The single server queue with mixing dependencies

We study a single server queue, where a certain type of dependence is introduced between the service times, or between the inter-arrival times, or both between the service times and the inter-arrival times. This dependence arises via mixing, i.e., a parameter pertaining to the distribution of the service times, or of the inter-arrival times, is itself considered to be a random variable. We give a duality result between such queueing models and the corresponding insurance risk models, for which the respective dependence structures have been studied before. For a number of examples we provide exact expressions for the waiting time distribution, and compare these to the ones for the standard M/M/1 queue. We also investigate the effect of dependence and derive first order asymptotics for some of the obtained waiting time tails. Finally, we discuss this dependence concept for the waiting time tail of the G/M/1 queue.