Large Deviations of the Waiting Time for Tandem Queueing Systems

We consider some queueing system with two sequential servers (a tandem queueing system). Let the ergodicity conditions be satisfied. In a stationary regime denote by $T_i$ the waiting time of the beginning of servicing at the $i$th, $i=1,2$, server. In the article we obtain some conditions for an integro-local version of the large deviation principle to hold for the vector $T=(T_1,T_2)$: given a square
$$ \Delta(x)=\bigl\{y=(y_1,y_2):x_i\le y_i<x_i+\Delta,\ i=1,2\bigr\}, $$
we have
$$ \lim_{|x|\to\infty,\,x/|x|\to\omega}\frac1{|x|}\ln{\mathbb P}\bigl(T\in\Delta(x)\bigr)=-{}\,\overline{\!D}(\omega), $$
with $|x|=(x_1^2+x_2^2)^{1/2}$ and ${}\,\overline{\!D}(\omega)$ the deviation function in explicit form.