Arrival first queueing networks with applications in kanban production systems

Abstract

In this paper, we introduce a new class of queueing networks called arrival first networks. We characterise its transition rates and derive the relationship between arrival rules, linear partial balance equations, and product form stationary distributions. This model is motivated by production systems operating under a kanban protocol. In contrast with the conventional departure first networks, where a transition is initiated by service completion of items at the originating nodes that are subsequently routed to the destination nodes (push system), in an arrival first network a transition is initiated by the destination nodes of the items and subsequently those items are processed at and removed from the originating nodes (pull system). These are similar to the push and pull systems in manufacturing systems. Our characterisation provides necessary and sufficient conditions for the network to possess linear traffic equations, and sufficient conditions for the network to have a product form stationary distribution. We apply our results to networks operating under a kanban mechanism and characterise the rate at which items are pulled as well as the routing and blocking protocols that give rise to a product form stationary distribution.