Philipp Afèche

Pricing and Priority Auctions in Queueing Systems with a Generalized Delay Cost Structure

This paper studies alternative price-service mechanisms for a provider that serves customers whose delay cost depends on their service valuations. We propose a generalized delay cost structure that augments the standard additive model with a multiplicative component, capturing the interdependence between delay cost and values. We derive and compare the revenue-maximizing and socially optimal equilibria under uniform pricing, preemptive, and nonpreemptive priority auctions with an admission price. We find that the delay cost structure has a paramount effect on system behavior. The classical result that the revenue-maximizing admission price is higher and the utilization lower than is socially optimal can be reversed under our generalized structure, and we identify the conditions driving this reversal under each mechanism. We show that the conditional bid equilibria are unique and induce the socially optimal allocations. The auctions yield gains in system net value and provider profit over uniform pricing, which are dramatically larger for the preemptive mechanism. Both auctions perform better under multiplicative compared to additive delay costs. The highest-value customers always gain under the preemptive, but may lose under the nonpreemptive auction. The lowest-value customers always gain in either auction.

Incentive-Compatible Revenue Management in Queueing Systems: Optimal Strategic Delay

How should a firm design a price/lead-time menu and scheduling policy to maximize revenues from heterogeneous time-sensitive customers with private information about their preferences? We consider this question for a queueing system with two customer types and provide the following results. First, we develop a novel problem formulation and solution method that combines the achievable region approach with mechanism design. This approach extends to menu design problems for other systems. Second, the work conserving cμ priority rule, known to be delay cost minimizing, incentive-compatible, and socially optimal, need not be revenue maximizing. A strategic delay policy may be optimal: It prioritizes impatient customers, but artificially inflates the lead times of patient customers. This suggests a broader guideline: Revenue-maximizing firms that lack customer-level demand information should also consider customer incentives, not only operational constraints, in their scheduling policies. Third, we identify general necessary and sufficient conditions for optimal strategic delay: a price, a lead-time, and a segment-size condition. We translate these into demand and capacity parameter conditions for cases with homogeneous and heterogeneous valuations for each type. In some cases strategic delay is optimal if capacity is relatively abundant, in others if it is relatively scarce.

Optimal Price/Lead-Time Menus for Queues with Customer Choice: Segmentation, Pooling, and Strategic Delay

How should a firm design a price/lead-time menu and scheduling policy to maximize revenues from heterogeneous time-sensitive customers with private information about their preferences? We consider a queueing system with multiple customer types that differ in their valuations for instant delivery and their delay costs. The distinctive feature of our model is that the ranking of customer preferences depends on lead times: patient customers are willing to pay more than impatient customers for long lead times, and vice versa for speedier service. We provide necessary and sufficient conditions, in terms of the capacity, the market size, and the properties of the valuation-delay cost distribution, for three features of the optimal menu and segmentation: pricing out the middle of the delay cost spectrum while serving both ends, pooling types with different delay costs into a single class, and strategic delay to deliberately inflate lead times. This paper was accepted by Assaf Zeevi, stochastic models and simulation.

Pricing Time-Sensitive Services Based on Realized Performance

Services such as FedEx charge up-front fees but reimburse customers for delays. However, lead-time pricing studies ignore such delay refunds. This paper contributes to filling this gap. It studies revenue-maximizing tariffs that depend on realized lead times for a provider serving multiple time-sensitive customer types. We relax two key assumptions of the standard model in the lead-time pricing literature. First, customers may be risk averse RA with respect to payoff uncertainty, where payoff equals valuation, minus delay cost, minus payment. Second, tariffs may be arbitrary functions of realized lead times. The standard model assumes risk-neutral RN customers and restricts attention to flat rates. We report three main findings: 1 With RN customers, flat-rate pricing maximizes revenues but leaves customers exposed to payoff variability. 2 With RA customers, flat-rate pricing is suboptimal. If types are distinguishable, the optimal lead-time-dependent tariffs fully insure delay cost risk and yield the same revenue as under optimal flat rates for RN customers. With indistinguishable RA types, the differentiated first-best tariffs may be incentive-compatible even for uniform service, yielding higher revenues than with RN customers. 3 Under price and capacity optimization, lead-time-dependent pricing yields higher profits with less capacity compared to flat-rate pricing.

Double-Sided Batch Queues with Abandonment: Modeling Crossing Networks

We study a double-sided queue with batch arrivals and abandonment. There are two types of customers, patient ones who queue but may later abandon, and impatient ones who depart immediately if their order is not filled. The system matches units from opposite sides of the queue based on a first-come first-served policy. The model is particularly applicable to a class of alternative trading systems called crossing networks that are increasingly important in the operation of modern financial markets. We characterize, in closed form, the steady-state queue length distribution and the system-level average system time and fill rate. These appear to be the first closed-form results for a double-sided queuing model with batch arrivals and abandonment. For a customer who arrives to the system in steady state, we derive formulae for the expected fill rate and system time as a function of her order size and deadline. We compare these system-and customer-level results for our model that captures abandonment in aggregate, to simulation results for a system in which customers abandon after some random deadline. We find close correspondence between the predicted performance based on our analytical results and the performance observed in the simulation. Our model is particularly accurate in approximating the performance in systems with low fill rates, which are representative of crossing networks.

Optimal Price-Lead Time Menus for Queues with Customer Choice: Priorities, Pooling & Strategic Delay

How should a firm design a price-lead time menu and scheduling policy to maximize revenues from heterogeneous time-sensitive customers with private information about their preferences? We consider a queueing system with multiple customer types that differ in two dimensions, their valuations for instant delivery and their delay cost rates. The distinctive feature of our model is that the ranking of customer preferences depends on lead times: patient customers are willing to pay more for long lead times than impatient ones, and vice versa for speedier service. We provide necessary and sufficient conditions, in terms of the capacity, market size, and properties of the valuation-delay cost distribution, for three features of the optimal menu and segmentation: 1. Pooling types with different delay costs into a single class; 2. Pricing out the middle of the delay cost spectrum while serving both ends; and 3. Strategic delay to deliberately inflate lead times.

Bayesian Dynamic Pricing in Queueing Systems with Unknown Delay Cost Characteristics

The revenue management literature for queues typically assumes that providers know the distribution of customer demand attributes. We study an observable M/M/1 queue that serves an unknown proportion of patient and impatient customers. The provider has a Bernoulli prior on this proportion, corresponding to an optimistic or pessimistic scenario. For every queue length, she chooses a low or a high price, or turns customers away. Only the high price is informative. The optimal Bayesian price for a queue state is belief-dependent if the optimal policies for the underlying scenarios disagree at that queue state; in this case the policy has a belief-threshold structure. The optimal Bayesian pricing policy as a function of queue length has a zone or, nested-threshold structure. Moreover, the price convergence under the optimal Bayesian policy is sensitive to the system size, i.e., the maximum queue length. We identify two cases: prices converge 1 almost surely to the optimal prices in either scenario or 2 with positive probability to suboptimal prices. Only Case 2 is consistent with the typical incomplete learning outcome observed in the literature.

Delay performance in stochastic processing networks with priority service

We evaluate the delay performance of an open multi-class stochastic processing network of multi-server resources with preemptive-resume priority service. We show that the stationary distribution of aggregate queue lengths has product form. For each service class we derive explicit expressions for the following stationary performance measures: The mean and, under feedforward routing, the Laplace transform of the delay distribution at each resource. We show that these measures are the same as if the resources were operating in isolation.

Rational Abandonment from Priority Queues: Equilibrium Strategy and Pricing Implications

Observable priority queues are prevalent in practice and create incentives for utility-maximizing customers to abandon after joining. However, the literature has so far ignored this behavior and the resulting system control issues. This paper studies the rational abandonment behavior of utility-maximizing customers in the context of an observable two-class priority queue, and identifies novel implications. We characterize the equilibrium abandonment strategy of low-priority customers and show that it has a threshold structure that depends on the fee structure. We then consider pricing as a means to control the balking and abandonment behavior, both under welfare maximization and revenue maximization. Our pricing results highlight the importance of the timing of payments. We show that welfare-maximization requires charging only a service fee and no entrance fee. In contrast, revenue maximization generally requires a combination of both an entrance and a service fee. This two-fee structure is equivalent to charging only upon entrance but offering a partial cancellation refund. Moreover, charging only an entrance fee may generate more or less revenue than charging only a service fee, but the performance of the latter policy is more robust. This appears to be the first paper that (i) gives an analytical characterization of equilibrium abandonment behavior in observable priority queues, and (ii) studies pricing for any queueing system in presence of rational customer abandonment.

Customer Acquisition, Retention, and Service Access Quality: Optimal Advertising, Capacity Level, and Capacity Allocation

Problem definition We provide guidelines on three fundamental decisions of customer relationship management (CRM) and capacity management for profit-maximizing service firms that serve heterogeneous repeat customers, whose acquisition, retention, and behavior depend on their service access quality to bottleneck capacity: how much to spend on customer acquisition, how much capacity to deploy, and how to allocate capacity and tailor service access quality levels to different customer types. Academic/ practical relevance These decisions require a clear understanding of the connections between customers’ behavior and value, their service access quality, and the capacity allocation. However, existing models ignore these connections. Methodology We develop and analyze a novel fluid model that accounts for these connections. Simulation results suggest that the fluid-optimal policy also yields nearly optimal performance for large stochastic queueing systems with abandonment. Results First, we derive new customer ...