The Streaming Reservoir Convergence Theorem: A Prospect-Theoretic Framework for Multi-Provider Adaptive Streaming

We present the Streaming Reservoir Convergence Theorem (SRCT), a novel mathematical framework for multi-provider adaptive bitrate streaming that addresses three fundamental structural weaknesses in current systems: linear provider probing, reactive failover, and cold standby transitions. SRCT models stream acquisition as a concurrent reservoir filling problem$-$probing all $N$ providers simultaneously rather than in batches$-$and maintains $k$ pre-verified, pre-fetched standby streams alongside the active stream to enable sub-second failover with zero user-visible disruption. We prove four principal results: (1) a harmonic lower bound on reservoir safety showing that $k$ independent streams provide $H_k / \bar{\lambda}$ expected uptime where $H_k$ is the $k$-th harmonic number; (2) a concurrent acquisition speedup $S(N,b) = (N/b) \cdot (1-F^b)/(1-F^N)$ over batched probing, yielding $3$-$5\times$ practical improvement; (3) monotonic non-decreasing quality under lazy-refill with convergence to the Pareto-optimal frontier; and (4) a prospect-weighted switching rule$-$using Kahneman-Tversky value functions with $\alpha=\beta=0.88$, $\lambda=2.25$ $-$ that provably eliminates thrashing between similar-quality streams via a no-thrash bound on the expected switch count. We implement SRCT across two production streaming pipelines: a primary movie/TV system serving 12+ HLS providers with $k=3$ reservoir slots, and a live sports system with multi-format DASH/HLS failover. Empirical verification via Monte Carlo simulation (5000 trials) confirms all four theorems across 22 independent checks. The reservoir of $k=3$ streams achieves $9.15\times$ mean time to depletion versus a single stream, and concurrent probing of 12 providers at 40% failure rate yields a $4.27\times$ speedup over the current batched-by-3 default.