On Value-at-Risk Based Queueing Systems

Abstract

In this research, we investigate the economics of queueing systems with strategic non-risk-neutral customers, emphasizing the value-at-risk (VaR) framework. It is distinct from the moment-based utility structure of Naor’s model by introducing the maximum loss customers can bear with a certain level of confidence. We consider both homogeneous and heterogeneous cases of risk preferences under different information levels. Our findings demonstrate that optimal strategies in observable queueing systems with non-risk-neutral customers exhibit threshold-type behaviors that differ significantly from those observed in risk-neutral customer settings. For unobservable queueing systems, we derive an equilibrium joining probability for homogeneous risk preferences and a multidimensional equilibrium joining probability for heterogeneous risk preferences. Interestingly, under the VaR criterion, there is an indifferent action region in the observable queueing model. Customers with risk preferences falling within this region will adopt the same joining strategy. And, in the unobservable scenario, there is an indifference curve for risk preference. Customers on the same indifference curve have the same level of risk preference. Furthermore, we provide a conversion formula that facilitates comparison between two risk preferences with different confidence levels. When exploring social welfare, we use conditional value-at-risk (CVaR) to characterize the customer’s excessive losses that lead to potential negative social utility. To distinguish from the classic social welfare function, we term our proposed social welfare function the “CVaR-Based Extended Social Welfare Function (CVaR-ESW)”. In observable systems with homogeneous risk preferences, the socially optimal CVaR-ESW-based risk preference falls within a “ribbon” region, while in unobservable systems, it forms a curve. We also uncover intriguing insights about the information disclosure. In unobservable situations, the throughput in the model with naively assessed (NA) customers is always greater than that in the model with precisely assessed (PA) customers. Based on the differences in risk tolerance values among customers with different risk preferences, we found that, under certain circumstances, NA customers can generate more social welfare, while in other cases, the opposite is true. Overall, this research sheds light on the interplay between VaR-based queueing economics, risk preferences, and strategic customer behavior.