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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.
Author: Managing Customer Expectations and Priorities in Service Systems?4 Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!)
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
Author: Managing Customer Expectations and Priorities in Service Systems?Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!) 5
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
Author: Managing Customer Expectations and Priorities in Service Systems?6 Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!)
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
Author: Managing Customer Expectations and Priorities in Service Systems?Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!) 7
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.
Author: Managing Customer Expectations and Priorities in Service Systems?8 Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!)
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’
Author: Managing Customer Expectations and Priorities in Service Systems?Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!) 9
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
Author: Managing Customer Expectations and Priorities in Service Systems?10 Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!)
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.
Author: Managing Customer Expectations and Priorities in Service Systems?Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!) 11
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.
Author: Managing Customer Expectations and Priorities in Service Systems?12 Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!)
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.
Author: Managing Customer Expectations and Priorities in Service Systems?Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!) 13
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.
Author: Managing Customer Expectations and Priorities in Service Systems?14 Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!)
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.
Author: Managing Customer Expectations and Priorities in Service Systems?Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!) 15
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
Author: Managing Customer Expectations and Priorities in Service Systems?16 Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!)
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.
Author: Managing Customer Expectations and Priorities in Service Systems?Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!) 17
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.
Author: Managing Customer Expectations and Priorities in Service Systems?18 Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!)
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.
Author: Managing Customer Expectations and Priorities in Service Systems?Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!) 19
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.
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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.
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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.
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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.
Author: Managing Customer Expectations and Priorities in Service Systems?Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!) 23
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.
Author: Managing Customer Expectations and Priorities in Service Systems?24 Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!)
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
Author: Managing Customer Expectations and Priorities in Service Systems?Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!) 25
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
Author: Managing Customer Expectations and Priorities in Service Systems?26 Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!)
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
Author: Managing Customer Expectations and Priorities in Service Systems?Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!) 27
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.
Author: Managing Customer Expectations and Priorities in Service Systems?28 Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!)
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
Author: Managing Customer Expectations and Priorities in Service Systems?Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!) 29
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.
Author: Managing Customer Expectations and Priorities in Service Systems?30 Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!)
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
Author: Managing Customer Expectations and Priorities in Service Systems?Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!) 33
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.
Author: Managing Customer Expectations and Priorities in Service Systems?34 Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!)
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.
Author: Managing Customer Expectations and Priorities in Service Systems?Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!) 35
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.
Author: Managing Customer Expectations and Priorities in Service Systems?36 Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!)
(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
Author: Managing Customer Expectations and Priorities in Service Systems?Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!) 37
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
Author: Managing Customer Expectations and Priorities in Service Systems?38 Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!)
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).
Author: Managing Customer Expectations and Priorities in Service Systems?Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!) 39
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 .
Author: Managing Customer Expectations and Priorities in Service Systems?40 Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!)
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
T1Vk(i, j) = maxvH +Vk(i+ 1, j), Vk(i, j)
≥ vH +Vk(i+ 1, j)
≥ vH +Vk(i+ 2, j) = T1Vk(i+ 1, j)
Author: Managing Customer Expectations and Priorities in Service Systems?Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!) 41
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
T1Vk(i, j) = maxvH +Vk(i+ 1, j), Vk(i, j)
≥ Vk(i, j)
≥ Vk(i, j+ 1) = T1Vk(i, j+ 1),
where the second inequality is based on the condition given by (15).
— Similar, when y= 1, we have
T1Vk(i, j) = maxvH +Vk(i+ 1, j), Vk(i, j)
≥ 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),
where the second inequality is based on the condition given by (16).
— 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.
Author: Managing Customer Expectations and Priorities in Service Systems?42 Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!)
— When we have y1 = y2 = 1,
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.
• We now show that T1 preserves the property given by (17). Similarly, we let y1 and y2 denote the optimal
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:
— When we have y1 = y2 = 0,
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).
— When we have y1 = y2 = 1,
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).
— When we have y1 = 1 and y2 = 0,
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).
— When we have y1 = 0 and y2 = 1,
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).
Author: Managing Customer Expectations and Priorities in Service Systems?Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!) 43
• We now show that T1 preserves the property given by (18). Similarly, we let y1 and y2 denote the optimal
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:
— When we have y1 = y2 = 0,
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).
— When we have y1 = 1 and y2 = 0,
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.
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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),
where the inequality is due to (18).
Author: Managing Customer Expectations and Priorities in Service Systems?Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!) 45
— 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),
where the inequality is due to (19).
• 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
Author: Managing Customer Expectations and Priorities in Service Systems?46 Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!)
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.
Author: Managing Customer Expectations and Priorities in Service Systems?Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!) 47
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)
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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
Author: Managing Customer Expectations and Priorities in Service Systems?Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!) 49
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.
Proof of Proposition 2:
Author: Managing Customer Expectations and Priorities in Service Systems?50 Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!)
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.
Proof of Proposition 3:
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∅.
Author: Managing Customer Expectations and Priorities in Service Systems?Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!) 51
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).
Author: Managing Customer Expectations and Priorities in Service Systems?52 Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!)
— 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.
Author: Managing Customer Expectations and Priorities in Service Systems?Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!) 53
Proof of Proposition 4:
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.
Proof of Proposition 5:
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.
Author: Managing Customer Expectations and Priorities in Service Systems?54 Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!)
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.
Proof of Proposition 6:
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
Author: Managing Customer Expectations and Priorities in Service Systems?Article submitted to Management Science; manuscript no. (Please, provide the mansucript number!) 55
not have an incentive to deviate either. Q.E.D.
Proof of Proposition 7:
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.
Proof of Proposition 8:
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.
Proof of Proposition 9:
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.