A queueing-theoretic model for resource allocation in one-dimensional distributed service network.

We consider assignment policies that allocate resources to requesting users, where everyone is located on a one-dimensional line. First, we consider a unidirectional problem where a resource can only be allocated to a user located to its left as exemplified here. Imagine a one-way city street with traffic flowing from right to left; a ride-sharing company has distributed its vehicles along the street, which are ready to pick up users waiting at various locations. Users equipped with smartphone ride-hailing apps can register their requests on a ride allocation system. The latter then attempts to service each user by assigning a vehicle with spare capacity located to the user's right such that the average "pick up" distance is minimized. We propose the Move to Right (MTR) policy which assigns the nearest available resource located to the right, and contrast it with the Unidirectional Gale-Shapley (UGS) Matching policy known in the literature. While both these policies are optimal, we show that they are equivalent with respect to the expected distance traveled by a request (request distance), although MTR tends to be fairer, i.e., has low variance. Moreover, we show that when the locations of users and resources are modeled by statistical point processes, the spatial system under unidirectional policies can be mapped to a temporal queuing system, thus allowing the application of a plethora of queuing theory results that yield closed form expressions. We also consider the bidirectional problem where there are no directional restrictions on resource allocation and give an optimal policy that has lower expected time complexity than known algorithms in literature in resource rich regimes. Finally, numerical evaluation of performance of unidirectional and bidirectional allocation schemes yield design guidelines beneficial for resource placement.