A Probabilistic Approach to Growth Networks

Widely used closed product-form networks have emerged recently as a primary model of stochastic growth of subcellular structures, for example, cellular filaments. The baseline bio-molecular model is equivalent to a single-class closed queueing network, consisting of single-server and infinite-server queues. Although this model admits a seemingly tractable product-form solution, explicit analytical characterization of its partition function is difficult due to the large-scale nature of bio-molecular networks. To this end, we develop a novel methodology, based on a probabilistic representation of product-form solutions and large-deviations concentration inequalities, which identifies distinct operating regimes and yields explicit expressions for the marginal distributions of queue lengths. The parameters of the derived distributions can be computed from equations involving large-deviations rate functions, often admitting closed-form algebraic expressions. From a methodological perspective, a fundamental feature of our approach is that it provides exact results for order-one probabilities, even though our analysis involves large-deviations rate functions, which characterize only vanishing probabilities on a logarithmic scale.