Polling systems with permanent and transient jobs

Almost all polling models studied in the literature deal with 'open' systems where all jobs are transient, i.e., they arrive to, are served by, and leave the system. Recently Altman & Yechiali [1994] introduced and analyzed models for 'closed' polling networks in which a fixed number of permanent jobs always reside in the system, such that each job, after being served in one station, is immediately (and randomly) routed to another station where it waits to be served again by the rotating server. Boxma and Cohen [1991] analyzed a single-node M/G/1 configuration with regular (transient) Poisson jobs and with a fixed number of permanent jobs who immediately return to the end of the queue each time they receive a service. In this work we study hybrid multi-node polling systems with both transient and permanent jobs, operated under the Gated, Exhaustive, or Globally-Gated service regime. We define the laws of motion governing the evolution of such systems and derive the multi-dimensional generating functions of the number of jobs at the various queues at polling instants, and at arbitrary points in time. For each regime we investigate the interaction between the two populations of jobs and derive formulae for the means, as well as expressions for calculating the second moments, of the number of jobs at the various queues. Waiting times and throughput rates are calculated and the systems are compared with each other.