This study establishes limiting distributions for customer waiting times and queue lengths in treelike networks with single-server nodes. The main result characterizes the limiting distributions when the network data (interarrival times, service times and routes) is "asymptotically stationary." This is a weak condition covering a variety of networks including standard ones where the network data is stationary, regenerative, Markovian, satisfies coupling, and so on. The dependencies in the network data may be customer centered or node centered. The proof is based on two preliminary results that are of interest by themselves. The first one justifies the existence of the waiting time and queue length processes on the entire time axis for any network whose service capacity has been adequate to handle all the customers as one looks back to the "beginning of time." This is a sample-path generalization of a result of Loynes for a queueing system with stationary data. The second preliminary result is a characterization of functionals of sequences that preserve the asymptotic stationarity property. This is somewhat analogous to continuous-mapping principles for weak convergence. We also present functional central limit theorems for the waiting time processes in a network when the partial sums of the network data obey a heavy-traffic functional limit property. The limiting waiting time sequence is a functional of a process that is typically a multivariate Brownian motion, or a process with stationary increments and long range dependence such as a fractional Brownian motion.
"Treelike queueing networks: asymptotic stationarity and heavy traffic." Ann. Appl. Probab. 8 (2) 541 - 568, May 1998. https://doi.org/10.1214/aoap/1028903538