Correlated Queues with Inter-Arrival Times Depending on Service Times

We consider a type of correlated queue in which the inter-arrival time between two consecutive customers linearly depends on the service time of the first customer. We first derive infinite systems of linear equations for the moments of the waiting time in two special cases, based on which we then develop several methods to calculate the moments of the waiting time by using MacLaurin series approximation, Padé approximation and truncation method. In addition, we show how the moments and covariances of the inter-departure times of the correlated queue can be calculated based on the moments of the waiting time. Finally, extensive numerical examples are provided to validate our methods.