Call Centers with Delay Information:
Models and Insights
Oualid Jouini† • Zeynep Akşin‡ • Yves Dallery†
†
Laboratoire Genie Industriel, Ecole Centrale Paris, Grande Voie des Vignes, 92290
Châtenay-Malabry, France
‡
College of Administrative Sciences and Economics, Koç University, Rumeli Feneri Yolu,
34450 Sariyer-Istanbul, Turkey
[email protected] • [email protected] • [email protected]
In this paper, we analyze a call center with impatient customers. We study how informing
customers about their anticipated delays affects performance. Customers react by balking
upon hearing the delay announcement, and may subsequently renege, particularly if the realized waiting time exceeds the delay that has originally been announced to them. The balking
and reneging from such a system are a function of the delay announcement. Modeling the
call center as an M/M/s+M queue with endogenized customer reactions to announcements,
we analytically characterize performance measures for this model. The analysis allows us
to explore the role announcing different percentiles of the waiting time distribution, i.e.,
announcement coverage, plays on subsequent performance in terms of balking and reneging.
Through a numerical study we explore when informing customers about delays is beneficial,
and what the optimal coverage should be in these announcements. It is shown how managers of a call center with delay announcements can control the tradeoff between balking
and reneging, through their choice of announcements to be made.
Keywords queues; telephone call centers; customer behavior: impatient customers, balking;
state-dependent analysis; predicting and announcing delays.
1.
Introduction
As call centers are growing in numbers and importance, they are also becoming more sophisticated in their use of technology (Gans et al. (2003); Akşin et al. (2007)). Announcing
anticipated delays to customers in order to modulate their use of the call center is one such
use of technology, which motivates the research in this article. The main reason for informing customers about their queueing delays is to alleviate congestion and to reduce customer
dissatisfaction with waiting.
Information about anticipated delays is especially important in service systems with
invisible queues (tele-queue) of which call centers are a prime example (Bitran et al. (2008),
Zohar et al. (2002). In such systems, the uncertainty involved in waiting is higher than
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that in systems with visible queues. Upon arrival and during their wait, customers have
no means to estimate queue lengths or progress rate. ”Uncertain waits are perceived to be
longer than known, finite waits” (Maister (1985) p. 118) and have been related with lower
satisfaction (Taylor (1994)). Providing delay information is shown to improve satisfaction
(see for example Taylor (1994), Katz et al. (1991), Hui and Zhou (1996)). Several reasons
have been attributed to this improvement: a sense of better control on the part of the
customer (Katz et al. (1991),Taylor (1994), Hui and Tse (1996)), a reduction in perceived
waiting time (Hui and Zhou (1996)), or alternatively an improvement in the acceptability of
a wait as a result of delay information (Hui and Zhou (1996)).
In call center settings, satisfaction with waiting experiences affect customers’ reactions
in terms of balking and reneging behavior. Delay announcements, through their effects
on customers, may further modulate these customer reactions. When a new arrival thinks
that her anticipated delay is too long, she can balk without joining the system. A delay
announcement will instigate further balking, leading to a reduced number of customers in
queue and shorter waits. For customers who enter the queue, delay announcements may
further have the effect of increasing their patience by reducing the uncertainty. However,
since perfect announcements are not possible in reality, some customers may experience
longer delays than what has been announced to them. This in turn may have a negative
effect on satisfaction and resulting patience behavior. This loose discussion indicates a close
relationship between delay announcements and resulting customer behavior. We explore this
relationship in the subsequent analysis and propose a model to analyze it. We aim to answer
the following questions: how can delays be predicted and once the predicted waiting time
distribution is obtained, what should be announced to customers? Is it always in the service
provider’s best interest to announce delays, and if so should the announcement be as close
to reality as possible?
Predicting delays for arriving customers is system state-dependent. This is different from
estimating stationary performance measures and usually makes the analysis intractable.
In the context of prediction and announcement of delays, an extra layer of complexity is
introduced. The analysis becomes more difficult since we have to take into account the
description of the system in addition to the announcements given to each waiting customer in
queue. Existing research often looks for approximations, such as announcing the stationary
mean waiting time, or announcing the actual delay of the last customer (motivated by
large systems in an overload regime, see Armony et al. (2005)). In this work, we analyze
Markovian models. Paralleling Whitt (1999a), we consider a single class multi-server call
center model with impatient customers and working under the first come, first served (FCFS)
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assumption. Customer reactions to delay announcements are modeled as balking and a form
of abandonment. Our main insights are derived from the analysis of this simple model.
Starting from a model with perfect delay announcements and contrasting it with a model
with no delay announcement, we detail and justify the quantitative building of the new model
with potentially imperfect delay information. In our model, customers have the opportunity
to balk in response to their anticipated delay. Further to the balking reaction, it has been
shown using real call center data that the reneging behavior may also change in response to
delay information (Feigin (2005)). We model that effect by extending the model of Whitt
(1999a), letting already informed customers renege, even after having chosen to join the
queue. We propose a method for the modeling of the new reneging experience by relating it
to the delay announcement that is made.
Once we obtain the predicted waiting time distribution of a new arrival, we investigate
how the service manager should profit from that information to make the announcement.
The manager has a choice in terms of choosing a particular percentile of the waiting time
distribution to announce to the customer. In making that choice, the manager considers the
following tradeoff. Informing the customer of short waiting times, which is likely to underestimate the actual waiting, might lead to less balking but excessive reneging and reduce the
reliability of the service provider in the eyes of the customers. On the other hand, informing
the customer of large waiting times increases the number of balking customers, but as a
result leading to a system that might allow to serve customers within shorter and reasonable
delays. Through a numerical analysis, we investigate how the ideal percentile should be chosen as a function of costs associated with balking and reneging customers, system size, and
different modeling assumptions about customer reaction. The idea of controlling the tradeoff
between balking and reneging through appropriate manipulation of customer reactions that
we analyze herein, is similar to the idea of selecting the size of a finite waiting space in a
queueuing system to control this tradeoff (Kolesar (1984), Koole and Mandelbaum (2002)).
Our main contributions can be summarized as follows: We propose a model that explicitly
takes customer reactions to delay announcements into account, both in terms of balking and
reneging behavior. This distinction in reactions allows us to model reality more closely, where
we observe both types of reactions. It further enables us to differentiate between the two
types of departure from queue from a customer standpoint, where a customer may prefer an
informed balking decision when delays are announced over a misinformed reneging decision
when the announced time is exceeded in practice. We analyze the resulting model as a birthdeath process, and characterize its performance measures. The analysis allows us to explore
the role announcing different percentiles of the waiting time distribution, i.e., announcement
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coverage, plays on subsequent performance in terms of balking and reneging. We show under
what conditions providing information on delays to customers will be beneficial, and illustrate
that more likely delay information is not necessarily better in terms of system performance.
We demonstrate that while the service provider will want to announce a delay that is typically
different from the expected waiting time, the relationship between performance and different
announcement coverages is not monotonic due to the tradeoff between balking and reneging.
The idea of controlling this tradeoff through a choice of announcement coverages is new, and
constitutes our main managerial contribution.
The paper is structured as follows. In Section 2 we provide a review of related literature.
Section 3 describes the setting and possibilities regarding types of delay announcements
and reactions to these by customers. The models without and with delay announcement
are formulated in Section 4. Sections 5 and 6 characterize their performance measures,
respectively. We also establish in Section 6 the basic relationship between announcement
coverage and customer balking and reneging through a stochastic comparison. The numerical
analysis in Section 7 compares the models with and without announcements, and explores
the role the announcement coverage plays. The paper ends with concluding remarks. In the
appendix, we present support and proofs of some results derived in the core of the paper.
2.
Literature Overview
The literature related to the subject of this paper spans mainly three areas. The first
one deals with queueing models incorporating reneging. The second area pertains to the
prediction and announcement of delays seen from a queueing perspective. The third area
is related to the psychology of waiting and the qualitative impact of announcing delays on
customer behavior.
In the following, we highlight some of the literature with regard to the first area, without
providing a complete overview. The presence of customers that renege subsequent to delay
announcements is a feature often found in call centers that makes this study of value in
practice. The importance of modeling abandonments in call centers is emphasized by Garnett
et al. (2002), Gans et al. (2003) and Mandelbaum and Zeltyn (2006). Empirical evidence
regarding abandonments in call centers can be found in Brown et al. (2005) and Feigin (2005).
We refer the reader to Ancker and Gafarian (1962), Garnett et al. (2002), and references
therein for simple models assuming exponential impatience, as we do in this paper. In Jouini
and Dallery (2007a), the authors consider performance evaluation of a Markovian multiserver
queue with two types of impatient customers, having different priorities. Other papers have
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allowed reneging times to follow a general distribution. Related studies include those by
Baccelli and Hebuterne (1981), Brandt and Brandt (2002) and Zeltyn and Mandelbaum
(2005).
As for the second area, we refer the reader to the relevant work of Whitt (1999b). The
author focuses on estimating state-dependent delays and presents both accurate methods
and approximations of waiting times in different situations. Models under consideration are
queueing systems with different customer classes, characterized by different service rates,
exponential and non-exponential service times. Nakibly (2002) reviews several classical results, and extends the analysis to some other complex models, with priorities but without
abandonments. Announcement of these delay estimates to customers and subsequent reactions of customers are not considered in these studies. As already mentioned, estimating
state-dependent delays is known to be difficult. It is related to the transient analysis of birthdeath processes, and in general of Markov chains. The literature on birth-death processes is
extensive and growing, see Keilson (1964a), Kijima (1997), Jouini and Dallery (2007b), and
references therein.
The problem of predicting and announcing delays has recently received a lot of attention in the field of call centers. Whitt (1999a) models and quantifies the effect of giving
state information to customers on the performance in a single class Markovian call center
model. The analysis shows that average waiting times can be reduced with accurate announcements. Jouini et al. (2007) extend the analysis of Whitt (1999a) to a setting with
two customer classes with priorities, incorporating delay announcements. Guo and Zipkin
(2007) consider a simple queueing model where three levels of information can be provided
to customers, namely no information, partial information and full information (the exact
waiting time). The authors investigate how information about delays can enhance system
performance. None of these papers allow for reneging subsequent to joining the queue, which
is a restriction that is relaxed in Armony et al. (2005) as well as in the current paper. Taking
general distributions for service times and times before reneging, Armony et al. (2005) develops methods based on fluid approximations to study the effect of announcing delays. The
performance of one particular delay announcement, that of announcing the last customer’s
realized delay, is explored. Motivated by this type of delay announcement, Ibrahim and
Whitt (2007) explore the performance of different real-time delay estimators based on recent
delay experience by customers. Armony and Maglaras (2004), and Armony and Maglaras
(2004a) consider a slightly different model. Based on his anticipated delay information, a
customer may balk, elect to wait, or leave a message. When a message is left, the service
provider calls the customer back within a guaranteed time. The authors show that system
5
performance improves, both in terms of throughput and in terms of average waiting time.
Note that although the modeling approaches differ from one work to another, the findings
usually confirm the benefits of communicating delays to customers. Guo and Zipkin (2007)
is one exception, where conditions are identified under which more information may hurt the
customer or the service provider. In a more general service setting, where delay information
provided to customers is allowed to take different forms including qualitative and vague
statements, Allon et al. (2007) show the possibility of a strategic self-interested firm that
might choose to provide intentionally vague information to strategic customers to induce
desired behavior from them. Our modeling approach, which allows us to capture the link
between the announcement that is made and resulting system performance explicitly, enables
us to similarly explore different effects on system performance that different announcements
may have and as a result propose that the service provider could choose delay announcements
to induce particular balking and reneging reactions by customers.
The third area of literature close to this paper is related to the psychology of waiting.
The literature on customers influenced by delay begins with Naor (1969). Focusing on
customer psychology in waiting situations Maister (1985) proposes a set of hypotheses some
of which are tested in the subsequent literature. We rely on the recent review by Bitran et al.
(2008) and references therein to highlight results from the literature on which we draw in
formulating our model. It is apparent from this review that modeling waiting experiences and
generalizing customer reactions are difficult to do due to the presence of many moderating
effects including personal differences and service context. Given that call center specific
evidence is very limited, we also rely on results that emerge from other contexts.
One of the first models is by Osuna (1985) constructing a direct relationship between waiting time and dissatisfaction or stress experienced by customers during the wait. Elsewhere
it is argued that it is not time but the perception thereof that drives customer satisfaction
(see for example Hornik (1984), Zakay (1989)). It is shown that in settings where customers
lack information on the duration of the wait, i.e. in settings with uncertainty concerning
queue length and progress as in tele-queues, people tend to overestimate waiting durations
(Taylor (1994)). This implies higher dissatisfaction with the wait. Information on delays on
the other hand shorten perceived waiting time by customers (Katz et al. (1991), Hui and Tse
(1996)). Many have attributed this to the sense of control that the customer feels about the
wait once uncertainty about it has been removed through an announcement (Taylor (1994),
Katz et al. (1991), Hui and Tse (1996)). But this does not seem to be the only effect: told
that they will wait for a certain amount, people are documented to adjust their acceptance
level for waiting (Hui and Zhou (1996)). More recent results emphasize the importance of
6
a sense of progress to customers Rafaeli (2002), Munichor and Rafaeli (2007)). In physical
queues, people observe that they are getting closer to service and this has an improving effect
on their mood while waiting (Rafaeli (2002)). Lab experiments simulating a tele-queue in
Munichor and Rafaeli (2007) show that customers prefer queue position announcements over
music or apologies, and react by being more persistent in holding the line, providing further
evidence to the idea that progress matters to customers. An improvement in satisfaction
level as customers approach service is also consistent with the check-in phase of the five
phase service encounter profile detailed in Bitran and Lojo (1993), of which an example in a
call center context is provided in Bitran et al. (2008). While a delay announcement can act
like a time guarantee thereby increasing satisfaction as the wait proceeds to the announced
time, exceeding this time will have a negative effect on customers, reducing their satisfaction
(Katz et al. (1991), Kumar et al. (1997)).
Results from this literature motivate some of our modeling assumptions, as detailed in
the following section. We note that many of the results pertain to satisfaction levels in
general waiting settings, and links to customer reactions like balking and reneging in call
center settings have not been established. Some exceptions are papers that directly analyze
call center data (Brown et al. (2005), Feigin (2005)) and the lab experiments in Munichor
and Rafaeli (2007).
3.
Announcing Delays in Call Centers
This section lays out basic characteristics of the setting being modeled and describes possible
impact of the announcement of delays on customer behavior.
3.1
Preliminaries
Consider a call center with a single group of agents, serving a single class of customers. The
model consists of one infinite queue, and a set of s parallel, identical servers representing
the set of customer representatives (agents). Customers waiting in queue are served under
the FCFS discipline. Interarrival times as well as successive service times are random, and
assumed to be independent and identically distributed (i.i.d.).
Upon arrival, a customer is addressed by one of the available agents, if any. If not, the
customer may decide to immediately leave the system (balking) or to wait, at least for a
certain amount of time (waiting). In addition, this decision is likely to depend on whether
or not the call center is providing information about anticipated delays in the queue. In this
paper, we will study and compare the behavior of a call center without delay announcements
7
and of a call center with delay announcements. In order for the comparison to make sense, we
need to specify customer patience in these different environments. To do so, let us consider
the following idealized situation with perfect delay announcements first.
Consider a “perfect information call center” in which the waiting time of a customer could
be predicted exactly and given to the customer upon his arrival as information. Further
assume that the customers of this call center are fully aware of such an announcement and
its exactness. In this situation, each customer, say customer k, would arrive with a given
(deterministic) patience time tk . Note that this patience time could possibly be equal to
0 representing a customer who is not willing to wait (infinitely impatient customer). This
time represents an ideal construct which we may call patience threshold under perfect delay
information. It would represent the customer’s willingness to wait including all dimensions
of the service (like for example the importance attributed to the service by the customer),
however excluding the customer’s preferences concerning uncertainty in the duration of the
hold. The behavior would then simply be as follows. If the announced delay time, say
dk , exceeds the patience time of the customer, i.e., dk > tk , the customer would leave the
system immediately (balking); otherwise, the customer would join the queue and stay until
entering service. Because of the accuracy of the delay announcement and the definition
of tk , the customer would never abandon the queue. For simplicity, we assume here and
throughout the paper that the customer’s discovery of a busy system, listening to a delay
announcement, and making a decision on balking all happen in zero time. In reality we
expect these durations to be very short, nevertheless still positive.
Since customers will vary in their preferences, each customer may have a different tk
value. We assume that we can statistically characterize the patience behavior of the customers by a distribution of patience times. This distribution will in general consist of an
atom at the origin, corresponding to very impatient customers, and a remaining continuous
distribution. An atom close to the origin is also present in the Israeli call center data analyzed in Brown et. al. (2005) and the U.S. Bank call center data analyzed in Feigin (2005).
Let α0 denote the probability that a customer, arriving to a busy system, will immediately
balk. This feature models a non-negligible portion of the customers who call with the idea to
hang up immediately once they know that they have to wait for service. On the other hand,
customers who find a busy system will accept to join the queue with probability 1 − α0 .
For these customers, we assume that their patience thresholds are random and i.i.d. under
a given continuous distribution. The patience of a particular customer is thus a random
outcome of this distribution and is known upon arrival in the queue. We will assume that
these patience times are exponentially distributed with parameter γ. We continue to impose
8
the exponentiality assumption on patience times in the models with (imperfect) delay announcement and no delay announcement that we present below. This is mainly to preserve
tractability in our models. Indeed, empirical evidence from real call centers suggests that
patience times may not be exponential (Brown et al. (2005), Feigin (2005)).
3.2
The Impact of Announcing Delays
In this section we consider a call center that provides information on delays to its customers.
Unlike what is assumed in Section 3.1, delay announcements are not perfect, thus allowing
us to model reality where indeed it is impossible to provide exact delay announcements. This
brings with it the added complexity that different delay predictions are possible and hence
that different durations may be announced to customers.
From a modeling perspective, an important feature is that customers are expected to
react to such announcements by changing their behavior (Katz et al. (1991), Hui and Tse
(1996), Taylor (1994), Hui and Zhou (1996), Munichor and Rafaeli (2007)). Different types
of announcements might instigate different types of reactions in terms of balking or reneging. When we inform a customer about her anticipated delay, she will decide right away,
either to hang up immediately because she estimates that her delay is too long, or to start
waiting in the queue. In the latter case, there are two further possibilities. The first is that
customers never abandon thereafter. The second possibility is that they are still impatient,
i.e., customers may abandon even if they have chosen to start waiting. It is natural to expect
that customers would abandon in a different way than in the model without announcements
(Feigin (2005)).
Similarly, the balking reaction is expected to depend on the information provided. For
customers who find the system busy, the balking with probability α0 is still present. The
remaining customers (with probability 1 − α0 ) will get delay information. Thus, their subsequent balking decisions will depend on the delay information. Below, we discuss possible
forms of delay information that could be given to customers, and present the different modeling possibilities that arise from these.
3.2.1
What Information Do We Provide?
It is possible to provide different types of information regarding delays. A common one
is to announce the number of customers ahead. This is not very meaningful in a setting
where the number of servers are unknown to customers, service times are random and where
customers may abandon the queue. A more sophisticated version announces the number
9
of customers ahead upon arrival, and updates the announcement each time that number
changes. However, in addition to creating operational complexity, this type of information
may create customer confusion: due to randomness, equivalent decreases in queue length
may take very different lengths of time. A further possibility is to announce some realtime delay estimators based on recent delay experience by customers, as in Armony et al.
(2005) or Ibrahim and Whitt (2007). This type of announcement is shown to work well
when delays are relatively long, i.e., for highly loaded systems. Since we want to announce
state dependent queueing delays, we do not consider this type of announcement any further
herein.
For call center settings where the state-dependent waiting time of each new arrival can
be derived, we distinguish between three additional types of delay announcement. The first,
as in Whitt (1999a), is to give the whole distribution of the anticipated waiting time to each
new customer. A customer will make use of the entire delay distribution to decide wether to
join the queue or not. This is like saying that each customer has her own coverage probability
(precision) which may be different from that of the other customers. Then, the customer will
pick up the realization from the distribution, that corresponds to her coverage probability,
and will make her balking decision in response. This first type of announcement is at odds
with reality. In practice it is unrealistic to give a customer the distribution function of her
anticipated delay.
For that reason, a second type of announcement, which is to communicate the expected
value of the delay distribution, has been suggested by Whitt (1999a). This is simple and easy
to understand by customers. On the other hand, when the number of customers already in
queue is small the state-dependent waiting time can be quite different from its mean, since
the variability of state-dependent delays will be high in that case. The expected delay will
only be a good estimate of the state-dependent delay when the number of customers ahead
in queue is large, since the delay distribution is in that case highly concentrated around its
mean (see the remark on page 196 of Whitt (1999a)).
In the third type of announcement, the call center manager specifies a unique tail probability for everybody, say β. The parameter β is a coverage probability based on which we
determine a time x from the waiting time distribution of a newly arriving customer. The
actual waiting time of this customer will be less than that x with probability β. It will be
larger than that x with probability 1 − β. Typically, the announcement information would
be “You will wait less than x minutes”. This third type of delay announcement is more plausible in a real setting, since a typical customer will understand a given delay announcement,
and the service provider is able to control the desired reliability of the announcement by
10
choosing β. Our analysis in this paper is for the third form of announcement with a given
coverage probability.
3.2.2
How Does a Customer React?
The first model we can consider, that captures the change in customer behavior with respect
to delay information, is the model analyzed by Whitt (1999a). Accordingly, a customer balks
with a probability that depends on her anticipated delay. Once she decides to join the queue,
she never abandons thereafter. Replacing all abandonments by balking is quite meaningful
in an environment with perfect delay announcements, as described in Section 3.1, but when
predictions and therefore announcements are not exact and customers know about this, we
can envisage many other things happening. In particular, we expect customers to abandon
the queue, even after having chosen to join initially, and further expect this abandonment
behavior to depend on the announced delay. For example a customer who does not balk,
may perceive the subsequent wait as longer than it really is and lose patience as a result.
Customers may take the announced time like a time guarantee, or time progressing towards
the announced wait as a sign of progress and persist in holding until the announced delay,
but if the wait starts exceeding the announced time they will be frustrated and may lose their
patience as a result. Other customers may not experience any dissatisfaction from uncertain
waits, or may not take the delay announcements by the firm to be credible, and thus may
not change their initial patience behavior. These are only some possibilities, motivated
by observations from the earlier literature, and we can easily envisage these effects being
combined with others to yield customer responses to imperfect delay announcements. What
we conclude from these possibilities is that a model that considers customer abandonments
subsequent to a decision to join the waiting line seems more realistic. We thus consider
a model that incorporates customer reactions both in terms of balking that depends on a
customer’s anticipated delay, and subsequent reneging. Given the complexity of reality, we
opt for a heuristic approach and model the customer reaction in terms of reneging by making
use of several extreme cases.
One possibility is that of a customer who ignores the delay information and does not
update her patience threshold tk . We call this the no-update case. Among other things,
this might be attributed to a form of duration negligence, as discussed and reviewed in
Bitran et al. (2008). Another possibility is a customer who completely updates her patience
threshold and replaces tk by the announced delay. We call this the update case. One possible
explanation for this type of behavior is the presence of an anchoring effect (see for example
Kahneman et al. (1982)) on the announced delay. Given the definition of tk in a world
11
with perfect delay information, we do not expect a delay announcement to affect customer
patience by prolonging it beyond this value. In other words, the patience prolonging effect
of delay announcements that is suggested by the data in Feigin (2005) and by evidence in
Hui and Zhou (1996), is already incorporated in tk . Clearly, other external factors may
prolong a customer’s patience but we do not model these herein, so these possibilities are
not considered. This suggests that the no-update case represents one extreme case. Similarly,
some customers may update their patience threshold to a value that is below the announced
delay. While we recognize this as a possibility, we think such customers will be limited and
can be disregarded in the model. The earlier cited work suggesting an increase in satisfaction
as time progresses and approaches the announced delay subsequent to joining the queue
(Bitran et al. (2008), Munichor and Rafaeli (2007)), lends support to this view. Thus, we
consider the update case as another extreme case. In reality, the most likely scenario is that
of customers that combine these effects, resulting in a patience threshold that lies somewhere
between the announced delay and tk . We propose to model this case by assuming that each
customer has a probability q to be an update type and 1 − q to be a no-update type, and call
the resulting situation the mixed case.
For each of the three cases that we consider in modeling customer behavior (no-update,
update, and mixed), the desired patience behavior is approximated by an exponential distribution. The new reneging experience in the system with delay announcement is characterized
in Section 6.1.
4.
Description of the Models
In Sections 4.1 and 4.2, we formulate the models without and with delay information, respectively. In the first model we consider, we assume that no delay information is given
to customers. After entering the queue, a customer will wait a random period of time for
service to start. If service has not begun by this time she will renege. In the second model,
each customer gets information about her anticipated delay and reacts to that.
We formulate both models based on comparisons with the perfect delay information call
center we introduced in section 3.1. As before, in both models there is a single class of
customers and s statistically identical servers in parallel. We also assume, in both models,
that arrivals follow a Poisson process with rate λ, and service times are exponentially distributed with rate µ. Another similarity is that in all models, we consider the proportion α0
of customers who are not willing to wait, irrespective of available information.
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4.1
The Model without Delay Announcement
The model with perfect delay information represents a setting with no uncertainty about
waiting duration, once the customer has chosen to wait. The model without delay announcement on the other hand represents the other extreme with the highest level of uncertainty
concerning the waiting duration. Given risk-averse behavior exhibited by customers in the
domain of time Leclerc et al. (1995) and based on earlier cited evidence, as uncertainty increases, we expect lower customer satisfaction. Due to this, we consider the proportion of
customers who are unwilling to wait to be higher in a call center with no delay announcement
when compared to an ideal call center with perfect delay information.
Among the customers who find all agents busy, a proportion α0 immediately balks. This
is the base proportion (customers who are unwilling to wait any amount of time). For the
remaining customers (proportion 1−α0 ), there is a proportion α1 who choose to immediately
balk too. We attribute this proportion to the frustration a customer may anticipate in case
she will join an invisible queue with no information about anticipated delays. Even if these
customers were willing to wait some given amount of time, they would feel afraid that
their waiting time will be too long. So, they would prefer to immediately leave the system
upon their arrival instead of experiencing the uncertainty of waiting in queue. In practice,
distinguishing the two proportions (α0 , α1 ) is difficult and we only differentiate between the
two proportions to clarify the comparisons.
The uncertain wait is also expected to affect the resulting patience of the customers that
decide to wait, potentially due to overestimating the delay Katz et al. (1991), or because
uncertainty negatively effects the acceptability of a wait Hui and Zhou (1996). In the model
with perfect delay announcements customers have been modeled as having patience times
that are exponentially distributed with parameter γ. Since delay announcements are exact,
there is no reneging by customers in this model, and all reneging is replaced by balking
subsequent to delay announcement. We model the setting with no delay announcements
as having patience times that are exponentially distributed with parameter γ + γ1 where
γ1 ≥ 0. This allows us to consider the possibility of higher impatience in the case with no
delay information.
While a comparison with a perfect information call center is not possible in practice, the
comparison of hazard function estimates for customers who have and have not heard delay
announcements in a call center (in Figure 12 of Feigin (2005)) seems to lend support to the
idea that initial abandonment (modeled as balking in our setting) is higher and subsequent
patience seems to be lower when there is no announcement.
13
The resulting model referred to as Model 1 is shown in Figure 1.
λα 0
λ
λ (1 − α 0 )α1
λ (1 − α 0 )
γ + γ1
λ (1 − α 0 )(1 − α1 )
s
Figure 1: Model without delay announcement, Model 1
We note that in Model 1 there is no time difference between the two balking decisions.
The only difference is the reason for balking, or in other words the type of balking. Customers
of both types of balking will immediately leave the system. So in Model 1, a customer who
finds all servers busy will balk with probability α0 + (1 − α0 )α1 . She will join the queue
with the complementary probability (1 − α0 )(1 − α1 ). In the latter case, she is willing to
wait in queue only a certain amount of time. If service has not begun by this time she will
renege and be lost. We assume that her patience threshold is a realization derived from an
exponentially distributed random variable with rate γ + γ1 . Finally, we assume that retrials
are ignored, and reneging is not allowed once a customer starts being served. Model 1 can
be viewed as an M/M/s+M queueing system with balking. The symbol M after the + is to
indicate the Markovian assumption for times before reneging. Note that reneging makes the
system unconditionally ergodic for any γ > 0.
4.2
The Model with Delay Announcement
Based on the discussion in Section 3.2, we describe in what follows the call center model in
which we add delay announcement.
Upon arrival, if less than s customers are in system, no action is taken because the new
customer gets service immediately. If all agents are busy, we again model the fact that a
proportion of new arrivals come being unwilling to wait any duration of time. We assume
that each new arrival to a busy system has a probability α0 to immediately balk, before even
hearing his anticipated delay. It is reasonable to assume that the proportion α0 is identical
to that in the model with perfect information and in Model 1 since these customers do not
experience any difference between the two systems.
Contrary to Model 1, we believe that balking stemming from waiting uncertainty (no
information) has no reason to be present here, i.e., there is no α1 parameter. In addition, we
assume that customers have trust in the information given by the call center. All customers
who were initially willing to wait (with probability 1 −α0 ) will give the call center the chance
to provide them with delay information. This information is assumed to be delivered in zero
14
time. These customers may however balk for another reason, namely, they may decide to
balk as a function of the announced delay.
Consider a customer who finds all servers busy and n (n ≥ 0) waiting customers ahead
of her in queue. With probability 1 − α0 , she will accept to hear the information provided
to her about her anticipated delay. If she does so, we derive the distribution of her virtual
delay, say Dn . The quantity Dn is a conditional random variable, given the queue state n.
It is the waiting time in queue for a customer that will wait until service begins (in case she
elects to join the queue). Then, we communicate to her the delay which corresponds to a
given coverage probability β. Let gn (t) be the probability density function (pdf) of Dn , and
let Gn (t) be its Cumulative Distribution Function (CDF), t ≥ 0.
Let T be the random variable measuring the initial random patience threshold of customers, (with probability 1 − α0 : an exponential distribution with rate γ). Let dn be the
delay we communicate to our customer, dn = G−1
n (β). It means that the queueing delay of
the new customer does not exceed dn with a chance β. For that customer, we stipulate that
she decides to balk if her random patience threshold, T , does not exceed her anticipated delay dn . Let pB (n) denote the probability of this event. Next assuming that balking decisions
of successive customers are independent, we get
pB (n) = P (T < dn ) = 1 − e−γdn , n ≥ 0.
(1)
So, given that a customer does not balk with probability α0 , she may balk with probability
pB (n) in response to the delay announcement. Once the new customer elects to wait with
probability 1 − pB (n), she may renege within a random delay. This new random threshold is
statistically different from the initial one and is characterized in Section 6.1 as a function of
the delay announcement made. Again, we assume that retrials are ignored, and reneging is
not allowed once a customer starts being served. The resulting model referred to as Model
2 is shown in Figure 2. The parameter of reneging γ 0 in this figure will be defined later in
Section 6.1.
λα 0
λ
λ (1 − α 0 ) p B (n)
λ (1 − α 0 )
λ (1 − α 0 )(1 − p B (n))
γ'
s
Figure 2: The new model incorporating delay announcement, Model 2
15
5.
Analysis without Delay Announcement
In this section, we derive several performance measures related to Model 1. The approach
here is based on system state probabilities seen by a randomly chosen new arrival. From the
classical PASTA property (Poisson Arrivals See Time Averages), these probabilities coincide
with those seen by an outside random observer, i.e., simply the probabilities that the system
is in a given state at a random instant. The PASTA property is based on the memoryless
property of the Poisson process, which allows to generate a sequence of arrivals that take a
random look at the system. We refer the reader to Kleinrock (1975) for further explanation,
and Wolff (1982) for a rigorous proof. While our approach is similar to that of Whitt (1999a),
the model we consider as well as details of how we derive the performance expressions are
different.
To avoid clutter in the notation, we will use the same notation for Models 1 and 2. Also,
we derive the performance measures while ignoring the parameter γ1 defined in Section 4.1.
In the analysis below, the reneging rate γ might be simply replaced by γ + γ1 if we have
more reneging as explained in Section 4.1.
We denote the system state by a random variable E(t) taking non-negative integer values
representing the number of customers in the system at t ≥ 0. Let p(i) be the steady state
probability that i customers are present in Model 1 at a random instant, i ≥ 0. The process
{E(t), t ≥ 0} is a birth-death process, see Figure 3. When the number of customers present
in system, i, is less than or equal to s−1, all arrivals enter the system. So, the birth rates are
λ. In addition, all departures are service completions. Then, the death rates are iµ, i < s.
Otherwise if i ≥ s, an arrival has a probability α0 + (1 − α0 )α1 to immediately balk. So, the
birth rates are λ(1 − α0 )(1 − α1 ). Departures may be service completions or abandonments.
Thus, the death rates are sµ + (i − s)γ.
λ
0
1
µ
λ
λ
…
(1 −
α0 )
(1 −
sµ
λ
(1 −
α0 )
(1 −
s+1
s
s-1
α1 )
sµ+ γ
α1 )
s+2
…
sµ+2 γ
Figure 3: Birth-death process for Model 1
In the steady state, one has a set of infinite recursive relations relating the steady state
probabilities p(i), i ≥ 0. We go on to solve by iteration and get the following solutions
p(i) =
λi
λi (1 − α0 )i−s (1 − α1 )i−s
p(0) for i > s,
p(0)
for
0
≤
i
≤
s,
and
p(i)
=
Q
i! µi
s!µs i−s
j=1 (s µ + j γ)
16
(2)
where p(0) is the stationary probability to have no customers in system. Using the normalization condition, it follows that
!−1
à s
∞
X λi
X
λi (1 − α0 )i−s (1 − α1 )i−s
p(0) =
.
+
Q
i! µi i=s+1 s! µs i−s
j=1 (s µ + j γ)
i=0
(3)
Equations (2) and (3) enable us to compute all system states’ stationary probabilities
at a random instant. Define now P I as the probability of immediate service for a newly
arriving customer. Applying PASTA, we get
I
P =
s−1
X
p(i).
(4)
i=0
We denote by Lq the expected number of customers waiting in queue, and by Ls the
expected number of customers present in system. They are given by
Lq =
∞
X
ip(s + i),
and Ls =
i=1
∞
X
ip(i).
(5)
i=1
Let us now proceed to compute the probability for a new arrival to balk, say P B , that to
enter service, say P S , and that to renege, say P R .
When a new customer finds less than s customers in the system, she gets service immediately. This is equivalent to finding at least one server idle, and it occurs with the probability
P I given in Equation (4). Let us now consider the complementary event, i.e., assume that a
new customer finds all servers busy. This customer will balk with probability α0 +(1−α0 )α1 .
Applying PASTA we get
¡
¢
P B = (α0 + (1 − α0 )α1 ) 1 − P I .
(6)
Let us compute the stationary probability to renege, P R . This quantity can be viewed as
the stationary proportion of customers who renege, seen at the epoch of a new arrival. Then,
it is the fraction of the stationary mean rate of reneging customers (seen by a new arrival)
over that of arrivals. Applying PASTA, the latter fraction is equal to that seen at a random
instant. Moreover, since times before reneging are i.i.d. and exponentially distributed, the
stationary mean rate of reneging customers is simply γLq . So, P R may be computed as
PR =
γLq
.
λ
(7)
A customer who does not balk upon arrival and does not renege thereafter (in case she
waits in queue) will necessarily enter service. Thus, the stationary probability for a new
arrival to enter service is simply given by P S = 1 − P B − P R .
17
We consider the analysis of queueing delays next. The sojourn time of a customer in queue
will end either as a result of reneging or start of service. Here, we focus on the conditional
waiting time, given that service is completed. We compute any kth order moment, k ≥ 0,
of this conditional random variable, say E(X k | S). To do so, we use a similar approach to
that used in Jouini and Dallery (2007a), which is originally inspired from Whitt (1999a).
Consider a customer who finds all servers busy and n waiting customers in queue, n ≥ 0.
Assume that this customer does not balk, and let us define a pure-death process with statedependent death rates as shown in Figure 4. The process is derived from that in Figure 3.
We only consider states ranging from s to s + n + 1 (the n + s customers already in system
plus the new arrival). We do not consider birth rates because future arrivals have no priority
over the customer of interest (the discipline of service is FCFS).
s
s+1
…
sµ + γ
s+n
sµ + (n+1) γ
s+n+1
Figure 4: Pure death process for the customer s + n + 1
Conditioning on the system state, this new arrival will start her sojourn at position
s + n + 1 in system (position n + 1 in queue). There are two possibilities: the customer of
interest will wait until service begins (with probability Ψn+1 ) or renege before service begins
(with probability 1−Ψn+1 ). The process moves from state i to state i−1, s+1 ≤ i ≤ s+n+1,
in the case of a departure event, i.e., in the case of either a service completion with rate sµ,
or a reneging event with a rate equal to the number of waiting customers times the reneging
rate, (i − s)γ. The memoryless property of service times as well as times before reneging
allows us to state the following claim. When being in state i, the probability that the process
moves down due to the event of reneging of our customer of interest is given by
γ
.
sµ+iγ
Next,
the conditional probability, Ψn+1 , that our customer does not renege while waiting in queue,
given that she finds n waiting customers upon arrival and does not balk, may be written as
Ψn+1 =
n+1
Yµ
i=1
γ
1−
sµ + iγ
¶
.
(8)
In other words, the latter event means that our customer does not renege in all possible
queue positions she will occupy, starting from position n + 1 until position 1 and enters
service afterwards.
Let us now define the conditional random variable Xn+1 , given that our customer finds
the system busy and n waiting customers, n ≥ 0. The random variable Xn+1 denotes the
time it takes to empty the queue of n + 1 waiting customers (without considering eventual
18
future arrivals). Our customer will enter service with probability (1 − α0 )(1 − α1 )Ψn+1
in Xn+1 units of time. Conditioning on a state seen by a new arrival and averaging over
all possibilities, we state using PASTA that the kth order moment of the sojourn time in
P
k
queue and being served afterwards is given by ∞
n=0 (1 − α0 )(1 − α1 )p(s + n)Ψn+1 E(Xn+1 ),
k
where E(Xn+1
) denotes the kth order moment of Xn+1 . Thus, the kth order moment of the
conditional waiting time in queue, given service is completed, is
P∞
k
)
(1 − α0 )(1 − α1 )p(s + n)Ψn+1 E(Xn+1
k
E(X | S) = n=0
.
S
P
(9)
It remains for us to derive the expression of E(Xnk ), n ≥ 1. The random variable Xn
can be viewed as the first passage time to state s starting from state s + n in the pure
death process of Figure 4. Then, the distribution of Xn is the convolution of n independent
exponential distributions with parameters sµ + γ, sµ + 2γ, ..., and sµ + nγ, which is an
hypoexponential distribution, see Ross (1997). So, all moments of Xn may be easily derived
in closed form using for example the Laplace transform of the hypoexponential distribution.
Here, we only give the first two moments of Xn . They are
à n
! Ã n
!2
n
X
X
X
1
1
1
E(Xn ) =
, and E(Xn2 ) =
+
.
2
sµ
+
iγ
(sµ
+
iγ)
sµ
+
iγ
i=1
i=1
i=1
(10)
Finally, the variance and standard deviation of the conditional waiting time given service, are
p
given by V ar(X | S) = E(X 2 | S) − E(X | S)2 and σ(X | S) = V ar(X | S), respectively.
The performance measures we derive in this section will be used later in Section 7 for the
numerical comparison between the models with and without announcement.
6.
Analysis with Delay Announcement
In this section, we consider the analysis of the model with delay information, Model 2. We
first characterize in Section 6.1 the new patience behavior of customers as a function of the
delay announcement made. Performance measures of Model 2 are then derived in Section
6.2. In Section 6.3, some results about the relationship between announcement coverage and
customer related performance measures are investigated.
6.1
Characterization of the New Reneging Behavior
For a customer waiting in queue in Model 2, we assume that her random patience threshold is
exponentially distributed with rate γ 0 , γ 0 6= γ. While we need the exponentiality assumption
(which is an approximation) for an exact analysis of the corresponding queue to be feasible,
19
we argue later in this section (after Equation (16)) that the same procedure can be applied
as a good approximation in the case of some general impatience distributions. A plausible
alternative to the exponential distribution with rate γ 0 is a delayed exponential distribution
where each customer waits for a given delay and then reneges with an exponential rate. In
the appendix, we present simulation experiments that give further support regarding the
quality of the exponential approximation.
In what follows, we compute the distribution of the virtual delay Dn as a function of
γ 0 . The time until our new arrival is scheduled to start service is the time it takes for
the n customers ahead to leave the queue (either abandon or enter service) plus the time
required for a service completion (when all servers are busy). So, the virtual delay Dn can
be characterized by the pure-death process as shown in Figure 5.
s-1
s
…
s+n
s+n-1
sµ+
sµ
n γ’
Figure 5: The random variable Dn
The random variable Dn is the downcrossing time from state n + s until absorption in
state s−1. It represents the time it takes to empty the queue for our customer, plus a service
completion. Thus, the distribution of Dn is the convolution of n + 1 independent exponential
distributions with parameters sµ, sµ + γ 0 , ..., and sµ + nγ 0 , which is an hypoexponential
distribution. From Ross (1997), the pdf and CDF of Dn are given by
!
à n
n
X
Y sµ + jγ 0
0
· (sµ + iγ 0 ) · e−(sµ+iγ )t ,
gn (t) =
0
(j − i)γ
i=0
j=0, j6=i
Gn (t) = 1 −
n
X
i=0
Ã
n
Y
sµ + jγ 0
(j − i)γ 0
j=0, j6=i
(11)
!
0
· e−(sµ+iγ )t ,
t ≥ 0.
(12)
(Note that by convention, an empty product is equal to 1).
To answer the question of how γ 0 should be computed, it is natural to relate the new
customer patience to her original one and to the announced delay. To do so, we define for each
new arrival finding n customers in queue, the conditional probability that this customer will
renege, given that she does not balk after hearing her anticipated delay. Recall that we will
consider three situations in our analysis: the no-update case where we denote this conditional
probability by rnN , and the update case where we denote it by rnU , and the mixed case where
we denote it by rnM . The no-update case is one where the patience threshold of our customer
(denoted by customer k) does not change, that is, she was willing to wait up to a duration
20
tk and she still has that threshold even after being informed about her anticipated delay.
The update case is one where the delay information affects the customer’s initial patience,
and the delay value dn we announce to her substitutes her initial patience threshold tk . The
mixed case combines the behavior in the other two cases, and is analyzed in the appendix.
We show there that the resulting conditional probability that the customer will renege, rnM ,
will lie between the corresponding probabilities of the update and no-update cases.
For the no-update case, the quantity rnN is the conditional probability that the queueing
delay, Dn , exceeds the random initial patience, T , given that our customer elects to join the
queue, i.e., given that the random patience exceeds the announced delay, dn . It is given by
rnN = P (T < Dn | T ≥ dn ).
(13)
For the update case, the quantity rnU is the conditional probability that the queueing delay,
Dn , exceeds the announced value, dn , given that our customer elects to join the queue, i.e.,
given that the random patience exceeds the announced delay, dn . It is given by
rnU = P (dn < Dn | T ≥ dn ).
(14)
With a little thought, it should be clear that rnU is equal to 1−β, which is the probability that
the call center misinforms the customer. We will present necessary details for calculating
both rnN and rnU in Section 6.2, and will prove that rnN is lower than rnU . Intuitively, this
can be seen as follows: Whenever we announce dn based on a given coverage probability β,
we shall make at worst a mistake with a chance of 1 − β. Consider a new arrival finding n
customers in queue. Assume that the customer is willing to wait tk for service to begin. The
duration tk is a random realization of the random variable T . If tk > dn , then the customer
joins the waiting line. After joining the queue, the probability that the time it takes for a
server to become free (for the customer of interest) exceeds dn is 1 − β. Since the customer
is initially willing to wait up to tk , hence knowing that tk > dn , the probability that the
duration tk passes before a server becomes free for our customer is less than 1 − β.
In what follows, we give a method to compute the new reneging rate γ 0 for both cases.
For presentation issue, we will differentiate between the notations of the quantities rnN and
rnU only if necessary. Otherwise, i.e., if what we are considering holds for both cases, we will
denote both of them by simply rn . Assume we reach the stationary regime, and let λR be
the mean rate of abandoning customers. Applying PASTA, this quantity equals that seen
by a new arrival. We get, by averaging on all possibilities,
λR =
∞
X
λ(1 − α0 )(1 − pB (n))p(s + n)rn ,
n=0
21
(15)
where p(n), n ≥ 0, are the stationary probabilities of the number of customers in system
seen at a random instant. We will compute these quantities in Section 6.2.
On the other hand, if we denote by Lq the expected number of customers in queue, we
can write using the exponentiality of customer impatience
λR = γ 0 Lq .
(16)
We shall give the expression of Lq in Section 6.2. Note that Equation (16) implies a linear
relationship between the reneging probability and average waiting time in queue, which is
exact in the case of an exponential patience distribution as assumed here. While this relationship does not hold theoretically for general patience time distributions, Mandelbaum and
Zeltyn (2004) show that practically this linearity is preserved for many general distributions,
including the delayed exponential, under moderate loads. This suggests that our approach
to calculate γ 0 provides a good approximation under some non-exponential patience times.
Combining Equations (15) and (16) we get
γ0 =
∞
λ X
(1 − α0 )(1 − pB (n))p(s + n)rn .
Lq n=0
(17)
The quantities Lq , p(s + n), pB (n) and rn are functions of γ 0 . So, denoting the right hand
side in Equation (16) by a continuous function f in γ 0 , we may write γ 0 = f (γ 0 ). As a
consequence, we state that γ 0 is a point mapped to itself by the function f . In mathematical
terms, γ 0 is said to be a fixed point of f . To numerically compute γ 0 , we propose the following
fixed point algorithm.
Fixed point algorithm()
Initialization:
γ 0(0) ← γ, i ← 0, ²
Do
i←i+1
P
B
0(i−1)
)) · p(s + n)(γ 0(i−1) ) · rn (γ 0(i−1) )
λR (i) ← ∞
n=0 λ · (1 − α0 ) · (1 − p (n)(γ
P
(i)
0(i−1)
)
Lq ← ∞
n=1 n · p(s + n)(γ
(i)
γ 0(i) ← λR (i) /Lq
While | γ 0(i) − γ 0(i−1) |> ²
γ 0 ← γ 0(i)
End Algorithm.
We note that in the algorithm above, p(s + n)(γ 0(i−1) ) is a functional form with γ 0(i−1) as
the argument. We do not prove the convergence and uniqueness of the solution of the fixed
22
point algorithm, but a large numerical experience with the algorithm is encouraging in this
respect.
Clearly the reneging parameter γ 0 depends on the choice of β. On the one hand if β
increases, the announced delays will increase. Then, for both the no-update and update
cases, the set of customers that will elect to wait is smaller, by excluding the customers with
low patience thresholds. As β increases, the rate γ 0 will thus decrease. This is confirmed
later in the numerical experiments in the appendix.
In the particular case, β = 0, one may see that Model 2 under the no-update case coincides
with Model 1 in terms of the distribution of times before reneging (γ 0 = γ). Consider Model
2 under the no-update case and let β = 0. Then, we will announce dn = 0 for any new arrival
(finding a busy system and n waiting customers in queue, n ≥ 0). No customer will balk,
i.e. pB (n) = P (T < dn ) = 0. In addition, a new customer who chooses to join the queue
has a probability rnN = P (T < Dn | T ≥ dn ) = P (T < Dn ). In this particular case, the
expression of the mean abandoning rate λR , see Equation (15), coincides with that in the
case of an exponential patience with parameter γ. We then deduce that γ is the fixed point
of f (solution of γ 0 = f (γ 0 )). In words, it means that we are accepting all arrivals in Model
2. Since these arrivals will keep their initial patience thresholds, we exactly reproduce the
initial reneging distribution (exponential with rate γ).
6.2
Performance Analysis
We start by defining the birth-death process as shown in Figure 6. Birth and death rates
are both state-dependent. The new element here is that we have to take into account the
state-dependent balking decisions when the process moves from state i to state i + 1, for
i ≥ s, i.e., all servers are busy.
λ
µ
λ
λ
1
0
…
(1 − α ) (1 − p
0
s
s-1
sµ
B
λ
(0))
(1 − α ) (1 −
0
B
p (1))
s+2
s+1
sµ+ γ’
…
sµ+2 γ’
Figure 6: Birth-death process for Model 2
In the stationary regime, we get the following steady state probabilities seen at a random
instant
λi
λi (1 − α0 )i−s
p(i) =
p(0),
for
0
≤
i
≤
s,
and
p(i)
=
i! µi
s!µs
à i−s
!
Y 1 − pB (j − 1)
j=1
s µ + j γ0
p(0), for i > s,
(18)
23
where p(0) is the stationary probability to have no customers in the system. It is given by
à s
à i−s
!!−1
∞
X λi
X
λi (1 − α0 )i−s Y 1 − pB (j − 1)
p(0) =
+
.
(19)
i
s
0
i!
µ
s!
µ
s
µ
+
j
γ
j=1
i=0
i=s+1
Hence, the probability of immediate service, P I , the mean number of customers in queue,
Lq , and the mean number of customers in system, Ls , can be calculated as in Relation (20).
I
P =
s−1
X
p(i),
Lq =
∞
X
i=0
ip(s + i),
and Ls =
i=1
∞
X
ip(i).
(20)
i=1
Having the expression for Lq , it only remains for us to compute the quantity rn so that we
can apply the fixed point algorithm, shown in Section 6.1, to get γ 0 . In what follows, we give
closed-form expressions of rn in the update and no-update cases.
Let us start by the update case, rnU . It is the simpler case. Equation (14) may be rewritten
as
P (Dn < dn and T ≥ dn )
.
(21)
P (T ≥ dn )
Since the random variables T and Dn are independent, we have P (Dn < dn and T ≥ dn ) =
rnU =
P (Dn < dn ) · P (T ≥ dn ). Thus
rnU = 1 − β,
(22)
which agrees with intuition as explained in Section 6.1.
For the no-update case, we use Equation (13) to rewrite rnN as follows.
rnN =
P (dn ≤ T < Dn )
.
P (T ≥ dn )
(23)
Since T is exponentially distributed with rate γ, the denominator in the right hand side of
Equation (23) is simply
P (T ≥ dn ) = e−γdn .
(24)
As for the numerator, it is given by
Z ∞
Z
P (dn ≤ T < Dn ) =
gn (t) · P (dn ≤ T < t) dt =
dn
∞
¡
¢
gn (t) · e−γdn − e−γt dt.
(25)
gn (t) · e−γt dt.
(26)
dn
Calculating further, we get
Z
−γdn
P (dn ≤ T < Dn ) = (1 − β) · e
∞
−
dn
Next, observing that
à n
Z ∞
n
X
Y
gn (t) · e−γt dt =
dn
!
Z ∞
sµ + jγ 0
0
0
· (sµ + iγ ) ·
e−(sµ+γ+iγ )t dt
0
(j − i)γ
dn
i=0
j=0, j6=i
Ã
!
n
n
X
Y
sµ + jγ 0
sµ + iγ 0
−(sµ+γ+iγ 0 )dn
=
·
·
e
,
0
0
(j
−
i)γ
sµ
+
γ
+
iγ
i=0
j=0, j6=i
24
(27)
and coming back to Equation (23), we finally state that
!
à n
n
X
Y sµ + jγ 0
sµ + iγ 0
0
rnN = 1 − β −
·
· e−(sµ+iγ )dn .
0
0
(j − i)γ
sµ + γ + iγ
i=0
j=0, j6=i
(28)
Also, we show through Proposition 1 our intuitive claim about the comparison between
the probabilities rnN and rnU . All proofs can be found in the Appendix.
Proposition 1 For β ∈ [0, 1], the conditional probability to renege in queue given the system
state under no-update, rnN , is lower or equal to that under update, rnU . The equality only
holds for β = 1. In the latter case, we have rnN = rnU = 0.
As a conclusion, a system with announcement in which the original patience times of customers do not change (no-update) leads to less reneging than that where customers substitute
their original patience times by the delays we announce to them (update). A mixture of these
two behaviors, as modeled by the mixed case leads to an intermediate level of reneging.
We want to note here that under the no-update case there is really no change in customer
behavior, although the new patience distribution is different from the initial one. Customers
who self-select to wait have higher thresholds of patience. These thresholds are identical
to the original ones. So, times before reneging in the no-update case are simply sampled
from a subset of the original ones. Assuming further that the new times before reneging
are exponentially distributed, we conclude that γ 0 < γ. This will be confirmed later in the
numerical experiments.
The probability to balk, to enter service, and to renege are P B , P S , and P R , respectively.
A customer who finds at least one server idle will immediately start service (balks with
probability 0). If she finds all servers busy and n waiting customers, n ≥ 0, she balks with
probability α0 because she is not willing to wait any duration of time. If not, she will balk
with probability pB (n). Then, a customer who finds all servers busy and n waiting customers,
will balk with probability α0 + (1 − α0 )pB (n). Averaging on all possibilities and applying
PASTA, we get
P
B
∞
X
=
(α0 + (1 − α0 )pB (n))p(s + n).
(29)
n=0
In the same manner as in Section 5, the stationary probability to renege is given by
PR =
γ 0 Lq
.
λ
(30)
Finally, we remark that a customer who does not balk and does not renege will necessarily
enter service. Hence, the stationary probability to enter service is P S = 1 − P B − P R .
25
We carry on computing the performance measures of interest, namely the analysis of the
conditional waiting time in queue, given that service is completed. We denote by E(X k | S)
the kth order moment of that random variable. Following again a similar analysis as that in
Section 5, we have
P∞
k
E(X | S) =
n=0 (1
k
− α0 )(1 − pB (n))p(s + n)Ψn+1 E(Xn+1
)
,
S
P
(31)
where Ψn+1 is the probability that a new arrival who joins the queue and finds n waiting
customers ahead of her in queue does not renege until the start of service. It is given by
Ψn+1 =
n+1
Yµ
i=1
γ0
1−
sµ + iγ 0
¶
,
(32)
and Xn , n ≥ 1, denotes the time it takes to empty the queue of n waiting customers (without
considering eventual future arrivals), and E(Xnk ) denotes its kth order moment, k ≥ 1. As in
Model 1, the distribution of Xn is the convolution of n independent exponential distributions
with parameters sµ + γ 0 , sµ + 2γ 0 , ..., and sµ + nγ 0 . The expected value and the second
moment of Xn are respectively given by the same expressions as in 10 where γ is replaced
by γ 0 .
6.3
Stochastic Comparison
Understanding the relationship between announcement coverage and customer related performance measures is important. We next provide a stochastic comparison result, that allows
us to state the direction of this relationship.
Proposition 2 Consider Model 2 under any case, update or no-update. For two systems
a and b having identical parameters but with different announcement coverage βa > βb , we
have PaB > PbB and PaR < PbR .
In words, increasing the announcement coverage reduces the reneging of customers at the
expense of additional balking. The numerical analysis below will help us understand the role
the announcement coverage β plays on performance further, and compare to performance
under Model 1 when no delay announcements are made.
7.
Numerical Analysis
In this section we analyze the sensitivity of the optimal announcement coverage to various
system parameters and compare the performance of the models with and without delay
26
information. The parameters of Model 1 are λ, s, µ, α0 , α1 , γ and γ1 , while those of Model
2 are λ, s, µ, β, α0 and γ. Throughout this section, we set α0 to 0. In other words, we
consider numerical examples where the initial balking probability α0 is negligible. We expect
this choice to have no effect on the results below in the qualitative sense. As a consequence,
we only consider the second type of balking. Also in Model 1, we do not consider the
possibility of higher impatience, i.e., γ1 = 0. We choose to capture customers’ dislike for
uncertainty, in case of a system with no information about anticipated delays, only by a
higher initial abandonment (modeled as balking in our setting).
To analyze and compare Models 1 and 2, we build two frameworks of comparison. The
first is based on system throughput and is investigated in Section 7.1. The second is based
on an economic model and is presented in Section 7.2.
7.1
Flow Framework
In what follows, we compare the two models using a flow framework. We compare the
throughput, or equivalently the probability to enter service. A coherent comparison requires
computation of the optimal announcement coverage, β ∗ , that maximizes the probability to
enter service, P2S in Model 2.
Let us first consider Model 2 under the no-update case. Consider two systems. The
first is Model 2 with β = 0 (all customers join the system) and the second is Model 2 with
β > 0. Then more customers are accepted in the first system because some customers will
immediately balk in the second one. In the first system, some of these customers (who have
balked in the second system) may not renege (due to uncertainty the system may be available
to serve them before they would renege). So, more customers enter service in the first system
than in the second. In Model 2, β = 0 maximizes P2S , β ∗ = 0. As already mentioned at
the end of Section 6.1, the threshold patience of a customer who elects to wait (in Model
2 under the no-update case and with β = 0) is exponentially distributed with rate γ. As a
conclusion, Models 1 and 2 coincide for α1 = 0 and Model 2 performs better than Model 1
for any α1 > 0.
For a more interesting comparison, we next focus for Model 2 on the customer reaction
following the update case. If β is small, customers tend to join the queue but they will
renege very quickly after. So, P2S will be very low. In the extreme case (β approaches 0),
only customers who find an idle system will be served. On the other hand if β is large, only a
few customers will elect to join the queue. Again in this second extreme case (β approaches
100%), only customers who find an idle system will be served. As a consequence, there
27
should an optimal β in between, β ∗ ∈ (0, 100%). In other words, β ∗ is the tradeoff between
two situations. In the first situation, we provide low anticipated delays to the detriment
of having a lot of reneging. In the second situation, we decrease the reneging: we ensure
high coverage probability by providing high values of anticipated delays to the detriment of
having a lot of immediate balking.
Under a flow framework, we ask whether we really need to provide information about
delays. We consider Model 2 working under the optimal value of β. We vary the balking
parameter α1 in Model 1 from 0% to 100%, and compute the associated probability of being
served P1S . It is not surprising that P1S is continuous (from Equations (6) and (7), and
knowing that P1S = 1 − P1B − P1R ) and strictly decreasing in α1 . The maximum of P1S is
reached for α1 = 0, and its minimum is reached for α1 = 100% (all customers finding a
busy system will balk, i.e., Model 1 have no queue). Furthermore for α1 = 0, P1S > P2S
for any β > 0 because we accept fewer customers in Model 2. For β < 100%, P2S > 0. As
a consequence, there exists a unique threshold value α10 belonging to (0, 100%) such that
any value lower or equal to that value will imply P1S ≥ P2S , and above that value will lead
to P1S < P2S . In other words, α10 is the threshold below which Model 1 has not that much
balking. For this range of values, one would prefer not to announce delays because Model 2
performs worse.
For Models 1 and 2, we consider six systems with increasing levels of pooling. The
common parameters are (s, λ) = (3,3), (5,5), (10,10), (20,20), (50,50), and (100,100). For
all cases, µ = 1, γ = 0.5, α0 = 0, and γ1 = 0 (for Model 1). We purposely choose highly
loaded systems in order to see the effect of reneging. For each set of parameters, we consider
Model 2 working under the optimal announcement coverage β ∗ , found numerically, given in
percentages and rounded up to integer values. The results of the comparison are presented
in Table 1. For each example, we give for Model 1 the corresponding threshold value α10 ,
found also numerically, given in percentages and rounded up to integer values.
Table 1: Comparison of throughput
s, λ = 3
s, λ = 5
s, λ = 10
s, λ = 20
s, λ = 50
s, λ = 100
Model 1
α10
P1S
31% 75.110%
24% 80.630%
17% 86.260%
11% 90.460%
6% 94.140%
4% 95.899%
Model 2
β∗
P2S
83% 75.241%
89% 80.637%
94% 86.360%
97% 90.503%
99% 94.155%
99% 95.939%
From Table 1, we see that α10 is relatively small for large systems. The reason is that large
28
systems have good performance due to the pooling effect. By increasing pooling in Model
2, we accept more customers because waiting times and particulary their variances become
shorter and shorter. The latter allows us, indeed, to provide higher coverage probability in
our announcements while improving at the same time the probability to enter service. This
explains the increase of β ∗ in the system size. The minimum value of the balking probability
α10 in Model 1, that makes Model 2 better in terms of throughput, decreases as a consequence
in the system size.
These examples illustrate that from a flow perspective, managers of small call centers may
prefer to have some balking, and if the balking is sufficiently low, may prefer not to announce
anticipated delays to customers. On the other hand, they would prefer large call centers with
delay information. The latter has less lost customers than that without delay information
under realistically achievable non-zero immediate balking probabilities in settings with no
announcement.
7.2
Economic Framework
In this section, we use a more robust framework of comparison by considering the loss of
goodwill costs. We define an economic framework in which we introduce different penalty
costs for lost customers, those who balk and those who renege, and differentiate between
these based on whether the customers were being informed about delays or not. The penalty
costs can be seen as a valuation of the customer loss of goodwill (and potentially loss of
business) generated by balking and reneging customers.
Consider a customer in Model 1 who is not immediately served, and let us define the
cost parameters that the service provider may incur. This customer may balk because of
non immediate service (with probability α0 ). The service provider then incurs cB,0
1 . If not,
she may balk because of no delay announcement (with probability α1 ). The service provider
R
then incurs cB,1
1 . If not, she may renege while queueing. The service provider then incurs c1 .
The second type of balking, i.e., due to no delay announcement in Model 1, should be
more costly than the first one due to non immediate service. Lost customers under the second
type of balking decide to leave the system because they anticipate that they will experience
a long delay of uncertain duration before entering service. At the epoch of their call, they
desire to obtain the service and are willing to wait for it, albeit not for long and possibly
not for an unknown duration. This is different from lost customers’ perception under the
first type of balking. The latter have no idea about the system state with regard to its
availability for them, or do not consider this in their decision. They are simply unwilling to
29
wait at the epoch of their call. It is the difference in initial willingness to wait (irrespective
of subsequent delays) that distinguishes the two types of balking. We assume cB,0
< cB,1
1
1
since the second type of customer is foregoing more by balking. While this assumption is
consistent with our distinction between α0 and α1 , it remains to be tested and verified in
practice.
A customer who balks (under any type of balking) in Model 1 should have a higher
probability to call back than that of a customer who reneges. A reneging customer leaves
the system with frustration due to the time lost, potentially also losing trust in the service
provider. However, a balking customer leaves the system without waiting. This customer
may think that the system will be able to serve her later within a reasonable delay, whereas
the customer who leaves the system by reneging has already experienced a delay perceived
to be too long. Consistent with these observations, we assume that cB,0
< cB,1
< cR
1
1
1.
The second inequality implies that customers attach an opportunity cost to the time spent
waiting before reneging, which is lost once they renege. While a consistent valuation of time
by customers has been challenged in the literature (as reviewed by Bitran et al. (2008), page
67), our qualitative results which only depend on the direction of the relationship and not
on the absolute values of these costs should not be affected by these findings.
Similarly for Model 2, we pay cB,0
for a customer who balks because of non immediate
2
service. If she balks based on the delay announcement we provide her, we pay cB,1
2 . If she
reneges while queueing, we pay cR
2 . With the same reasoning as that for Model 1, the second
type of balking in Model 2 (based on delay information) should be more costly than the first
one (with probability α0 ), cB,0
< cB,1
2
2 . Also in Model 2, a reneging customer should cost
more than a balking customer (any type of balking). A balking customer leaves the system
based on information provided. This information would avoid loss of business because it is
perceived as an invitation to call back later. A reneging customer leaves because the wait
is perceived as too long, possibly because the announced delay has been exceeded. That
such a customer should feel less satisfaction is consistent with evidence in Katz et al. (1991)
and Kumar et al. (1997). So, we assume cB,0
< cB,1
< cR
2
2
2 . In Figures 7 and 8, we give a
presentation of the cost parameters in Models 1 and 2, respectively.
Let us now compare the two models for each type of cost. In a real setting, customers
who are unwilling to wait any amount of time (first type of balking with probability α0 in
both models) should have very similar loss of goodwill when being lost. Their experience
will not lead them to distinguish between Model 1 and 2. Hence, it should be plausible to
B,1
choose cB,0
= cB,0
should
1
2 . By focusing on the second type of balking, one may see that c1
B,1
be higher than cB,1
> cB,1
2 , c1
2 . The reason is basically related to a higher uncertainty
30
Immediate service (IS)
Immediate service (IS)
Non IS
Balking because of non IS (α0), c1B,0
Non IS
Balking because of no delay
announcement (α1), c1B,1
Balking because of non IS (α0), c2B,0
Balking because of delay
announcement, c2B,1
Reneging while queueing, c1R
Reneging while queueing, c2R
Being served
Being served
Figure 7: Costs in Model 1
Figure 8: Costs in Model 2
(or lack of information) in Model 1 when comparing this type of lost customers to that in
Model 2 (see for example Katz et al. (1991), Hui and Zhou (1996)). However, consistent
with Kumar et al. (1997), a customer who reneges in Model 2 would be unhappier than that
in Model 1 because she has been misinformed about her queueing delay. To reflect this, we
R
assume cR
1 < c2 . In Table 2, we summarize the comparison between the costs parameters
we described above.
Table 2: Ordering relations between the cost parameters of Models 1 and 2
Model 1
Model 2
Balking
cB,0
< cB,1
1
1W <
k
B,0
c2 < cB,1
<
2
Reneging
R
cV
1
cR
2
Recall that we set α0 to 0, see Section 7.1. For both models, costs related to the first
type of balking are nonexistent. For presentation issues, we then substitute the notations
B
by cB
and cB,1
cB,1
2
1
1 and c2 , respectively. Also we set α1 to 10%.
In the following analysis, we consider Model 2 under the no-update case and investigate
four issues. The first three only concern Model 2: the effect of the announcement coverage
β on the total loss of goodwill cost of the call center (balking + reneging costs) leading
to the determination of the optimal announcement coverage, β ∗ , which minimizes the total
operating cost. The second issue is the effect of the assumed cost parameters on β ∗ . The
third one is the effect of pooling on the performances of Model 2 and on β ∗ . The last issue
deals with the comparison between Models 1 and 2, where we assume that Model 2 is being
operated under the optimal announcement coverage.
Effect of β on total cost: This section numerically illustrates the stochastic comparison
result in Section 6.3, and introduces the notion of an optimal announcement coverage β ∗
under an economic framework.
31
As we will see, increasing β is not always better. So, there is a need to control this
parameter. Let s = λ = 5, the service rate be µ = 1, and the abandonment rate be γ = 0.5.
We label this example as the base case. We vary the announcement coverage β from 50%
up to 99%. For each β, we build up the model with announcement by computing the new
reneging rate γ 0 , as explained in Section 6.1. We assume that the call center incurs cB
2 = 1
for each customer who balks, and three times the latter cost for each customer who reneges
cR
2 = 3.
We compute for each β the stationary probabilities of balking and reneging. We also
compute the stationary cost (per unit of time) of balking, that of reneging and the total
B
one, say C2B , C2R and C2T , respectively. These quantities are calculated as C2B = cB
2 λP ,
R
C2R = cR
and C2T = C2B + C2R . In Figure 9, we plot the curves of the different costs as a
2 λP
function of β.
Costs for Model 2
1.40
1.20
1.00
0.80
B
C2
C(BK)
0.60
C(R)
C2
R
T
C(TOT)
C
2
0.40
0.20
beta
β
0.00
50%
60%
70%
80%
90%
100%
Figure 9: Costs for Model 2 as a function of β, s = λ = 4.5
As shown in Proposition 2, we see from Figure 9 that P B (P R ) is strictly increasing
(decreasing) in β. A better announcement coverage leads to more balking upon arrival and
less reneging for customers who elect to join the queue. The total cost is unimodal and its
minimum is reached at β ∗ = 83% with C2T ∗ = 0.916.
Effect of the costs parameters on the optimal announcement coverage β ∗ : In
Figure 10, we plot for Model 2 the total cost as a function of β, for different cost parameters.
R
We keep the balking cost unchanged, cB
2 = 1, and vary the reneging cost, c2 = 1.25, 1.5, 2,
3, 5 and 10. From Figure 10, we see that the optimal announcement coverage is increasing
in the reneging cost, varying between 75% and 95%. The balking cost does not change
since cB
2 is held constant. However the reneging cost and its relative weight in the total
cost increases in cR
2 . The higher the cost of reneging, the more the announcement coverage
should be increased. The analysis illustrates how β controls the tradeoff between balking
32
and reneging, and points out the importance of controlling this parameter as a function of
customer characteristics (captured through costs herein).
1,75
Total Cost for Model 2
c 2 =1.25, βbeta*=75%
* =75%
C'2/C2=1.25,
R
c 2 =1.5, βbeta*=80%
* =80%
C'2/C2=1.5,
R
1,50
c 2 =2, βbeta*=81%
* =81%
C'2/C2=2,
R
* =83%
c 2 =3, βbeta*=83%
C'2/C2=3,
R
R
β
c 2 =5, beta*=84%
* =84%
C'2/C2=5,
1,25
R
c 2 =10, β * =95%
C'2/C2=10, beta*=95%
1,00
0,75
70%
β
beta
80%
90%
100%
Figure 10: Total costs for Model 2 as a function of β, s = λ = 5
Effect of the system size on the optimal announcement coverage β ∗ : We again
consider (as in Section 7.1) six systems with increasing size (s, λ) = (3,3), (5,5), (10,10),
R
(20,20), (50,50), and (100,100). For all cases, µ = 1, γ = 0.5, α0 = 0, cB
2 = 1 and c2 = 3.
Recall that α1 = 10% and γ1 = 0. For each set of parameters, we consider the model corresponding to the optimal announcement coverage β ∗ , found numerically, given in percentages
and rounded up to integer values.
The results are displayed in Tables 3 and 4. We compute various performance measures
for Model 2 but also for the corresponding Model 1, which will be used in the subsequent
comparison of the two models. For simplicity, we choose equal costs for the second type of
B
balking customers, i.e., cB
1 = c2 . Note that this plays in favor of Model 1.
Observe that the optimal announcement coverage is increasing in system size. Even
though β ∗ is increasing, the delay that is announced with n customers waiting in queue decreases in system size. This is shown in Figure 11, where we use the virtual delay distribution
in Section 6.1 with the corresponding γ 0 and β ∗ from Tables 3 and 4 to obtain the delay announcement for a given queue size n. We observe that as the number of servers increases, β ∗
approaches 100% with both the probability of balking and reneging going to zero. However
the relative decrease in the reneging probability (when moving from a system to a larger
one) is greater than that in the balking probability. The shorter waiting times coupled with
higher coverage probability in the announcements as we pool, ensure that fewer customers
exceed the announced delays, thus reducing the probability of reneging. The resulting γ 0 is
decreasing. Balking is also affected by shorter delay announcements, but does not benefit
from the increased coverage probability since customers who balk do not elect to wait in
the first place. In fact as β ∗ increases the decrease in the announced delay is much slower
33
(as seen from the shape of the curves in Figure 11) so that the relative benefit in terms of
reduced balking is also slower. Due to this relative difference, the minimum cost is reached
for a higher value of β that favors a faster decrease in reneging probability as system size
B
increases. The increase in β ∗ as a function of system size would be higher for lower cR
2 /c2
ratios.
Table 3: (s, λ) = (3,3), (5,5) and (10,10)
PI
Lq
Ls
PS
PR
PB
E(X | S)
σ(X | S)
CB
CR
CT
Decrease in C T
s = 3, λ = 3
β ∗ = 82%, γ 0 = 0.033
Model 1
Model 2
0.393
0.431
0.884
0.619
3.260
2.935
0.792
0.772
0.147
0.007
0.061
0.221
0.283
0.263
0.444
0.471
0.182
0.664
0.884
0.061
1.066
0.725
—
31.996%
s = 5, λ = 5
β ∗ = 83%, γ 0 = 0.024
Model 1
Model 2
0.419
0.439
1.089
0.881
5.254
5.006
0.833
0.825
0.109
0.004
0.058
0.171
0.214
0.211
0.336
0.359
0.290
0.853
1.089
0.063
1.379
0.916
—
33.552%
s = 10, λ = 10
β ∗ = 83%, γ 0 = 0.020
Model 1
Model 2
0.456
0.443
1.400
1.388
10.156
10.178
0.876
0.879
0.070
0.003
0.054
0.118
0.140
0.157
0.225
0.250
0.544
1.182
1.400
0.083
1.945
1.265
—
34.943%
Table 4: (s, λ) = (20,20), (50,50) and (100,100)
PI
Lq
Ls
PS
PR
PB
E(X | S)
σ(X | S)
CB
CR
CT
Decrease in C T
s = 20, λ = 20
β ∗ = 84%, γ 0 = 0.018
Model 1
Model 2
0.496
0.446
1.727
2.101
19.855
20.418
0.906
0.916
0.043
0.002
0.050
0.082
0.087
0.114
0.146
0.174
1.009
1.647
1.727
0.111
2.736
1.758
—
35.753%
s = 50, λ = 50
β ∗ = 86%, γ 0 = 0.012
Model 1
Model 2
0.557
0.447
2.098
3.545
48.836
50.931
0.935
0.948
0.021
0.001
0.044
0.051
0.043
0.075
0.078
0.110
2.213
2.572
2.098
0.126
4.311
2.698
—
37.417%
s = 100, λ = 100
β ∗ = 86%, γ 0 = 0.010
Model 1
Model 2
0.612
0.446
2.248
5.215
97.239
101.563
0.950
0.963
0.011
0.0005
0.039
0.036
0.023
0.054
0.046
0.078
3.885
3.604
2.248
0.142
6.133
3.746
—
38.910%
Comparison between Models 1 and 2: We again consider Tables 3 and 4, and compare
Models 1 and 2. The latter is assumed to be operating under the optimal announcement
coverage. The rows C B , C R and C T are to indicate the costs for balking, reneging and the
total one, respectively. The last row gives the relative decrease in the total cost in going from
Model 1 to Model 2,
C1T −C2T
C1T
. From the numerical experiments, we see that all performance
34
Figure 11: Delay announcements as a function of s, for given n
measures are improving as a function of system size for both models. This is the well known
pooling effect in queues.
Since the optimal β ∗ for our examples here are quite high and α1 is not that high, we
have more balking in Model 2 in all cases except for s = λ = 100. In the latter, waiting times
are low because of the pooling effect. Then, balking as a response to delay announcement
(in Model 2) is low so that the balking related to α1 (in Model 1) becomes relatively more
important. We also see that the probability of reneging is lower for Model 2 throughout.
The reason is that Model 2, as it were, “refuses” entry for customers who potentially may
renege.
If α1 were chosen to be 0, one would see for all system sizes, but especially for small
systems, that the probability of service is a bit better in Model 1. The reason is that
some of the customers who balk in Model 2 might wait in Model 1 until service begins,
without reneging. Furthermore, the conditional waiting time would be better in Model 2
(less customers are served in Model 2). For our numerical examples here (α1 = 10%), we
see that for small call centers, the probability of service (P S ) is better for Model 1. For
large call centers, the opposite is true. The reason is that balking is high in Model 2 for
small call centers. As call centers size grows, waiting times improve. So, balking decreases
in Model 2 because the delays we are announcing are decreasing. The conclusions about
the comparison of conditional waiting times are the opposite to those of the probability of
service. This is expected since when the probability of service is the best in one model, then
more customers are present in the latter (in the statistical sense). While having the same
capacity, this model will perform the worst in terms of waiting times before being served.
For the cost parameters chosen here, we see that the balking cost is lower in Model 1
for all cases except for the case s = λ = 100. The explanation is the same as that for the
comparison between balking probabilities (same balking cost for both models). Although the
35
3.00
Costs for Model 2, update
2.50
B
C(BK)
C2
R
C(R)
C2
2.00
T
C(TOT)
C
2
1.50
1.00
0.50
0.00
50%
beta
β
60%
70%
80%
90%
100%
Figure 12: Costs for Model 2, update
reneging cost parameter is higher for Model 2, reneging costs are always better for Model
2 because β ∗ is high (so little reneging). We observe that the total cost is always better
for Model 2 than that in Model 1. In addition, increasing the pooling effect makes Model
2 further preferable. The relative difference between the two models’ total costs is indeed
increasing in the system size. We conclude that Model 2 is profiting more from the pooling
effect than Model 1 is.
Numerical Analysis under the update case: The results presented up to this point
are still valid in the update case, where customers substitute their patience threshold by the
delay we announce to them. Let us again consider Model 2 in the base case with update type
customers. In Figure 12, we plot the costs as a function of β. From Figure 12, we see the
same qualitative results as those for the no-update case. The difference here is that γ 0 may
be greater than γ for low values of β. The conditional probability for a customer to renege
given that she elects to wait is 1 − β, which can approach 1 for low β. The new reneging
rate γ 0 will converge to 0 as β goes to 100%.
From Figure 12, we deduce that the optimal announcement coverage is β ∗ = 98%, higher
than the one found under the no-update case. In general the optimal value β ∗ is larger for
the update case than that of the no-update case. This is due to the fact that the reneging
rate in the update case is higher than that of the no-update case, see Proposition 1.
8.
Concluding Remarks
Our analysis has formulated and characterized the performance measures of a call center
model where anticipated delays are announced to customers upon arrival, who may then balk
in response. Announcements are made as a chosen percentile of the delay distribution. For
36
customers who opt to wait, if realized waiting times exceed those that have been announced,
a customer further reacts by reneging, where this reneging rate is modeled to be different
from the one in an equivalent system without delay announcement. This latter feature, as
well as the possibility of announcing a percentile of the delay distribution, different than the
mean delay, are key distinguishing features of the model.
The numerical analysis illustrates tradeoffs between reneging and balking that have to
be made in choosing the announcement percentile, and illustrates the role cost and system
size parameters play on this choice. It is shown that more coverage is not necessarily better
for the service provider and that announcement coverage has to be carefully controlled in
the presence of different customer reactions. This additional control, if properly used, provides managers with a means of improving costs and reaping higher benefits from pooling.
Considering different performance measures as comparison points, conditions are established
under which the model with delay information is preferred. This analysis highlights that in
some settings, particularly in smaller call centers, managers may prefer not to provide delay
information.
The fundamental tradeoff between balking and reneging in a call center with delay information (selection of the announcement percentile) is similar to that in queueing systems
with finite waiting space (selection of the queue size). In such systems, less waiting space
means more balking customers; but at the same time it also leads to lower waiting time for
those who get access (and if reneging is present, it leads to lower reneging customers).
In future work, it would be interesting to empirically describe customers’ reactions in
response to delay announcements, in order to validate our modeling of that reaction herein.
In particular the heuristic modeling of the reneging reaction, through customers of type
no-update, update and mixtures of the two, remains to be tested. Lab experiments that
control for everything else and allow a direct comparison between the models with perfect,
imperfect, and no delay information would be valuable in supporting assumptions pertaining
to balking and reneging made in the analysis herein. Providing direct evidence from a call
center setting for the cost relationships in the economic framework of Section 7.2 is another
issue that remains to be explored.
An ambitious extension of the current analysis is to consider non-stationary arrivals which
would be an important issue in practice. Another extension worth exploring is one where
balking or reneging customers may call back. Rather than capturing different implications
of balking and reneging through assumptions on the costs, as was done in the current paper,
one could concretely model the differences as different probabilities to call back for customers
who have balked or abandoned in a setting with retrials.
37
Appendix
Validation of the Exponential Approximation
In this section, we investigate the quality of the approximation we consider for new patience
times in Model 2. To get a tractable analysis, we assumed in Section 6.1 that new patience
times are exponentially distributed with rate γ 0 and we developed a method to compute the
parameter γ 0 . To assess the exponential assumption, we compare in what follows the performance measures of Model 2 derived by the numerical method (exponential approximation)
with those given by simulation. We conduct this study for both the no-update and update
cases.
Consider the proposed Model 2 with update or no-update behavior, which we will call
the ideal simulation. Under the no-update case, a customer who joins the queue will renege
as soon as her waiting time reaches tk (initial random threshold). Under the update case,
a customer who joins the queue will renege as soon as her waiting time reaches dn (the
delay estimate). We underline a complexity associated with the delay we announce, dn . It
is very hard to derive the exact distribution of Dn , based on which we determine the value
dn with coverage probability β. We need to compute the convolution of the distributions
of the remaining patience times of the customers waiting in queue for each customer. To
simplify the analysis, we simulate an intermediate Model 2 in which we assume that Dn is
exponentially distributed with rate γ 0 . To determine the value of γ 0 , one would think to
apply a fixed point algorithm using simulation at each step. This is again very heavy to do.
Instead we use the value of γ 0 derived from the numerical computation using the fixed-point
algorithm from Section 6.1. We label the latter as the exponential approximation method
in this appendix. In summary, we will simulate Model 2 such that we announce the same
delay dn as in the exponential approximation model. Customers who join the queue will then
renege exactly as assumed either under the no-update (after tk ), or the update case (after dn ).
The resulting Model 2 that we simulate is intermediate between the ideal simulation and the
exponential approximate model. We believe that the comparison between this model and
the approximate exponential one (developed in Section 6.1) will give good indications about
the quality of the exponential approximation made in the latter.
We run simulations for several values of the system parameters. For all cases, µ = 1,
γ = 0.5, α0 = 0 and γ1 = 0 (for Model 1). We consider (s, λ) = (5,5), (10,10), and (50,50).
For each system, we choose the coverage probability β = 50% and β = 80%. For each β, the
reneging rate, based on which we compute dn , is derived from the numerical approximation
of Section 6.1. In Tables 5 and 6, we give the results of the comparison for the no-update
38
and update cases, respectively. In these tables, we present the performance measures derived
by simulation and those derived by the numerical method (approximation) developed in the
core of the paper.
Table 5: Simulation vs exponential approximation, no-update case
β
γ0
PS
PR
PB
E(X | S)
σ(X | S)
s = 5,
50%
0.258
Sim
App
0.833 0.844
0.050 0.030
0.117 0.126
0.211 0.254
0.336 0.372
λ=5
80%
0.033
Sim
App
0.827 0.827
0.006 0.008
0.167 0.165
0.217 0.206
0.363 0.339
s = 10,
50%
0.318
Sim
App
0.879 0.890
0.043 0.020
0.078 0.090
0.141 0.186
0.221 0.264
λ = 10
80%
0.029
Sim
App
0.880 0.880
0.004 0.007
0.116 0.113
0.160 0.156
0.253 0.244
s = 50,
50%
0.454
Sim
App
0.941 0.952
0.023 0.020
0.036 0.028
0.051 0.088
0.081 0.119
λ = 50
80%
0.017
Sim
App
0.949 0.946
0.001 0.013
0.050 0.042
0.078 0.061
0.112 0.088
Table 6: Simulation vs exponential approximation, update case
β
γ0
PS
PR
PB
E(X | S)
σ(X | S)
s = 5,
50%
3.528
Sim
App
0.769 0.775
0.189 0.181
0.043 0.044
0.037 0.033
0.094 0.072
λ=5
80%
0.668
Sim
App
0.803 0.808
0.077 0.065
0.121 0.127
0.118 0.114
0.226 0.200
s = 10,
50%
6.762
Sim
App
0.820 0.824
0.161 0.156
0.019 0.020
0.015 0.014
0.045 0.036
λ = 10
80%
1.106
Sim
App
0.853 0.859
0.079 0.066
0.068 0.075
0.069 0.074
0.133 0.130
s = 50,
50%
32.464
Sim
App
0.905 0.906
0.093 0.091
0.002 0.002
0.002 0.002
0.007 0.006
λ = 50
80%
4.170
Sim
App
0.922 0.927
0.065 0.056
0.013 0.017
0.014 0.019
0.031 0.037
From Tables 5 and 6, we observe that the exponential approximation gives good results
in both the no-update and update cases. This can be explained using the results derived in
Zeltyn and Mandelbaum (2005). Under the Quality-Efficiency-Driven regime where delays
are short, it is pointed out that the patience distribution near the origin determines the
behavior of the system. Similar results are found for patience distributions with a positive
density at the origin (Theorem 4.1 in Zeltyn and Mandelbaum (2005)), as well as for patience
distributions with an atom at the origin (Theorems 4.2 and 4.3 in Zeltyn and Mandelbaum
(2005)). In Theorem 4.3 of Zeltyn and Mandelbaum (2005), the authors consider a scaled
balking increasing in the queue state, which is somewhat similar to what we consider in this
paper, pB (n). They conclude that approximate performance measures depend only on the
patience distribution near the origin.
For small systems as those in our experiments, waiting times are also relatively short
thus giving the main importance to the behavior of the patience distribution at the origin
rather than in the tail. Since in addition to that the balking behavior is the same for the
simulated and exponential approximated models, the exponential approximation has a good
39
quality. Recall that for both the no-update and the update cases the simulated model takes
the random variable Dn to be the same as that in the approximation (based on the same
value of γ 0 ). So, pB (n) is the same in the simulated and approximated models. Then, the
patience distributions in both models have the same atom at the origin.
Based on the results in Tables 5 and 6 we make some observations about the value of the
new reneging rate γ 0 . For the no-update case, γ 0 is always lower than γ (As already explained
in Section 6.2). We observe from 6 that for the update case γ 0 > γ in these examples. From
Table 5, γ 0 increases in the system size for β = 50%, and decreases in the system size for
β = 80%. This example illustrates how beta controls the tradeoff between reneging and
balking, and how a low coverage probability choice as represented by the β = 50% case here
can lead to an increased rate of reneging from the system despite the pooling effect. In the
larger systems, we are accepting more customers with low patience thresholds, and having
announced delays with β = 50%, more of these customers will have been misinformed which
will thus lead to higher reneging rates from the system. For the update case in Table 6, even
for β = 80% the system reneging rate cannot be reduced through pooling. This is because
customers who decide to wait are updating their patience to the times we announce, which
are lower than their original thresholds, thus leading to a situation resembling that described
for the β = 50% case under no-update.
Analysis of the Mixed Case
In this section, we consider the mixed case. To analyze this case, it suffices to compute
for each new arrival finding n customers in queue, the conditional probability that this
customer will renege, given that he does not balk after hearing her anticipated delay. We
denote this probability by rnM . We then apply the same analysis as that in Section 6 to get
the performance analysis of Model 2 under the mixed case.
We assume that we have a Bernoulli process. Each customer electing to join the queue
has a probability q (q ∈ [0, 1]) to be an update type and 1 − q to be a no-update type,
independently of all other events. This leads to
rnM = q rnU + (1 − q) rnN ,
(33)
where rnU and rnN are given in Equations (22) and (28), respectively. Using Proposition 1
and Equation (33), one may easily state the following inequality
rnN ≤ rnM ≤ rnU .
40
(34)
Proof of Proposition 1
We use the expressions of rnU and rnN given in Equations (22) and (28), respectively. For
0 ≤ i < n + 1, we have
1
1
>
.
(35)
0
sµ + γ + iγ
sµ + γ + (n + 1)γ 0
Then, we derive for β < 1 the following inequalities for the last term in the right hand side
of Equation (28).
!
à n
n
X
Y sµ + jγ 0
sµ + iγ 0
gn (dn )
0
·
· e−(sµ+iγ )dn >
> 0.
0
0
(j − i)γ
sµ + γ + iγ
sµ + γ + (n + 1)γ 0
i=0
j=0, j6=i
(36)
where gn is the pdf of the hypoexponential distribution given in Equation (11). This finishes
the first part of the proof about the strict inequality.
In the extreme case β = 1, Equations (22) and (28) lead to rnU = rnN = 0. This means
that all customers in both situations will not renege. The reason is that no customers enter
the systems. All of them balk because we announce to them an infinite anticipated delay,
dn = ∞ for n ≥ 0, so, pB (n) = P (T < dn ) = 1 for n ≥ 0.
2
Proof of Proposition 2
Let Ni (t), i = {a, b} be the number of customers in System i at time t. We use uniformization
and coupling, and show by induction that Na (t) ≤ Nb (t), implying the desired orderings on
the probability of balking and reneging in Systems a and b. All events are generated from the
same Poisson process with rate Γ = λ + sµ + Kγ where K is chosen as a very large number,
ensuring that the probability of having more than K customers in queue is negligible. The
birth death rates provide the corresponding event probabilities. For example, an arrival
occurs with probability λ/Γ, or a service completion with kµ/Γ in state k < s. Both systems
are assumed to start out empty, Na (0) = Nb (0) = 0. For all states k < s arrivals and service
completions are identical in both systems, and there is no balking or reneging from either
system. So until the time when k ≥ s, we have Na (t) = Nb (t).
Now consider a state k > s: βa > βb implies that da,k−s > db,k−s . Since the probability of
balking is given by P (T < dk−s ), we note that every time a customer balks from System b, she
will also balk in System a and there will be additional customers who balk in System a, thus
leading to Na (t) ≤ Nb (t). Whenever Na (t) ≤ Nb (t) there will be more service completions
from System b.
Under the no-update case, the probability that the k − sth customer in queue reneges is
given by P (T < Dk−s |T > dk−s ), where the delay in queue increases as a function of the
length of the queue. Using the fact that reneging is exponential, we can state that whenever
41
a customer reneges from System a, she will also renege in System b with additional reneging
occurring in System b. More specifically, all customers in System a will have their identical
counterparts in System b. These identical customers will renege in a similar fashion. In
addition, System b will have some additional customers, non-identical to any in System a.
If all of these non-identical customers do not renege at time t, one has Na (t) < Nb (t), if
on the other hand all of them renege the system states can revert back to Na (t) = Nb (t).
Under the update case, the probability that the k − sth customer in queue reneges is given
by P (dk−s < Dk−s |T > dk−s ). Once again, when Na (t) ≤ Nb (t), there will be a customer
in System b corresponding to each customer in System a, plus some additional customers.
Since da,k−s > db,k−s for all k > s, for the customers that are common in both systems a
renege is more likely in System b. In other words every time one of the common customers
reneges in System a she will also renege in System b, but it is possible to have the customer
in a stay while the corresponding customer in b departs. The additional customers in System
b may stay or renege, ensuring all the time that either Na (t) < Nb (t) or the system states
revert back to Na (t) = Nb (t).
Whenever Na (t) = Nb (t) either due to extra service completions or extra reneging in
System b, the two systems can be coupled again and the same arguments ensuring the sample
path ordering Na (t) ≤ Nb (t) will hold. In steady state we can state that E[Na (t)] < E[Nb (t)],
implying the desired relationships PaR < PbR and PaB > PbB .
2
References
Akşin, O., Armony, M., and Mehrotra, V. 2007.
The Modern Call-Center: A Multi-
Disciplinary Perspective on Operations Management Research.
Production and Op-
erations Management, 16:665–688.
Allon, G., Bassamboo, A., and Gurvich, I. 2007. We Will Be Right with You”: Managing
Customers with Vague Promises. Working paper.
Ancker, C. J. and Gafarian, A. 1962. Queueing with Impatient Customers Who Leave at
Random. Journal of Industrial Engineering, 13:84–90.
Armony, M. and Maglaras, C. 2004a. Contact Centers with a Call-Back Option and RealTime Delay Information. Operations Research, 52:527–545.
Armony, M. and Maglaras, C. 2004.
On Customers Contact Centers with a Call-Back
Option: Customer Decisions, Routing Rules, and System Design. Operations Research,
52:271–292.
42
Armony, M., Shimkin, N., and Whitt, W. 2005. The Impact of Delay Announcements in
Many-Server Queues with Abandonment. Operations Research. To appear.
Baccelli, F. and Hebuterne, G. 1981.
On Queues With Impatient Customers.
Perfor-
mance’81 North-Holland Publishing Company, pages 159–179.
A Framework for Analyzing the Quality of the Customer Interface. European Management
Journal, 11:385-396.
Bitran, G.R., Ferrer, J-C., and Rocha e Oliviera, P. 2008. Managing Customer Experiences:
Perspectives on the Temporal Aspects of Service Encounters. Manufacturing & Service
Operations Management, 10:61–83.
Brandt, A. and Brandt, M. 2002. Assymptotic Results and a Markovian Approximation for
the M(n)/M(n)/C + GI System. Queueing Systems, 41:73–94.
Brown, L., Gans, N., Mandelbaum, A., Sakov, A., Shen, H., Zeltyn, S., and Zhao, L. 2005.
Statistical Analysis of a Telephone Call Center: A Queueing-Science Perspective. Journal
of the American Statistical Association, 100:36-50.
Feigin, P. 2005. Analysis of Customer Patience in a Bank Call Center. Working Paper.
Gans, N., Koole, G., and Mandelbaum, A. 2003. Telephone Call Centers: Tutorial, Review,
and Research Prospects. Manufacturing & Service Operations Management, 5:73–141.
Garnett, O., Mandelbaum, A., and Reiman, M. 2002. Designing a Call Center with Impatient
Customers. Manufacturing & Service Operations Management, 4:208–227.
Guo, P. and Zipkin, P. 2007. Analysis and Comparison of Queues with Different Levels of
Delay Information. Management Science, 53:962–970.
Hornik, J. 1984. Subjective vs. Objective Time Measures: a Note on the Perception of
Time in Consumer Behavior. Journal of Consumer Research, 11:615–618.
Hui, M. and Tse, D. 1996.
What to Tell Customer in Waits of Different Lengths: an
Integrative Model of Service Evaluation. Journal of Marketing, 60:81–90.
Hui, M. and Zhou, L. 1996. How Does Waiting Duration Information Influence Customers’
Reactions to Waiting for Services? Journal of Applied Social Psychology, 26:1702–1717.
Ibrahim, R. and Whitt, W. 2007. Real-Time Delay Estimation Based on Delay History.
Manufacturing & Service Operations Management. To appear.
Jouini, O. and Dallery, Y. 2007a. Analysis of a Multiple Priority Queue with Impatient
Customers. Working paper.
Jouini, O. and Dallery, Y. 2007b. Moments of First Passage Times in General Birth-Death
43
Processes. Mathematical Methods of Operational Research, 68:49–76.
Jouini, O., Dallery, Y., and Akşin, O. Z. 2007. Queueing Models for Full-Flexible Multi-class
Call Centers with Real-Time Anticipated Delays. International Journal of Production
Economics. to appear.
Kahneman, D., Slovic, P., and Tversky, A., eds. 1982.
Judgement Under Uncertainty:
Heuristics and Biases Cambridge university Press, Cambridge.
Katz, K., Larson, B., and Larson, R. 1991.
Prescription for the Waiting-in-Line Blues:
Entertain, Enlighten, and Engage. Sloan Management Review, pages 44–53.
Keilson, J. 1964a. A Review of Transient Behavior in Regular Diffusion and Birth-Death
Processes. Part I. Journal of Applied Probability, 1:247–266.
Kijima, M. 1997. Markov Processes for Stochastic Modeling. Chapman & Hall. 1st Edition.
Kleinrock, L. 1975. Queueing Systems, Theory, volume I. A Wiley-Interscience Publication.
Kolesar, P. 1984. Stalking the Endangered CAT: A Queueing Analysis of Congestion at
Automatic Teller Machines. Interfaces, 14:16–26.
Koole, G. and Mandelbaum, A. Queueing Models of Call Centers: An Introduction Annals
of Operations Research, 113:41–59.
Kumar, P., Kalwani, M.U., and Dada, M. 1997. The Impact of Waiting Time Guarantees
on Customers Waiting Experiences. Marketing Science, 16:295-314.
Leclerc, F., Schmitt, B., and Dube, L. 1995. Waiting Time and Decision Making: Is Time
Like Money? Journal of Consumer Research, 22:110-119.
Maister, D. 1985. The Psychology of Waiting Lines. Czepiel, J., Solomon, M., Suprenant,
C. eds. The Service Encounter: Managing Employee/Customer Interaction in Service
Businesses. Lexington Books, Lexington MA, 113-123.
Mandelbaum, A. and Zeltyn, S. 2004. The Impact of Customers Patience on Delay and
Abandonment: Some Empirically-Driven Experiments with the M/M/N+G Queue. OR
Spectrum, 26:377–411.
Mandelbaum, A. and Zeltyn, S. 2006. Staffing Many-Server Queues with Impatient Customers: Constraint Satisfaction in Call Centers. Working paper.
Munichor, N. and Rafaeli, A. 2007. Numbers or Apologies? Customer Reactions to Telephone Waiting Time Fillers. Journal of Applied Psychology, 92:511-518.
Nakibly, E. 2002. Predicting Waiting Times in Telephone Service Systems. Ph.D. Thesis,
Technion.
44
Naor, P. 1969. The Regulation of Queue Size by Levying Tolls. Econometrica, 37:15–24.
Osuna, E. 1985. The Psychological Cost of Waiting. Journal of Mathematical Psychology,
29:82–105.
Rafaeli, A., Barron, G., and Haber, K. 2002. The Effects of Queue Structure on Attitudes.
Journal of Service Research, 5:125–139.
Ross, M. 1997. Introduction to Probability Models. Academic Press, San Diego. 7th Edition.
Taylor, S. 1994. Waiting for Service: The Relationship Between Delays and Evaluations of
Service. Journal of Marketing, 58:56–69.
Whitt, W. 1999a. Improving Service by Informing Customers about Anticipated Delays.
Management Science, 45:192–207.
Whitt, W. 1999b. Predicting Queueing Delays. Management Science, 45:870–888.
Wolff, R. 1982. Poisson Arrivals See Time Averages. Operations Research, 30:223–231.
Zakay, D. 1989. An Integrated Model of Time Estimation. Times and Human Cognition:
A Life Span Perspective. Iris Levin and Dan Zakay, eds, Amsterdam: North Holland.
Zeltyn, S. and Mandelbaum, A. 2005. Call Centers with Impatient Customers: Many-Servers
Asymptotics of the M/M/n+G Queue. Queueing Systems, 51:361–402.
Zohar, E., Mandelbaum, A., and Shimkin, N. 2002. Adaptive Behavior of Impatient Customers in Tele-Queues: Theory and Empirical Support. Management Science, 48: 566–
584.
45