= Many-source queue scaling
{title2=$L^{-1}\log\mathbb P(Q_L>Lb)\to-\inf_{n\ge1}I_n(b+cn)$}
Scale independent source count, service and buffer by the same factor $L$. For lookback horizon $n$, the source-average cumulative input has <rate function> $I_n$, and reaching buffer level $b$ requires input above $b+cn$. Optimizing this horizon gives the displayed exponent under uniform control of the infinite stationary supremum. For independent slots, $I_n(x)=nI_A(x/n)$; discrete time keeps $n$ an integer. Temporal dependence instead enters through the cumulative-input <cumulant-generating function>.
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