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 $H_k / \barλ$ 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 $α=β=0.88$, $λ=2.25$ $-$ 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.

翻译：我们提出流式水库收敛定理（SRCT），这是一个针对多提供商自适应比特率流传输的新型数学框架，用于解决当前系统中的三个根本性结构缺陷：线性提供商探测、被动式故障转移以及冷备用过渡。SRCT 将流获取建模为一个并发水库填充问题——同时探测所有 $N$ 个提供商而非分批进行——并在活动流旁边维护 $k$ 个预先验证、预取备用的流，以实现亚秒级故障转移且用户零可见中断。我们证明了四个主要结果：（1）水库安全性的调和下界，表明 $k$ 个独立流提供 $H_k / \barλ$ 的预期正常运行时间，其中 $H_k$ 是第 $k$ 个调和数；（2）相对于分批探测的并发获取加速比 $S(N,b) = (N/b) \cdot (1-F^b)/(1-F^N)$，在实际中达到 $3$-$5$ 倍的改进；（3）在惰性填充下质量单调非递减，并收敛至帕累托最优前沿；（4）一个前景权重切换规则——使用 Kahneman-Tversky 价值函数，参数 $α=β=0.88$，$λ=2.25$——通过预期切换次数的无颠簸界，被证明能消除相似质量流之间的颠簸现象。我们在两条生产级流传输管道上实现了 SRCT：一个面向 12 个以上 HLS 提供商、配备 $k=3$ 个水库插槽的主要电影/电视系统，以及一个具有多格式 DASH/HLS 故障转移功能的体育直播系统。通过蒙特卡洛模拟（5000 次试验）进行的实证验证确认了所有四个定理，共涉及 22 项独立检查。含有 $k=3$ 个流的水库相对于单个流实现了 9.15 倍的平均耗尽时间，而在 40% 的失败率下对 12 个提供商进行并发探测相比当前默认按 3 个一批的探测方式实现了 4.27 倍的加速。