Strategic Arrivals to Queues Offering Priority Service

TL;DR

This paper analyzes strategic customer behavior in queue systems with fixed start times and priority levels, characterizing equilibria and optimizing revenue through pricing and priority structuring.

Contribution

It introduces a model for strategic arrivals with heterogeneous customers and multiple priority levels, providing equilibrium analysis and revenue optimization strategies.

Findings

Unique Nash equilibria for heterogeneous arrivals are characterized.

Customers naturally divide into priority intervals at equilibrium.

Near-maximum revenue achieved with only three priority queues.

Abstract

We consider strategic arrivals to a FCFS service system that starts service at a fixed time and has to serve a fixed number of customers, e.g., an airplane boarding system. Arriving early induces a higher waiting cost (waiting before service begins) while arriving late induces a cost because earlier arrivals take the better seats. We first consider arrivals of heterogeneous customers that choose arrival times to minimize the weighted sum of waiting cost and and cost due to expected number of predecessors. We characterize the unique Nash equilibria for this system. Next, we consider a system offering L levels of priority service with a FCFS queue for each priority level. Higher priorities are charged higher admission prices. Customers make two choices - time of arrival and priority of service. We show that the Nash equilibrium corresponds to the customer types being divided into L intervals and customers belonging to each interval choosing the same priority level. We further analyze the net revenue to the server and consider revenue maximizing strategies - number of priority levels and pricing. Numerical results show that with only three queues the server can attain near maximum revenue.

Equations

$$c_v(t)=NF(t)+n_{v}g(t)$$

$$F(t)=\int \text{mathbbm}{1}_{T(v)\ge t}d\mathcal{G}(v).$$

$$T^{\text{NE}}(v)=\underset{t\ge 0}{\mathrm{argmin}}{c}_v(t),$$

$$T^{\text{NE}}(v)=g^{-1}\left({\int }_v^{B}N\frac{d\mathcal{G}(x)}{n_x}\right).$$

Full text

Next, we consider a system offering $L$ levels of priority service with a FCFS queue for each priority level. Higher priorties are charged higher admission prices. Customers make two choices—time of arrival and priority of service. We show that the Nash equilibrium corresponds to the customer types being divided into $L$ intervals and customers belonging to each interval choosing the same priority level. We further analyze the net revenue to the server and consider revenue maximising strategies—number of priority levels and pricing. Numerical results show that with only a small number of queues (two or three) the server can obtain nearly the maximum revenue.

1 Introduction

Consider an airline that starts boarding a plane at time [math] and the $N$ customers that are booked to fly on the plane all arrive before boarding starts and wait in a FCFS queue. A customer that has a better rank in the queue gets a better choice of seats and luggage space. However, a better queue rank is achieved by arriving early and hence incurring a cost due to the waiting time before service (boarding) begins. Thus, with each customer we can associate a total inconvenience cost that has two components—(1) the waiting cost from the inconvenience due to arriving early (before boarding time), and (2) the boarding cost from the inconvenience due to customers that are served before the focal customer, i.e., inconvenience due to its rank or position in the boarding queue. An immediate model would be to assume that these costs are separable, and that the total cost is a weighted sum of the waiting time and the rank in the boarding sequence. (Note that the second component accounts for possibly non negligible service time once boarding begins.) Each customer then has to choose the time at which to arrive into the queue, i.e., the waiting time before service begins. Of course, for a given customer the queue rank that it obtains depends on the choice of arriving (and hence waiting) times of the other customers. In this paper we assume that the customers are strategic and that they choose their arrival time to minimize their individual cost. We also assume that none of the customers renege and that all $N$ take service. Thus we have the rudiments of the airplane boarding game which we will define more generally in the following.

Now assume that the customers are from a heterogeneous population and that the different types of customers calculate their total costs differently. Specifically, for the linear cost function described in the preceding, different types of customers have different weights for the queue rank and for the waiting time before boarding. In this case, rather than have a single queue and hence provide egalitarian service, the airline could reduce the social cost, and possibly increase its revenue, by providing service differentiation as follows. Maintain two FCFS queues with queue $2$ having strict boarding priority over queue $1,$ i.e., board customers from queue $2$ before boarding customers from queue $1.$ As before, assume that all the customers arrive before boarding begins and that there is no reneging by any of the $N$ customers. The airline could now charge a fixed premium to the customers who join the higher priority queue. To obtain better seats, i.e., to make their boarding costs lower, the customers can now choose to join the high priority queue and trade off some waiting time (before boarding begins) for payment of the premium in the higher priority queue. In such a system the customers make two choices—the arrival time and the queue that they plan to join and hence the premium that they will be paying. Once again, the actual total cost for each customer depends on the choice of these parameters by the other customers. We thus have the outline of the basic airplane boarding game in which customers strategically choose two parameters to minimize their individual total cost.

Several budget airlines in fact follow a simple version of the system described above and motivates this paper. In these airlines, seat numbers are not issued at check-in and typically two FCFS queues are maintained at the boarding gate. The premium queue allows service differentiation to a heterogeneous population where different types of customers have different relative values for their time and money and provides an opportunity for the airline to make additional revenue. If there is significant heterogeneity in the customer population then increased differentiation can provide increased revenue. Thus, rather than provide just two priorities, the airline could provide $L$ priority queues with customers in priority $l$ having strict priority over those of strictly less than $l.$ In this paper we consider this general case of the following airplane boarding game.

$N$ customers are to be served by a single server starting service at time $0.$ Different types of customers will be served through $L$ FCFS priority queues with higher priorities charging higher premiums. Associated with each type of customer is a total cost function that depends on the waiting time, the rank in the boarding sequence, and the premium paid to receive service from the queue of choice. The distribution of the types in the customer population is assumed known to the server and to the other customers. Each customer strategically chooses two parameters, the arrival time and the priority level of the queue from which it wants to receive service, to minimize its expected total cost. None of the $N$ customers renege and all of them obtain service and this is known to all the customers.

1.1 Preview of the Results

In this paper our objectives are twofold: (1) studying customers’ strategic behavior in their choice of the priority and of the arrival time, and (2) analyzing how the server may tune the service parameters, the number of priority levels and the charge for each of these priority levels, to maximize its revenue. The following is the summary of the key results in the paper.

In Section 3, we consider a system where customers choose a single priority parameter that we call the grade of service. The service provider offers a continuum of grades of service and defines a pricing function that maps the service grade to a price. Higher service grades have strictly higher priority in boarding. Customers are strategic and choose the service grade and hence the price. For this case we show that the revenue is independent of the pricing function.

Interestingly, we will see that the model considered in this section is very general in that it is applicable to ‘finite duration’ games like in the airline boarding game where a finite number of customers are to be allocated a finite resource. For this case our results should be interpreted in the spirit of Gueant10. The results are also applicable to ‘infinite duration’ games where the customer arrivals and service processes are in a stationary regime; in this case the results are to be interpreted in the spirit of, e.g., Lui85; Glazer86; Doroudi13. 2. 2.

In Section LABEL:sec:multiple-queues, we consider the system where the server maintains several FCFS queues at different priority levels. Customers pay a higher price to join a queue with higher priority. The number of priority levels and the price of service from these priority levels is assumed known to all the customers. In this case the customers have to select two parameters—the arrival time before boarding starts, and the priority of the queue from which they will receive service. We show that when the prices are fixed, the game admits a unique Nash equilibrium that we characterize exactly. We also numerically illustrate the strategic choices of the priority parameters at equilibrium. 3. 3.

Finally, in Section LABEL:sec:revenue, we investigate how the service provider may set the prices to join the various queues to maximize its revenue. When the number of priority queues are fixed, we show that these prices can be computed by solving a simple dynamic program. Numerically, we observe that the revenue increases with the number of available queues. However, the marginal gain of operating additional queues vanishes rapidly with increasing number of priorities. In fact we see that for several examples, maintaining a small number number of queues (two or three) may actually yield a revenue that is almost optimal, i.e., two or three levels priority levels is enough to extract a ‘close to maximum’ revenue.

All proofs are carried in the appendix.

In the next section we provide a brief overview of the relevant literature and delineate them from the results of this paper.

2 Related Work

The meeting game and the concert queueing game have similarities to the airplane boarding game . We first describe these briefly.

A meeting is scheduled to start at time $t$ but the participants arrive at random times and it usually starts at a random time $T$ when quorum is achieved. Knowing that the meeting does not necessarily start at time $t,$ participants $i$ chooses to arrive at $\tau_{i}$ to minimize her costs due to waiting (a function of $(T-\tau_{i})^{+}$), due to inconvenience (a function of $(\tau_{i}-T)^{+}$), and due to loss of reputation (a function of $(\tau_{i}-t)^{+}$). The knowledge of the structure and distribution of these costs among the participants can be strategically used by participant $i$ to choose $\tau_{i}.$ This meeting game has been introduced and studied in Gueant10. A variant of the meeting game is the ‘concert queueing game’. Here the concert hall opens at time $t$ and the attendees queue up for FCFS service; each attendee requires a random service time. Arriving late has the drawback that the better seats will be taken by those who came earlier, whereas arriving early induces a cost of waiting. Attendees strategically choose their arrival time distribution to mitigate these opposing costs. Several versions of this game have been studied in Jain11; Juneja12a; Juneja12b.

In the airplane boarding game all customers arrive before the start of service which is unlike in the meeting game and in the concert queueing game. Further, all analyses on the concert queueing game consider a single queue while we consider multiple priority queues providing differentiated service to a heterogeneous customer population that values time, money and queue rank (boarding costs) differently. A finite number of customers being served in a finite time is also considered in Lariviere04. They consider a discrete time system in which the customers incur a congestion cost due to other customers arriving in the same slot. This congestion cost could also be time-dependent. The airline boarding game that we describe here does not consider such a congestion cost.

Queueing systems with customers who value delays differently have been investigated in the queueing and economics literature. Specifically, providing lower delays for customers with higher delay costs and extracting a commensurate payment from them has been of interest for a while now. In an early work on pricing based on differential service, Kleinrock67 considered a queue in which customers bribe the server and the server provides a ‘highest bribe first’ service. Strategic customers in such a queue has been considered in Balachandran72; Lui85; Glazer86. In fact, our Theorem 3.1 in Section 3 is a generalization of the equilibrium analysis in Lui85; Glazer86. Providing differentiated service to a heterogeneous population through priority queues and pricing the priorities has also been considered in, among others, Marchand74; Rao98. More recent work on priority pricing is in Afeche04; Kittsteiner05 where only the expected delay contributes to the cost and the weights have a specific form. Priorities may also be auctioned like in Lui85; Glazer86.

The models in all of the preceding studies have considered what can be termed a ‘simple’ system where there is only one parameter that the customer chooses. It is the arrival time in Gueant10, the bid value in Kleinrock67; Balachandran72; Lui85; Glazer86 and the priority value in Marchand74; Rao98; Afeche04; Kittsteiner05. In contrast, in our model, the customer has to choose two parameters—the waiting time and the priority level. The closest system to the one that we consider is that in Honnappa11 where the attendees choose both the arrival time and the queue to join. The key difference with our model is that in Honnappa11 the queues start their service at different times and could be serving in parallel. Thus in this case there is ‘isolation’ between the customers that choose different queues.

A more recent work is that of Doroudi13 which considers a priority queueing system with a pricing menu. Here the customers have to declare their delay valuation and the focus is on deriving incentive compatible revenue maximizing pricing functions. In our model, type is private information and does not have to be revealed and the choice of the grade of service is made by the customer and not by the service provider.

3 Single Queue: Choosing the Arrival Time

A plane is to board $N$ customers starting at time $0.$ The $N$ customers arrive at different times before the boarding starts and form a FCFS queue. The population from which the $N$ customers are drawn is heterogeneous in that different customers value their waiting time differently. Let $\mathcal{V}=[A,B]\subset\mathbb{R}^{+}$ be the set of all customer types, and let it be endowed with a probability measure that has a cumulative distribution function (CDF) given by $\mathcal{G}.$ We assume $\mathcal{G}$ to be a continuous distribution.

A customer has to decide how much before the boarding time she has to arrive; we will denote this choice of waiting time (before boarding begins) by $T(v),$ for every $v\in\mathcal{V}.$ Therefore, waiting time of $T(v)$ implies that the customer arrives at time $-T(v)$. We assume that all customers of the same type make the same decision. Further, we assume that none of the $N$ customers renege and all of them join the queue before time $t=0.$ If a customer of class $v\in\mathcal{V}$ arrives at time $-t$ then her cost is given by

[TABLE]

where $F(t)$ denotes the fraction of customers that choose to arrive before $-t$ on average, $g(t)$ is the cost of waiting for $t$ time units, and $n_{v}$ is the weight that a type $v$ customer assigns to it’s waiting time cost. Notice that $F(t)$ is a monotonically decreasing function. Further, $g(t)$, being the cost of waiting for time $t$, is a monotonically strictly increasing function in $t$. We also assume that $g$ is continuous and differentiable function with $g(0)=0.$ Note that $g(\cdot)$ is independent of the customer type $v$. Without loss of generality we let $n_{v}$ to be a monotonically strictly increasing function in $v.$ Thus $NF(t)$ represents boarding cost, or the cost corresponding to the customer’s position in the queue. This cost includes the inconvenience due to the position in the boarding sequence and the service time of customers ahead of the focal customer after boarding begins. $n_{v}g(t)$ is the total waiting cost for a customer of type $v.$ Now, $F(t)$ is given by

[TABLE]

An arriving customer of type $v$ chooses $T(v)$ to minimize its expected cost where the expectation is taken over the customer type distribution $\mathcal{G}$. We seek an arrival profile $v\mapsto T(v)$ that minimizes the individual expected cost for all customers. This is also the Nash equilibrium (NE) policy as defined below.

Definition 1

$T^{NE}:\mathcal{V}\rightarrow[0,+\infty)$ is a Nash equilibrium policy if

[TABLE]

for every $v\in\mathcal{V}.$ 111This definition of NE is consistent with that found in the literature on non-atomic games AliKhan97.

Since customers with larger $v$ value their time more than those with smaller $v,$ we would expect that customers with larger $v$ should arrive later in a NE policy. We prove this in Lemma 1. Further, in Theorem 3.1 we show that there is a unique NE and also characterize this policy.

Lemma 1

Assume that a NE policy $T^{NE}$ exists. Then $T^{\text{NE}}(v)$ is a non-increasing function in $v.$

Proof

See Appendix LABEL:app:pf:lem:ne_property_single_p. ∎

Theorem 3.1

If $g(\cdot)$ is such that there exists a NE policy then it is unique and is given by

[TABLE]