A push–pull queueing system

We consider a two node multiclass queueing network given by two machines each with two classes. There are two streams of jobs. One stream originates in machine 1, which feeds it for further processing to machine 2, and the other stream moves in the opposite direction. We describe a policy for this system which is stable and which keeps both machines busy at all times. We obtain explicit expressions for its steady-state behavior under M/M/· and under M/G/· assumptions. We also describe a similar M machine system.