$\alpha $-stable limits for multiple channel queues in heavy traffic
We consider a sequence of renewal processes constructed from a sequence of random variables belonging to the domain of attraction of a stable law ($1<\alpha <2$). We show that this sequence is not tight in the Skorokhod $J_1$ topology but the convergence of some functionals of it is derived. Using the structure of the sample paths of the renewal process we derive the convergence in the Skorokhod $M_1$ topology to an $\alpha $-stable Lévy motion. This example leads to a weaker notion of weak convergence. As an application, we present limit theorems for multiple channel queues in heavy traffic. The convergence of the queue length process to a linear combination of $\alpha $-stable Lévy motions is derived.