hashes.

Summary
This chapter unifies time semantics → windows → caches → backpressure → panels for streaming execution. With P714-* / S714-* / C70-14xx / I70-14* / M7-14, runtime behavior becomes contracted and traceable, ensuring stable publication of x̂ / u_c under resource limits and out-of-order arrivals—and providing a plug-and-play framework for the use cases in Ch. 15.