A zero-or-one law in aggregated closed queueing networks

For a closed queueing network with single-server nodes, we prove that if the total number of requests, the number of servers in one of the nodes, and service rates in all other nodes are made $n$ times as large, then the stationary number of requests in the multiserver node divided by $n$ converges in probability as $n\to\infty$ to a positive constant, determined by parameters of the original network, with geometric convergence rate. Single-server nodes in the constructed network can be interpreted as repair nodes, the multiserver node as a set of workplaces, and requests as elements in a redundancy-with-repair model.