Pricing in Queues with Abandonments: Optimal Policies and Practical Heuristics

TL;DR

This paper studies optimal dynamic pricing in queues with customer abandonment, characterizing the structure of optimal policies and proposing heuristics that perform near optimally under various conditions.

Contribution

It fully characterizes the structure of optimal pricing policies in queues with abandonments and introduces two practical heuristics with near-optimal performance.

Findings

Optimal prices do not always increase with queue length.

Conditions are provided for when the optimal policy is increasing in queue size.

Two heuristics are proposed that simplify pricing and achieve near-optimal profits.

Abstract

We investigate the optimal pricing strategy in a service-providing framework, where customers can leave the system prior to service completion. In this setting, a price is quoted to an incoming customer based on the current number of customers in the system. When the quoted price is lower than the price the incoming customer is willing to pay (which follows a fixed probability distribution), then the customer joins the system and a reward equal to the quoted price is earned. A cost is incurred upon abandonment and a holding cost is incurred for customers waiting to be served. Our goal is to determine the pricing policy that maximizes the long-run average profit. Unlike traditional queueing systems without abandonments, we show that the optimal quoted prices do not always increase with the queue length in this setting. We fully characterize the possible structure of the optimal dynamic pricing policy and provide conditions guaranteeing that the optimal policy is increasing in the number of customers in the system. Moreover, we introduce two heuristics that simplify the optimal dynamic pricing policy. Both heuristics admit customers until the number of customers in the system reaches a certain threshold. The cutoff-static policy charges all admitted customers a fixed price while the two-price policy charges one price when the arriving customer can enter service immediately and another price if the customer needs to wait. By selecting the price(s) and threshold that maximize the long-run average profit, both heuristics achieve near optimality in general and the two-price policy provides more robustness compared to the cutoff-static policy.