Sample path large deviations for marked point processes in the many sources asymptotic with small buffers: Heavily and lightly loaded systems

Consider a queueing system fed by traffic from $N$ independent and identically distributed marked point processes. We establish several novel sample path large deviations results in the scaled uniform topology for such a system with a small buffer. This includes both the heavily loaded case (the load grows as $N\to \mathrm{\infty }$) and the previously unexplored lightly loaded case (the load vanishes as $N\to \mathrm{\infty }$); this latter case requires the introduction of novel speed scalings for such queueing systems. Alongside these sample path large deviations results, we introduce a new framework to explore the range of scalings in the many sources asymptotic for these systems.