Extreme values in closed networks

For a widely used hub-and-spoke closed product-form network consisting of an infinite-server node and several single-server queues, we characterize the maximum queue-length distribution in various operational regimes by leveraging a novel probabilistic representation of the joint queue-length distribution and scaling where the number of customers grows. In these limiting regimes, we derive explicit characterizations of the maximum that are asymptotically equivalent to the maximum of independent random variables with the same geometric marginal distribution as queue lengths. In particular, when both the number of customers and queues grow, the parameters of the marginal distribution depend on the global characteristics of the network and are explicitly computed from a quadratic equation that arises from the corresponding large-deviation rate functions. Explicit computation of global characteristics of product-form distribution beyond the marginals, e.g., the maximum, appears novel, and our methodology may apply to other global measures of interest.