Duality relations for queues with arrival and service control

We consider finite buffered queues with service or arrival control. In the case of service control, service may be stopped and restarted depending on the queue length. In the case of arrival control, the arrival stream can be turned off and on or arrivals may be rejected depending on the queue length. We give duality relations for various systems with arrival and service control that enables us to relate their stationary queue length distributions. We use physical coupling arguments which imply the stochastic coupling necessary to relate the queue lengths. We also discuss special cases for which queue length relationships can be obtained by analyzing the underlying Markov process. Two examples are provided to demonstrate the application of the duality property. The first example is a case where the existing queue length distribution for a given model can be used to obtain the queue length distribution of another model. In the second example, we obtain the previously unknown queue length distributions for two related models at once.