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# Eindhoven Stochastics Seminar

## Dec 16, 2019, 15:45 - 16:45

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.

Date:
Dec 16, 2019
Time:
15:45 - 16:45
Event Category:

## Venue

MF 11-12 (4th floor MetaForum Building, TU/e)