Quasi-product form to a multinode queueing system with a common exponential set-up server

We consider a K-node queueing system sharing a set-up server. Each node has a node server, a waiting position, and a service position, and it behaves as an M/G/1/2 type queue except that each job in the waiting position requires a set-up by the set-up server to move to the service position. The service discipline of the set-up server is nonpreemptive work-conserving random selection. The set-up times have a common exponential distribution. The main purpose of this paper is to derive the stationary distribution for the K-node system. For each node, we construct a corresponding set-up server queue (CSQ). A product form of the stationary distributions of CSQs gives the stationary distribution of the K-node system. This result enables us to obtain the stationary distribution of a K-node system by analysing individual CSQs.