Sample path large deviations for single and multi class queues in the many sources asymptotic
Toh, Bemsibom C.
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In this thesis we consider prove large deviations results for two kinds of queuing systems. In the first case, we consider a queuing system fed by traffic from N independent and identically distributed marked point processes. We establish novel one-dimensional large deviations results for such a system in the previously unexplored lightly loaded case (the load vanishes as N → ∞). This case requires the introduction of novel speed scalings for such queueing systems. We also prove some important properties about the sample paths of such systems in the scaled uniform topology. However, we are unable to prove sample path large deviations principles in this case because the log moment-generating function in this case is not steep, and we are unable to find tools in the literature that enable us to deal with such scenarios. This part of the work is done using the framework introduced by Cruise  and Cruise et al.  to explore this scaling. In the second case, we consider a two-class queuing network, with each class fed by traffic from N independent and identically distributed marked point processes. We introduce a new, probabilistic interpretation of state-space collapse, and show that under a given scaling of the system, the probability of the vector of stationary queue lengths being a given distance from the identity line in R2 decreases exponentially as the distance increases, and therefore the most likely sample paths are those which stay close to the identity line in R2.