Note on a tandem queue with delayed server release

We consider a tandem queue with two stations. The first station is an $s$-server queue with Poisson arrivals and exponential service times. After terminating his service in the first station, a customer enters the second station to require service at a single server, while in the meantime he is blocking his server in station 1 until he completes service in station 2, whereupon the server in station 1 is released. Two cases are considered. In the first case it is assumed that the service times in the second station are exponentially distributed. The solution of this model can be formulated using the matrix-geometric method. In the second case $s=2$ and, the service times are generally distributed. The model is analyzed using the supplementary variable technique, obtaining the relevant probability generating functions.