Quasi-product form to a multinode queueing system sharing a state-dependent setup server (Q2711560)

The paper deals with a K-node queueing system sharing a setup server, where each node has a node server, a finite buffer, and a service position. Each job in the buffer at a node requires a setup by the setup server to receive service from the node server at the service position. At each node there arrive jobs according to a Poisson process. The distribution of node service times at each node is general, and the setup times have a common exponential distribution. The setup server can simultaneously process up to \(M\) jobs. Two setup mechanisms (Model I, Model II) are considered for the setup sever: In Model I for a setup of a job at each node, both the waiting position (occupied by the job) and service position are used. In Model II only the service position is used for a setup. For each node in Model I or II a corresponding setup server queue (CSQ) is constructed.\ The aim of the paper is to show that the stationary distribution of Model I and II is given by a product form of the stationary distributions of CSQs.