Staffing Call Centers with Uncertain Arrival Rates

Buy ready-to-submit essays. No Plagiarism Guarantee!

Note: Our papers are 100% human-written. 

Check before you submit. Get Turnitin Score Report in 15 Minutes.

Don't risk the 'Red' score. Get the exact same Turnitin report your professor uses. Join 50,000+ students who submitted their essays with confidence this semester.

Get Turnitin Score Report Request Human Rewrite

Staffing Call Centers with Uncertain Arrival Rates and
Co-sourcing
Yasar Levent Kocaga Sy Syms School of Business, Yeshiva University, Belfer Hall 403/A, New York City, New York 10033, USA, kocaga@yu.edu
Mor Armony
Stern School of Business, New York University, KMC 862, New York City, New York 10012, USA, marmony@stern.nyu.edu
Amy R. Ward
Marshall School of Business, University of Southern California, Bridge Hall 401H, Los Angeles, California 90089, USA
amyward@marshall.usc.edu
I
n a call center, staffing decisions must be made before the call arrival rate is known with certainty. Once the arrival
rate becomes known, the call center may be over-staffed, in which case staff are being paid to be idle, or understaffed, in which case many callers hang-up in the face of long wait times. Firms that have chosen to keep their call center
operations in-house can mitigate this problem by co-sourcing; that is, by sometimes outsourcing calls. Then, the required
staffing N depends on how the firm chooses which calls to outsource in real time, after the arrival rate realizes and the
call center operates as a M/M/N + M queue with an outsourcing option. Our objective is to find a joint policy for staffing
and call outsourcing that minimizes the of this when there is a linear
staffing cost per unit time and linear costs associated with abandonments and outsourcing. We propose a policy that uses
a rule, and outsources calls in accordance with a threshold rule that characterizes when the system is too crowded. Analytically, we establish that our proposed policy is asymptotically optimal, as the mean arrival
rate becomes large, when the level of uncertainty in the arrival rate is of the same order as the inherent system fluctuations in the number of waiting customers for a known arrival rate. Through an extensive numerical study, we establish
that our policy is extremely robust. In particular, our policy performs remarkably well over a wide range of parameters,
and far beyond where it is proved to be asymptotically optimal.
Key words: call center operations; co-sourcing; staffing; overflow routing; parameter uncertainty
History: Received: July 2013; Accepted: September 2014 by Michael Pinedo, after 2 revisions.
1. Introduction
Call centers have become ubiquitous in business.
Today, every Fortune 500 company has at least one
call center, and the average Fortune 500 company
employs 4500 call center agents (who may be distributed across more than one site) (Gilson and Khandelwal 2005). For many companies, the call center is a
primary point-of-contact with their customers. Hence,
a well-run call center promotes good customer relations, and a poorly managed one hurts them. But call
center management is difficult.
A call center manager faces the classical operational
challenge of determining appropriate staffing levels
throughout the day and week in order to meet a random and time-varying call volume. This is extremely
difficult especially when call arrival rate is itself random, as was empirically shown in Brown et al. (2005)
and Maman (2009), among others. When staffing levels are too low, customers are put on hold, and many
hang up in frustration while waiting for an agent to
take their call. But when staffing levels are too high,
the call center manager ends up paying staff to be
idle.
One option in managing this uncertainty is for a
company to outsource its call center operations. Then,
the challenges of call center management can be handled by a vendor firm whose primary focus is call
center operations. That vendor can pool demand
amongst various companies, thereby lowering variability, which should allow for more accurate demand
forecasts, and so better staffing decisions. However, it
is also true that many companies are reluctant to
relinquish control of their call center operations. This
is evidenced by a recent survey from the Incoming
Call Management Institute (ICMI 2006): only 7.9% of
279 call center professionals used an outside vendor
to handle most or all of their calls. One reason for that
are the hidden costs of outsourcing (Kharif 2003),
which include service quality costs that are hard to
Vol. 24, No. 7, July 2015, pp. 11011117 DOI 10.1111/poms.12332
ISSN 1059-1478|EISSN 1937-5956|15|2407|1101 2014 Production and Operations Management Society
1101
explicitly quantify. As a result, many of these companies prefer to co-source; that is, to outsource some,
but not all, of their calls.
We study a co-sourcing structure in which the vendor charges the company a fee per call outsourced,
which is consistent with the pay-per-call (PPC) cosourcing structure analyzed in Aksin et al. (2008). We
assume the company can decide on a call-by-call basis
which calls to answer in-house and which calls to
route to the vendor. This is helpful because call centers typically make their daily staffing decisions at
least a week in advance, before the actual arrival rate
to the call center for a given day is known. If the
planned staffing is sufficient to handle the mean arrival rate, then the company needs to outsource only a
small fraction of calls in order to handle the inherent
variability that results in congestion every so often.
On the other hand, if the planned staffing is insufficient to handle the mean arrival rate, then the company can outsource a large fraction of its calls,
thereby preventing high congestion levels.
The relevant question for this study is: how do we
decide on staffing levels when the arrival rate is
uncertain and the aforementioned co-sourcing option
is present? To answer this question, we begin with
one simple and widely used queueing model of a call
center, the Erlang A or M/M/N + M (see, e.g., section
4.2.2 in Gans et al. 2003), and add uncertainty in the
arrival rate and an outsourcing option. Then, our
model for the call center is a multi-server queue with
a doubly stochastic time-homogeneous Poisson arrival process, exponential service times, and exponential times to abandonment. Although this model
ignores the time-varying nature of the arrival rate
over the course of each day, there is call center literature that discusses how to use the Erlang A model to
make staffing decisions for time-varying arrival rates,
using the stationary independent period by period
approach (SIPP); see Green et al. (2001), Gans et al.
(2003), Aksin et al. (2007), and Liu and Whitt (2012)
for more discussion. We suppose that a similar
approach can be adopted here to accommodate the
added feature of arrival rate uncertainty.
The control decisions in our model are (i) an
upfront staffing decision and (ii) real-time call outsourcing (routing) decisions. Recall that staffing decisions are made on a much longer time horizon and
well before the timing of the control decisions. In particular, these decisions are made on two different time
scales. This means that we have a two stage stochastic
program: The staffing decisions are made in the first
stage, before the arrival rate is known, and the outsourcing decisions are made in the second stage, after
the arrival rate is known. Then, the outsourcing decisions can depend on the actual arrival rate even
though the staffing decisions cannot.
Our objective is to propose a policy for staffing and
outsourcing under the assumption of linear staffing
cost and linear abandonment and outsourcing costs.
For each arriving customer that cannot be immediately served, there is a tension between choosing to
outsource that customer (and paying the outsourcing
fee) or having that customer wait for an in-house
agent (and risking incurring an abandonment cost).
In summary, we are solving a joint staffing and routing control problem for a (modified) Erlang A model
with an uncertain arrival rate and an outsourcing
option.
The three main contributions in this study are:
The modeling contribution is the formulation of
a joint staffing and outsourcing problem for a
call center that has access to co-sourcing, and
must make staffing decisions when there is
arrival rate uncertainty. This modeling framework can be used to study more general joint
staffing and control problems in call centers
that have been previously studied in the literature under the assumption of a known arrival
rate.
The application contribution is the development
of a square-root safety staffing and threshold
outsourcing policy that we numerically show
to be extremely robust over the entire parameter space. This robustness may come as no surprise for readers who are familiar with related
literature such as Borst et al. (2004) and Gurvich et al. (2014). However, the existing literature has not addressed the issue of robustness
in the context of random arrival rates and
dynamic control, nor can this robustness be
readily explained using existing results.
The technical contribution is the proof that our
proposed square-root safety staffing and
threshold outsourcing policy is asymptotically
optimal, as the mean arrival rate becomes
large, when the level of uncertainty in the arrival rate is of the same order as the inherent
system stochasticity (which is of the order of
the square-root of the mean of the arrival rate).
The remainder of this paper is organized as follows.
First, we review the most relevant literature. Next, in
section 2, we describe our model in detail. In section
3, we present the exact (non-asymptotic) analysis
which leads to an algorithm to compute the optimal
policy numerically. However, that algorithm does not
provide insight into the structure of an optimal policy,
and so, in section 4, we perform an asymptotic analysis under the assumption that the level of uncertainty
in the arrival rate is of the same order as the inherent
system stochasticity. That asymptotic analysis motivates us to propose, in section 5, a square-root safety
Kocaga, Armony, and Ward: Staffing and Co-sourcing for Call Centers
1102 Production and Operations Management 24(7), pp. 11011117, 2014 Production and Operations Management Society
staffing and threshold outsourcing policy that is universal in the sense that there is no assumption on the
level of uncertainty in the arrival rate. We evaluate
the performance of our universal policy numerically
in section 6. We make concluding remarks in section
7. All proofs and additional numerical results can be
found in the electronic companion (EC).
1.1. Literature Review
Previous work on joint staffing and routing problems
in call centers includes Gurvich et al. (2008) who
study staffing and dynamic routing in call centers
with multiple customer classes and a single server
pool, Armony and Mandelbaum (2011) who consider
the symmetric case of a single customer type and a
heterogeneous server pool, and Gurvich and Whitt
(2010) who consider multiple customer classes and a
heterogeneous server pool. These papers study the
staffing and dynamic routing problems within the
HalfinWhitt heavy traffic regime, pioneered by
Halfin and Whitt (1981), and extended to include
abandonments by Garnett et al. (2002). This is also
known as the quality and efficiency driven (QED)
heavy traffic regime. The key idea is to approximate
the behavior of call centers that are modeled as multiserver queues with that of their limiting diffusions.
The limiting diffusion arises from a specific relationship between the arrival rate and the staffing level as
both grow large without bound. Our work is different
from the aforementioned papers in that our model is
pertinent to situations where the arrival rate is not
known when staffing decisions are made, and thus
has to be inferred or forecasted using available
historical data.
Given a staffing level and a realized arrival rate,
our dynamic outsourcing decision is equivalent to the
admission control problem studied in Kocaga and
Ward (2010). The equivalence follows because we do
not explicitly model the vendor firm and assume it
has ample capacity to handle the outsourced calls.
Kocaga and Ward (2010) show that a threshold
admission control policy is optimal, and characterize
a simple form for the threshold level that is asymptotically optimal when the staffing level is assumed to be
such that the system operates in the QED regime. In
contrast, this study explicitly models the staffing decisions and has a random arrival rate.
There is a growing body of literature that studies
staffing for call centers with uncertain arrival rates
including (in chronological order) Chen and Henderson (2001), Jongbloed and Koole (2001), Ross (2001),
Bassamboo et al. (2005), Whitt (2006), Bassamboo
et al. (2006), Steckley et al. (2009), Maman (2009),
Bassamboo and Zeevi (2009), Gurvich et al. (2010),
Robbins and Harrison (2010), Bassamboo et al. (2010),
Mehrotra et al. (2010), Gans et al. (2012), and Zan
et al. (2014). The two works most closely related to
ours are Maman (2009) and Bassamboo et al. (2010),
and we discuss each in turn.
The focus of Maman (2009) is to extend the QED
staffing formula under a general form of arrival uncertainty. Our asymptotic optimality result assumes a
special case of the form of the arrival rate uncertainty
presented in that paper. However, that paper does not
explicitly study the cost minimizing staffing and does
not model routing decisions, as we do.
Bassamboo et al. (2010) propose a staffing policy
for an M/M/N + M queue in which the arrival rate is
random, and there is no outsourcing option. They
establish that a simple newsvendor based staffing policy performs extremely well when the order of uncertainty in the arrival rate exceeds the order of the
inherent system stochasticity. In contrast, we establish
the asymptotic optimality of our proposed policy
when the aforementioned two magnitudes are the
same. In our numeric study, we adapt their policy to
our setting with outsourcing, in order to evaluate
policy robustness.
In relation to the literature on call center outsourcing (see, e.g., Zhou and Ren 2011), our paper is most
similar to Aksin et al. (2008). In contrast to most
papers in this literature, which assumes all calls will
be outsourced, Aksin et al. (2008) considers the contract design problem of a company which faces an
uncertain call volume, and can outsource part of its
calls by choosing between a capacity-based and volume-based contract that is pay-per-call. Although both
Aksin et al. (2008) and our model study co-sourcing
decisions which are driven by call volume uncertainty,
Aksin et al. (2008) focuses on the optimal contract
choice, whereas we focus on the in-house staffing and
dynamic routing decisions and assume the contract.
2. Model Description
We model the in-house call center (which we henceforth refer to as the call center or the system) as
an M/M/N + M queueing system in which the arrival rate is uncertain. We let denote the random arrival rate with known cdf FK, and we let l denote a
particular realization of . We assume that is a nonnegative random variable with mean E[] = k. For
ease of exposition, we assume the mean of the exponential service time is 1, so that we can think of measuring time in terms of the mean time to serve an
arrival. Each customer call put on hold has an exponential patience time with mean 1/c and abandons
the system if not answered within this time. We
assume that is independent of the service and
patience times, and that, given a realization l of , the
inter-arrival, service, and patience times are independent and identically distributed.
Kocaga, Armony, and Ward: Staffing and Co-sourcing for Call Centers
Production and Operations Management 24(7), pp. 11011117, 2014 Production and Operations Management Society 1103
The call center manager must make two decisions:
the upfront staffing level N, and the dynamic outsourcing decision. The staffing level N : NFK
must be set before the arrival rate is realized, based
on the knowledge of its distribution. After the arrival
rate is realized as l, every arriving call can be either
accepted into the system, or routed to the outsourcing
vendor. The routing control policy p := p(N,l) is in
general a function of the staffing level N and l. The
notation p(N,) refers to a routing policy that may
depend on the actual realization l of . Any stationary
routing control policy p pn : n 2 f0; 1; …g is a
vector, where pn 2 0; 1 denotes the probability that a
customer is accepted into the system when there are n
customers currently present there. We let be the set
of all such vectors. An admissible policy
u : N; pN; K N;pnN; K : n 2 f0; 1; …g
sets the staffing level as a non-negative integer N,
and, after the arrival rate realizes as l, controls
outsourcing decisions dynamically by routing calls
according to the policy p(N,l) 2 .
After the arrival rate l realizes, the system operates
as a birth and death process with birth rate lpn and
death rates
ln minN; n cn N
:
It is straightforward to solve the balance equations
for the steady-state distribution for the number-insystem process, from which it follows that the following performance measures are well defined:
Ppab; N; l the probability an entering customer
abandons;
Ppout; N; l the probability an arriving customer
is routed to the outsourcer:
The objective of the system manager is to minimize
the expected long-run average cost, when there are
costs due to customer abandonment, routing customers to the outsourcing vendor, and staffing costs.
Every customer that abandons the system before
receiving service costs a and the per call cost of routing to the outside vendor is p (which can also include
indirect costs such as the hidden costs of outsourcing). The long-run average operating cost associated
with p 2 when the arrival rate realizes as l and the
staffing level is N is
zp : zpN; l plPpout; N; l alPpab; N; l: 1
This is expressed as a random variable by replacing
the realized arrival rate l in Equation (1) with the
random arrival rate . For a given realization l of ,
zoptN; l denotes the minimum cost, and
poptN; l 2 P is a policy that achieves that minimum
cost. We let zoptN; K denote the random variable
associated with the minimum cost and let poptN; K
denote an optimal routing policy that may depend
on the actual realization l of . The expected longrun average cost under the policy u = (N,p) = (N,
p(N,)), with respect to the random arrival rate , is
Cu cN EzpN; K: 2
We would like to find a staffing level Nopt and a
routing control policy poptN; K that achieves the
minimum long-run average cost
Copt : inf
u Cu min
N2f0;1;2;…g
cN E zoptN; K : 3
REMARK 1. (INCLUDING WAITING COSTS). The objective
function in Equation (2) can be modified to include
a customer waiting cost at the in-house call center
by modifying Equation (1) as follows. Suppose the
cost for one customer to wait one time unit is w 0.
Then, Equation (1) becomes
zp : zpN; l
plPpout; N; l alPpab; N; l
wl1 Ppout; N; lWpl;
where WpN; l is the steady-state average waiting
time, including both abandoning and served customers. Letting QpN; l denote the steady-state
average number of customers waiting in queue to
be served, it follows from Littles law that
l1 Ppout; N; lWpN; l QpN; l;
and so
zp plPpout; N; l alPpab; N; l wQpN; l:
Also, since the steady-state rate at which abandoning customers arrive must equal the steady-state
abandonment rate
lPpab; N; l cQpN; l;
and so
zp plPpout; N; l a
w
c
lPpab; N; l:
The analysis in this paper is valid with a replaced
by a0 : a w=c. Therefore, to include a customer
waiting cost, the only change is to replace a in Equation (1) by a0
.
Kocaga, Armony, and Ward: Staffing and Co-sourcing for Call Centers
1104 Production and Operations Management 24(7), pp. 11011117, 2014 Production and Operations Management Society

Get Professional Assignment Help Cheaply

Are you busy and do not have time to handle your assignment? Are you scared that your paper will not make the grade? Do you have responsibilities that may hinder you from turning in your assignment on time? Are you tired and can barely handle your assignment? Are your grades inconsistent?

Whichever your reason is, it is valid! You can get professional academic help from our service at affordable rates. We have a team of professional academic writers who can handle all your assignments.

Why Choose Our Academic Writing Service?

• Plagiarism free papers
• Timely delivery
• Any deadline
• Skilled, Experienced Native English Writers
• Subject-relevant academic writer
• Adherence to paper instructions
• Ability to tackle bulk assignments
• Reasonable prices
• 24/7 Customer Support
• Get superb grades consistently

Online Academic Help With Different Subjects

Literature

Students barely have time to read. We got you! Have your literature essay or book review written without having the hassle of reading the book. You can get your literature paper custom-written for you by our literature specialists.

Finance

Do you struggle with finance? No need to torture yourself if finance is not your cup of tea. You can order your finance paper from our academic writing service and get 100% original work from competent finance experts.

Computer science

Computer science is a tough subject. Fortunately, our computer science experts are up to the match. No need to stress and have sleepless nights. Our academic writers will tackle all your computer science assignments and deliver them on time. Let us handle all your python, java, ruby, JavaScript, php , C+ assignments!

Psychology

While psychology may be an interesting subject, you may lack sufficient time to handle your assignments. Don’t despair; by using our academic writing service, you can be assured of perfect grades. Moreover, your grades will be consistent.

Engineering

Engineering is quite a demanding subject. Students face a lot of pressure and barely have enough time to do what they love to do. Our academic writing service got you covered! Our engineering specialists follow the paper instructions and ensure timely delivery of the paper.

Nursing

In the nursing course, you may have difficulties with literature reviews, annotated bibliographies, critical essays, and other assignments. Our nursing assignment writers will offer you professional nursing paper help at low prices.

Sociology

Truth be told, sociology papers can be quite exhausting. Our academic writing service relieves you of fatigue, pressure, and stress. You can relax and have peace of mind as our academic writers handle your sociology assignment.

Business

We take pride in having some of the best business writers in the industry. Our business writers have a lot of experience in the field. They are reliable, and you can be assured of a high-grade paper. They are able to handle business papers of any subject, length, deadline, and difficulty!

Statistics

We boast of having some of the most experienced statistics experts in the industry. Our statistics experts have diverse skills, expertise, and knowledge to handle any kind of assignment. They have access to all kinds of software to get your assignment done.

Law

Writing a law essay may prove to be an insurmountable obstacle, especially when you need to know the peculiarities of the legislative framework. Take advantage of our top-notch law specialists and get superb grades and 100% satisfaction.

What discipline/subjects do you deal in?

We have highlighted some of the most popular subjects we handle above. Those are just a tip of the iceberg. We deal in all academic disciplines since our writers are as diverse. They have been drawn from across all disciplines, and orders are assigned to those writers believed to be the best in the field. In a nutshell, there is no task we cannot handle; all you need to do is place your order with us. As long as your instructions are clear, just trust we shall deliver irrespective of the discipline.

Are your writers competent enough to handle my paper?

Our essay writers are graduates with bachelor’s, masters, Ph.D., and doctorate degrees in various subjects. The minimum requirement to be an essay writer with our essay writing service is to have a college degree. All our academic writers have a minimum of two years of academic writing. We have a stringent recruitment process to ensure that we get only the most competent essay writers in the industry. We also ensure that the writers are handsomely compensated for their value. The majority of our writers are native English speakers. As such, the fluency of language and grammar is impeccable.

What if I don’t like the paper?

There is a very low likelihood that you won’t like the paper.

Reasons being:

• When assigning your order, we match the paper’s discipline with the writer’s field/specialization. Since all our writers are graduates, we match the paper’s subject with the field the writer studied. For instance, if it’s a nursing paper, only a nursing graduate and writer will handle it. Furthermore, all our writers have academic writing experience and top-notch research skills.
• We have a quality assurance that reviews the paper before it gets to you. As such, we ensure that you get a paper that meets the required standard and will most definitely make the grade.

In the event that you don’t like your paper:

• The writer will revise the paper up to your pleasing. You have unlimited revisions. You simply need to highlight what specifically you don’t like about the paper, and the writer will make the amendments. The paper will be revised until you are satisfied. Revisions are free of charge
• We will have a different writer write the paper from scratch.
• Last resort, if the above does not work, we will refund your money.

Will the professor find out I didn’t write the paper myself?

Not at all. All papers are written from scratch. There is no way your tutor or instructor will realize that you did not write the paper yourself. In fact, we recommend using our assignment help services for consistent results.

What if the paper is plagiarized?

We check all papers for plagiarism before we submit them. We use powerful plagiarism checking software such as SafeAssign, LopesWrite, and Turnitin. We also upload the plagiarism report so that you can review it. We understand that plagiarism is academic suicide. We would not take the risk of submitting plagiarized work and jeopardize your academic journey. Furthermore, we do not sell or use prewritten papers, and each paper is written from scratch.

When will I get my paper?

You determine when you get the paper by setting the deadline when placing the order. All papers are delivered within the deadline. We are well aware that we operate in a time-sensitive industry. As such, we have laid out strategies to ensure that the client receives the paper on time and they never miss the deadline. We understand that papers that are submitted late have some points deducted. We do not want you to miss any points due to late submission. We work on beating deadlines by huge margins in order to ensure that you have ample time to review the paper before you submit it.

Will anyone find out that I used your services?

We have a privacy and confidentiality policy that guides our work. We NEVER share any customer information with third parties. Noone will ever know that you used our assignment help services. It’s only between you and us. We are bound by our policies to protect the customer’s identity and information. All your information, such as your names, phone number, email, order information, and so on, are protected. We have robust security systems that ensure that your data is protected. Hacking our systems is close to impossible, and it has never happened.

How our Assignment  Help Service Works

1.      Place an order

You fill all the paper instructions in the order form. Make sure you include all the helpful materials so that our academic writers can deliver the perfect paper. It will also help to eliminate unnecessary revisions.

2.      Pay for the order

Proceed to pay for the paper so that it can be assigned to one of our expert academic writers. The paper subject is matched with the writer’s area of specialization.

3.      Track the progress

You communicate with the writer and know about the progress of the paper. The client can ask the writer for drafts of the paper. The client can upload extra material and include additional instructions from the lecturer. Receive a paper.

4.      Download the paper

The paper is sent to your email and uploaded to your personal account. You also get a plagiarism report attached to your paper.

PLACE THIS ORDER OR A SIMILAR ORDER WITH US TODAY AND GET A PERFECT SCORE!!!

The post Staffing Call Centers with Uncertain Arrival Rates appeared first on UK Superior Essays.