Gianmarco Bet (University of Florence)

Weighted Dyck paths for non stationary queues

We consider a model for a transitory queue in which only a fixed number $N$ of customers can join. Each customer joins the queue independently of the other ones at a random time, which we assume to be exponentially distributed. Assuming further that the service times also follow an exponential distribution, this system can be described as a two-dimensional Markov process on a triangular subset of $\mathbb Z^2$. The resulting random walk with state-dependent transition rates has a rich but intricate combinatorial structure which we study by introducing appropriate generating functions that exploit the recursive structure of the model. We derive a fully explicit expression for the probability density function of the number of customers served in the first busy period (and thus in any busy period) in terms of a certain balls-in-urns combinatorial scheme. This formula can be interpreted as a decomposition of weighted Dyck paths. We also derive an explicit expression for the joint probability distribution of the maximum queue length and the number of customers served during a busy period.