Product Form Models for Queueing Networks with an Inventory

Abstract

We investigate a new class of stochastic networks that exhibit a product form steady state distribution. The stochastic networks developed here are integrated models for networks of service stations and inventories. We integrate a server with attached inventory under (r, Q)- or (r, S)-policy into Jackson or Gordon-Newell networks. Replenishment lead times are non-zero and random and depend on the load of the system. While the inventory is depleted the server with attached inventory does not accept new customers (lost sales regime), but we assume that the lost sales are not lost to the system. We pursue three different approaches to handle routing with respect to this node during the time the inventory is empty.

We derive stationary distributions of joint queue lengths and inventory processes in explicit product form. The stationary distributions are then used to calculate performance measures of the respective systems. We discuss the advantages and disadvantages of product form modeling in the context of service-inventory systems.

Finally, we sketch two network models where several nodes may have an attached inventory.

Keywords:

• Inventory level
• Inventory policy
• Inventory systems
• Jackson networks
• Performance analysis
• Product form solution
• Service level
• Stationary distributions

MSC 2000 Subject Classification:

• 60K25
• 90B22
• 90B05

ACKNOWLEDGMENT

We thank the referee for her or his careful reading of the paper and for helpful comments and suggestions.