Set the Group Call participant cap

We must set a maximum participants cap K for Group Calls. You have telemetry at the call level: calls(call_id, start_ts, participants_count, video_on_ratio, median_mos_score, p95_jitter_ms, packet_loss_pct, bitrate_kbps, cpu_load_pct, network_type, region, device_tier, dropped_call_flag). Propose a statistical method to choose K that maximizes total successful participant‑minutes subject to guardrails: P95 median_mos_score ≥ 3.8 and dropped_call_rate ≤ 2%. Include: (a) how you’d fit and validate a piecewise/segmented regression or change‑point model of QoE vs participants_count; (b) how you’d compute uncertainty on K and report a 95% CI that is within ±1 participant; (c) whether to output a single global K or dynamic caps by device/network, and the decision rule to collapse to a single product limit; (d) how much data (number of calls) you need to estimate K with the required precision and how you’d check robustness to heavy‑tail outliers.