Generalized Product-Form Solutions for Stationary and Non-Stationary Queuing Networks with Application to Maritime and Railway Transport

The paper advances the theory of queuing networks by presenting generalized product-form solutions that explicitly take into account the service intensity depending on the number of customers in the network nodes, including the presence of multiple service channels and multi-threaded nodes. This represents a significant extension of the classical results on the Jackson network by integrating graph-theoretic methods, including basic subgraphs with service rates depending on the number of requests. The originality of the article is in the combination of stationary and non-stationary approaches to modeling service networks within a single approach. In particular, acyclic networks with deterministic service time and non-stationary Poisson input flow are considered. Such systems present a significant difficulty, which is noted in well-known works. A stationary model of an open queuing network with service intensity depending on the number of customers in the network nodes is constructed. The stationary network model is related to the problem of marine linear navigation along a strictly defined route and schedule. A generalization of the product theorem with a new form of stationary distribution is developed for it. It is shown that even a small increase in the service intensity with a large number of requests in a queuing network node can significantly reduce its average value. A non-stationary model of an acyclic queuing network with deterministic service time in network nodes and a non-stationary Poisson input flow is constructed. The non-stationary model is associated with irregular (tramp) sea transportation. The intensities of non-stationary Poisson flows in acyclic networks are represented by product formulas using paths between the initial node and other network nodes. The parameters of Poisson distributions of the number of customers in network nodes are calculated. The simplest formulas for calculating such queuing networks are obtained for networks in the form of trees.