  Widely used closed product-form networks have emerged recently as a primary
model of stochastic growth of sub-cellular structures, e.g., cellular
filaments. In the baseline model, homogeneous monomers attach and detach
stochastically to individual filaments from a common pool of monomers,
resulting in seemingly explicit product-form solutions. However, due to the
large-scale nature of such networks, computing the partition functions for
these solutions is numerically infeasible. To this end, we develop a novel
methodology, based on a probabilistic representation of product-form solutions
and large-deviations concentration inequalities, that yields explicit
expressions for the marginal distributions of filament 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.
