Stability analysis of cascade networks via fluid models

In this work, the fluid approach methodology is first applied to find a sufficient stability condition for a two-station cascade network: customers that are awaiting service at the first queue can move to the second station, whenever it is free, to be served there immediately, but the opposite is not allowed. Each station is fed by a renewal input with general i.i.d. inter-arrival times and general i.i.d. service times (possibly different in the two stations). Then we extend the stability analysis to cascade networks with an arbitrary number N of stations and find a sufficient stability condition. In such networks, an awaiting customer from the queue j ≤ N − 1 jumps to station j + 1 if free, to be served there. Finally, we obtain a necessary condition for stability of the N -station cascade network, which matches the sufficient condition when N = 2.