Equilibrium, social welfare, and revenue in an infinite-server queue

TL;DR

This paper analyzes equilibrium, social welfare, and revenue in an infinite-server queue, revealing threshold relationships and effects of information levels on system performance through theoretical and numerical methods.

Contribution

It introduces a comprehensive analysis of thresholds and revenue in infinite-server queues, extending cost functions and comparing observable and unobservable settings.

Findings

Optimal revenue threshold is smaller than the socially optimal threshold.

Social welfare and revenue differ significantly between observable and unobservable settings.

Extended cost functions to polynomial forms with non-negative coefficients.

Abstract

Motivated by the impact of emerging technologies on toll parks, this paper studies a problem of equilibrium, social welfare, and revenue for an infinite-server queue. More specifically, we assume that a customer's utility consists of a positive reward for receiving service minus a cost caused by the other customers in the system. In the observable setting, we show the existence, uniqueness, and expressions of the individual threshold, the socially optimal threshold, and the optimal revenue threshold, respectively. Then, we prove that the optimal revenue threshold is smaller than the socially optimal threshold, which is smaller than the individual one. Furthermore, we also extend the cost functions to any finite polynomial function with non-negative coefficients. In the unobservable setting, we derive the joining probabilities of individual and optimal revenue. Finally, using numerical experiments, we complement our results and compare the social welfare and the revenue under these two information levels.