Two limit theorems for queueing systems around the convergence of stochastic integrals with respect to renewal processes

Two limit theorems on asymptotic behaviors of some processes related to some queueing systems are investigated. In the first result (Theorem 1), sticky diffusions appear as limit processes for queues with vacations. In the second result (Theorem 2), limiting behavior of occupation times and counting processes related to open queueing networks is discussed. The core of the arguments for obtaining our results is to discuss the convergence of stochastic integrals with respect to renewal processes.