Verification of the Stability of a Two-Server Queueing System With Static Priority

In this work, we use simulation to verify the stability conditions of the so-called N -model, which consists of two servers and two classes of external customers, both generated by Poisson inputs. Service times are server-dependent and, in each server, are i.i.d. When server 1 is occupied, and there are waiting customers in queue of server 1, then a class-1 customer jumps to server 2, thereby becoming a class-(1,2) customer. We consider a non-preemptive service priority: a class-1 customer starts service in server 2, when a class-2 customer, if any, finishes his service. Thus, server 2 assists server 1, while the reverse interaction is impossible. The purpose of this research is to verify the tightness of the stability condition found in [8] by fluid a approach, and to deduce a simpler sufficient stability condition, which is obtained in an explicit form by a regenerative approach. Moreover, our analysis includes verification of the conditions when the 1st server is stable, while the 2nd server is unstable. In addition, we verify by simulation a monotonicity property of this model: the idle stationary probability of server 1 attains a minimum when the 2nd server is permanently occupied by class-2 customers.