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Managing Customer Expectations and Priorities inService Systems

Qiuping YuKelley School of Business, Indiana University, [email protected]

Gad AllonThe Wharton School, University of Pennsylvania, [email protected]

Achal BassambooKellogg School of Management, Northwestern University, [email protected]

Seyed IravaniIndustrial Engineering and Management Sciences, Northwestern University, [email protected]

We study how to use delay announcements to manage customer expectations while allowing a firm to

prioritize among customers with different sensitivities to time and value. We examine this problem by

developing a framework which characterizes the strategic interaction between the firm and heterogeneous

customers. When the firm has information about the state of the system, yet lacks information on customer

types, delay announcements play a dual role: they inform customers about the state of the system, while they

also have the potential to elicit information on customer types based on their response to the announcements.

The tension between these two goals has implications for the type of information that can be shared credibly.

To explore the value of the information on customer types, we also study a model where the firm can

observe customer types. We show that having information on the customer type may improve or hurt the

credibility of the firm. While the creation of credibility increases the firm’s profit, the loss of credibility does

not necessarily hurt its profit.

Key words : delay announcements; heterogenous customers; priority queue; information asymmetry; cheap

talk

1. Introduction

Delay announcements are common practice in service systems. Firms use delay announcements to

inform customers about the congestion level of the system. Among others, ComEd provides delay

announcements to its customers, such as “your waiting time is about 4 minutes.” In service sys-

tems where the queue is not visible to customers, delay announcements may influence customers’

expectation about their waiting time and thus their decisions of whether to join the system or not.

1

Author: Managing Customer Expectations and Priorities in Service Systems?2 Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!)

Consequently, to maximize the service provider’s value and minimize the costs, it is important for

the firm to understand how to use delay announcements to influence customer behavior. However,

this is a complex problem, which depends on the dynamic of the underlying service system, the

structure of the delay announcements, customers’ strategic behavior and their heterogeneity. Note

that, in the service industry, the customer population is often heterogeneous along various dimen-

sions and the firm may have limited capability to segment customers. While many call centers

may request customers to reveal service types by choosing one of the options provided through the

Interactive Voice Response (IVR) system, they may still have limited capability to differentiate

the types of their customers due to various reasons. First of all, the call centers can only provide a

limited number of options in the IVR system which cannot provide sufficient information for the

call center to identify the specific reasons for the calls. Even if a customer has chosen a particular

option, there is still remaining unknown information to be elicited about the customers’ value of

service and patience. Moreover, many customers may simply skip the IVR system and request to

speak to the agent immediately. In this paper, we study how the firm should use delay announce-

ments to manage customers’ expectations and priorities, when it does not directly observe the

types of its heterogeneous customers. To study the value the firm may gain or lose by observing

customer types, we will also explore a model where we allow the firm to observe customer types.

Given customers are heterogeneous, our model has two important features: 1) The customers

may have different patience times and may value the service offered by the firm differently, which is

private information to the customers; and 2) Given that the customers value the service differently

and have different patience times, the firm may want to prioritize the customers. To this end, our

work is related to the literature on delay announcements with strategic firm and strategic cus-

tomers and the literature on priority queues. Note that previous works on the strategic interaction

between the customers and the firm through delay announcements focus on the case with homoge-

neous customers. Thus, in this literature, customers’ identity is known to the firm. The firm uses

delay announcements only to inform customers about the system congestion, as a way to manage

customers’ expectations regarding their anticipated delay. However, in our context, customers are

heterogeneous and the firm does not directly observe the different values of obtaining the service for

the customers or the customers’ patience times. Thus, for announcements to be effective, the firm

may also use delay announcements to elicit information about customers’ preference, besides using

it to inform customers about their anticipated delay. The firm can then prioritize the customers

based on the elicited customer information. Note that in the literature of priority queues, it is

often assumed that customers are pre-segmented. Thus, the firm can prioritize customers based on

the segments of the customers. However, in our model, the firm can only prioritize the customers

Author: Managing Customer Expectations and Priorities in Service Systems?Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!) 3

based on the customer information elicited through delay announcements. We show that the ten-

sion between the two roles played by delay announcements and the interdependence between the

firm’s announcement and priority policies in our context, lead to unique insights about how the

firm should manage customers’ expectation and priorities through delay announcements.

In our model, we consider a queuing system in which customers arrive to seek the rewards of

service, while they incur costs due to waiting in the system. There are two types of customers, who

differ in their rewards of being served and their waiting cost per unit time. As for the firm, it obtains

values by serving customers and incurs costs for holding customers in the system. The value that the

firm obtains by serving a customer is different for customers of different types. Both the customers

and the firm have private information of their own: the customers have private information on

their own types, while the firm has private information about the system state. When customers

arrive, the firm provides announcements to the customers. Customers decide on whether or not to

join the system based on the announcements and their own types to maximize their own utility. As

the firm does not observe customer types, delay announcements play a dual role: they inform the

customers about their expected delay, while they may also help the firm elicit information about

the types of the customers based on their response. Consequently, the firm may be able to prioritize

customers based on the elicited customer type information. The firm is strategic in choosing its

announcement and priority policies to maximize its profit, while anticipating customers’ response.

Note that the firm’s priority policy highly depends on its announcement policy, given that firm

can only schedule the customers based on the customer type information elicited through delay

announcements. We examine the ability of the firm to sustain an equilibrium with influential cheap

talk in such settings.

We next summarize our main contribution and insights in this paper:

1. We have analytically characterized the structures of the firm’s optimal announcement and

priority policies under the following two benchmark cases: 1) the benchmark case where the firm

has full information on customer types and full control over customers’ admission; and 2) the

benchmark case where the firm has full information on customer types and control over customers’

admission but is subjected to non-negative expected utility for both customer types.

2. We have explicitly characterized the influential equilibria emerging between the firm and its

heterogeneous customers in a complex cheap talk game. Based on the equilibrium analysis, we

show that it may not be necessary for the firm to fully differentiate customers of different types to

achieve the unconstrained first best, when the per unit holding cost is the same for all customers.

Partially separating customers could be sufficient to achieve the unconstrained first best solution.

Furthermore, we show that under certain conditions, a pooling equilibrium, where the firm does

not elicit information on customer types at all, may perform the best in firm’s profit among all

equilibria.


3. When the per unit holding costs are different for customers of different types, we show that

the firm cannot achieve the unconstrained first best through delay announcements. To improve

firm’s profit, we find that it is optimal for the firm to give absolute priority to customers who

receive announcements associated with the higher expected per unit holding cost. As for the firm’s

announcement policy, we show that the firm cannot improve its profit by using more than three

announcements.

4. We have characterized the equilibria where no credible information is shared between the cus-

tomers and the firm. We refer to such equilibria as babbling equilibria. By comparing the babbling

equilibria and the influential equilibria where credible delay information is provided, we find that

providing announcements always improves the firm’s profit compared to the case when announce-

ments are not provided. However, from customers’ perspective, in contrast to Allon et al. (2011)

which shows that providing delay announcements always improves customers’ utility when cus-

tomers are homogeneous, we show that it may improve or hurt customers’ utility when customers

are heterogeneous.

5. To study the value that the firm may gain or lose by observing customer types, we have also

studied a model where the firm observes the types of customers upon their arrival. Through the

comparison between the equilibria emerging when the firm observes customer types and the ones

emerging when the firm does not, we find that information on customer types may expand or

contract the region where the firm can credibly share the delay information. We also find that the

creation of credibility improves the firm’s profit. Similarly, one may expect the loss of credibility

to hurt the firm’s profit. However, we show that the loss of credibility may improve or hurt the

firm’s profit.

From a practical point of view, this paper can guide system managers to make a decision on

whether the firm should use delay announcements to manage the system congestion. More specifi-

cally, what delay information should be provided, and what information about the system status

and customer types should be collected to provide such delay information? Our results show that

providing delay announcements is most valuable for the firm when it directly observes the types

of the customers or when the costs incurred by the firm for keeping the customers waiting are of

similar order of magnitude for all customers. Surprisingly, providing delay announcements is not

very helpful for the firm if the costs of delaying customers for the firm are vastly different across

different customer types and the firm does not directly observe the types of these customers. In

this case, other tools (such as pricing) or refined customer segmentation may be required to further

improve the firm’s profit. When providing delay announcements is valuable for the firm, we show

that it may not be necessary to (fully) segment customers through delay announcements or to

provide announcements with very refined granularity. This is in line with the strategies used by


many firms which have no intentions to differentiate customers through delay announcements or

use announcements with very refined granularity.

2. Literature and Positioning

As we study the use of delay announcements to manage customers, we divide the relevant literature

into the following branches: queuing models with delay announcements, admission control, pricing

in priority queue, and cheap talk games.

Queuing Models with Delay Announcements. One of the first papers that discusses the

question of whether to reveal the queue length information to customers is Hassin (1986), which

studies the problem of whether a price-setting, and revenue-maximizing service provider should

provide the queue length information to arriving customers when it has the option to do so. Whitt

(1999) brings the concept of information revelation to the specific setting of call centers, where

call centers communicate with their customers about the anticipated delay by providing delay

announcements. Guo and Zipkin (2007) extends the model above by studying the impact of delay

announcements with different information accuracy. All the above papers focus on the cases when

customers are either all provided with information or none of the customers are provided with

information. A recent paper Hu et al. (2015) has studied how information heterogeneity among

customers impacts the throughput and social welfare. Armony et al. (2009) extends the works above

by accounting for customer abandonment in the model. Motivated by various delay announcements

used in practice, Ibrahim and Whitt (2009) explores the performance of different real-time delay

estimators based on recent delay experienced by customers, allowing for customer abandonment.

All the aforementioned works assume that the information is credible and is treated as such

by customers. To this end, it is important to note that Yu et al. (2014) has provided empirical

evidence indicating that customers may be able to strategically interpret the announcement. Allon

et al. (2011) has accounted for such strategic customers. Specifically, the authors study the problem

of information communication by considering a model in which both the firm and the customers

act strategically: the firm in choosing its delay announcement while anticipating the customer

response, and the customers in interpreting these announcements and in making the decision on

whether to join the system. Note that Allon et al. (2011) focuses on the setting where customers

are homogeneous. Thus the sole role of delay announcements is to inform customers about their

estimated delay to manage their expectations. However, we consider the case with heterogeneous

customers where the firm does not directly observe the types of the customers. To this end, delay

announcements play a duel role in our context. In particular, besides using delay announcements

to inform customers about their expected delay, the firm may also use delay announcements to

elicit information on customer types. The firm can then prioritize customers based on the elicited


customer type information to optimize its profit.The tension between the roles played by the

announcements, and the interdependence between the announcement and priority policies lead to

unique insights in how the firm should provide delay information. It is also worth mentioning that

there have been very few studies in priority queues with imperfect information on customer types,

see Chan et al. (2013) and Argon and Ziya (2009). In these studies, the imperfect customer type

information is exogenously given, while it is endogenously elicited through delay announcements

in our study.

Admission Control. Our paper is related to the literature of admission control, which starts

from Naor (1969). The author shows customers are more patient than what a social planner would

like them to be. The imposition of tolls may lead to the attainment of social optimality. Rue and

Rosenshine (1981) extends the model above to the setting with multiple customer classes who are

first-come, first-served. While none of the works mentioned above consider service priorities, Chen

and Kulkarni (2007) takes one step further and studies the admission control problem for queuing

system serving two customer classes with priority. Our model significantly differs form the models

above, given that, in our setting, the firm does not have control over customers’ decisions and

customers terminate their calls based on their assessment of the waiting time.

Pricing in Priority Queue. In the presence of multiple customer classes and when the firm

does not observe customer types or does not have direct control on customers’ priorities, pricing is

one of the commonly used tools to differentiate customers and then prioritize them when necessary.

Mendelson and Whang (1990) suggests a pricing mechanism to optimize the overall social welfare in

an M/M/1 system with multiple types of customers. Afeche (2004) extends the model in Mendelson

and Whang (1990) to study how the firm should design an incentive compatible pricing-scheduling

mechanism to maximize its revenue, given that there are two types of customers. The papers above

show that one may design a direct revelation mechanism to achieve the optimal result with pricing

strategies. However, there are organizations where pricing strategies are not preferred or allowed,

e.g., Disneyland, DMV or IRS. To address the problems that arise in these contexts, our paper

aims to explore how to manage customer expectations and priorities using delay announcements.

Cheap Talk Game. The framework used in this paper is inspired by the classical cheap talk

model proposed in Crawford and Sobel (1982). The authors introduced a cheap talk game model

of strategic communication between a sender and a receiver. In this model, the sender, who has

private information, sends possibly noisy information to the receiver, who then takes payoff-relevant

actions. It is important to note that the distribution of the sender’s private information is given

exogenously and does not depend on the equilibria of the game. However, in our endogenous cheap

talk setting, the distribution of the private information depends on the equilibrium of the game.

Driven by the specific queuing application, our model has two novel features: first, the game is


played with multiple and different types of receivers (customers) whose actions have externalities

on other receivers; and second, the stochasticity of the state of the system is not exogenously given

but is determined endogenously. Allon et al. (2011) appears to be the first paper in the operations

management literature to consider a model in which a firm provides unverifiable real time dynamic

delay information to its customers.

3. Model with No Information on Customer Types

We consider a service system with a single service provider1, where customers arrive according to

a Poisson process with rate λ and the service time is exponentially distributed with rate µ2. We

assume that there are two types of customers , which we refer to as low and high type customers

denoted by L and H, respectively. With probability βi, for i ∈ H,L, an arriving customer is a

type i customer. Customers arrive to seek service and get rewards from the service, while they

incur costs due to their waiting in the system. The reward of being served for type i customers

is denoted as ri, while the waiting cost per unit time is denoted as ci, for i ∈ H,L. From the

firm’s perspective, the firm obtains values from serving customers, while it incurs costs for holding

customers in the system. Let us denote the value that the firm obtains from serving a type i

customer as vi > 0, for i ∈ H,L. Without loss of generality, we let vH > vL. As for the holding

costs that the firm incurs, they include, among others, the loss of goodwill for keeping the customers

waiting and the cost of providing the telephone connection. In particular, we denote the per unit

time holding cost of a type i customer as hi, for i∈ H,L. In practice, the holding cost of the more

valuable customers is often higher than, if not equal to, that of the less valuable customers. Thus,

we focus on the case with hH ≥ hL throughout the paper. We assume all the above parameters

are known to both the customers and the firm.3 When customers arrive, the firm provides delay

announcements to customers possibly based on the current congestion in the system. We focus on

the scenario where the firm cannot observe customer types before it provides announcements in

this section. To explore the value that the firm may gain or lose by directly observing customer

types, we will relax this assumption in Section 6. Based on the announcements received, customers

decide on whether or not to join the system by trading off between their rewards of being served

1 While we assume there is only one agent in the system, we conjecture that the structural results of the firm’s policiesand our main insights will continue to hold if we consider an M/M/c system where there are multiple agents.

2 The assumptions of Poisson arrival process and exponential service times allow us to formulate the problem andcharacterize the structure of the optimal announcement policy of the firm. It is worth mentioning that Brown et al.(2005) shows that a Poisson process can characterize the arrival process of callers in call centers extremely well. Aboutthe service time, Carr and Duenyas (2000) studies a similar exponential model but investigates optimal productionand admission control policies in manufacturing systems. They extend it with Erlang distributed inter-arrival andproduction times, and they show that the structural results obtained with the exponential model continue to hold.Similar intuition applies to our model.

3 We will discuss the robustness of our main insights to this assumption at the end of this section.


and their waiting costs. To characterize the interactions between customers and the firm through

delay announcements, we next define the game that both the customers and the firm engage in.

The expected utility of a type i customer, for i∈ H,L, is given by

ui(ai,w) =

ri− ciw if ai = join0 if ai = balk,

(1)

where ai is type i customers’ decision on whether or not to join the system and w is customers’

expected sojourn time in the system. Note that to maximize utility, customers of type i, i∈ H,L,

would like to join the system when the expected waiting time in the system is smaller than rici

,

balk otherwise. To this end, we refer to rici

as the patience of type i customers with i ∈ H,L.

Throughout the paper, we assume that rici> 1

µ, for i∈ H,L, so that customers of both types are

better off by joining the system when there is no waiting. Otherwise, customers would not join

the system even when there is no delay, and it is not necessary to provide delay announcements at

all. Meanwhile, the firm’s expected profit by serving a customer of type i, for i ∈ H,L, is given

by vi− hiw. We assume hi > 0 for all i ∈ H,L, so that the firm would have an incentive not to

admit either customer type beyond a certain finite threshold.

We assume customer types are private information of the customers, while the current state of

the system, i.e., the number of customers in the system, is private information of the firm. To

investigate how delay announcements impact customers’ behavior and what announcements the

firm should provide to maximize its profits, we next formally describe the game played between the

firm and the customers. The equilibrium concept that we use is the Markov Perfect Bayesian Nash

Equilibrium (MPBNE). In our case, it is simply a set of strategies of the firm and the customers

at Nash Equilibrium that describes how customers incorporate delay announcements and their

own types into their decisions about whether or not to join the system, and how the firm chooses

announcements to maximize its profits. MPBNE only allows actions to depend on pay-off relevant

information, which rules out strategies that depend on non-substantive moves by the opponent.

We will formally define MPBNE later in this section.

To describe the announcements, let M = m1,m2,m3... be the set of possible discrete announce-

ments provided by the firm. The announcements can be in a wide variety of forms. They can be

quantitative announcements, such as the one: “Your estimated waiting time is less than 2 minutes.”

They can also be qualitative announcements, such as the one: “All agents are currently serving

other customers, please hold.” To characterize the interaction between the customers and the firm,

we start from how customers respond to announcements. Once customers receive announcements

from the firm, they decide whether or not to join the system based on the announcements received

and their own types. Customers of different types may respond differently to the same announce-

ment due to different waiting costs and rewards that they receive from being served. Customers’


action rule is given by a function ai :M 7→ 1, 0, for i ∈ H,L. In particular, ai(m) = 1 means

the type i customer joins the system when she receives the announcement m, while ai(m) = 0

represents that the customer balks.

We next turn to define the strategy of the firm. Note that the firm’s optimal strategy is comprised

of two components in our model: 1) the firm decides what announcements to provide based on the

number of customers of each type to induce the desired customer response, and 2) given that there

are two types of customers in the system, the firm may want to prioritize them when necessary.

Let us start from the announcement policy. To make a better decision on what announcements to

provide, the firm may want to elicit as much information on customer types as possible. However,

given that the firm does not directly observe customer types, it can only differentiate customers

if they respond to announcements differently. Thus, instead of differentiating customers based on

their types, the firm can only classify customers based on the announcements that they receive.

According to the action rule defined above, there are two different customer reactions, i.e., join and

balk, for each customer type. Since there are two types of customers in the system, we can classify

the announcements into four categories. In particular, the first category includes announcements

under which both customer types balk. The second category includes announcements under which

only the high type customers join the system but not the low type. The third category includes

announcements under which only the low type customers join the system but not the high type,

while the fourth category includes the announcements under which both customer types join the

system. To represent these four categories of announcements, we let MO be the set of announcement

where customers of type i ∈O join and customers of type i ∈Oc balk. Thus, we have M∅, MH,

ML and MH,L denoting the four categories of announcement sets mentioned above, respectively.

One can see that M∅, MH, ML and MH,L are all subsets of M , which is the set of all possible

announcements provided by the firm. Moreover, the announcement subsets M∅, MH, ML and

MH,L are mutually exclusive. To this end, the firm can classify the customers in the system into

three categories: customers receiving announcements from MH, ML or MH,L. Note that the

system state can be fully characterized by the number of customers from each of these types in

the system. We let nH , nL and nHL denote the number of customers in the system that received

announcements from the subsets MH, ML and MH,L, respectively. To this end, the set of

system states, denoted by S, is given by S = (nH , nL, nHL)|(nH , nL, nHL)∈N30.

We next formally define the announcement policy of the firm. Note that the announcement

policy of the firm can be characterized by a function A : S 7→M , where S is the set of system

states. In particular, we have A(nH , nL, nHL) = m, if the firm provides the announcement m to

the next arriving customer at the system state (nH , nL, nHL). For the ease of notation, we let


n = (nH , nL, nHL)4, 1H = (1,0,0), 1L = (0,1,0), and 1HL = (0,0,1) throughout the paper. As for

the scheduling policy of the firm, it is a function which maps the current system state to the next

customer to be served. As we mentioned earlier, the firm can only distinguish the customers based

on the announcements they receive. In particular, the firm can sort the customers in the system

into three categories: customers receiving announcements from MH, ML or MH,L. To this end,

the firm can only schedule the customers based on the announcements they receive. In particular,

the firm’s scheduling policy is given by a function g : S 7→X, where S is the set of system states

and X is the set of announcement types with X = M∅,MH,ML,MH,L. For example, we

have g(n) =ML, if the next customer to be served at state n is the first customer in the system

who received an announcement from the announcement subset ML.5

Note that the underlying stochastic process can be modeled as a birth-death process. To this

end, we assume λ < µ6 so that there exists a unique steady state for the underlying birth-death

process. The steady-state probability distribution of the system state n depends on both the

customer strategy, ai with i ∈ L,H, the firm’s scheduling policy g and announcement policy A.

Let p(n|a, g,A) represent the steady-state probability of the system state n, conditional on the

type i customers’ strategy ai, the firm’s announcement policy A and scheduling policy g with

i∈ H,L. Meanwhile, we let wgm(n) denote the expected waiting time of the customer who receives

the announcement m and joins the system at state n. To this end, the expected utility of type i

customers who receive the announcement m∈M is given by E[ri− ciwgm(n)|A(n) =m], if they join

the system. In particular, we have

E[ri− ciwgm(n)|A(n) =m] =

∑(n):A(n)=m [ri− ciwgm(n)]p(n|a, g,A)∑

(n):A(n)=m p(n|a, g,A)

We next formally define the pure strategy MPBNE in the following definition. For simplicity, we

focus on pure strategy equilibria throughout the paper.

Definition 1 (Markov Perfect Bayesian Nash Equilibrium). We say that the firm’s

announcement policy A(.), scheduling policy g(.) and customers’ action rule ai(.) with i∈ H,L,

form a Markov Perfect Bayesian Nash Equilibrium (MPBNE), if they satisfy the following condi-

tions:

1. For each m∈M and i∈ L,H, we have

ai(m) =

1 if E[ri− ciwgm(n)|A(n) =m]≥ 00 otherwise,

(2)

4 Note that we obtain the system state n = (nH , nL, nHL) excluding the next arriving customer throughout the paper.

5 We have g(n) =M∅, if the firm stays idle at state n.

6 Note that λ< µ is a sufficient condition for a steady state to exist, given there are balking customers.


2. There exists relative value functions V (n) with n∈N30, constant γ, together with the schedul-

ing policy g(n) and the announcement policy m=A(n), that solve the following Bellman equation:

V (n) +γ

Λ

=1

Λ

− ((hHβH +hLβL)nHL +hLnL +hHnH)

+λmaxm∈M

(V (n + 1HL) +βHvH +βLvL)aH(m)aL(m)

+ (βHV (n + 1H) +βLV (n) +βHvH)aH(m)(1− aL(m))

+ (βHV (n) +βLV (n + 1L) +βLvL)aL(m)(1− aH(m))

+V (n)(1− aH(m))(1− aL(m))

+µmaxg(n)

(V (n−1H)InH>0+V (n)InH=0)Ig(n)=MH

+ (V (n−1L)InL>0+V (n)InL=0)Ig(n)=ML

+ (V (n−1HL)InHL>0+V (n)InHL=0)Ig(n)=MH,L

+V (n)Ig(n)=M∅

, (3)

with Λ = λ+µ.

The first condition in the above MPBNE definition, see (2), describes the customers’ decision rule.

In particular, customers join the system if the expected utility, conditional on the announcements

received and the firm’s scheduling policy, is non-negative, and balk otherwise. The second condition

given by (3) is the Bellman optimality condition for the firm’s scheduling and announcement

policies conditional on customers’ response.7

We assume that the reward cost ratio (ri/ci) of the customers are known to the firm conditional

on the types of the customers. As we explained in the beginning of this section, the reward cost

ratio of customers is tied to customer patience. It is important to note that, as part of the common

practice, call centers conduct extensive research to understand their customers’ willingness to wait

to make better operational decisions. In fact, Yu et al. (2014) provides one possible approach that

call centers can take to estimate the reward cost ratio of a given customer type. This indicates that

the firm should be able to figure out the reward cost ratios of the customers once it observes the

type of the customer. Moreover, we also assume that the customers know the value and holding

cost parameters of the firm. We impose this assumption for technical convenience to solve the

model mathematically. In fact, all of our insights will continue to hold, provided that customers

7 Note that the expected proportion of the high type customers among the nHL customers is same as the expectedproportion of the high type customers among the arriving customers. This is due to the following reasons: 1) thearrival process is Poisson; 2) the service time is the same for both customer types; 3) the firm cannot differentiatethese nHL customers and they are served in a first-come, first-served manner within themselves; and 4) we focus onpure strategy equilibria.


can form a correct belief about their waiting time based on the announcements. To this end,

it is worth mentioning that Yu et al. (2014) provides strong empirical evidence supporting the

assumption that customers are capable of forming a correct belief on the system dynamic based

on the announcements.

Another implicit assumption made in Definition 1 is that customers only decide on whether

to join or balk the system, and customers will not renege after joining the queue. To this end,

it is worth mentioning that Allon et al. (2011) shows that, even if the customers are allowed

to update their beliefs and renege the queue, it is in the customers’ best interest to stay in the

system until service after joining the system when their waiting cost is linear. This is because

the hazard rate of the waiting time is increasing over time. While Allon et al. (2011) focuses on

the case when customers are homogeneous, the same logic applies to our setting where customers

are heterogeneous. Thus, customer abandonment will not arise endogenously in our model and

the characterization of equilibria remains the same even when we allow customers to abandon.

Moreover, note that it is common in the literature to assume that providing delay announcements

tends to cause balking instead of reneging, see Whitt (1999).

3.1. Terminologies

We next introduce a few important terminologies that we will use throughout the paper.

Definition 2 (Influential and Non-influential Equilibrium). 1. We say that an

MPBNE (aL, aH ,A, g) is influential if, ∀i ∈ H,L, there exists two announcements mi1 and mi

2

which are used with positive probability8 in the equilibrium so that we have ai(mi1) 6= ai(m

i2).

2. We say that an MPBNE (aL, aH ,A, g) is non-influential, if we have ai(m1) = ai(m2), ∀m1,m2 ∈

M and i∈ H,L.

Definition 3 (Pooling, Semi-separating and Separating Equilibrium). 1. We say

that an MPBNE (aL, aH ,A, g) is a pooling equilibrium if, ∀m ∈M , which are used with positive

probability in equilibrium, we have aL(m) = aH(m).

2. We say that an MPBNE (aL, aH ,A, g) is a semi-separating equilibrium, if ∃i, j ∈ H,L with

i 6= j, ∀m ∈ M that is used with positive probability in equilibrium, we have ai(m) ≥ aj(m);

moreover, there exists at least one announcement m ∈M which is used with positive probability

in equilibrium, such that ai(m)>aj(m) holds.

3. We say that an MPBNE (aL, aH ,A, g) is a separating equilibrium if ∃m1,m2 ∈M and m1 6=m2,

which are used with positive probability in equilibrium, such that aL(m1)>aH(m1) and aL(m2)<

aH(m2) both hold.

8 We say that an announcement m is used with positive probability under an equilibrium (aL, aH ,A, g), if∑(n):A(n)=m p(n|aL, aH ,A, g)> 0.


Following the above definition, we refer to an influential equilibrium, where any given announce-

ment influences customers of different types identically, as a pooling equilibrium. We refer to an

influential equilibrium as a separating equilibrium, if there exists one announcement that induces

low type customers to join and high type customers to balk, while another announcement that

induces the exact opposite reactions from these two types of customers. Moreover, we refer to

an influential equilibrium between a pooling equilibrium and a separating equilibrium as a semi-

separating equilibrium.

From the cheap talk literature, one may expect that an equilibrium in cheap talk games is not

unique even when it exists. This is because, one can always relabel the announcements to induce

other equilibria with the same outcomes and pay-offs for the firm and the customers. Similar

to Allon et al. (2011), we introduce the definition for MPBNE being Dynamic and Outcome

Equivalent9 (DOE) as follows.

Definition 4 (Dynamic and Outcome Equivalent (DOE)). We say that two MPBNE

(a1L, a

1H ,A

1, g1) and (a2L, a

2H ,A

2, g2) are DOE, if a1i (A

1(n)) = a2i (A

2(n)), ∀i∈ H,L and ∀ n∈N30.

It is important to note that the utility of each customer type and the profit of the firm are identical

under any two MPBNEs which are Dynamic and Outcome Equivalent.

4. Benchmarks: Unconstrained First Best & Constrained First Best

Note that the main goal of the paper is to study how the firm should use delay announcements

to manage the expectation and priorities of its heterogeneous customers. To do so, we need to

characterize the equilibria emerging between the firm and its heterogeneous customers. However,

given the complexity of the model, it is difficult to directly construct the equilibrium. To this end,

we start from a benchmark case where the firm has not only full control over customer admission,

but also full information on their types. We refer to the solution to this problem as the firm’s

unconstrained first best solution. It characterizes the firm’s pure preference under no constraints.

Given that, in our model, the firm has no control over customers’ joining behavior and lacks the

ability to differentiate customers of different types, the firm’s unconstrained first best solution may

not always be sustained in equilibrium. However, it plays a critical role in constructing equilibria

for our model as it will become clear in Sections 5 and 6.

One may also envision an alternative first best solution where the firm has information on cus-

tomer types and control over customer admission but is subjected to non-negative expected utility

for the arriving customers of both types. We refer to it as the firm’s constrained first best solution.

While the constrained first best solution does account for customers’ incentives, we will show later

9 Note that “Dynamic” in “Dynamic and Outcome Equivalent” refers to the underlying stochastic process of thequeuing system.


that it cannot be sustained in an influential equilibrium when it differs from the unconstrained

first best solution. To this end, throughout the paper, we use the unconstrained first best solution

to help construct equilibria. Meanwhile, we use both the constrained and unconstrained first best

solutions as benchmarks with which we compare the equilibria emerging in our model. We next

characterize both the unconstrained and constrained first best solutions.

4.1. Unconstrained First Best Solution

Let us start with the unconstrained first best solution. Note that the firm’s strategy is comprised

of two components: the firm’s admission policy and its scheduling policy. In particular, one will

see that the firm’s optimal admission policy may depend on the system states. When the firm

observes the types of the customers upon their arrivals, the system states can be characterized

by the number of customers of each type. To characterize the system state, we let n0H and n0

L be

the number of high and low type customers in the system, respectively. Thus, the total number of

customers in the system is given by nT = n0H + n0

L. Moreover, we let SI be the set of the system

states when the firm observes the type of the customers. In particular, the set of the system states

is given by SI = (n0H , n

0L)|(n0

H , n0L) ∈ N2

0.10 Other than the admission policy, the firm may also

want to schedule customers appropriately to optimize profits. The first two results in the following

lemma show that the firm’s optimal admission policy can be characterized by two monotonically

non-increasing switching curves. Furthermore, the last result of the lemma characterizes the firm’s

optimal scheduling policy. In particular, we find that, when we have hH > hL and the service

requirement is the same for both the low and high type customers, it is optimal for the firm to

give absolute priority to the high type customers. This shows that the cµ rule, which was first

established in Smith (1956), continues to hold in our setting. All proofs including the one for the

following lemma are relegated to Appendix C.

Lemma 1. The unconstrained first best solution of the firm is characterized as follows:

1. For each n0L ≥ 0, there exists a threshold SH(n0

L), such that a high type customer is accepted

if and only if n0H ≤ SH(n0

L). Furthermore, SH(n0L) is monotonically non-increasing in n0

L.

2. For each n0H ≥ 0, there exists a threshold SL(n0

H), such that a low type customer is accepted

if and only if n0L ≤ SL(n0

H). Furthermore, SL(n0H) is monotonically non-increasing in n0

H .

3. When hH > hL, the firm gives preemptive resume priority to the high type customers in the

system. When hH = hL, the order of service does not impact the profit of the firm.

10 Recall that, in the model with no information described in Section 3, the firm does not observe customer typesand can only differentiate customers based on the announcements that they receive. To this end, the system statesare characterized by the number of customers receiving each type of the announcements. In particular, the set ofthe system states S is given by S = (nH , nL, nHL)|(nH , nL, nHL) ∈N3

0, where nH , nL and nHL are the number ofcustomers in the system receiving announcements from the announcement sets MH, ML and MH,L, respectively.The total number of customers in the system is given by nT = nH +nL +nHL.


It is worth mentioning that, if there exists i ∈ H,L so that we have Si(0)< 0, to achieve the

unconstrained first best solution, the firm will not admit type i customers regardless of the number

of customers in the system. In this case, the system dynamic will be identical to the one discussed

in Allon et al. (2011) where there is only one customer class. To this end, throughout this paper,

we focus on the cases with SH(0)≥ 0 and SL(0)≥ 0.

The following lemma shows that, when the per unit holding cost is the same for all customers,

we can simplify the firm’s optimal admission policy characterized in Lemma 1.

Lemma 2. When hH = hL, the two switching curves given in Lemma 1, i.e., SL(n0H) and SH(n0

L),

are given by the following equations:

SL(n0H) = nfL−n0

H and SH(n0L) = nfH −n0

L.

Moreover, nfL and nfH are two finite constants with nfL ≤ nfH . These two constants are independent

of the system state given by (n0H , n

0L).

The above lemmas imply that if the firm has full control over customers’ admission to the system

and has full information about the customer types, it is optimal for the firm to adopt the threshold-

based policy characterized by the two switching curves SH(n0L) and SL(n0

H). Moreover, when the

per unit holding cost is the same for all customers, we can simplify these switching curves and

characterize the firm’s optimal admission policy by the two finite thresholds nfL and nfH . Note that,

when hH = hL and vH > vL, the firm will admit the high type customers whenever it admits the

low type to maximize profits. In terms of the firm’s optimal scheduling policy, when hH = hL, the

order of service does not impact the total waiting cost and thus does not impact the firm’s profit.

To this end, when hH = hL, we focus on the scheduling policy where the firm serves the customers

in a first-come, first-served manner, regardless of their types.

4.2. Constrained First Best Solution

We next turn to characterize the firm’s constrained first best solution. It is important to note that

the constrained first best solution is identical to the unconstrained first best solution if customers’

expected utility from joining the system under the unconstrained first best solution is non-negative

for both customer types. We will explicitly provide the conditions under which the two first best

solutions are equivalent in Appendix A.

When the constrained first best solution differs from the unconstrained first best solution, we

show that the constrained first best solution can be characterized by a mixed strategy where the

firm randomizes between two threshold based pure strategy policies. To characterize such a mixed

strategy, we define yj, with j = 1,2, as a threshold based pure strategy of the firm which can be


characterized by two switching curves SjH(n0L) and SjL(n0

H). In particular, under the policy yj with

j = 1,2, high type customers are admitted if and only if n0H ≤ S

jH(n0

L), while low type customers are

admitted if and only if n0L ≤ S

jL(n0

H). Moreover, the firm gives preemptive resume priority to high

type customers under both policies y1 and y2. We next formally characterize the constrained first

best solution when it is different from the unconstrained one in the following lemma. The proof to

the following lemma is inspired by Gans and Zhou (2003), Sennott (2001) and Beutler and Ross

(1985).

Lemma 3. Assuming hH >hL and rici≥ 1

µ−βiλwith i∈ L,H11, there exists monotonically non-

increasing switching curves, SjH(n0L) and SjL(n0

H) with j = 1,2, and a probability p ∈ (0,1), such

that the constrained first best solution can be characterized by the mixed strategy θ(p) where the

firm chooses the policy y1 with probability p and the policy y2 with probability 1− p.

Note that, while the constrained first best solution guarantees non-negative expected utility for

the arriving customers of both types, it cannot be sustained in an influential equilibrium when

it differs from the unconstrained first best solution. In particular, for an influential equilibrium

to exist, for any given customer type, the firm should be able to signal “High” when the system

congestion level is beyond a switching curve, and customers of the corresponding type who receive

the announcement “High” should balk. If the switching curve is not the one characterized in the

unconstrained first best solution, it is always better off for the firm to deviate to the optimal

policy characterized in the unconstrained first best solution. Thus, any strategy emerging in the

constrained first best solution (that is different from the one in the unconstrained first best solution)

can never be sustained in an influential equilibrium.

5. Equilibria and Insights: Model with No Information on CustomerTypes

Note that, in our model, customers have no information about the system status, while the firm

not only has no control over customer behavior, but also lacks the ability to differentiate customers

of different types. The key questions now are whether and how the firm can credibly communicate

with the customers using delay announcements in our model. To address these questions, we next

characterize the equilibria emerging between the firm and its heterogeneous customers based on

the results in Section 4. In particular, we start by characterizing the influential equilibria where

the firm provides credible delay announcements to induce the desired responses from its heteroge-

neous customers. We will then explore the equilibria where the firm provides no announcements

11 In Lemma 3, we assume rici≥ 1

µ−βiλwith i ∈ L,H for technical convenience. Note that this condition implies

that, for the customer type who receives absolute priority, the expected utility of joining the system is non-negative.We believe this is a reasonable assumption in most service systems in practice.


or announcements which are independent from the system states. We refer to such equilibria as

babbling equilibria. By comparing the babbling equilibria and the influential equilibria, we provide

insights on whether and how the firm should provide announcements from both the customers’ and

the firm’s perspectives.

Recall that there are two different actions, i.e., join and balk, for each customer type. Thus, there

are four possible customer reactions when there are two customer types: both customer types join

the system, only the high type customers join the system, only the low type customers join the

system, and both customer types balk. One may expect the firm to use four different announcements

to induce the desired customer reactions in equilibrium. However, the following theorem shows

that, for any given pure strategy equilibrium, we can find a pure strategy equilibrium where the

firm uses at most three announcements which is DOE to the given equilibrium. The main reason

is that the second and the third reactions mentioned above, i.e., only the high type customers join

the system, and only the low type customers join the system, are mutually exclusive in equilibrium.

Theorem 1. Given any pure strategy MPBNE for the two-class cheap talk game, there exists a

pure strategy MPBNE which is DOE to the given equilibrium and in which the firm uses at most

three announcements.12

The above result implies that it may not be necessary for the firm to provide announcements with

very refined granularity. A simple announcement system comprised of three different announce-

ments “Low”, “Medium” and “High” may perform equally well if not better than announcement

systems with more refined granularities. This is consistent with the empirical results in Yu et al.

(2014). It is also in line with the announcement policies used by many firms, such as Delta and

American Airline. They provide delay estimates in the form of intervals rather than precise point

estimates. Following the theorem above, without loss of generality, we focus on the pure strategy

equilibria where the firm uses at most three announcements.

5.1. Influential Cheap Talk: homogeneous holding cost

Given that the firm’s unconstrained first best solution is different when the per unit holding cost is

the same for all customers and when it is different for customers of different types, we consider these

two cases separately. We focus on the case with hH = hL in this section, while we will investigate

the case with hH >hL in Section 5.2.

Note that the firm obtains a higher value by serving a high type customer than by serving a

low type customer. Thus, in the case with hH = hL, the firm would prefer admitting a high type

12 Note that one can easily extend the result in Theorem 1 to the setting where there are d customer classes withd > 2. In particular, one can show that the firm cannot improve its profit by using more than d+ 1 announcementswhen there are d customer types.


customer to a low type customer. When the high type customers are more patient than the low type

customers, i.e., rHcH> rL

cL

13, the high type customers will join the system whenever the low type join.

Due to such incentive alignment between the customers and the firm, we show that the firm may

be able to achieve the unconstrained first best solution when high type customers are more patient

than the low type. In order to characterize such an equilibrium, we let nL be the expected number

of customers in the system conditional on the number of customers in the system being less than

nfL under the unconstrained first best solution. Similarly, we define nH as the expected number of

customers in the system conditional on the number of customers in the system being between nfL

and nfH under the unconstrained first best solution. We next formally construct the equilibrium

where the firm achieves the unconstrained first best solution in the following proposition.

Proposition 1. When hH = hL and nfH > nfL, there exists an equilibrium with influential cheap

talk, in which the firm achieves its unconstrained first best solution, if and only if,

nL + 1≤ rLµ

cL< nH + 1, (4)

nH + 1≤ rHµ

cH< nfH + 2. (5)

Furthermore, one such equilibrium is defined as follows: the announcement policy of the firm is

given by

A(nT ) =

m1 if nT ≤ nfLm2 if nfL <nT ≤ n

fH

m3 otherwise,(6)

customers are served in a first-come, first served manner, and the action rules of low type and high

type customers are given by

aL(m) =

join if m=m1

balk otherwise,aH(m) =

join if m=m1 or m=m2

balk otherwise.

where nfL and nfH are the thresholds identified in Lemma 2.

The equilibrium above shows that the firm may be able to achieve the unconstrained first best

solution without fully separating the customers. In particular, the firm uses three announcements

to signal three different levels of congestion, i.e., Low, Medium, and High. When the congestion

level is low, all customers join the system. When the congestion level is medium, only the high type

customers join but not the low type. Meanwhile, when the congestion level is high, neither type

of customers join the system. The solution above is clearly incentive compatible to the firm, as it

13 Note that, while the high type customers may obtain higher value from the service than the low type customers,they are also more likely to incur a higher per unit waiting cost. Thus, the high type customers may be more or lesspatient than the low type customers in practice, see Figure 17 in Mandelbaum and Zeltyn (2013). To this end, weconsider both cases rH

cH> rL

cLand rL

cL> rH

cHin our analysis.


allows the firm to achieve its unconstrained first best solution. From the customers’ point of view,

as long as their reward-cost ratios are between the four thresholds given in Proposition 1, customers

have no incentives to deviate from the unconstrained first best solution either. In particular, the

low type customers obtain non-negative expected utility by joining the system when they receive

the announcement m1, while obtaining negative expected utility by joining the system when they

receive the announcements m2 or m3. Similarly, the high type customers obtain positive expected

utility by joining the system when they receive the announcements m1 or m2, while obtaining

negative expected utility by joining the system when they receive the announcement m3. While

this is an influential equilibrium, it is also a semi-separating equilibrium. This is because one of

the announcements, i.e., m2 triggers different reactions from customers of different types, while the

announcements m1 and m3 trigger the same reactions from both customer types.

Note that for the firm to achieve the unconstrained first best solution, it requires the high type

customers to be more patient than the low type, i.e., rHcH> rL

cL. The next question is that whether

the firm can replicate the unconstrained first best solution when the low type customers are more

patient than the high type, i.e., rLcL> rH

cH. In this case, the low type customers are willing to join

the system whenever the high type customers are. However, based on the unconstrained first best

solution, the firm is willing to admit the high type customers whenever it admits the low type when

it does not have any constraints. Due to this conflicting preferences of the firm and the customers,

the firm cannot achieve the unconstrained first best solution through delay announcements. In fact,

the best the firm can do is to induce an influential pooling equilibrium, where customers of both

types react to announcements identically.

According to Definition 3, in a pooling equilibrium, the firm treats customers of different types

identically, and customers of different types also respond to the announcements in the same manner.

Hence, similar to Allon et al. (2011), we can construct the pooling equilibrium as if there is only one

type of customers, by using one single threshold which we refer to as nf . We denote the expected

number of customers in the system conditional on the number of customers in the system being

not larger than nf under the pooling equilibrium as n. The pooling equilibrium is characterized

in the following proposition. We show that given rLcL> rH

cH, there are no other equilibria, where the

firm achieves a higher profit.

Proposition 2. When hH = hL14, the firm may achieve a pooling equilibrium, if and only if,

nf + 2>rLµ

cL>rHµ

cH≥ n+ 1 (7)

14 As we will discuss in Section 5.2, when hH > hL, Proposition 2 with (7) replaced by nf + 2> rHµcH

> rLµcL≥ n+ 1

continues to hold.


One such equilibrium is defined as follows: the announcement policy of the firm is given by

A(nT ) =

m1 if nT ≤ nfm2 otherwise

and the action rules of the customers are given by

aL(m) =

join if m=m1

balk otherwise,aH(m) =

join if m=m1

balk otherwise.

As for the firm’s scheduling policy g, the firm serves customers in a first-come, first-served manner.

Furthermore, the firm’s profit under other equilibria are bounded by the profit under the above

pooling equilibrium.

As one may expect, given that there are two types of customers, the firm may want to elicit

information from customers regarding their types at least to a certain extent in order to maximize

profits. However, we show that, the pooling equilibrium, where the firm does not elicit information

on customer types at all, may perform the best in the firm’s profit among all other equilibria. In

fact, we observe that many call centers have no intentions to differentiate customers of different

types through announcements at all. In particular, many call centers simply provide the same

generic announcement, (e.g., “all agents are currently busy, we will be with you shortly”), to all

customers regardless of their types when the system is congested. Our result provides theoretical

support for such practice in the service industry.

5.2. Influential Cheap Talk: heterogeneous holding cost

We next turn to the case when the holding cost is different for customers of different types. Recall

that the order of service does not impact the firm’s profit when we have hH = hL. To this end,

the firm focuses on the problem of what announcement to provide to induce the desired customer

responses. However, when hH >hL, besides providing delay announcements to influence customers’

decision on whether or not to join the system, the firm may also like to prioritize the customers

who have joined the system appropriately to reduce its overall cost.

Note that we have shown that the firm can achieve the unconstrained first best solution through

delay announcements without observing customer types or fully separating the customers when the

per unit holding cost is homogeneous among customers in Section 5.1. However, we show that the

firm cannot achieve its unconstrained first best solution via delay announcements when the per unit

holding costs for customers of different types are different. Note that the firm can only prioritize

the customers, whose types it knows. Meanwhile, the firm can only elicit information on customer

types, when customers of different types respond to announcements differently.15 We next argue

15 Note that, given the holding cost includes the loss of goodwill due to long wait in our setting, the firm will not beable to observe customers’ holding cost before knowing their types.


that the firm cannot fully separate customers of different types through delay announcements.

Based on Lemma 1, to achieve the unconstrained first best solution, the firm would like to admit

both customer types when there are no customers in the system for any non-degenerate case with

Si(0)≥ 0,∀i∈ H,L.16 As a result, to achieve the unconstrained first best, the firm must provide

at least one announcement which induces both customer types to join the system. This prevents

the firm from fully separating the customers and thus fail to achieve the unconstrained first best.

We next present this result formally in the following theorem.

Theorem 2. When hH > hL and the firm does not observe customer types, the firm cannot

achieve the unconstrained first best solution by only using delay announcements.

While the firm cannot fully separate the customers or achieve the unconstrained first best, it

may be able to partially separate customers in equilibrium. As a result, the firm can prioritize

the customers, whose types it elicits through their different reactions towards announcements,

to optimize the profit. To explore how to use delay announcements to optimize firm’s profit, we

next characterize the influential equilibrium emerging between the firm and the customers when

hH >hL. Based on Theorem 1, we consider the setting where the firm uses at most three different

announcements without loss of generality.

As we will show in Proposition 3, when hH >hL, there exists no mH ∈MH which induces the

high type customers to join and the low type to balk in any influential equilibrium. We next explore

the intuition for this result. Recall that, in this paper, we focus on the non-degenerate cases where

it is optimal for the firm to admit both customer types and its optimal for both customer types to

join when there are no customers in the system. Thus, for any given influential equilibrium, there

exists an announcement mHL ∈MH,L which induces both customer types to join the system.

Let us first assume there also exists an announcement mH ∈MH which induces the high type

customers to join but the low type to balk in an influential equilibrium. To this end, one can show

that the firm would like to prioritize customers receiving the announcement mH over the customers

receiving the announcement mHL in the equilibrium. This is because the expected per unit holding

cost of customers receiving the announcement mH is larger than that of customers receiving the

announcement mHL when hH >hL. To this end, the expected waiting time of customers receiving

the announcement mH is shorter than that of customers receiving the announcement mHL. Given

16 Note that when the system is extremely congested, we may have Si(0) < 0 for i = L or i = H. Recall that Si(.)is the switching curve for the type i customers characterized in Lemma 1. In this case, the firm will be better offnot to accept the type i customers regardless of the system status. As a result, the system dynamic will be identicalto the one discussed in Allon et al. (2011) where there is only one customer type. Thus, when the system is reallycongested, i.e.,Si(0)< 0 for i= L or i=H, the firm may achieve the first best even when hH > hL. This indicatesthat delay announcements are more valuable when the system is more congested not only for the customers but alsofor the firm.


it is better off for the low type customers to join the system when they receive the announcement

mHL, it should also be better off for them to join the system upon receiving the announcement mH

in the given influential equilibrium. This contradicts to the definition of mH . Thus, when hH >hL,

the customer response that only the high type customers join but the low type balk cannot be

sustained in any influential equilibrium.

Based on the above discussion, one can see that there exists no mH ∈MH in any influential

equilibrium for the case with hH > hL. Meanwhile, as we mentioned above, there exists at least

one announcement mHL ∈MH,L which induces both customer types to join the system in any

influential equilibrium. One can also see that, in any influential equilibrium, the firm would like

to provide an announcement m∅ ∈M∅ to induce both customer types to balk when the system is

really congested. When the gain due to the lower holding cost of the low type customers (compared

to the high type) more than compensates for the loss due to the lower value obtained by serving

the low type customers, the firm may like to provide an announcement mL ∈ML to induce the

low type customers to join but the high type to balk in an influential equilibrium. Note that such

customer response can only be sustained when the low type customers are more patient than the

high type customers. In fact, we find that, under certain incentive compatibility conditions on

customers’ patience time, there exists a semi-separating equilibrium where the firm provides the

announcements mHL, mL and m∅ to induce the corresponding customer response described above.

Moreover, we show that under this semi-separating equilibrium, it is optimal for the firm to priori-

tize the customers who receive the announcement mHL over customers receiving the announcement

mL. Note that the expected per unit holding cost of customers receiving the announcement mHL

is higher than that of the customers receiving the announcement mL, when hH > hL. Thus, pri-

oritizing customers receiving the announcement mHL over customers receiving the announcement

mL minimizes the overall cost.

Above we have described the firm’s announcement policy and scheduling policy under the semi-

separating equilibrium. To characterize the corresponding customer incentive compatibility condi-

tions, we let wm∅ , wmL , and wmHL denote the expected waiting time of customers receiving the

announcement m∅, mL and mHL, respectively, under the semi-separating equilibrium. We next

formally present the semi-separating equilibrium in the following proposition.

Proposition 3. When hH > hL, there exists a semi-separating equilibrium with influential

cheap talk, if and only if,

wmHL ≤rHcH

< wmL ≤rLcL< wm∅ . (8)

Furthermore, one such equilibrium is defined as follows: the action rules of the low and high type

customers are given by

aH(m) =

join if m=mHL

balk otherwise,aL(m) =

join if m=mL or m=mHL

balk otherwise.


In terms of the firm’s strategy, the firm provides three distinct announcements m∅, mL and mHL

which satisfy the condition given by (8). However, we cannot explicitly characterize the announce-

ment policy. The optimal scheduling rule of the firm is given by

g(n) =

MH,L if nHL > 0ML if nHL = 0 and nL > 0m∅ if nHL = nL = 0

with n = (nH , nL, nHL) and nH = 0.

It is important to note that the equilibrium above requires the low type customers to be more

patient than the high type customers, i.e., rLcL> rH

cH. When the high type customers are more

patient than the low type customers, following a similar argument on the misalignment between

the firm’s and the customers’ preferences in Section 5.1, one can show that the firm achieves the

best profit in a pooling equilibrium among all equilibria. The pooling equilibrium is identical to

the one characterized in Proposition 2 but with the incentive compatibility condition (7) replaced

by nf + 2> rHµ

cH> rLµ

cL≥ n+ 1, when hH >hL.

5.3. Babbling Equilibria

We have focused on the influential equilibria where the firm provides credible information and

customers take the announcements into account when they make joining decisions. However, in

practice, there are many service providers that share no information whatsoever with the customers

or information uncorrelated with the state of the system. To this end, we explore whether these

systems are in equilibrium. We show that such an equilibrium where no meaningful information is

provided by the firm and customers disregard the announcements may indeed exist. In line with the

cheap talk literature, we refer to it as a babbling equilibrium, which is formally defined as follows.

Definition 5 (Babbling Equilibrium). A pure strategy MPBNE (aL, aH ,A, g) is a babbling

equilibrium if the two random variables, i.e., the announcement given by the firm A(Q(aL, aH ,A, g))

and the system state Q(aL, aH ,A, g), are independent, and ai(m1) = ai(m2) ∀i ∈ H,L and

m1,m2 ∈M .

Note that there are two different actions for a customer from either class, i.e., join or balk.

As a result, one may expect that there exists four types of pure strategy babbling equilibria.

However, one can show that it cannot be an equilibrium when customers of both types balk. Thus,

there are only three types of pure strategy babbling equilibria that may exist: 1) a pure strategy

babbling equilibrium where both low and high type customers join the system regardless of the

announcements; 2) a pure strategy babbling equilibrium where only high type customers join the

system, while all the low type customers balk; 3) a pure strategy babbling equilibrium where only

low type customers join the system, while all high type customers balk.


The question now is under what conditions these babbling equilibria may exist. To address this

question, we start by exploring the conditions under which the babbling equilibrium where both

types of the customers join the system regardless of the announcements may exist. If customers of

both types indeed join the queue irrespective of the announcements received, the system becomes

an M/M/1 system with the arrival rate and the service rate being λ and µ, respectively. Thus, one

can show that the average waiting time in the system is given by 1µ−λ . Since customers would join

the system if and only if their expected utility is non-negative in equilibrium, we have ri− ciµ−λ ≥

0 ∀i ∈ H,L. Given that the firm cannot differentiate customer types in any way through a

babbling equilibrium, we focus on the case when the firm serves the customers in a first-come,

first-served manner. Following a similar logic, we can characterize the other two types of pure

strategy babbling equilibria. We formalize the characterization in the following proposition.

Proposition 4. 1. The pure strategy babbling equilibrium where both low and high type cus-

tomers join the system exists, if and only if, rici≥ 1

µ−λ ,∀i∈ H,L.

2. The pure strategy babbling equilibrium where all high type customers join the system but none

of the low type customers do exist, if and only if, rLcL< 1

µ−βHλ≤ rH

cH.

3. The pure strategy babbling equilibrium where all low type customers join the system but none

of the high type customers do exists, if and only if, rHcH< 1

µ−βLλ≤ rL

cL.

Based on the proposition above, one can see that none of these pure strategy babbling equilibria

can co-exist. Moreover, neither the firm’s value of serving customers nor its holding cost impacts

the existence of any of the babbling equilibira.

5.4. Should the firm provide announcements?

We have shown that both the babbling equilibria and the influential equilibria may simultaneously

exist. We next explore which equilibrium the firm and the customers would prefer. To this end, we

compare the influential equilibria with the babbling ones in the regions where they both exist, in

terms of both customers’ utility and the firm’s profit. Note that there exists two types of influential

equilibria, i.e., the semi-separating equilibrium and the pooling equilibrium. Meanwhile, we have

three types of babbling equilibria characterized in Proposition 4.

Let us start with the comparison between the pooling equilibrium and the babbling equilibrium.

Given that the babbling equilibria are mutually exclusive, there is at most one babbling equilibrium

which may co-exist with the pooling equilibrium for given parameters. To this end, we let ΠIP

and U oIP denote the profit of the firm and the expected total customer utility under the pooling

equilibrium, respectively. Moreover, we refer to ΠNI and U oNI as the profit of the firm and the

expected total customer utility in the corresponding babbling equilibrium, respectively, which co-

exists with the pooling equilibrium for the given parameters. We show that the firm achieves a


higher profit under the pooling equilibrium compared to the one achieved in the corresponding

babbling equilibrium. Similarly, one may expect providing announcements to improve the expected

total customer utility. However, we find that providing announcements may improve or hurt the

expected total utility of the customers compared to the case when announcements are not provided.

In particular, when the pooling equilibrium co-exists with the babbling equilibrium where both

customer types join the system, customers always obtain a higher expected total utility in the

pooling equilibrium compared to that in the babbling equilibrium. However, when the pooling

equilibrium co-exists with the babbling equilibrium where only one customer type joins the system,

customers may achieve a higher expected total utility in the babbling equilibrium than that in the

pooling equilibrium under certain conditions.

We next characterize the conditions under which delay announcements may hurt the overall

customer utility. We start with the case when hH = hL. We can characterize the conditions for the

case with hH > hL in a similar manner. It is important to note that, when hH = hL, the pooling

equilibrium can only co-exist with the babbling equilibrium where both customer types join the

system or the babbling equilibrium where only low type customers join the system. Moreover, the

pooling equilibrium co-exists with the babbling equilibrium where only low type customers join

the system, if and only if n+1µ≤ rH

cH< 1

µ−βLλ≤ rL

cL< nf+2

µ, see Propositions 2 and 4. Conditional on

the coexistence, customers achieve higher expected total utility in the babbling equilibrium than

that in the pooling equilibrium, if and only if (1−P )rLcL

+ (1 + βHcHβLcL

) (n+1)P

µ> βHrHP

βLcL+ 1

µ−βLλ, where

P is the stationary probability that there are less than nf + 1 customers in the system under the

pooling equilibrium. It is given by P = 1− (1−ρ)ρnf+1

1−ρnf+2with ρ = λ

µ. We now formally present the

above results in the following proposition.

Proposition 5. Assuming that both a pure strategy pooling equilibrium with influential cheap

talk and a pure strategy babbling equilibrium exist, we have:

1. ΠNI <ΠIP ;

2. when hH = hL, U oNI > U o

IP if and only if n+1µ≤ rH

cH< 1

µ−βLλ≤ rL

cL< nf+2

µand (1−P )rL

cL+ (1 +

βHcHβLcL

) (n+1)P

µ> βHrHP

βLcL+ 1

µ−βLλ; When hH >hL, U o

NI >UoIP if and only if n+1

µ≤ rL

cL< 1

µ−βHλ≤ rH

cH<

nf+2µ

and (1−P )rHcH

+ (1 + βLcLβHcH

) (n+1)P

µ> βLrLP

βHcH+ 1

µ−βHλ.

We obtain similar results when we compare the semi-separating equilibrium and the babbling

equilibria. To this end, we show that providing delay announcements always increases the firm’s

profit. This is due to the strategic nature of the firm in the cheap talk game who chooses the

announcement and scheduling policy to maximize its own profit. However, from the customers’ per-

spective, in contrast to the results in Allon et al. (2011) which shows that providing announcements

always benefits the customers compared to the case with no announcements, our results show that


providing delay announcements may improve or hurt the expected total customer utility. To explore

the intuition why providing delay announcement may improve customer utility, it is important to

mention the main insights in Naor (1969). In particular, Naor (1969) shows that customers are

more willing to join the system than what the social planner would like them to. This is because

customers decide on whether to join only to maximize their own utility, while ignoring the negative

externalities that they may impose on other customers by joining the system. The threshold that

the firm induces through the delay announcements helps reduce such externalities and thus may

improve the expected total customer utility. Meanwhile, providing delay announcements may also

hurt the expected total customer utility. This is because more of the less patient customers would

join the system when announcements are provided compared to the case when announcements are

not provided. To this end, providing announcements may impose more negative externalities on the

more patient customers due to the increased number of the less patient customers in the system.

This is consistent with the empirical results shown in Yu et al. (2014). Such negative externalities

on the more patient customers may be larger than the gain in the utility of the less patient cus-

tomers. This is most likely to happen when the system is very congested and the utility of the less

patient customers by joining the system is positive but rather low.

It is worth noting that the result that providing delay announcements may improve or hurt

customers’ utility is consistent with the conclusion in Guo and Zipkin (2007). Recall that Guo

and Zipkin (2007) and our paper study completely different issues about delay announcements.

In particular, Guo and Zipkin (2007) explores the impact of given announcement policies on cus-

tomer behavior where the firm’s announcement policy is fixed and the firm serves customers in a

first-come, first-served manner. However, our paper studies how the firm should use delay announce-

ments to manage the expectations and priorities of its heterogeneous customers, where both the

firm and the customers are strategic. The firm is strategic in both its announcement and scheduling

policies to maximize profits, while the customers are strategic in interpreting the announcements

and making decisions on whether or not to join the system to maximize their utility. This implies

that this result may not be driven by considering delay announcements as a cheap talk but rather

by the role the announcements play in encouraging customers to join or not.

6. Value of Customer Type Information

So far, we focus on the case where the firm cannot observe customer types. As we discussed

earlier, it is often difficult if not impossible for call centers to differentiate all of their customers,

especially the newly acquired customers or customers who skip the IVR system. However, it is

worth noting that many call centers can obtain fairly refined information about a proportion of

its recurrent customers. Given that it is often costly for firms to elicit information from customers


and keep track of it in practice, it is important to understand the value the firm may gain or

lose by observing customer types compared to the case when it does not. Recall that we have

focused on the case when the firm does not directly observe customer types so far. To study the

value of customer type information for the firm, we next extend our model by allowing the firm

to observe customer types before it provides announcements. We refer to this model as the model

with information.17 This model is identical to the model with no information presented in Section 3

with two key modifications: 1) the firm can now decide on whether to provide announcements and

what announcements to provide to customers based on their types; and 2) the firm can schedule

customers based on their types instead of the announcements that they receive.

We next characterize the equilibria that emerge between the customers and the firm when the

firm observes customer types upon their arrivals. We employ the same equilibrium concept as the

one used in the model with no information on customer types, see Definition 1. However, due to

the unique features that the firm can provide announcements and schedule customers based on

their types in the model with information, the specific mathematic formulation of the MPBNE in

the model with information is slightly different from the one in Definition 1. We refer the readers

to Appendix A for more details on the MPBNE formulation for the model with information. Recall

that we have characterized the unconstrained first best solution of the firm in Section 4, where the

firm has full information about customer types and full control over customer admission. Although

in the model with information, the firm does not have control over customers’ admission, the

following theorem shows that the queuing dynamic observed under any MPBNE with influential

cheap talk (if it exists) corresponds to the one where the firm achieves its unconstrained first best

solution. Note that, we say an MPBNE is influential if the announcements are influential for both

customer types in the model with information. This is in line with the definition on influential

cheap talk for the model with no information, see Definition 2.

Theorem 3. When the firm observes customer types, the firm achieves its unconstrained first

best solution under any MPBNE with influential cheap talk.

The intuition for the above result is that, in any influential equilibrium, the firm should be able to

signal “High” when the system state is beyond certain switching curves. If these switching curves

are different from the ones characterized in the unconstrained first best solution, the firm will always

17 Note that we consider the model with observable customer types is mainly to explore the value the firm may gainor lose by obtaining customer type information. However, the analysis on the model with observable customer typehas its own contribution to the literature. In particular, the literature on how the firm and customers interact throughdelay announcements has mostly focused on the case with one customer type. It seems that Jouini et al. (2015) isamong the few exceptions which consider multiple observable customer types. However, Jouini et al. (2015) focuseson evaluating the accuracy of different delay estimators using NewsvendorLike performance criterion, while we focuson how the firm should provide delay announcements in the presence of heterogeneous and strategic customers.


be better off to deviate to the policy prescribed in the unconstrained first best solution. Meanwhile,

we show that, under certain incentive compatibility conditions on the customers’ patience, the firm

can achieve the unconstrained first best solution in equilibrium. We refer the readers to Appendix A

for the detailed characterization of this equilibrium. It is worth noting that when the unconstrained

first best solution can be sustained in an influential equilibrium, the constrained and unconstrained

first best solutions are equivalent.

6.1. Impact on Firm’s Credibility and Profit

Based on the above results for the model with information, we are now ready to study the value

the firm may gain or lose by directly observing customer types. Let us start by exploring whether

the firm can improve its capability to influence customers by observing customer types upon their

arrivals and if so, under what conditions. We say that the firm can credibly communicate with

the customers through delay announcements if there exists an influential equilibrium. To this end,

we refer to the firm’s capability of inducing influential equilibria as the firm’s credibility. To do

so, we compare the equilibria that emerge when the firm observes customer types with the ones

emerging when the firm does not. We show that information on customer types may expand or

contract the region where the firm achieves influential equilibria. It is intuitive that information

on customer types may enhance the credibility of the firm by extending the region where the firm

achieves equilibria with influential cheap talk. This is because when the firm observes customer

types, the firm can provide information to customers based on their types to better match their

expectations. However, we also find that customer type information may hurt the credibility of

the firm by contracting the region where the firm achieves influential equilibria. One possible

explanation is that when the firm observes customer types, it will intend to extract more profit from

the customers. This may lead to misalignment between the incentive of the firm and that of the

customers, which may cause the firm failing to achieve an influential equilibrium when it observes

customer types. We refer the readers to Appendix B.1 for the detailed and rigorous analysis which

leads to the above insights.

We have explored the question of whether information on customer types would improve or hurt

the credibility of the firm. We next discuss if the creation of credibility translates into the creation

of profit for the firm. We find that the creation of credibility always improves the firm’s profit,

while the loss of credibility may improve or hurt the firm’s profit. Given that the firm achieves the

unconstrained first best solution in any influential equilibrium when it observes the type of the

customer, one can see that the creation of credibility by observing customer types always leads to

improvement of firm’s profit. Following a similar logic, one may expect the loss of credibility to

hurt the profit of the firm. However, our results show that loss of credibility may improve or hurt


the firm’s profit when hH >hL, while it always hurts the profit of the firm when hH = hL. We refer

the readers to Appendix B.2 for the detailed analysis which supports this result. To this end, it is

worth noting that information on customer types allows the firm to better prioritize the customers

in the babbling equilibria. The improvement in firm’s profit from the prioritization may more than

compensate for the loss due to the firm’s lack of ability to induce the desired customer response in

the contraction region. Thus, the loss of credibility may not necessarily hurt the firm’s profit. We

refer the readers to Appendix B.2 for the detailed analysis which supports the above results.

7. Conclusion

In this paper, we study how to use delay announcements to manage customer expectations and

priorities in the presence of heterogeneous customers. We examine this problem by developing

a framework which characterizes the strategic interaction between the self-interested firm and

heterogeneous selfish customers. We first explored a model where both the customers and the firm

have private information of their own. The customers have private information on their types, while

the firm has private information on the system status. To study the value that the firm may gain

or lose by observing customer types, we have also investigated a model where the firm can observe

customer types. We characterize the equilibria that emerge between the firm and its heterogeneous

customers in both models.

The analysis of the emerging equilibria demonstrates the role of suppressed information in sus-

taining an equilibrium with influential cheap talk. Our analysis also underscores that the het-

erogeneity among the customers raises interesting issues about the firm’s ability to influence the

different types of customers differently through delay announcements. We show that the firm can-

not fully separate the customers of different types through delay announcements. This prevents

the firm from achieving the unconstrained first best solution when the per unit holding costs are

different for customers of different types. However, the ability to partially separate among the

different customer types through delay announcement allows the firm to sustain a semi-separating

equilibrium with influential cheap talk to improve profits. Under such semi-separating equilibrium,

we show it is optimal for the firm to give absolute priority to customers receiving announcements

corresponding to the larger expected per unit holding cost over customers receiving announce-

ments associated with the smaller expected per unit holding cost. It is also worth mentioning that,

when the per unit holding cost is the same for customers of both types, the firm can achieve the

unconstrained first best solution without fully separating the customers but by only partially sepa-

rating the customers. Moreover, we show that providing announcements always improves the firm’s

profit compared to the case when announcements are not provided. However, from the customers’

perspective, providing announcements may improve or hurt the expected total customer utility.


To explore the value that the firm may gain or lose by observing the type of the customer, we

have also studied a model where the firm can directly observe the types of customers. We show

that the information on customer types may enhance the firm’s credibility by extending the region

where the firm can achieve equilibria with influential cheap talk. However, such information may

also hurt the credibility of the firm by contracting the region where the firm achieves influential

equilibria. We show that the creation of credibility always improves the firm’s profit, while the loss

of credibility may not necessarily hurt the firm’s profit.

Our study has certain limitations that should be explored in future research. We assume that

there are two customer classes in our model. While our framework can be easily extended to the

case with more than two customer classes, the equilibrium analysis will be more complicated due

to the increasing dimensionality. We believe that the structural results and the main insights in our

paper will continue to hold for the case where there are n customer types with n> 2. However, it

may worth confirming our conjecture through a more comprehensive study. In addition, there are

more and more call centers providing the option to call the customers back when the system gets

really congested. We focus on the setting where there is no such call-back option in this paper. We

think it will be interesting to investigate the problem of how the firm should strategically use the

call-back option together with delay announcements to manage customers’ abandonment behavior.

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Appendix A: Model with Information

In Section 6, we have briefly described the model with information and summarized the corresponding

main results. We next provide more detailed and rigorous description about the model and the results. In

particular, we will formally present the MPBNE definition for the model with information. We will then

explicitly characterize the influential equilibria that emerge in the model.

Recall that the model with information is identical to the model with no information presented in Section 3

with two key modifications: 1) the firm can now decide on whether or not to provide announcements and what

announcements to provide to customers based on their types; and 2) the firm can schedule customers based

on their types instead of the announcements that they receive. To incorporate the above unique features of

the model with information, we let SI represent the set of system states in the model with information. Given

the firm has perfect information on customer types in this model, the system states can be characterized

by the number of low type customers n0L and the number of high type customers n0

H . Thus, the set of the

system states SI is given by SI = (n0H , n

0L)|(n0

H , n0L) ∈ N2

0, which coincides with the set of system states

for the full information and full control case presented in Section 4. We then let the announcement policy

of the firm for the type i customers be a function given by Ai : SI 7→M with i∈ H,L. To account for the

new feature on the firm’s scheduling policy, we let the scheduling policy of the firm be given by a function

gI : SI 7→ ∅,L,H. In particular, we have gI(n0H , n

0L) = i∈ H,L, if the next customer to be served is a type

i customer in the system at state (n0H , n

0L). Meanwhile, we have gI(n

0H , n

0L) = ∅, if the firm decides to be idle

at state (n0H , n

0L). It is worth mentioning that the subscription I in SI and gI indicates the condition that

the firm has information on customer types.

We employ the equilibrium concept of MPBNE which is the same as the one used in the model where the

firm does not directly observe the types of the customers. However, due to the unique features that the firm

can schedule and provide announcements to customers based on their types in the model with information,

the MPBNE formulation for the model with information is slightly different from Definition 1. To formally

define the equilibrium, we let pI(n0H , n

0L|aH , aL, gI ,AH ,AL) be the probability that there are n0

H high type

and n0L low type customers in the system at the steady state given the customer strategy ai, the firm’s

scheduling rule gI and announcement policy Ai with i ∈ H,L. Meanwhile, we define wgIi (n0H , n

0L) as the


expected waiting time of the type i customer who joins the system at state (n0H , n

0L), for i∈ H,L. To this

end, the expected utility of the type i customers who receive the announcement m∈M and have joined the

system is given by EI [ri− ciwgIi (n0H , n

0L)|Ai(n0

H , n0L) =m]. It can be further expressed as

EI [ri−ciwgIi (n0H , n

0L)|Ai(n0

H , n0L) =m] =

∑(n0

H,n0

L):Ai(n0

H,n0

L)=m [ri− ciwgIi (n0

H , n0L)]p(n0

H , n0L|aH , aL, gI ,AH ,AL)∑

(n0H,n0

L):Ai(n0

H,n0

L)=m p(n

0H , n

0L|aH , aL, gI ,AH ,AL)

We next formally define the pure strategy MPBNE for the model with information below.

Definition 6. We say that (aH , aL, gI ,AH ,AL) forms a Markov Perfect Bayesian Nash Equilibrium

(MPBNE), if and only if, it satisfies the following conditions:

1. For each m∈M and i∈ H,L, we have

ai(m) =

1 if EI [ri− ciwgIi (n0

H , n0L)|Ai(n0

H , n0L) =m]≥ 0

0 otherwise.(9)

2. There exists relative value functions VI(n0H , n

0L) with (n0

H , n0L) ∈ N2

0, constant γI , together with the

scheduling policy gI(n0H , n

0L) and the announcement policy mi =Ai(n0

H , n0L), that solve the following Bellman

equation:

VI(n0H , n

0L) +

γIΛ

=1

Λ

−hLn0

L−hHn0H

+βHλ maxmH∈M

VI(n

0H , n

0L)(1− aH(mH)) + (VI(n

0H + 1, n0

L) + vH)aH(mH)

+βLλ maxmL∈M

VI(n

0H , n

0L)(1− aL(mL)) + (VI(n

0H , n

0L + 1) + vL)aL(mL)

+µmax

gI

(VI(n

0H − 1, n0

L)In0H>0+VI(n

0H , n

0L)In0

H=0

)IgI(n0

H,n0

L)=H

+(VI(n

0H , n

0L− 1)In0

L>0+VI(n

0H , n

0L)In0

L=0

)IgI(n0

H,n0

L)=L

+VI(n0H , n

0L)IgI(n0

H,n0

L)=∅

, (10)

with Λ = λ+µ.

The above definition is related to the one defined for the model with no information, see Definition 1.

The key difference is that, in the model with information, the firm can provide announcements and schedule

customers based on the type of the customers. These unique features in the firm’s announcement and

scheduling policies are captured in (10).

We next explore the equilibria that emerge when the firm observes customer types upon their arrivals.

Recall that Theorem 3 shows that the queuing dynamic observed under any MPBNE with influential cheap

talk (if it exists) corresponds to the one where the firm achieves its unconstrained first best solution. Thus,

to construct the equilibrium when the firm observes customer types, we consider the system where the firm

implements the unconstrained first best solution. Note that we have characterized the unconstrained first

best solution, where the firm has full information on customer types and full control over customer admission

in Lemmas 1 and 2. In particular, to achieve the unconstrained first best, the firm would like the high type

customers to join the system when the number of high type customers in the system is not larger than

the threshold SH(n0L), i.e., n0

H ≤ SH(n0L). Otherwise, the firm would like the high type customers to balk.


Similarly, the firm would like to accept the low type customers when n0L ≤ SL(n0

H). Otherwise, the firm would

like the low type customers to balk.

We let,¯wi and wi, with i ∈ H,L, denote the expected waiting time of the arriving type i customer (if

she joins the system) given that the firm wants her to join and balk the system under the unconstrained first

best solution, respectively. For convenience of notations, we let −i represent the customer type which is not

i, for i,−i∈ H,L. To this end, for i∈ H,L, we have

¯wi =EUFB[wgIi (n0

H , n0L)|n0

i ≤ Si(n0−i)] and wi =EUFB[wgIi (n0

H , n0L)|n0

i >Si(n0−i)]. (11)

Note that we have wH =nfH

+2

µand

¯wL = nL+1

µfor the case with hH = hL, where nL and nfH are the thresh-

olds given in (4) and (5), respectively. We next characterize the equilibrium where the firm achieves the

unconstrained first best while observing customer types upon their arrivals in the following proposition.

Proposition 6. There exists an equilibrium with influential cheap talk where the firm achieves the uncon-

strained first best, if and only if,

¯wi ≤

rici< wi, ∀i∈ H,L (12)

Furthermore, one such equilibrium is defined as follows: The announcement policy of the firm is given by

AH(n0H , n

0L) =

mH

1 if n0H ≤ SH(n0

L)mH

2 otherwiseAL(n0

H , n0L) =

mL

1 if n0L ≤ SL(n0

H)mL

2 otherwise.

Moreover, the action rules of low and high type customers are given by

aH(m) =

join if m=mH

1

balk if m=mH2

aL(m) =

join if m=mL

1

balk if m=mL2 .

As for the scheduling policy of the firm, it serves customers with the same per unit holding cost in a first-come,

first-served manner. When hH >hL, the scheduling policy of the firm is given as follows:

gI(n0H , n

0L) =

H if n0H > 0

L if n0H = 0 and n0

L > 0∅ if n0

H = n0L = 0.

Note that, in the equilibrium above, the firm gives preemptive resume priority to the high type customers

when hH >hL. This is because the holding cost of the high type customers is higher than that of the low type.

Thus, prioritizing the high type customers over the low type reduces the overall holding cost. Given that

the firm achieves the unconstrained first best solution in the equilibrium above, it clearly has no incentives

to deviate. As for the customers, due to incentive compatibility conditions given in (12), it is optimal for

them to follow the unconstrained first best solution prescribed by the firm. To this end, one can see that the

constrained first best solution is equivalent to the unconstrained first best solution if and only if¯wi ≤ ri

cifor

i ∈ H,L. This also implies that, if the unconstrained first best solution can be sustained in an influential

equilibrium, the constrained and unconstrained first best solutions are equivalent.

Appendix B: Impact of Customer Information on Firm’s Credibility andProfitability

In Section 6.1, we have discussed the main insights on the impact of customer type information on the firm’s

capability to influence customers’ behavior through delay announcements and thus its profitability. We next

present the rigorous analysis to better support our discussion there.


B.1. Impact on Firm’s Credibility

To study the impact of customer type information on firm’s capability to influence customers, we characterize

the conditions under which the firm can induce influential equilibria in the model with information and the

model with no information, respectively. Following Theorem 3 and Proposition 6, the necessary and sufficient

condition for the existence of equilibria with influential cheap talk can be written as rici∈ [

¯wi, wi),∀i∈ H,L

in the model with information. We can view rici

as the type i customers’ perspective on their willingness to

wait, while¯wi, wi as the firm’s perspective on the desired congestion level of the system for type i customers

with i∈ H,L. In studying the impact on firm’s credibility, we shall fix the firm’s perspective and vary the

customers’. In particular, we introduce the following terminology: for given fixed firm’s cost parameters, the

percentage of each customer type, the service rate and the arrival rate, we let DI and DNI be the set of

the patiences of both customer types for which the firm can achieve influential equilibria with and without

information on customer types, respectively. Based on the above discussion, we have DI = ( rLcL, rHcH

)| rici∈

[¯wi, wi),∀i ∈ H,L. Figure 1a shows the region DI for the case with hH = hL, where the horizontal and

vertical axises represent the patiences of the low and high type customers, respectively.

Note that when the firm cannot observe customer types, the firm can achieve influential equilibria through

either a semi-separating equilibrium or a pooling equilibrium. To this end, we let DSSNI and DP

NI be the set

of patiences of both customer types for which the firm achieves the semi-separating equilibrium and the

pooling equilibrium without observing customer types, respectively. To this end, we have DNI =DSSNI ∪DP

NI .

Following Propositions 1 and 3, we have

DSSNI =

( rL

cL, rHcH

)|wmHL≤ rH

cH< wmL

≤ rLcL< wm∅ if hH >hL

( rLcL, rHcH

)| nL+1µ≤ rL

cL< nH+1

µ≤ rH

cH<

nfH

+2

µ if hH = hL

Moreover, based on Proposition 2, we have

DPNI =

( rL

cL, rHcH

)| n+1µ≤ rL

cL< rH

cH< nf+2

µ if hH >hL

( rLcL, rHcH

)| n+1µ≤ rH

cH< rL

cL< nf+2

µ if hH = hL

Figure 1b shows the regions DSSNI and DP

NI , juxtaposed with the region DI depicted in Figure 1a when

hH = hL.

We next define the expansion region due to the information on customer types as DI ∩DcNI , where Dc

NI

represents the complement of the set DNI . Similarly, we define the contraction region due to customer type

information as DNI ∩DcI , where Dc

I is the complement set of DI . Lastly, we define the neutral region due

to the information on customer types as DI ∩DNI . We say that information on customer types leads to a

contraction if the expansion region is empty. Similarly, we say that information on customer types results in

an expansion if the contraction region is empty. Lastly, we say that information on customer types leads to a

mixed contraction-expansion if neither of these sets is empty. In fact, Figure 1 depicts a case where customer

type information results in a mixed contraction-expansion when hH = hL.

The following proposition shows that information on customer types may lead to an expansion or a mixed

contraction-expansion when we have hH = hL. In particular, the expansion region is never empty when we

have hH = hL, while the contraction region may be empty under certain conditions.


(a)

𝑟𝐿𝑐𝐿

𝑟𝐻𝑐𝐻

𝑤𝐻 = 𝑛𝐻𝑓+ 2

𝜇

𝑤𝐻

𝑤𝐿 = 𝑛𝐿 + 1

𝜇 𝑤𝐿

𝐷𝐼

(a)

(b)

𝐷𝑁𝐼𝑆𝑆

𝑟𝐿𝑐𝐿

𝑟𝐻𝑐𝐻

𝑤𝐻 = 𝑛𝐻𝑓+ 2

𝜇

𝑛𝐻 + 1

𝜇

𝑤𝐻

𝑤𝐿 = 𝑛𝐿 + 1

𝜇

𝑛 + 1

𝜇 𝑤𝐿 𝑛𝑓 + 2

𝜇

𝐷𝑁𝐼𝑃

Figure 1 (a)Sets of customer patience time DI where the firm achieves influential equilibria with information

on customer types; (b)Sets of customer patience time DSSNI and DP

NI where the firm achieves influential equilibria

without observing customer types.

Proposition 7. When hH = hL, we have:

1. DcNI ∩DI 6= ∅.

2. DcI ∩DNI = ∅, if and only if, we have µ

¯wH ≤ n+ 1≤ nf + 2≤ µwL.

As we have discussed in Section 6.1, it is intuitive that information on customer types may enhance

the credibility of the firm by extending the region where the firm achieves the equilibria with influential

cheap talk. This is because when the firm observes customer types, the firm can provide information to

customers based on their types to better match their expectation. However, we find that there might also

be a contraction region. The key reason is that when the firm observes customer types, it will intend to

extract more profit from the customers. This may lead to the misalignment between the incentive of the firm

and that of the customers. As a result, the firm fails to achieve an influential equilibrium when it observes

customer types in the contraction region.

Above we focused on the case with hH = hL, where we show that the expansion region is never empty,

while the contraction region may. However, when hH > hL, our results show that the contraction region is

never empty, while the expansion region may, see Proposition 8.]]

Proposition 8. Assuming hH >hL, we have

1. DcNI ∩DI = ∅, if and only if, we have wmHL

≤¯wH ≤ wH ≤ wmL

≤¯wL ≤ wL ≤ wm∅ .

2. DcI ∩DNI 6= ∅.

We next explore the intuition why the results for the case with hH >hL differ from the ones for the case

with hH = hL. Note that, the extent to which the firm can use information to steer customers to its preferred

behavior hinges on the degree of misalignment between the incentives of the firm and the customers. When

hH >hL, there exists a higher degree of misalignment between the incentives of the two parties compared to


the case with hH = hL. In the case with hH = hL, the firm and the customer may disagree in whether or not

the customer should join the system, while the order of the service is fixed and will not create additional

misalignment between the incentives of the two parties. However, in the case with hH >hL, the firm would

like to elicit information on customer types through delay announcements and prioritize customers based on

the elicited customer type information. This will induce additional misalignment between the incentives of

the customers and the firm along with the misalignment in whether or not the customers should join the

system. This leads to qualitatively different results for the case with hH = hL and the case with hH >hL. In

particular, it explains why information on customers types always leads to a non-empty contraction region

when hH >hL, while the contraction region may be empty when hH = hL.

B.2. Impact on Firm’s Profitability

We have discussed the key insights on whether the creation of credibility can be translated to the creation

of profit for the firm in Section 6.1. One of the key results is that, the loss of credibility may hurt or improve

the firm’s profit when hH > hL, while it always hurts the firm’s profit when hH = hL. We next present the

detailed analysis to support this result.

We first formalize the results for the case with hH = hL in the following proposition.

Proposition 9. When hH = hL, the firm achieves a higher profit when it does not observe customer types

compared to the case when it does in the contraction region.

Note that, when hH =HL, the firm achieves the pooling equilibrium (characterized in Proposition 2) in the

contraction region, when it does not directly observe customer types. Meanwhile, the firm achieves babbling

equilibria when it observes customer types in the contraction region. When hH = hL, the babbling equilibria

emerging when the firm observes customer types are equivalent to the ones characterized in Proposition 4.

Based on Proposition 2, the firm achieves the highest profit in the pooling equilibrium among all equilibria

including the babbling ones characterized in Proposition 2. Thus, when hH = hL, loss of credibility always

hurts the profit of the firm.

When hH > hL, one can show that the loss of credibility may improve or hurt the profit of the firm. We

next illustrate this result with the following numerical example.

Example 1: In this example, we let the total arrival rate λ be 6.7 customers per unit time. There is a

single agent whose service rate is 7.5 customers per unit time, i.e., µ = 7.5. We let the value for the firm

by serving a high type customer be 15, while the value by serving a low type customer be 10, i.e., vH = 15

and vL = 10. Meanwhile, the per unit holding cost incurred to the firm for the high and low type customers

are 1 and 2, respectively. We assume 50% of the customers are low type customers, i.e., βL = 50%. As for

customers’ parameters, we let the service value obtained by each of the high and low type customers be 1.3

and 2.1, i.e., rH = 1.3 and rL = 2.1, respectively. Meanwhile, the per unit time waiting costs for the high and

low type customers are assumed to both equal 1. One can show that, given the parameters above, when the

firm does not directly observe customers types, the firm can achieve the pooling equilibrium characterized in

Proposition 2. When the firm has information on customer types, it cannot induce any influential equilibrium.

Instead, there may exist a babbling equilibrium where both customer types join the system regardless of


the announcements received, while the firm gives absolute priority to the low type customers over high type

customers.18 To this end, one can see that the given patiences of the customers lie in the contraction region.

We next evaluate the firm’s profits under both the pooling equilibrium and the babbling equilibrium. Our

results show that the firm’s profit under the pooling equilibrium is 75 per unit time, while the firm’s profit

under the babbling equilibrium is 81 per unit time. Thus, in this example, we show that information on

customer types may even improve the profit of the firm in the contraction region.

Example 2: In this example, we use the same parameters as the ones in Example 1 with the following

modification: hL = 3, βL = 90% and rH = 0.67. Similar to Example 1, one can show that, if the firm does not

observe customer types, the firm can induce the pooling equilibrium characterized in Proposition 2 under the

given parameters. The firm’s profit under this pooling equilibrium is 58 per unit time. Meanwhile, when the

firm directly observes the customer types, the firm cannot achieve any equilibrium with influential cheap talk

for the given parameters. However, there may exist a babbling equilibrium where only the low type customers

join regardless of the announcements while all the high type customers balk. This babbling equilibrium is

characterized in Proposition 4. The firm’s profit under this babbling equilibrium is 48 per unit time. Based

on the above discussion, we can also see that the given customer patiences lie in the contraction region. Thus,

this example shows that information on customer types could also hurt the firm’s profit in the contraction

region.

Appendix C: Proofs

Proof of Lemma 1:

We let V (i, j) be the relative profit of the firm when there are i high type and j low type customers in

the system. In order to characterize the unconstrained first best solution of the firm, we let the scheduling

policy of the firm be given by a function gI : SI 7→ ∅,L,H. In particular, we have gI(i, j) = k ∈ H,L, if

the next customer to be served is a type k customer in the system, when there are i high type customers

and j low type customers in the system. Meanwhile, we have gI(i, j) = ∅, if the firm decides to be idle.

It is worth noting that the definition of the firm’s scheduling policy gI for the full information and full

control case is consistent with the one defined in the model with information. Meanwhile, we let the firm’s

admission policy on type k customers be given by the function Ok : SI 7→ 0,1, for k ∈ H,L. In particular,

for k ∈ H,L, we have Ok(i, j) = 1 if the firm would like to admit type k customers. Otherwise, we have

Ok(i, j) = 0. To this end, the firm’s optimal scheduling policy gI and admission policy Ok with k ∈ H,Lcan be characterized by the following Bellman optimality condition.

V (i, j) +γIΛ

=C(i, j) +λ1

ΛT1V (i, j) +

λ2

ΛT2V (i, j) +

µ

ΛT3V (i, j), (13)

with

C(i, j) =− (hHi+hLj)

Λ

T1V (i, j) = maxOH∈0,1

(vH +V (i+ 1, j))OH +V (i, j)(1−OH)

18 When the firm observes customer types, a babbling equilibrium where customers of both types join the systemregardless of the announcements exists, if and only if, rL

cL≥ 1

µ−βLλand rH

cH≥ µ

(µ−βLλ)(µ−λ).


T2V (i, j) = maxOL∈0,1

(V (i, j+ 1) + vL)OL +V (i, j)(1−OL),

T3V (i, j) = maxgI∈H,L,∅

(V (i− 1, j)Ii>0+V (i, j)Ii=0)IgI=H

+ (V (i, j− 1)Ij>0+V (i, j)Ij=0)IgI=L

+V (i, j)IgI=∅ ,

λ1 = βHλ and λ2 = βLλ. Note that γI is the long run average profit of the firm per unit time.

We next show that the optimal relative value function V (i, j) is in V , which is a set of functions defined

as follows.

Definition 7. We define V as the set of functions such that if V ∈ V , then V satisfies the following

conditions:

V (i, j)≥ V (i+ 1, j) (14)

V (i, j)≥ V (i, j+ 1) (15)

V (i, j+ 1) +V (i+ 1, j)≥ V (i, j) +V (i+ 1, j+ 1) (16)

V (i, j+ 1) +V (i+ 1, j+ 1)≥ V (i+ 1, j) +V (i, j+ 2) (17)

V (i+ 1, j) +V (i+ 1, j+ 1)≥ V (i, j+ 1) +V (i+ 2, j) (18)

V (i, j+ 1)≥ V (i+ 1, j) if hH >hL; (19)

V (i, j+ 1) = V (i+ 1, j) if hH = hL.

Before we show V ∈ V , we first prove the following three lemmas, i.e., Lemma 4, 5 and 6. For exposition

purposes, we present the proofs for Lemma 4, 5 and 6 at the end of the proof of this Proposition.

Lemma 4. If V ∈ V , then T1V ∈ V .

Lemma 5. if V ∈ V , then T2V ∈ V .

Lemma 6. if V ∈ V , then T3V ∈ V .

We now ready to show V ∈ V . Consider a value iteration algorithm to solve for the optimal policy in which

V0(i, j) = 0 for all i and j, and

Vk+1(i, j) =C(i, j) +λ1

ΛT1Vk(i, j) +

λ2

ΛT2Vk(i, j) +

µ

ΛT3Vk(i, j) (20)

Based on Proposition 4.1.7 in Bertsekas (2012), we have limk−>∞ Vk = V . Thus, to show V ∈ V , we only

need to show Vk ∈ V for any k ∈ N0. We do so by induction. Given that V0(i, j) = 0,∀i, j ∈ N0, one should

see V0 ∈ V . We next show if Vk ∈ V , we have Vk+1 ∈ V . Based on Lemma 4, 5 and 6, if Vk ∈ V , we have

T1Vk(i, j) ∈ V , T2Vk(i, j) ∈ V and T3Vk(i, j) ∈ V . One should also see that C(i, j) ∈ V . To this end, we have

Vk+1 ∈ V if Vk ∈ V . Hence, by induction, we have Vk ∈ V for all k ∈ N0. Given limk−>∞ Vk = V , we have

V ∈ V .


Let us get back to the question of the firm’s optimal admission policy. We know that it is optimal for

the firm to accept the high type customers when we have V (i+ 1, j)− V (i, j) > −vH . Due to V ∈ V , one

should see that V (i+1, j)−V (i, j) is a non-increasing function in j based on property (16). Moreover, based

on (16)+(18), one can see that V (i+ 1, j)− V (i, j) is a non-increasing function in i. To this end, one can

show that the firm’s optimal admission policy to the high type customers can be characterized by a finite

switching curve SH(j) defined as follows

SH(j) = maxi : V (i+ 1, j)−V (i, j)>−vH |i, j ∈N0, (21)

where i is the number of high type customers in the system and j is the number of low type customers. In

particular, for any given number of low type customers in the system j, the firm would like to accept high

type customers if and only if i < SH(j). Moreover, SH(j) is monotonically non-increasing in j. Similarly, one

can show that the firm’s optimal admission policy to the low type customers can be characterized by a finite

switching curve SL(i) defined as follows

SL(i) = maxj : V (i, j+ 1)−V (i, j)>−vL|i, j ∈N0. (22)

In particular, for any given number of high type customers in the system i, the firm would like to accept low

type customers if and only if j < SL(i). Moreover, SL(i) is monotonically non-increasing in i.

As for the firm’s optimal scheduling policy, based on (14), (15) and (19), one should see that, when we

have hH >hL, it is optimal for the firm to give preemptive resume priority to the high type customers over

the low type. When we have hH = hL, the order of service does not impact the profit of the firm. (Please

see the proofs for Lemmas 4, 5 and 6 as follows.) Q.E.D.

Proof of Lemma 4:

To show T1Vk(i, j)∈ V if Vk(i, j)∈ V , we show the following:

• We next show T1 preserves the properties given by (14). We let y denote the optimal action for the

firm in the state (i+ 1, j). In particular, y = 0 means that it is optimal for the firm to reject the high type

customer when the system state is (i+ 1, j), while y = 1 means that it is optimal for the firm to accept the

high type customer:

— when y= 0, we have

T1Vk(i, j) = maxvH +Vk(i+ 1, j), Vk(i, j)

≥ Vk(i, j)

≥ Vk(i+ 1, j) = T1Vk(i+ 1, j),

where the second inequality is based on the condition given by (14).

— Similar, when y= 1, we have


≥ vH +Vk(i+ 1, j)

≥ vH +Vk(i+ 2, j) = T1Vk(i+ 1, j)


Thus, we have shown that the operator T1 preserves the property given by (14).

• We next show that T1 preserves the property given by (15). Similarly, we let y denote the optimal action

for the firm in the state (i, j+ 1). In particular, y= 0 means that it is optimal for the firm to reject the high

type customer when the system state is (i, j+ 1), while y= 1 means that it is optimal for the firm to accept

the high type customer:

— when y= 0, we have


≥ Vk(i, j)

≥ Vk(i, j+ 1) = T1Vk(i, j+ 1),


— Similar, when y= 1, we have


≥ vH +Vk(i+ 1, j)

≥ vH +Vk(i+ 1, j+ 1) = T1Vk(i, j+ 1)

• We now show that T1 preserves the property given by (16). Similarly, we let y1 and y2 denote the optimal

action for the firm in the state (i, j) and (i+ 1, j+ 1). In particular, y1 = 0 means that it is optimal for the

firm to reject the high type customer when the system state is (i, j), accept otherwise. Moreover, y2 = 0

means that it is optimal for the firm to reject the high type customer when the system state is (i+ 1, j+ 1),

accept otherwise:

— When we have y1 = y2 = 0,

T1Vk(i, j+ 1) +T1Vk(i+ 1, j)≥ Vk(i, j+ 1) +Vk(i+ 1, j)

≥ Vk(i, j) +Vk(i+ 1, j+ 1)

= T1Vk(i, j) +T1Vk(i+ 1, j+ 1),


— When we have y1 = 1 and y2 = 0,

T1Vk(i, j+ 1) +T1Vk(i+ 1, j)≥ vH +Vk(i+ 1, j+ 1) +Vk(i+ 1, j)

= T1Vk(i, j) +T1Vk(i+ 1, j+ 1),

— When we have y1 = 0 and y2 = 1, we show below it leads to contradiction. Given that y1 = 0, we have

Vk(i, j)−Vk(i+ 1, j)≥ vH ; Similarly, given that we have y2 = 1, hence, Vk(i+ 1, j+ 1)−V (i+ 2, j+ 1)≤ vH .

Therefore, we have

Vk(i, j) +Vk(i+ 2, j+ 1)≥ Vk(i+ 1, j+ 1) +Vk(i+ 1, j) (23)

However, it is important to note that summing (16) with i replaced by i+ 1, (16) and (18), we get Vk(i, j) +

Vk(i+ 2, j+ 1)≤ Vk(i+ 1, j+ 1) +Vk(i+ 1, j). This contradict to (23) above.



T1Vk(i, j+ 1) +T1Vk(i+ 1, j)≥ vH +Vk(i+ 1, j+ 1) + vH +Vk(i+ 2, j)

≥ vH +Vk(i+ 1, j) + vH +Vk(i+ 2, j+ 1)

= T1Vk(i, j) +T1Vk(i+ 1, j+ 1),

where the second inequality is based on the condition given by (16) with i replaced by i+ 1.


action for the firm in the state (i+ 1, j) and (i, j+ 2). In particular, y1 = 0 means that it is optimal for the

firm to reject the high type customer when the system state is (i+ 1, j), accept otherwise. Moreover, y2 = 0

means that it is optimal for the firm to reject the high type customer when the system state is (i, j + 2),

accept otherwise:


T1Vk(i, j+ 1) +T1Vk(i+ 1, j+ 1)≥ Vk(i, j+ 1) +Vk(i+ 1, j+ 1)

≥ Vk(i+ 1, j) +Vk(i, j+ 2)

= T1Vk(i+ 1, j) +T1Vk(i, j+ 2),

where the second inequality is based on (17).


T1Vk(i, j+ 1) +T1Vk(i+ 1, j+ 1)≥ vH +Vk(i+ 1, j+ 1) + vH +Vk(i+ 2, j+ 1)

≥ vH +Vk(i+ 2, j) + vH +Vk(i+ 1, j+ 2)

= T1Vk(i+ 1, j) +T1Vk(i+ 1, j+ 2),

where the second inequality is based on (17).


T1Vk(i, j+ 1) +T1Vk(i+ 1, j+ 1)≥ Vk(i+ 1, j+ 1) + vH +Vk(i+ 1, j+ 1)

≥ Vk(i+ 2, j) + vH +Vk(i, j+ 2)

= T1Vk(i+ 1, j) +T1Vk(i, j+ 2),

where the second inequality is based on the summation of (17) and (18).


T1Vk(i, j+ 1) +T1Vk(i+ 1, j+ 1)≥ Vk(i+ 1, j+ 1) + vH +Vk(i+ 1, j+ 1)

≥ Vk(i+ 1, j) + vH +Vk(i+ 1, j+ 2)

= T1Vk(i+ 1, j) +T1Vk(i, j+ 2),

where the second inequality is based on the summation of (16) and (17).



action for the firm in the state (i, j+ 1) and (i+ 2, j). In particular, y1 = 0 means that it is optimal for the

firm to reject the high type customer when the system state is (i, j+ 1), accept otherwise. Moreover, y2 = 0

means that it is optimal for the firm to reject the high type customer when the system state is (i+ 2, j),

accept otherwise:


T1Vk(i+ 1, j) +T1Vk(i+ 1, j+ 1)≥ Vk(i+ 1, j) +Vk(i+ 1, j+ 1)

≥ Vk(i, j+ 1) +Vk(i+ 2, j)

= T1Vk(i, j+ 1) +T1Vk(i+ 2, j),

where the second inequality is due to (18).


T1Vk(i+ 1, j) +T1Vk(i+ 1, j+ 1)≥ Vk(i+ 2, j) + vH +Vk(i+ 1, j+ 1)

= T1Vk(i+ 2, j) +T1Vk(i, j+ 1)

— When we have y1 = 0 and y2 = 1, we show that it is not feasible. Given that we have y1 = 0 and

y2 = 1, we get

Vk(i, j+ 1) +Vk(i+ 3, j)≥ Vk(i+ 1, j+ 1) +Vk(i+ 2, j) (24)

To this end, it is important to note that by replacing i with i+1 in (18), we get Vk(i+2, j)+Vk(i+2, j+1)≥

Vk(i+ 1, j + 1) + Vk(i+ 3, j). Similarly, by replacing i with i+ 1 in (16), we get Vk(i+ 1, j + 1) + Vk(i+

2, j) ≥ Vk(i + 1, j) + Vk(i + 2, j + 1). Summing up the above two inequalities together with (18), we get

Vk(i+ 2, j) +Vk(i+ 1, j+ 1)≥ Vk(i+ 3, j) +Vk(i, j+ 1), which contradicts to (24).

— When we have y1 = y2 = 1, the proof is similar to the case when we have y1 = y2 = 0.

• We now show that T1 preserves the property given by (19). Similarly, we let y1 denote the optimal

action for the firm in the state (i+ 1, j). In particular, y1 = 0 means that it is optimal for the firm to reject

the customer when the system state is (i+ 1, j), accept otherwise. Below, we start with the case hH > hL,

while the cases when hH = hL can be shown in a similar manner.

— When we have y1 = 0,

T1Vk(i, j+ 1)≥ Vk(i, j+ 1)≥ Vk(i+ 1, j) = T1Vk(i+ 1, j)

— When we have y1 = 1,

T1Vk(i, j+ 1)≥ vH +Vk(i+ 1, j+ 1)≥ Vk(i+ 2, j) + vH = T1Vk(i+ 1, j)

Thus, we have proved Lemma 4. Q.E.D.

Proof of Lemma 5:

The proof is similar to the proof of Lemma 4 above. Q.E.D.


Proof of Lemma 6:

We start with the proof for the case when we have hH >hL. Note that since Vk ∈ V , so when hH >hL,

T3Vk(i, j) is equivalent to

T3Vk(i, j) = Vk(i− 1, j)Ii≥1+Vk(0, j− 1)Ii=0,j≥1+Vk(0,0)Ii=j=0

• We now show that T3 preserves the property given by (14). If i ≥ 1 and j ≥ 0, we have T3Vk(i, j) =

Vk(i− 1, j)≥ Vk(i, j) = T3(i+ 1, j); If i= 0 and j ≥ 1, T3Vk(i, j) = Vk(i, j− 1)≥ Vk(i, j) = T3Vk(i+ 1, j); And

if i= j = 0, T3Vk(0,0) = Vk(0,0) = T3Vk(1,0).

• We now show that T3 preserves the property given by (15). It is similar to the proof above.

• We now show that T3 preserves the property given by (16), i.e., T3Vk(i, j + 1) + T3Vk(i + 1, j) ≥T3Vk(i, j) +T3Vk(i+ 1, j+ 1).

— if i≥ 1 and j ≥ 0,

T3Vk(i, j+ 1) +T3Vk(i+ 1, j) = Vk(i− 1, j+ 1) +Vk(i, j)

≥ Vk(i− 1, j) +Vk(i, j+ 1) = T3Vk(i, j) +T3Vk(i+ 1, j+ 1);

— if i= 0 and j ≥ 0,

T3Vk(i, j+ 1) +T3Vk(i+ 1, j) = Vk(0, j) +Vk(0, j)

≥ Vk(0, j− 1) +Vk(0, j+ 1) = T3Vk(i, j) +T3Vk(i+ 1, j+ 1);

where the inequality is based on condition given by summation of (16) and (17).

• We now show that T3 preserves the property given by (17), i.e., T3Vk(i, j + 1) + T3Vk(i+ 1, j + 1) ≥T3Vk(i+ 1, j) +T3Vk(i, j+ 2).

— if i≥ 1 and j ≥ 0,

T3Vk(i, j+ 1) +T3Vk(i+ 1, j+ 1) = Vk(i− 1, j+ 1) +Vk(i, j+ 1)

≥ Vk(i, j) +Vk(i− 1, j+ 2) = T3Vk(i+ 1, j) +T3Vk(i, j+ 2),

where the inequality is due to (17).

— if i= 0 and j ≥ 0,

T3Vk(0, j+ 1) +T3Vk(1, j+ 1) = Vk(0, j) +Vk(0, j+ 1)

= T3Vk(1, j) +T3Vk(0, j+ 2),

• We now show that T3 preserves the property given by (18), i.e., T3Vk(i+ 1, j) + T3Vk(i+ 1, j + 1) ≥T3Vk(i, j+ 1) +T3Vk(i+ 2, j).

— if i≥ 1 and j ≥ 0,

T3Vk(i+ 1, j) +T3Vk(i+ 1, j+ 1) = Vk(i, j) +Vk(i, j+ 1)

≥ Vk(i− 1, j+ 1) +Vk(i+ 1, j) = T3Vk(i, j+ 1) +T3Vk(i+ 2, j),



— if i= 0 and j ≥ 0,

T3Vk(1, j) +T3Vk(1, j+ 1) = Vk(0, j) +Vk(0, j+ 1)

≥ Vk(0, j) +Vk(1, j) = T3Vk(0, j+ 1) +T3Vk(2, j),


• We now show that T3 preserves the property given by (19), i.e., T3Vk(i, j+ 1)≥ T3Vk(i+ 1, j), assuming

hH >hL. If i≥ 1, we have T3Vk(i, j+ 1) = Vk(i−1, j+ 1)≥ V (i, j) = T3Vk(i+ 1, j), where the second equality

is due to (19); If i= 0, we have T3Vk(0, j+ 1) = Vk(0, j) = T3Vk(1, j).

We have shown the case when hH >hL. The cases with hH = hL can be shown in a similar manner, Q.E.D.

Proof of Lemma 2:

We know that the switching curves SH(.) and SL(.) given in Proposition 1 are defined as follows:

SH(j) = maxi : V (i+ 1, j)−V (i, j)>−vH |i, j ∈N0

SL(i) = maxj : V (i, j+ 1)−V (i, j)>−vL|i, j ∈N0

We let SH(0) = nfH , to show SH(j) = nfH − j, we only need to show SH(j+ 1) = SH(j)− 1. We know

SH(j+ 1) = maxi : V (i+ 1, j+ 1)−V (i, j+ 1)>−vH |i, j ∈N0

= maxi : V (i+ 2, j)−V (i+ 1, j)>−vH |i, j ∈N0

= maxi′− 1 : V (i′+ 1, j)−V (i′, j)>−vH |i′, j ∈N0

= maxi : V (i+ 1, j)−V (i, j)>−vH |i, j ∈N0− 1

= SH(j)− 1

The second equality is due to the property V (i+ 1, j) = V (i, j + 1), see (19) in the proof of Proposition 1.

Thus, we have shown SH(n0L) = nfH −n0

L. Similarly, we let SL(0) = nfL, we then can show SL(n0H) = nfL−n0

H .

Meanwhile, we have

SH(0) = maxi|V (i+ 1,0)−V (i,0)>−vH |i, j ∈N0

SL(0) = maxj|V (0, j+ 1)−V (0, j)>−vL|i, j ∈N0= maxj|V (j+ 1,0)−V (j,0)>−vL|i, j ∈N0

As it is shown in the proof of proposition 1, V (j + 1,0)− V (j,0) is decreasing in j. To this end, we have

nfH ≥ nfL if vH > vL. Q.E.D.

Proof for Lemma 3:

The overall procedure of the proof includes the following four steps: 1)we first define the original con-

strained optimization problem, whose objective is to maximize the long-run expected profit of the firm while

subjecting to the long-run expected utility of each customer type being non-negative. We refer to this con-

strained optimization problem as COP; 2)Secondly, we define the Lagrangian of the COP, which we refer to


as the Lagrangian; 3)Thirdly, we show that the Lagrangian can be solved using the dynamic programming

approach (DP). Moreover, the solution to the Lagrangian has the same structure as the unconstrained first

best solution characterized in Lemmas 1 and 2; 4)Lastly, we construct the solution to the COP using the

solutions to the Lagrangian.

We next introduce the notations which will be used throughout the proof. We let β represent the initial

state distribution. The system state x under the benchmarks can be characterized by the number of low

type customers n0L and the number of high type customers n0

H , i.e., x= (n0H , n

0L). The initial distribution β

and given policy u determine a unique probability measure P uβ over the space of trajectories of the states

and actions, see Hinderer (2012). We denote the corresponding expectation operator by Euβ. To put the

firm’s policy u in context, note that the firm’s policy is comprised of two components: the admission policy

and the scheduling policy. The admission and scheduling policies can be defined in a similar manner to the

ones defined for the unconstrained benchmark where the firm has full information on customer types and

full control over customers’ admission. In particular, we let the scheduling policy of the firm be a function

gI : SI 7→ ∅,L,H. We have gI(x) = i∈ H,L, if the next customer to be served is a type i customer in the

system at system state x. Meanwhile, we have gI(x) = ∅, if the firm decides to be idle. Meanwhile, we let the

firm’s admission policy on type i customers be a function Oi : SI 7→ 0,1, for i ∈ H,L. In particular, for

i∈ H,L, we haveOi(x) = 1 if the firm would like to admit the arriving type i customer at state x. Otherwise,

we have Oi(x) = 0. To this end, the policy of the firm at state x is given by u(x) = (gI(x),OH(x),OL(x)).

• Step 1: We first define the original COP. To formally present the COP, we first define the long-run

expected profit of the firm as follows:

K(β,u) = limn→∞

∑n

t=1 Euβkf (Xt, u)

n

where kf (Xt, u) is the firm’s expected profit at period t given the state Xt and policy u. In particular, for

the give state Xt and policy u, we have kf (Xt, u) =−h(Xt, u)+v(Xt, u), where h(Xt, u) and v(Xt, u) are the

firm’s holding cost and value obtained from admitting customers at time period t, respectively. Recall that

the holding cost per period is given by h(Xt) = n0HhH +n0

LhL with Xt = (n0H , n

0L). Meanwhile, we have

v(Xt, u) =

vH if there is an arriving type H customer at period t and OH(Xt) = 1

vL if there is an arriving type L customer at period t and OL(Xt) = 1

0 otherwise.

We next turn to formulate the constraints. The long-run expected utility of customer type i, for i∈ L,H,is given by

Di(β,u) = limn→∞

∑n

t=1 Euβdi(Xt, u)

n

where di(Xt, u) is the total expected utility of the type i customer at time period t given the state Xt and

policy u. In particular, we have di(Xt, u) = ri(Xt, u)− cin0i . Note that ri(Xt, u) is the total reward obtained

by the type i customers at time period t. In particular, for i∈ L,H, it is given by

ri(Xt, u) =

ri if there is an arriving type i customer at period t and Oi(Xt) = 1

0 otherwise.

We next formally define the COP in the following definition.


Definition 1 (COP). K(β) = maxuK(β,u), subject to Di(β,u)≥ 0 with i∈ L,H.

To simplify the COP in Definition 1, we assume

rici≥ 1

µ−βiλ∀i∈ L,H. (25)

The above assumption ensures that the expected utility of the type i customers when joining the system is

non-negative regardless of firm’s admission policy, if the firm gives absolute priority to the type i customers

over the other type, for i∈ H,L. We will show later, given (25), the constraint on the high type customers

in COP, i.e., DH(β,u)≥ 0, is never active. Thus, the COP defined in Definition 1 is equivalent to the following

problem which we refer to as COP1.

Definition 2 (COP1). K(β) = maxuK(β,u), subject to DL(β,u)≥ 0.

• Step 2: We next define the Lagrangian of the COP1.

Definition 3 (Lagrangian). The Lagrangian of the COP is defined as follows:

Jb(β,u) = limn→∞

∑n

t=1 Euβjb(Xt, u)

n=K(β,u) + bDL(β,u)

where we have

jb(Xt, u) = kf (Xt, u) + bdL(Xt, u) (26)

Definition 4 (Lagrangian Relaxation). For any constant b > 0, the Lagrangian Relaxation problem

is defined as follows:

Jb∗ = maxuJb(β,µ) (27)

We refer to the optimal solution(s) to the Lagrangian Relaxation Problem defined above as the b-optimal

solution(s).

• Step 3: Solution to the Lagrangian Relaxation Problem: Based on Lemmas 12.3 and 12.4 on

page 167 of Altman (1999), u∗(b) is the optimal solution to the Lagrangian Relaxation Problem defined in

(27) if and only if it satisfies the following Bellman optimality equation:

V b(x) +ψb = maxu

[jb(x,u) +

∑y∈X

Pxy(u)V b(y)

](28)

Based on (26) and (28), we obtain the following equivalent optimality condition.

VI(n0H , n

0L) +ψb

=1

Λ

−hLn0

L−hHn0H − bLcLn0

L

+βHλ maxOH∈1,0

VI(n

0H , n

0L)(1−OH) + (VI(n

0H + 1, n0

L) + vH)OH

+βLλ maxOL∈1,0

VI(n

0H , n

0L)(1−OL) + (VI(n

0H , n

0L + 1) + vL + bLrL)OL

+µmax

gI

(VI(n

0H − 1, n0

L)In0H>0+VI(n

0H , n

0L)In0

H=0)IgI=H

+ (VI(n0H , n

0L− 1)In0

L>0+VI(n

0H , n

0L)In0

L=0)IgI=L

+VI(n0H , n

0L)IgI=∅

, (29)


with Λ = λ+µ. Note that it is a well-known result that there exists a stationary and deterministic solution

which solves the MDP in (29) and thus is b-optimal, see Puterman (2014). Let yb be a stationary and

deterministic policy which is b-optimal. Note that we can obtain (29) by replacing hL and vL in the Bellman

optimality condition for the unconstrained first best solution (see (13)) with hL + bLcL and vL + bLrL,

respectively. Thus, following the same logic of the proof to Lemma 1, we show that yb can be characterized

by two monotonically non-increasing switching curves SbH(n0L) and SbL(n0

H). In particular, under policy yb,

the high type customers are accepted if and only if n0H ≤ SbH(n0

L), while the low type customers are accepted

if and only if n0L ≤ SbL(n0

H). Moreover, the firm gives preemptive resume priority to high type customers

under the policy yb if 0≤ b≤ hH−hL

cL. As we will show in step 4 that only the b-optimal solutions for b with

0≤ b≤ hH−hL

cLwill be relevant for constructing the solution to the COP1.

• Step 4: Solution to COP1. Let E be the set of all possible stationary and deterministic policies of the

firm. Given that both the state space and the action space are finite, the set E is finite. Following Theorem

2.1 in Sennott (2001), we have the following results:

—Jb∗ is a continuous and convex function in b, which consists finitely many linear segments. We denote

the number of linear segments as s.

— The unique break points 0 = b0 < b1 < · · ·< bs−1 define the s intervals [b0, b1], [b1, b2], · · · , [bs−2, bs−1]

and [bs−1, b∞], which correspond to the s linear segments.

— Let Jb∗ =Cj + bDj , for b∈ [bj , bj+1]. Then, DL0 <D

L1 < · · ·<DL

s−1 and C0 >C1 >C2 > · · ·>Cs−1.

To this end, we can divide this problem into the following four cases:

— If DLs−1 < 0, then the COP1 is infeasible. In our case, this is not possible. One can easily find a feasible

solution to COP1.

— If D0 ≥ 0, the constraint is never active and the constrained first best solution (i.e., the solution

to COP1) is equivalent to the unconstrained first best solution. We have analytically characterized the

unconstrained first best solution in Lemmas 1 and 2.

— If there exists j ∈ 0,1,2, · · · , s−1 such that Dj = 0, then there exists a stationary and deterministic

b-optimal solution with b ∈ [bj , bj+1] which solves the COP1 based on Theorem 4.3 in Beutler and Ross

(1985). However, given there are finite number of such j, the probability of this case happening is almost

surely 0.

— If Dj−1 < 0<Dj , we let y1 and y2 be stationary and deterministic b-optimal solutions for b∈ [bj−1, bj ]

and b ∈ [bj , bj+1], respectively. As we mentioned in Step 3, yl, with l = 1,2, can be characterized by two

monotonically non-increasing switching curves SlH(n0L) and SlL(n0

H). In particular, under the policy yl, with

l= 1,2, the high type customers are accepted if and only if n0H ≤ SlH(n0

L), while the low type customers are

accepted if and only if n0L ≤ SlL(n0

H). The firm’s priority policy under yl, with l= 1,2, depends on the value

of bj . Based on the results in Step 3, one can see that for b ≥ hH−hL

cL, the firm will give absolute priority

to the low type customers under the deterministic b-optimal solution. Thus, when rLcL≥ 1

µ−βLλ, we know

that the expected utility of the arriving low type customers is positive under any deterministic b-optimal

solution for b≥ hH−hL

cL. Given Dj−1 < 0<Dj , we know that, for b= bj , there exists a deterministic b-optimal

solution under which the expected utility of the arriving low type customers is positive and a deterministic


b-optimal solution under which the expected utility of the arriving low type customers is negative. Thus, we

have bj <hH−hL

cLand the firm gives preemptive resume priority to high type customers under the policy yl,

for l= 1,2. Note that for b≥ 0, every deterministic b-optimal policy induces a unichain Markov Chain with

aperiodic positive recurrent class. Thus, based on Proposition 3.3 of Sennott (2001), there exists p ∈ (0,1),

the mixed strategy θ(p) that chooses y1 with probability p and y2 with probability 1− p is bj-optimal with

DL(θ(p)) = 0, and thus solves COP1. Q.E.D.

Proof of Theorem 1:

Given that there are two different actions, i.e., join and balk, for each customer type, there are four pos-

sible reactions from customers: all customers joining the system, the high type customers joining the system

but not the low type customers, the low type customers joining the system but not the high type customers,

and all customers balking. However, the second and the third reactions, i.e., the high type customers joining

the system but not the low type customers, and the low type customers joining the system but not the high

type customers, are mutually exclusive in equilibrium. If there is an announcement m which induces the

outcome of the high type customers joining the system but not the low type customers, we must have

rH − cHWm > 0

and

rL− cLWm < 0,

where Wm is the expected waiting time of customers receiving the announcement m. Thus, we have

rHcH

> rLcL

. However, if there is a another announcement m′ which can induce the outcome of the low type

customers joining the system but not the high type customers, following similar arguments, we must have

rLcL> rH

cH. This leads to contradiction. Thus, the firm cannot improve its profit by using more than three

announcements when there are two customer types. Q.E.D.

Proof of Proposition 1:

It is clear that the proposed equilibrium achieves the unconstrained first best for the firm and hence

the firm does not have any profitable deviation. For the customer, one can see that if the announcement

provided is m1, the number of customers in the system is nL. Hence, the average waiting time experienced

by the customers who join the system when the firm announce m1 is nL+1µ

. Based on the (4) and (5) given

in the proposition, customers of both types are better off by joining the system when the announcement

received is m1. With similar arguments, one can show that only high type customers are better off by

joining the system when the announcement received is m2, while both high type and low type customers

are better off to balk when the announcement received is m3. Q.E.D.



The proof of (aL, aH ,A, g) is an equilibrium is similar to the proof of Proposition 1 above. We next show

that there does not exist any equilibrium which obtains a higher profit than (aL, aH ,A, g) characterized in

the proposition. Note that under any equilibrium (a′L, a′H ,A

′, g′), given rHcH

< rLcL

, we have a′L(m) ≥ a′H(m)

for all m that are used with positive probability in the equilibrium. To this end, let π denote the profit of

the firm when it cannot observe customer type and take the following actions: (1) allow both customers to

join the system; (2) allow only low type customers to join; and (3) allow neither type of customers to join.

It is worth mentioning that allowing only high type customers to join cannot be sustained in any equilibria.

Moreover, given vH > vL and hH = hL, we obtain that it is never optimal for the firm to allow only the low

type customers to join. Thus, π is the same as the profit of the firm when it treats customers of both types

identically. Hence, the firm’s profit is bounded by π when it does not observe customer types. Q.E.D.

Proof of Theorem 2:

Based on Lemma 1, we show that, when the per unit holding cost is different for customers of different

types, to achieve the unconstrained first best solution, the firm should give absolute priority to the type

of customers with a relatively higher per unit holding cost between the two types of customers. However,

the firm cannot directly observe the type of customers. As a result, it can only prioritize the customers

whose types it elicits based on their responses towards the announcements. Based on Proposition 1, one

can see that, to achieve the unconstrained first best, the firm would like to admit both customer types

when there are no customers in the system, for any non-degenerate case with Si(0) ≥ 0, ∀i ∈ H,L. As

a result, to achieve the unconstrained first best, the firm must provide at least one announcement which

induces both customer types to join the system. The firm cannot differentiate the customers who receive

such an announcement in the system. Hence, the firm cannot prioritize these customers appropriately which

prevents the firm from achieving the unconstrained first best. Q.E.D.


We start with the firm’s optimal strategy, which is comprised of the announcement policy and the

priority policy. Note that the firm’s optimal policy can be characterized by the following optimality equation.

V (i, j, k) +γ

Λ= C(i, j, k) +

λ

ΛT4V (i, j, k) +

µ

ΛT5V (i, j, k), (30)

with

C(i, j, k) =−−(hHβH +hLβL)k−hHi−hLjΛ

T4V (i, j, k) = maxm∈M

(V (i, j, k+ 1) +βHvH +βLvL)Im∈MH,L

+ (βHV (i+ 1, j, k) +βLV (i, j, k) +βHvH)Im∈MH

+ (βLV (i, j+ 1, k) +βHV (i, j, k) +βLvL)Im∈ML

+V (i, j, k)Im∈M∅.


and

T5V (i, j, k) = maxg

(V (i− 1, j, k)Ii>0+V (i, j, k)Ii=0)Ig=MH

+ (V (i, j− 1, k)Ij>0+V (i, j, k)Ij=0)Ig=ML

+ (V (i, j, k− 1)Ik>0+V (i, j, k)Ik=0)Ig=MH,L

+V (i, j, k)Ig=M∅,

where i, j, k are the numbers of customers receiving announcement mH ∈MH, mL ∈ML and mHL ∈MH,L, respectively.

We next show that the optimal priority policy of the firm is given by

g(i, j, k) =

MH if i > 0MH,L if i= 0 and k > 0ML if i= k= 0 and j > 0M∅ if j = k= i= 0,

(31)

with m∅ ∈M∅. To proceed with the proof, we define the set of functions G as follows.

Definition 8. If a function V ∈G, then the function V satisfies the following properties:

V (i, j, k)≥ V (i+ 1, j, k) (32)

V (i+ 1, j, k)≤ V (i, j+ 1, k). (33)

Note that with probability βi with i∈ H,L, a customer receiving announcement mHL is a type i customer.

Thus, we have V (i, j, k + 1) = βHV (i+ 1, j, k) + βLV (i, j + 1, k). As a result, the condition V (i+ 1, j, k) ≤V (i, j + 1, k) is equivalent to V (i+ 1, j, k)≤ V (i, j, k+ 1)≤ V (i, j + 1, k). To this end, to show the optimal

priority policy is given by (31), it is equivalent to show that the value function of the firm V belongs to the

set G. In order to show that V ∈G, following a similar logic to the one used in the proof for Lemma 1, it is

sufficient to show the following two lemmas.

Lemma 7. if V ∈G, then T4V ∈G.

Lemma 8. if V ∈G, then T5V ∈G.

• We now start proving Lemma 7:

— We next show that T4 preserves the property characterized by (33), which is equivalent to show that

if V ∈ G, then T4V (i+ 1, j, k) ≤ T4V (i, j + 1, k). In order to do so, we let m represent the optimal action

of the firm when the system state is (i+ 1, j, k). If m ∈MH,L, we have T4V (i, j + 1, k) ≥ V (i, j + 1, k +

1) + βHvH + (1−βH)vL ≥ V (i+ 1, j, k+ 1) +βHvH + (1−βH)vL = T4V (i+ 1, j, k); when m∈MH, we have

T4V (i, j+1, k)≥ βHV (i+1, j+1, k)+βHvH +(1−βH)V (i, j+1, k)≥ βHV (i+2, j, k)+βHvH +(1−βH)V (i+

1, j, k) = T4V (i+ 1, j, k); When m∈ML, we have

T4V (i, j+ 1, k)≥ βHV (i, j+ 1, k) +βLV (i, j+ 2, k) +βLVL

≥ βHV (i+ 1, j, k) +βLVk(i+ 1, j+ 1, k) +βLVL

= T4V (i+ 1, j, k);

When m ∈M∅, we have T4V (i, j + 1, k)≥ V (i, j + 1, k)≥ V (i+ 1, j, k) = T4V (i+ 1, j, k). To this end, we

have shown that if V ∈G, T4V satisfies condition (33).


— The proof for that T4 preserves the property given in (32) is similar to the one above.

• We next prove Lemma 8. We start by showing that T5 preserves the property characterized by (33),

which is equivalent to show that if V ∈ G, then T5V (i, j + 1, k) ≥ T5V (i + 1, j, k). In order to do so, we

let m represent the optimal announcement to provide for the firm when the system state is (i+ 1, j, k). If

i > 0, we have T5V (i, j + 1, k) = V (i− 1, j + 1, k) ≥ V (i, j, k) = T5V (i+ 1, j, k); If i = 0 and k > 0, we have

T5V (i, j + 1, k) = V (i, j + 1, k − 1) ≥ V (i, j, k) = T5V (i+ 1, j, k); If i = k = 0, T5V (i, j + 1, k) = V (i, j, k) =

T5V (i+ 1, j, k). To this end, we have shown T5 preserves the property (33). The proof for that T5 preserves

property (32) is similar.

Based on the proof above, we have shown that the optimal priority policy of the firm is given by (31). To

this end, T5V (i, j, k) defined in the optimality condition (30) can be simplified to be

T5V (i, j, k) = V (i− 1, j, k)Ii≥1+V (i, j, k− 1)Ii=0,k≥1+V (i, j− 1, k)Ii=k=0,j>0+V (0,0,0)Ii=j=k=0.

We next show that there exists no mH ∈MH and the response that only high type customers join but not

the low type cannot be sustained in any influential equilibrium. Recall that we focus on the non-degenerate

cases where it is optimal for the firm to admit customers of both types when there are no customers in

the system. Thus, in equilibrium, there must exist an announcement mHL ∈MH,L which induces both

customer type to join the system. If there also exists an announcement mH ∈MH which induces the high

type customers to join but low type customers to balk in an influential equilibrium, as we have shown

above that the firm would like to prioritize customers receiving the announcement mH over the customers

receiving the announcement mHL in any influential equilibrium. This is because the expected per unit holding

cost of customers receiving announcement mH is larger than that of customers receiving the announcement

mHL, when hH > hL. To this end, the expected waiting time of customers receiving announcement mH is

shorter than that of customers receiving the announcement mHL. Thus, given it is better off for the low

type customers to join the system when they receive the announcement mHL, it should also be better off

for them to join the system upon receiving the announcement mH in the given influential equilibrium. This

contradicts to the definition of mH . Thus, the customer response that only high type customers join but low

type customers balk cannot be sustained and there exists no announcement mH ∈MH in any influential

equilibrium.

Given hi > 0 for i∈ H,L, in any influential equilibrium, the firm would like to provide an announcement

with m∅ ∈M∅ to induce both customer types to balk when the system is really congested. Meanwhile, when

the gain due to the lower holding cost of the low type customers more than compensates for the loss due to

the lower value obtained by serving the low type customers, the firm may like to provide the announcement

mL ∈ML to induce the low type customers to join the system but not the high type in an influential

equilibrium. Above, we have shown firm’s announcement and scheduling policy in equilibrium. As for the

customers, given incentive compatibility conditions given in (8), it is better off for both customer types to

join when they receive the announcement mHL, while it is better off for low type customers but not high

type customers to join when they receive announcement mL. It is better off for both customer types to balk

when they receive announcement m∅. Q.E.D.



We start by exploring the conditions when the babbling equilibrium where both types of the customers

join the system regardless of the announcements may exist. If customers of both types indeed join the queue

disregard of the announcements received, the system becomes an M/M/1 system with the arrival rate and

the service rate being λ and µ, respectively. Given that the firm cannot differentiate customer types in

any way through a babbling equilibrium, we focus on the case when the firm serves the customers in a

first-come, first-served manner. Thus, one can show that the average waiting time in the system is given by

1µ−λ . Since customers would join the system if and only if their expected utility is positive in equilibrium,

we have ri − ciµ−λ ≥ 0, ∀i ∈ H,L. Following a similar logic, we can characterize the other two types of

babbling equilibria as described in Proposition 4. Q.E.D.


Firm’s Profit: We let ΠIP be the profit of the firm per unit time under the influential pooling equi-

librium, while let UIP be the utility of the customers per unit of time. We let the firm’s profit per unit time

under the system M/M/1/k be Ω(k). Based on Theory 1 in Knudsen (1972), Ω(k) is a unimodal in k. In

particular, there exist a finite k∗ ∈ Z+ such that the function Ω(k) is strictly increasing for k < k∗ and is

strictly decreasing for k ≥ k∗. To have the pooling equilibrium hold, we have k∗ = nf + 1. Meanwhile, the

system under the babbling equilibrium where both customer types join is equivalent to M/M/1/∞. Thus,

the firm’s profit under the pooling equilibrium is larger than the firm’s profit under the babbling equilibrium,

i.e, ΠIP >ΠNI .

Customer Utility: We next show that providing announcements may improve or hurt the expected total

customer utility. Let us first show that customers always achieve higher expected total customer utility in the

pooling equilibrium where both customer types join the system. Given that the firm could not differentiate

the customers of different types at all in the pooling equilibrium, we can consider that the system only

includes one customer type. For these customers, the value obtained by the firm through serving each

customer, the per unit holding cost, the reward of service for the customers and the per unit waiting cost

of the customers are given by βHvH + βLvL, βHhH + βLhL, βHrH + βLrL, and βHcH + βLcL, respectively.

To this end, when customers can observe the system state, they will join the system only if the number of

customers in the system is less than ncHL with ncHL = b (βHrH+βLrL)µ

βHcH+βLcLc. Note that the system dynamic under

the pooling equilibrium is the same as the system M/M/1/(nf +1). The system M/M/1/(nf +1) is identical

to the system M/M/1 with the modification that the total number of customers in the system is capped

by nf + 1. We now let the expected total customers utility per unit time under the system M/M/1/k be

Ωc(k). Based on on the results in Section 4 of Naor (1969), there exists k∗ ∈ Z+ such that the function

Ωc(k) is strictly increasing for k < k∗ and is strictly decreasing for k≥ k∗. Naor (1969) also shows k∗ <ncHL.

Meanwhile, one can show that ncHL ≤maxncH , ncL with ncH = b rHµcHc and ncL = b rLµ

cLc. Moreover, we have

nf + 1 > maxncH , ncL in order to have the pooling equilibrium to hold. Thus, we have k∗ < nf + 1. Note

that under the babbling equilibrium, the effective customer admission threshold is ∞. Thus, the expected

total customers utility under the pooling equilibrium is larger than the one under the babbling equilibrium

where both customer types join the system.


Now let us turn to the case when the pooling equilibrium coexists with the babbling equilibrium where

only one customer type joins the the system. We start with the case with hH = hL. Note that the pooling

equilibrium can only co-exist with the babbling equilibrium where only the low type customers join but not

the high type. We next show in this case, providing announcements may hurt the expected total customer

utility under certain conditions. Note that, the expected total customer utility per unit time under the

pooling equilibrium is given by UoIP = βLλP (rL− cL(n+1)

µ) +βHλP (rH − cH(n+1)

µ), where P is the stationary

probability that there are less than nf + 1 customers in the system under the pooling equilibrium. It

is given by P = 1 − (1−ρ)ρnf+1

1−ρnf+2with ρ = λ

µ. The expected total customer utility per unit time under the

babbling equilibrium where only low type customers join the system is given by UoNI = βLλ(rL − cL

µ−βLλ).

One can show that we have UoNI > Uo

IP if and only if (1−P )rLcL

+ (1 + βHcHβLcL

) (n+1)P

µ> βHrHP

βLcL+ 1

µ−βLλBased

on Propositions 2 and 4, the pooling equilibrium coexists with the babbling equilibrium where only low

type customers join the system if and only if n+1µ≤ rH

cH< 1

µ−βLλ≤ rL

cL< nf+2

µ. Thus, we have Uo

NI > UoIP

if and only if n+1µ≤ rH

cH< 1

µ−βLλ≤ rL

cL< nf+2

µand (1−P )rL

cL+ (1 + βHcH

βLcL) (n+1)P

µ> βHrHP

βLcL+ 1

µ−βLλ.We can

characterize the conditions for the case with hH >hL similarly. Q.E.D.

Proof of Theorem 3:

We start from the case when the holding cost is the same for both customers types, i.e., hH = hL.

Recall that Proposition 2 shows that, to achieve the unconstrained first best, the firm would like both types

of customers to join the system when the number of customers in the system is smaller than nfL, would

like high type customers to join but not the low type when the number of customers is between nfL and

nfH , and would like both customer types to balk otherwise. To this end, when the firm observes the type of

the customers, for any influential equilibrium to exist, the only threshold for the low type customers which

immunes from profitable deviations by the firm is nfL. Similarly, one can show that nfH is the only threshold

for the high type customers which prevents the firm from profitable deviations. To this end, we have shown

that, assuming hH = hL, under any influential equilibrium (if it exists), the firm achieves the unconstrained

first best. Similar arguments apply for the case with hH 6= hL. Q.E.D.


It is clear that the proposed equilibrium achieves the unconstrained first best for the firm and hence

the firm does not have any profitable deviations. For the high type customers, one can see that if the

announcement provided is mH2 , the number of high type customers in the system denoted by n0

H is larger

than the threshold given by SH(n0L). Hence, the expected waiting time of the arriving high type customer

who receives announcement mH2 is given by wH if they join the system. Given rH/cH < wH , the high

type customers would obtain negative utility in expectation by joining the system when they receive the

announcement mH2 . Hence, it is better off for the high type customers to balk the system when they receive

the announcement mH2 . Similarly, we can show that it is better off for the high type customers to join the

system when they receive the announcement mH1 . Thus, high type customers would have no incentives to

deviate from the equilibrium. Following a similar argument, we can show that the low type customers do


not have an incentive to deviate either. Q.E.D.


Note we have wH =nfH

+2

µfor the case with hH = hL. Thus, to show Dc

NI ∩DI 6= ∅, it is sufficient to

show that¯wH <

nH+1µ

. We know¯wH =

EFB [n|0≤n≤nfH

]+1

µ, while we have nH = EFB[n|nfL < n≤ nfH ]. To this

end, one can see¯wH <

nH+1µ

.

When we have hH = hL, we also have wL =EFB [n|n>nf

L]+1

µ. Thus, by definition, we have wL >

nH+1µ

.

Together with the result¯wH <

nH+1µ

, we have DSSNI ⊂DI . Thus, to show Dc

I ∩DNI = ∅, it is equivalent to

show DPNI ⊆DI . It is trivial to see that DP

NI ⊆DI is equivalent to µ¯wH ≤ n+ 1≤ nf + 2≤ µwL. Q.E.D.


When we have hH >hL, the high type customers have the absolute priority over the low type customers.

Thus, we have wH <¯wL. To this end, DP

NI ⊆ (DcI ∩DNI). We know DP

NI 6= ∅. Thus, we have DcI ∩DNI 6= ∅

when hH >hL.

DcNI ∩DI = ∅ is equivalent to DI ⊆DSS

NI . One can also see that DI ⊆DSSNI is equivalent to wmHL

≤¯wH ≤

wH ≤ wmL≤

¯wL ≤ wL ≤ wm∅ . Q.E.D.


When hH = hL, in the contraction region, the firm achieves the pooling equilibrium (characterized in

Proposition 2) when the firm does not directly observe customer types. However, the firm achieves babbling

equilibria when it observes customer types in the contraction region. Given that hH = hL, the babbling equi-

libria emerging when the firm observes customer types are equivalent to the ones characterized in Proposition

4. Based on Proposition 2, the firm achieves the highest profit in the pooling equilibrium among all equilibria

including the babbling ones characterized in Proposition 4. Thus, when hH = hL, loss of credibility hurts the

profit of the firm. Q.E.D.

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