Many-source queue scaling ($L^{-1}\log \mathbb{P}(Q_L>Lb)\to -{\inf }_{n\ge 1}{I}_n(b+cn)$)

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, $I_n(x)=n{I}_A(x/n)$; discrete time keeps $n$ an integer. Temporal dependence instead enters through the cumulative-input cumulant-generating function.