Charging more for priority via two-part tariff for accumulating priorities

Highlights

• We consider an M/G/1 accumulating priority (AP) queue with affine pricing of AP rates.

• Multiple equilibria, pure and mixed, may exist. In some cases, no evolutionary stable strategy (ESS) exists.

• If the server utilization level is less than 0.5, the optimal pricing achieves the highest possible revenue.

Abstract

We consider an unobservable M/G/1 queue with accumulating priorities and strategic customers who pay for priority accumulation rates. We show that when affine pricing is introduced, multiple equilibria may exist. This is in contrast to the standard linear pricing case where the equilibrium strategy is unique. Furthermore, a revenue-maximizing operator may generate more revenue under the optimal affine pricing than under linear pricing. In particular, we show that if the utilization level is not too high, no other combination of a priority scheme and pricing generates more revenue than the optimal affine pricing of accumulating priorities.

Introduction

Queues may form whenever a service facility of limited capacity serves a stream of incoming customers and the arrival process, the service process, or both of them are random. For example, in a single-server facility where customers are served one by one, a queue builds up whenever a customer with a relatively long service requirement is served while customers keep arriving with relatively short inter-arrival times. The larger the arrival rate is compared to the service rate, the utilization level (which is the ratio of these two rates) increases and so are the waiting times customers are experiencing. Classic queueing theory deals with modeling and analyzing such systems under various assumptions. This work belongs to a stream within this discipline where customers are assumed to be rational and strategic, such that they react to incentives. For example, customers may be able to decide whether to join a queue or balk, whether to pay for priority, how much to pay for priority, or when to arrive. A common ground of these decision problems is the idea that customers balance between their need for service and their desire to avoid the costs associated with congestion (typically in the form of waiting time). Since in queueing environment the congestion one experiences depends on the actions of other customers, such a strategic queueing decision model forms a non-cooperative game.

In order to balance between their need for service and the discomfort of waiting, customers make queueing-related decisions depending on the possible actions at their disposal. For example, in the seminal paper by Naor (1969), customers decide to join an observable M/M/1 queue if the queue is not too long, and balk otherwise. Similarly, in the unobservable counterpart of Naor’s model (Edelson & Hilderbrand, 1975), customers balk if the effective joining rate of the other customers is too high. However, in many queueing applications, balking is not an option. In such cases, customers’ desire to avoid congestion may be channeled towards attempts to gain priority over the other customers. In particular, an operator of such systems may offer a menu of levels of some priority mechanism, where each level comes at a different price. We henceforth refer to such a combination of a menu and a priority mechanism as a priority pricing mechanism. In the resulting priority queueing system, customers who pay more enjoy an absolute (see, e.g., p.83 in Hassin & Haviv, 2003, and Haviv & Winter, 2020) or non-absolute priority (see, e.g., Haviv & van der Wal, 1997 and Haviv & Ravner, 2016) over those who pay less or decide not to pay at all.

Considering strategic priority queues from the operator’s point of view, rises the following natural question: does there exist a revenue-maximizing priority pricing mechanism? As we show in this paper, the answer to this question for the M/G/1 queueing model is positive. The optimal revenue is achieved by auctioning absolute priority levels, as proposed in Glazer & Hassin (1986). The resulting equilibrium bidding strategy is a mixed strategy under which the bid of an arbitrary customer is a continuous random variable. Hence, the resulting service regime in equilibrium is random queue. The implementation of such a mechanism therefore requires constant monitoring of the queue and the exact bid value of each of the customers.

Other existing priority pricing mechanisms appear in the literature, including some mechanisms under which the resulting equilibrium strategy is pure, and therefore, the resulting queueing discipline is in fact first-come first-served (FCFS), which is much simpler to administer (although, the ability to enforce a different regime must exist regardless of the resulting equilibrium). However, non of these mechanisms achieves the optimal revenue. In this paper we propose such a mechanism that enjoys the simplicity of the FCFS discipline while achieving the optimal revenue when the utilization is low enough. This revenue-maximizing priority pricing mechanism is a two-part tariff for accumulating priority rates.

Accumulating priority is a non-absolute priority mechanism that takes into account not only the priority class of customers but also the time they have already spent in the system. In particular, customers accumulate priority with time spent in the system at a rate which is determined by their class. Whenever the server becomes available, the next customer to commence service is the one with the highest accumulated priority present in the queue at that time.

Accumulating priorities where introduced in Kleinrock (1964) for the M/M/1 queue followed by a more detailed analysis for the M/G/1 queue in Kleinrock (1976) (see p.126 there) where a recursion on the expected waiting times of different classes is derived. An extension of this recursion for a continuous distribution of accumulation rates is derived in Haviv & Ravner (2016). The waiting-time distributions of different priority classes (for the discrete case) are characterized in Stanford, Taylor, & Ziedins (2014). Revenue management in accumulating priority queues with non-strategic customers is studied in Sinha, Rangaraj, & Hemachandra (2010).

There is a vast literature on strategic behavior in queues. See Hassin & Haviv (2003) and Hassin (2016) for a comprehensive survey. The analysis in Balachandran (1972) proposes the term “stable” to describe a priority purchasing policy under which no customer can reduce his expected waiting cost by deviating from it, provided that all other customers follow it. It is shown that these stable policies are not socially optimal. This model is further studied in Glazer & Hassin (1986). A general model of delay cost is studied in Afèche & Mendelson (2004) where the revenue-maximizing and the socially-optimal pricing for preemptive and non-preemptive absolute priority is derived. Socially-optimal and incentive compatible pricing mechanisms of absolute priorities are studied in Mendelson & Whang (1990). The following model appeared in p.83 of Hassin & Haviv (2003). Two priority classes exist in an unobservable M/M/1 queue. Each customer may pay a price set by the operator in order to belong to the superior class. It is shown that as long as the price is low enough, paying for priority is a dominant strategy and as long as the price is high enough, not paying for priority is a dominant strategy. In the intermediate range of the price, three equilibria exist: two pure where all the customers pay or do not pay for priority, and one mixed equilibrium which is not evolutionary stable (see Smith, 1974). In an attempt to find a priority pricing mechanism that increases the operator’s revenue in that model, the following scheme is proposed in the recent paper (Haviv & Winter, 2020). Each customer is offered to pay a random amount for belonging to the superior class. The authors define the revenue-maximizing distribution from which the random prices are derived.

Strategic purchasing of accumulating priority rates were studied first in Haviv & Ravner (2016) where customers who arrive to a M/G/1 queue are offered a continuous menu of accumulating priority rates such that any non-negative rate may be purchased at a price that linearly increases with the rate. The coefficient of the linear pricing function may be considered as the price per unit of priory accumulation rate. The authors derive the unique symmetric equilibrium strategy, which is shown to be a pure strategy. Interestingly, the resulting revenue collected by the operator is invariant with the pricing coefficient. The discrete menu counterpart of this model is studied in Abeywickrama, Haviv, Oz, & Ziedins (2019). In this model, customers may select an accumulation rate out of a discrete menu (e.g., the non-negative integers), where the pricing is a linear function of the selected rate. It is shown that depending on the pricing coefficient value, multiple equilibria may exist. Nevertheless, the revenue is bounded from above by the constant revenue resulting from the continuous menu in Haviv & Ravner (2016).

The purpose of this paper is to extend the work in Haviv & Ravner (2016) to include a more complex pricing mechanism in the form of an affine function. This forms a two-part tariff (see, e.g., Lewis, 1941) for priority: a constant amount to be paid by any customer who wishes to have non-zero accumulating priority rate, and a linear variable cost per unit of accumulation rate. This pricing mechanism captures both attitudes towards priority purchasing: the absolute priority in p.83 of Hassin & Haviv (2003) and the accumulating priority in Haviv & Ravner (2016). Indeed, as we show, affine pricing of accumulating priority rates induces an equilibrium behavior under which the operator may gain (in systems that are congested enough) the sum of the revenue gained in the absolute priority case and the revenue collected in the case of the linear pricing of accumulating priority. Moreover, we show that when the utilization level is not too high, no other priority pricing mechanism gains more revenue than the mechanism proposed in this paper.