This paper presents a finite capacity queueing network model to evaluate congestion in protein synthesis networks. These networks are modeled as single server bufferless queues in a tandem topology. The model approximates the marginal stationary distributions of each queue. It consists of a system of linear and quadratic equations that can be decoupled. It is therefore a tractable and scalable method that is suitable for large-scale networks. This model proposes a detailed state space formulation, which provides a fine description of congestion and contributes to a better understanding of how the protein synthesis rate is deteriorated. This paper also generalizes the concept of blocking: blocking events can be triggered by an arbitrary set of queues. The numerical performance of this method is evaluated for networks with up to 100,000 queues, considering scenarios with various levels of congestion. Since tandem topology networks are of interest for a variety of application fields, the numerical efficiency and scalability of this model is of wide interest.