Limit Theorems for Queueing Systems with Doubly Stochastic Poisson Arrivals (Heavy Traffic Conditions)

We consider a single-server queueing system with a doubly stochastic Poisson arrival flow under heavy traffic conditions. We prove the convergence of the limiting stationary or periodic distribution to the exponential distribution. In a scheme of series, we consider the $C$-convergence of the waiting time process to a diffusion process with constant coefficients and reflection at the zero boundary. Examples of computation of the diffusion coefficient in terms of characteristics of the arrival flow and service time are given.