= Solution
The indexing defect also affects the downstream claim. For the two inputs in part (b), now choose $\lambda<d<c$. The constant input has zero downstream <queue workload>, while the input with $a_0=c+1$ has
$$
Q_0(b,d)=c+1-d,\qquad Q_0(b,c)=1,\qquad R_0(b,c,d)=c-d>0.
$$
Their printed distance is zero, so the literal post-slot downstream <continuity> assertion is false as well.
In the corrected <weighted cumulative-input topology for a slotted queue>, apply the finite-window proof of part (b) separately with services $d$ and $c$. Both have strictly negative long-window drift because $\lambda<d<c$. There are a neighborhood of $a$ and finite constants $C_d,C_c$ such that
$$
|R_0(a,c,d)-R_0(b,c,d)|
\leq |Q_0(a,d)-Q_0(b,d)|+|Q_0(a,c)-Q_0(b,c)|
\leq(C_d+C_c)d_\#(a,b).
$$
This proves <continuity>, and indeed local <Lipschitz continuity>, for the intended downstream function.
Suppose a family of random arrival paths $a^{(L)}$ has a <large deviation principle> in this corrected input space, at speed $L$ with <good rate function> $\mathcal I$. The <contraction principle for large deviations> then gives the downstream <good rate function>
$$
\boxed{J_R(r)=\inf\{\mathcal I(a):Q_0(a,d)-Q_0(a,c)=r\}.}
$$
It is infinity for $r<0$. The path <large deviation principle>, its topology, and its subcritical domain must actually be established for the chosen stochastic model: a collection of finite-dimensional <large deviation principles> alone does not control infinitely long workload windows. Nor should upstream departures be assumed <independent>; the path contraction avoids that unjustified assumption.
Back to article page
