Transient analysis of an affine Queue-Hawkes process

We investigate the transient behavior of a multi-server queue with a linearly increasing arrival rate, also called affine Queue-Hawkes process. We obtain expressions of the Laplace Transform of the transient probabilities involving confluent hypergeometric functions. When the arrival rate without congestion-attraction is higher than the service rate or with an initial condition with all servers busy, these Laplace Transforms can be numerically inverted. Our numerical investigations show that convergence to the stationary regime is slower than for an M/M/s queue.