On the behavior of stable subnetworks in non-ergodic networks with unreliable nodes

We consider Jackson networks with unreliable nodes, which randomly break down and are under repair for a random time. The network is described by a Markov process which encompasses the availability status and queue lengths vector. Ergodicity conditions for many related networks are available in the literature and can often be expressed as rate conditions. For (reliable) nodes in Jackson networks the overall arrival rate has to be strictly less than its service rate. If for some nodes this condition is violated, the network process is not ergodic. Nevertheless, it is known that in such a situation, especially in large networks, parts of the network (where the rate condition is fulfilled) in the long run stabilize. For standard Jackson networks without breakdown of nodes, the asymptotics of such stable subnetworks were derived by Goodman and Massey [J.B. Goodman, W.A. Massey, The non-ergodic Jackson network, Journal of Applied Probability 21 (1984) 860–869].In this paper, we obtain the asymptotics of Jackson networks with unreliable nodes and show that the state distribution of the stable subnetworks converges to a Jackson-type product form distribution. In such networks with breakdown and repair of nodes, in general, the ergodicity condition is more involved.Because no stationary distribution for the network exists, steady-state availability and performance evaluation is not possible. We show that instead assessment of the quality of service in the long run for the stabilizing subnetwork can be done by using limiting distributions. Additionally, we prove that time averages of cumulative rewards can be approximated by state-space averages.