QUASI-PRODUCT FORM TO A MULTINODE QUEUEING SYSTEM SHARING A STATE-DEPENDENT SETUP SERVER

Abstract

We consider a K-node queueing system sharing a setup server. 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. The arrival process of jobs at each node is Poisson, the distribution of node service times at each node is general, and the setup times have a common exponential distribution. The setup server behaves like M setup servers (i.e., the server can simultaneously process up to M jobs). We consider two setup mechanisms by the setup server. One is that for a setup of job at each node, both the waiting position (occupied by the job) and service position are used. The other is that only the service position is used for a setup. The model operating under the former or latter is referred to as Model I or II, respectively. For each node in Model I or II, we construct a corresponding setup server queue (CSQ). We show that the stationary distribution of Model I or II is given by a product form of the stationary distributions of CSQs.