Stability of a cascade system with multiple stations

This paper considers an m-station cascade system with renewal arrival processes and general service time distributions. Miyazawa and Morozov (Queueing Syst 100:225–227, 2022a; Stability of a cascade system with two stations and its extension for multiple stations, 2022b. arXiv:2203.14294v1) formulated a conjecture on the stability of this cascade system. In this paper, we show that this conjecture is not true for m≥ 3. We also show that when m≥ 3 , the stability condition is not determined solely by the moments of the interarrival and service times, but also depends on their distributions. In addition, we provide the necessary and sufficient condition for the stability of this cascade system by modifying a result of Miyazawa and Morozov (2022).