Spatial Queueing Processes

This study introduces a spatial queueing model for stochastic service systems in which customers or units move about and receive services in a region or a general space. The state of the system at any time is a point process that describes the numbers of units in subsets of the space, and the point process evolves over time as a Markov process. The system may be closed or open with a finite or unlimited capacity, and there may be multiple types of customers.

The main result is a closed form expression for the stationary distribution of a spatial queueing process. Sufficient conditions are given for the process to be ergodic. Further results include formulas for throughputs, Little laws for average waiting times in sectors of the space, and necessary and sufficient conditions for various flows in open systems to be Poisson processes in time and space. Examples are given of systems with sector-dependent service rates, local-regional service rates, and multiclass customers. A large family of spatial systems are multi-class queueing networks in which the customer classes are nondiscrete marks or quantities (e.g., resources for services, temperature, physical characteristics) that affect services and routings.