Greedy servers on a torus

Queuing systems in which customers arrive at a continuum of locations, rather than at a finite number of locations, have been found to provide good models for certain telecommunication and reliability systems as well as dynamic stochastic vehicle routing problems. In this paper the continuum is the unit square, where the opposite edges have been glued together to form a flat “torus”. Customers arrive according to a Poisson process with arrival rate λ and are removed by servers. We investigate properties of the system under various server strategies. We find that the greedy strategy, where a server simply heads for its closest point, results in a stable system and we analyze the equilibrium distribution. The greedy strategy is inefficient, in part because multiple greedy servers coalesce. We investigate the expected time until this occurs and identify improvements to the greedy strategy.