Computational algorithms and arrival theorem for non-conventional product-form solutions

Queuing networks with finite capacity are widely discussed in performance analysis literature. One approach to address the finite capacity of stations involves the implementation of a skip-over policy. Under this policy, when a customer arrives at a saturated station, service at that station is skipped, and the customer is rerouted based on the predefined network routing protocol. Skip-over networks have been extensively investigated, and they exhibit a product-form stationary distribution under the exponential assumptions of Jackson networks. However, a comprehensive understanding of the celebrated Arrival Theorem for this class of product-form models is still lacking and relies on certain conjectures. This paper makes three contributions: (i) it provides an in-depth comprehension of the Arrival Theorem for skip-over networks by offering a proof for the conjectures outlined in existing literature, (ii) it introduces a Mean Value Analysis (MVA) algorithm tailored for this type of queuing networks, and (iii) it explores the implications of these findings on the class of product-form queuing networks with fetching and repetitive service discipline.