Product Forms in Closed Order Independent Queues

There is a large literature on product form queueing models, but it is overwhelmingly focused on open systems. However, much less is known about product forms in closed queues, and existing results typically apply only to restrictive special cases. In this paper, we introduce the first broad family of closed queues with stationary product form distributions, which we call Front Index Order Independent (FIOI) queues. Closed models are of interest both as models of genuinely closed systems and as closed counterparts of open systems. In the latter case they are important because the throughput of the closed system determines the stability threshold of the corresponding open model.

Our first contribution is to identify this general FIOI family and derive its associated product form, in which the stationary probability of a state factors into a simple product of probabilities associated with the jobs present in the system. The basic model assumes uniform completion probabilities among jobs in service, and we then extend the framework in four directions: allowing the class of the arriving job to depend on the departing job; allowing completion probabilities to depend on position in the queue; allowing countably infinite class spaces; and allowing more general closed systems in which a completion may trigger multiple arrivals and the number of jobs in the system need not remain constant across states.

We then formulate Front Sum Order Independence (FSOI), a closed model analogue of assumptions from the open order independent queueing literature, and derive a corresponding product form for the closed system paralleling the open FSOI result.