Waiting time externality awareness in socially responsible queueing

Abstract

In service systems, customers tend to overutilize resources, resulting in high congestion. We study a non-coercive way to alleviate congestion based on customers’ awareness of the negative waiting externalities they impose. To this end, we revisit Naor’s rational-queue model by adding a term that captures the cost a customer associates with the waiting time generated for other customers encountered during their sojourn. First, we show that this modification preserves the threshold structure of customers’ joining strategy. Using lattice-path counting techniques, we derive a closed-form expression for the expected generated waiting time, conditional on the number of customers observed upon arrival. The generated wait is increasing and concave in the queue length at arrival. Viewing expected waiting time as allocated across arrivals, the queue length associated with an arbitrary unit of expected generated waiting time is first-order stochastically lower than the queue length associated with a unit of expected personal waiting time. We prove that the maximum generated wait is increasing and convex in the joining threshold and provide tight bounds with exact light/heavy traffic asymptotics. Internalizing social awareness (weight on generated wait) lowers the joining threshold and reduces waiting; in high-demand regimes, it can also increase social welfare. When the operator can also influence the weight on personal wait, that lever reduces congestion more strongly than waiting externality awareness. Finally, we quantify the awareness level needed to reach Naor’s social threshold and show that it is lowest at a non-extreme demand level.