Weak stationary solution of a G/G/1/∞ queue controlled by IPA-based SA with constant stepsize

We consider a single-server queue with stationary ergodic input and parameterized service time distribution. The real parameter of service time distribution is dynamically updated according to the Robbins-Monro stochastic approximation (SA) procedure coupled with infinitesimal perturbation analysis (IPA) type estimates, where the stepsize in stochastic approximation is constant. For both the cases where the parameter is updated per every customer and per every fixed number of customers, it is shown that there exists a stationary solution in a weak sense and some properties of the weak stationary solutions are derived.