Queueing games with an endogenous number of machines

TL;DR

This paper analyzes queueing problems where agents decide both how to queue and how many machines to activate, using game theory to study cost sharing and stability of allocations, including requeueing scenarios.

Contribution

It introduces a novel game-theoretic framework for queueing problems with endogenous machine numbers, providing bounds for stable cost sharing and conditions for stable allocations.

Findings

Bounds on machine costs ensure non-empty core

Stable allocations exist when machines are public goods

Stable solutions are guaranteed with prioritized processing

Abstract

This paper studies queueing problems with an endogenous number of machines with and without an initial queue, the novelty being that coalitions not only choose how to queue, but also on how many machines. For a given problem, agents can (de)activate as many machines as they want, at a cost. After minimizing the total cost (processing costs and machine costs), we use a game theoretical approach to share to proceeds of this cooperation, and study the existence of stable allocations. First, we study queueing problems with an endogenous number of machines, and examine how to share the total cost. We provide an upper bound and a lower bound on the cost of a machine to guarantee the non-emptiness of the core (the set of stable allocations). Next, we study requeueing problems with an endogenous number of machines, where there is an existing queue. We examine how to share the cost savings compared to the initial situation, when optimally requeueing/changing the number of machines. Although, in general, stable allocation may not exist, we guarantee the existence of stable allocations when all machines are considered public goods, and we start with an initial schedule that might not have the optimal number of machines, but in which agents with large waiting costs are processed first.