Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 79 / 3 / e / 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 $a_0=c+1$ hasTheir 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 thatThis 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 functionIt 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.