Sojourn time asymptotics in a parking lot network

For a two-class two-node bandwidth sharing network called parking lot network we investigate the tail behavior of the queue length and sojourn time under light-tailed assumptions. These results extend previous results in the literature obtained for a single-node network. Explicit conditions are given that indicate whether congestion at the second node influences the large deviations behavior or not. To overcome the complexities that arise when moving away from the single node case, we rely on recent results on overloaded bandwidth sharing networks obtained by Borst et al. (2009), and a comparison with the modified proportional fairness discipline, as introduced by Massoulié (2007). Specifically, our results include upper bounds for the distribution of the number of flows in the network, finiteness of the moment generating function of the workload, and large-deviations asymptotics for the sojourn time assuming flow size distributions having a bounded hazard rate.