# Machine learning-based agent staffing under uncertainty: The case of a relay call center

Samer Alsamadi, Cléa Martinez, Canan Pehlivan, Nicolas Cellier, Oualid Jouini, Yiping Fang, Benjamin Legros, Franck Fontanili

## ► To cite this version:

Samer Alsamadi, Cléa Martinez, Canan Pehlivan, Nicolas Cellier, Oualid Jouini, et al., Machine learning-based agent staffing under uncertainty: The case of a relay call center. Expert Systems with Applications, 2025, 281, pp.127385. 10.1016/j.eswa.2025.127385. hal-05025971

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Submitted on 28 May 2025

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# Machine learning-based agent staffing under uncertainty: The case of a relay call center

Samer Alsamadi<sup>a, ID</sup>, *, Cléa Martinez<sup>a ID</sup>, Canan Pehlivan<sup>a ID</sup>, Nicolas Cellier<sup>a ID</sup>, Oualid Jouini<sup>b ID</sup>, Yiping Fang<sup>b ID</sup>, Benjamin Legros<sup>c ID</sup>, Franck Fontanili<sup>a ID</sup>

## ARTICLE INFO

### Keywords:

Call center

Machine learning

Data-driven decision making

Agent staffing under uncertainty

## ABSTRACT

Classical queueing models fail to properly staff non-conventional call centers with complex internal structures. This is either due to the difficulty of finding suitable models whose underlying assumptions hold, or due to certain elements of the call center not being modeled such as caller patience times. Relay call centers, service providers that connect two different interested parties with one another through telecommunication channels, present a prime example of non-conventional call centers. Working on the study case of a relay call center for the deaf community, Erlang C, one of the most commonly used call center staffing formulae, fails to generate agent staffing that meets our target performance criteria for quality of service. We propose a machine learning-based approach leveraging an available log of historical data. Upon comparing the proposed approach's capability of performance evaluation and agent staffing to that of the Erlang C model and a baseline data-driven model, results indicate our approach's staffing superiority. Considering uncertainty within the system variable predictions carried out prior to the staffing phase, our approach generates agent staffing which enables us to meet our global quality of service objective.

## 1. Introduction

Call centers have long provided an interesting playground for scientists eager to study the operations that take place within these stochastically rich sociotechnical systems. Areas of study frequently visited by researchers include forecasting, agent staffing, queueing theory and agent shift scheduling as indicated in Aksin et al. (2007). Within the domain of forecasting, predictions are made regarding the volume of incoming calls and average call handling times to be then used within the planning process. Agent staffing consists of determining the number of agents required to receive these incoming calls during a future time interval while maintaining a certain level of performance. Analytical models existing within queueing theory allow us to approximate probabilities revolving around the waiting process of callers in queues. These probabilities can then be used to generate approximations on possible performance metrics, which can then assist us within the previously mentioned agent staffing process. Simulation models are also used to solve the agent staffing problem by providing simulated performance estimates. Agent shift scheduling is the process of transforming agent

staffing into a feasible agent schedule. The feasibility of this schedule is determined by adherence to the established workplace rules. These rules include protocols on agent shift durations, breaks and agent availability. Fig. 1 displays the connections between the different areas of research. As shown throughout the call center literature, successively executing the different agent planning steps is highly suboptimal as indicated in Koole and Li (2023), a recent overview of call center workforce planning. As the staffing levels resulting from the staffing process are usually hard to cover with agent shifts, a better approach would be the integration of the staffing and scheduling processes. However, as we aim throughout this article to test the performance of some more advanced performance evaluation models with respect to some more classical ones, we solely focus on the agent staffing process of agent planning within call centers. For an overall look into surveys on academic research dedicated to call center operations management, one may refer to Gans et al. (2003).

The agent staffing process requires the planning supervisors to adopt a certain staffing approach, in which two main points are to be

<footnote><sup>a</sup> Industrial Engineering Center, IMT Mines Albi, University of Toulouse, Albi, France</footnote>

<footnote><sup>b</sup> CentraleSupélec, Paris-Saclay University, Paris, France</footnote>

<footnote><sup>c</sup> SKEMA Business School, Côte d'Azur University, Paris, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail addresses: samer.alsamadi@mines-albi.fr (S. Alsamadi), clea.martinez@mines-albi.fr (C. Martinez), canan.pehlivan@yeditepe.edu.tr (C. Pehlivan), nicolas.cellier@elioz.fr (N. Cellier), oualid.jouini@centralesupelec.fr (O. Jouini), yiping.fang@centralesupelec.fr (Y. Fang), benjamin.legros@skema.edu (B. Legros), franck fontanili@mines-albi.fr (F. Fontanili).</footnote>
<figure><img src="image_3.png" /><figcaption>Fig. 1. Call center areas of study schema.</figcaption></figure>

addressed. The first point revolves around choosing the performance evaluation model capable of predicting future performance metrics based on a chosen agent staffing. Forecasts created within the previously mentioned forecasting process are fed into this chosen model to assist in the evaluation process. Classical analytical and simulation models constitute a pool of possible evaluation models to be chosen from. The second point focuses on choosing which optimization method is to be used in the process of searching for an optimized staffing. Applying a brute force grid search would simply be infeasible due to the problem of combinatorial explosion as explained in Garey and Johnson (1979). As computational resources required grow exponentially with the size of the optimization problem, search heuristics are used to explore the space of possible solutions without having to experiment every possible solution. Within the optimization algorithm, it may be chosen to execute the staffing procedure within a stochastic environment, considering the different uncertainties that circulate the call center's functionality including call arrivals. After choosing the desired staffing approach, supervisors must then decide on whether to consider time dependency between different planning intervals or not. The Stationary Independent Period by Period (SIPP) approach, whose term was coined by Green et al. (2001, 2003), ignores the impact of previously planned periods on latter ones. Constraints that the staffing method will be seeking to satisfy are also to be chosen. The choice typically lies between soft constraints where heavily staffed intervals (ex: 15 min or 30 min time slots) can compensate lightly staffed intervals in terms of performance, or hard constraints where the number of agents to be planned in each interval is strictly specified. Koole and Van Der Sluis (2003) was one of the first studies to use global service level constraints. They utilize the Erlang C model and use a local search algorithm to optimally staff a call center with the global service level modeled as a soft constraint.

Carefully chosen indicators allow analysts to properly assess the performance of the study case call center. Irrelevant indicators however would lead to an unrealistic or even inaccurate assessment, which could have unfavorable effects on the management of said study case. The choice of Key Performance Indicators (KPIs) to be optimized within a call center is crucial as indicated in Jouini et al. (2013). As two interested parties exist within call center operations: the caller and the call center itself, we decide to adopt: the Quality of Service (QoS) KPI defined as the percentage of answered incoming calls which are answered in less or equal to 1 min, and the Total Production (TP) service cost defined as the total number of work hours provided by the planned agents. Within our study, the QoS metric whose formula is shown in Eq. (1) as well as the TP metric are considered on a daily basis. Taking into account the uncertainties surrounding the call center environment, the objective lies in minimizing TP while maintaining a minimum level of performance denoted $QoS_{x,minimum}$ as shown in Eq. (2).

$$QoS_x = \frac{\text{Number of calls answered in } \le x \text{ minutes}}{\text{Number of calls answered}} \quad (1)$$

$$ \begin{array}{l} \text{minimize } TP \\ \text{subject to } QoS_x \ge QoS_{x,minimum} \end{array} \quad (2) $$

As uncertainties exist within the forecasting phase prior to the staffing process, errors in predicted demand could lead to planning intervals being understaffed or overstaffed, which would have a direct effect on the QoS and TP metrics we aim to optimize. Therefore, it must be taken into account when generating staffing under uncertainty.

Owing to its ease of use and fast computational execution, Erlang C is one of the most commonly used call center staffing formulae today. When it comes to our non-conventional call center study case however, the model fails when performing the agent staffing process. As underlying assumptions regarding call arrivals and handling time distributions are required for the model to properly evaluate the performance of staffed intervals, non-conventional call centers challenge the validity of these assumptions. In addition to the model not considering caller abandonment, complex internal structures of modern call centers make it difficult to find suitable queueing models. A relay call center is a prime example of non-conventional call centers, where two different interested parties are linked through the use of telecommunication channels. This relay call center therefore plays the man-in-the-middle role, providing service to callers who wish to connect with third party establishments.

Simulation models provide accurate performance evaluation and can be used to solve the agent staffing problem through simulation-based optimization if built using some fast-running coding language. However, if the model is not initially designed for the purpose of workforce optimization, it may fall short when it comes to the agent staffing process due to long execution run times when optimizing.

Despite the problem of resource planning in call centers being a very well studied subject of research throughout the years, not much work has been published regarding the usage of Machine Learning (ML)-based approaches for solving this problem. With the recent increase in the usage of ML models and the abundance of data due to advanced log recording systems, it is only natural to wonder how these models would advance the call center domain of research. With the aim of solving the agent staffing problem within a relay call center for the deaf community and learning purely from historical data, we propose an approach utilizing a feed-forward Artificial Neural Network (ANN) model. This model provides fast computational execution similar to the Erlang C model, accurate performance evaluation similar to simulation models and has no underlying assumptions. Non-conventional call centers within which assumptions of exponentially distributed inter-arrival times and handling times do not necessarily hold can therefore opt for these machine learning-based models. We then compare the performance of our approach to that of the classical Erlang C model and prove the former's staffing superiority.

In terms of this article's layout, Section 2 presents a literature review on the staffing problem within call centers, as well as insights into the more used ML models and sources of uncertainty within ML applications. Section 3 presents the functionality of ANGUS, our study
case call center, in addition to the available historical database and the
particularity of our study case in comparison with typical call center
studies. In Section 4, we introduce our ML-based approach for agent
staffing. Section 5 performs a benchmark for daily QoS predictions to
compare the predictive performance of our ML-based model to that
of Erlang C. In Section 6 and under no demand uncertainty, the poor
staffing performance of the Erlang C model with respect to the more
performant ML-based approach is portrayed through simulation. In
Section 7 and considering demand uncertainty, our model's generated
staffing are shown to meet our global QoS objective criteria. Finally,
Section 8 concludes on the findings of the article, as well as some future
perspectives of interesting lines of research that can be pursued.

## 2. Literature review

When it comes to call center analytical performance evaluation
models, one of the most currently used methods is the Erlang C queue-
ing model, which was first introduced in Erlang (1917). This model
was later modified in Palm (1957) to include caller abandonment,
leading to the so-called Erlang A model. Many different variations of
queueing models exist today. Some of these variations handle multi-
class systems, in which customers possess different levels of priority.
Other models assume different underlying assumptions on call arrival
processes, call handling times and customer abandonment times. All
these analytical models provide closed-form expressions for the service
level within single-skill call centers. No such expressions exist however
for the case of multi-skill call centers, which then necessitates the use
of another performance evaluation method: Discrete-Event Simulation
(DES). Tanir and Booth (1999) is one of the earlier articles which used
DES for the performance evaluation of a call center.

For the case of large call centers with several hundreds of agents,
researchers were able to develop many-server approximation models
for staffing. The well-known square-root safety staffing rule based on
the Erlang C model was first observed by Erlang (1948). This rule
was then formalized by Halfin and Whitt (1981). Later on, a body
of literature using many-server approximation models was pioneered
by Borst et al. (2004). Their work was then extended in Mandelbaum
and Zeltyn (2009) by including abandonments using the Erlang A
model. The optimality of different operating regimes is then subject
to study. The square-root staffing rule is later refined in Zhang et al.
(2012).

Another form of many-server approximation is that carried out
by Harrison and Zeevi (2005). They introduced the static and fluid
approximation model for staffing large call centers based on stochastic
fluid models. This approximation was then analyzed in Bassamboo et al.
(2006) and used in Bodur and Luedtke (2017), which incorporated it
into a two-stage stochastic IP problem, using an improved Benders cuts
algorithm by mixed-integer rounding for staffing a call center under
arrival rate uncertainty.

One of the works using simulation for performance evaluation is the
study carried out by Atlason et al. (2008), which uses DES with an an-
alytic center cutting-plane algorithm to search for improving solutions.
Their approach is then extended by Cezik and L'Ecuyer (2008) to the
case of multi-skill staffing. Avramidis et al. (2010) proceeds to extend
this same approach to staffing over multiple periods of time, such as
over a day.

In order to deal with the various uncertainties present within the
call center environment, stochastic programming can be used within
our optimization algorithm. Gurvich et al. (2010) proposes a new
chance-constrained formulation of the staffing problem, a variety of
stochastic programming, using the Erlang A model for evaluation. Rob-
bins and Harrison (2010) solve the staffing problem by using an Erlang
A model linear approximation for service level evaluation within a
two-stage mixed integer stochastic program. They also assume global
service level agreements modeled as soft constraints. Gans et al. (2015)

utilize the Erlang A model within a two-stage recourse program, an-
other variety of stochastic programming used for decision-making after
new information regarding the uncertain variables are obtained. Ak-
tekin and Ekin (2016) consider uncertainties within the arrival rate,
handling times and abandonment times. They introduce staffing strate-
gies for the Erlang A queueing system using augmented probability
simulation.

As for recent works within the agent staffing domain, Mattia et al. (2017) perform capacity allocation between inbound and back office using a two-stage robust integer program. A branch and cut algorithm is used to search the possible space of solutions. Zan et al. (2018) develop a methodology for optimally updating call forecasts and staffing levels in a later period based on newly obtained information. They utilize the Erlang C model within a two-stage stochastic optimization program. Ta et al. (2021) also proposes a two-stage stochastic program approach to modify staffing upon having more accurate forecasts. They however use simulation for evaluation, and linear cuts by mixed integer rounding inequalities and L-shaped decomposition to search the solution space. Alsamadi et al. (2022) use DES to simulate different agent staffing within a call center for the deaf community. Their results indicate the possibility of experienced planning supervisors improving the call center's performance metrics through manually editing existing staffing with the support of simulation-based predicted metrics.

Within the domain of call centers, we observe that usage of machine
learning algorithms has been limited to the demand forecasting phase
of agent staffing. In the work of Albrecht et al. (2021), different
statistical and machine learning models are experimented with to pre-
dict future incoming volumes of calls. A recent comparison between
classical statistical models and more advanced ML models is performed
in Chacón et al. (2023), where a benchmark is carried out. Classi-
cal models include Holt-Winters, exponential smoothing and SARIMA.
ML models include recurrent neural networks and gradient boosting
approaches.

Beyond the scope of call centers however, machine learning ap-
proaches such as reinforcement learning have proven competitive with
classical heuristics when it comes to resource planning. Chinchali et al.
(2018) proposes an RL-based scheduler which is capable of optimally
scheduling Internet of Things (IoT) traffic by dynamically adapting to
cellular traffic variation. Utilizing four weeks of real network data from
downtown Melbourne, Australia, they demonstrate the capability of
mobile networks to carry more data with minimal impact on existing
traffic. Their approach proves efficient as it outperforms other heuristic
approaches by more than a factor of two. In the field of job schedul-
ing, priority dispatching rules are widely used to address real-world
problems. In the work of Zhang et al. (2020), an approach using an
end-to-end reinforcement learning agent is proposed to automatically
learn these rules. Based on their experiments, the agent is capable of
learning rules of high quality and strongly performing with respect
to the currently best existing priority dispatching rules. Hubbs et al.
(2020), online dynamic scheduling in the presence of uncertainty is
carried out within the scheduling process of chemical production using
deep reinforcement learning. Based on their obtained results, they
advocate for the integration of data-driven methods with model-based
mathematical optimization approaches. As for specifically planning
human resources, Lazebnik (2023) employs an agent-based simulation
combined with deep RL to address the hospital staff and resource
allocation problem. Their results indicate the superiority of the pro-
posed approach over previous resource allocation algorithms in terms
of patient treatment success and cost-effectiveness.

Despite recent works further exploring the field of stochastic pro-
gramming and robust optimization, these works still utilize classical
Erlang and simulation models for evaluation. This serves as motivation
for our research study to identify other means of performance evalua-
tion that may compete with or even potentially outperform queueing
and simulation. Motivated by the impressive capabilities of data-driven
machine learning models throughout the literature and the abundance
of data throughout different service-providing industries including call centers, the contribution of this study is to carry out the call center agent staffing problem within our study case call center using both classical and advanced ML-based performance evaluation models.

With these machine learning models having no underlying assumptions on system variables compared to queueing models, or possibly complicated modeling processes compared to simulation models, they could possibly open new doors to agent workforce planning. Li et al. (2019) represents the current state of the art. An ML model for QoS prediction is trained on the results of a simulation model, after which a local search algorithm is used to solve the combined staffing and scheduling problem. Such ML-based approaches allow us to perform planning of multi-skill call centers of different sizes over different user-defined planning horizons. To position our research within this framework, we also adopt an ML-based approach with the focus being drawn to staffing. Instead of training on the output of a simulation model, we entirely rely on training using real data. Machine learning models rely heavily on having sufficiently large datasets for training, with the required dataset size depending on the complexity of the model being trained, the presence of seasonal patterns in the data and the level of noise within the dataset. We also do not operate on the daily level using the entire agent schedule as the input to our model, but we rather operate on the time slot level where we train the model on features including the assigned number of agents for each slot, the predicted number of incoming calls and the predicted average handling times. Indeed, an approximation of the QoS value as a function of the daily agent staffing vector is required to proceed with the optimization process. Multiple methods of approximating this value exist, including queueing models, simulation and machine learning-based models. Moreover, in lieu of a deterministic optimization procedure, we consider prediction uncertainty propagation within our search heuristic to generate agent staffing that meet our QoS targets. As shown in Robbins et al. (2010), the widely used classical Erlang C model works best in large call centers with relatively low agent utilization rates which might not be the case of the small ANGUS call center. Significant errors in predicting system performance could therefore result from the usage of this model. Our ML approach therefore presents an alternative solution which does not suffer from such errors and would therefore properly serve our small call center's needs.

In terms of choosing an ML-based model for performance evaluation, researchers can choose from a variety of possible supervised learning models. Starting off with simple linear regression as a baseline model, one can then proceed to some more advanced models such as decision tree models, for which an overview is provided in Kotsiantis (2013). Random Forests (RF) as explained in Liaw et al. (2002) are essentially an ensemble of decision tree models which can be used to carry out the regression task. Gradient boosting models such as the XGBoost model introduced in Chen and Guestrin (2016) have also proven to be very efficient. Haykin (2009) serves as an important source on artificial neural network architectures, such as the Multi-Layer Perceptron (MLP) model which utilizes the back-propagation algorithm. Some other possible options include K-Nearest Neighbors (KNN), Support Vector Regression (SVR) and Gaussian Process Regression (GPR) presented in Altman (1992), Awad et al. (2015) and Schulz et al. (2018) respectively.

Within machine learning, uncertainties are generally categorized as aleatoric or epistemic as shown in Kendall and Gal (2017). Aleatoric uncertainty stems from the data's inherent randomness. As predictions are performed prior to the planning process, the inescapable errors within these predictions introduce aleatoric uncertainties. In order to deal with this type of uncertainty, prediction intervals are to be generated in order to provide the planning supervisor with a sense of confidence in the provided performance metrics. Based on these prediction intervals, different planning techniques could be adopted, each of which provides different levels of probability in meeting our performance objectives. These aleatoric uncertainties within the training data used in the

staffing process can be managed using the Monte Carlo uncertainty propagation method based on the work of Metropolis and Ulam (1949). Epistemic uncertainty arises from model uncertainty as our chosen model would not perfectly align with the underlying dynamics of the call center's operation. To manage these epistemic uncertainties, user-defined hyperparameters which dictate the performance of the chosen ML model are optimized. As simple grid search techniques are dis-advised due to long running times, automatic hyperparameter optimization algorithms can be used. Akiba et al. (2019) introduces Optuna, a Bayesian optimization software capable of automatically choosing the set of hyperparameters to use, based on a user-defined metric of performance. Another Bayesian Optimization-based software is HyperOpt, which was introduced in Bergstra et al. (2013). Properly chosen hyperparameters would result in models being as close as possible to capturing the real-life system's dynamics. Selecting a sufficiently large dataset for the model to learn from is also essential to handling epistemic uncertainty.

## 3. ANGUS call center and data

ANGUS is an inbound relay call center for the deaf community. It occupies the man-in-the-middle role by connecting a calling deaf person to a third party public service office which is not equipped to handle incoming calls from the deaf community. Within this section, we explain how the services proposed by the ANGUS call center work, the available database of historical calls and the particularity of our study case call center.

### 3.1. ANGUS functionality

Having a pool of available sign language interpreters, also commonly known as agents within the call center research literature, ANGUS provides each caller with a dedicated agent to facilitate the process of communication, as shown in Fig. 2. As the deaf person tries to video call ANGUS, he or she is placed within a waiting queue if no agents are currently available to respond to their call. After a certain waiting time, callers may decide to abandon the queue. If they do not however, they are eventually assigned an agent to provide them with the service they require. Due to the nature of ANGUS being a relay call center, the conversation to be carried out naturally consists of three parties: the deaf caller, the ANGUS agent and the third-party public service employee. An initial discussion takes place between the deaf caller and the ANGUS agent using sign language, for the purpose of explaining who the deaf person would like to connect to and what they would like to achieve. The ANGUS agent would then proceed to call the third-party office, and translate in real-time the exchange between the third-party employee and the initial caller using both the French sign language and the French spoken language until the call terminates.

As for the planning process within ANGUS, a planning supervisor must each month prepare the planning of the next month. In order to do so, historical data regarding the volume of incoming calls, previously planned agents and resulting performance metrics is to be used.

### 3.2. Historical data

In addition to its role of properly routing calls from the waiting queue to available agents, the Automatic Call Distribution (ACD) system employed at ANGUS is used to record data regarding the current status of the call center. Arrival timestamps for incoming calls, call handling times, available agent levels and resulting performance metrics are all continuously recorded from 9:00 AM till 6:00 PM every weekday since the year 2020. Based on a planning interval of 15 min, the aggregated available database consists of around 40k data points up till the current date of this study, which is sufficiently large enough to train our ML neural network models. With the previously mentioned opening hours of the call center, a working day would thus consist of
<figure><img src="image_4.png" /><figcaption>Fig. 2. ANGUS services schema.</figcaption></figure>

thirty-six 15 min time slots. Leveraging this database, we are able to analyze the incoming volume of calls per time slot as shown in Fig. 3. We notice a first peak in call arrivals at around 11:00 AM, followed by a sharp decline until 12:30 PM, and then climbing back up to a second peak at around 2:00 PM, followed by another decline until the end of the day. It is this fluctuating arrival rate that motivates the process of agent staffing at different levels throughout the workday.

### 3.3. Call center particularity

The study case provided by ANGUS features some characteristics which are not usually present within the classical call center literature study cases. As seen in the schema of Fig. 1, due to the nature of ANGUS being a relay call center, an additional waiting queue is present on the third party side of the service resulting in potentially longer waiting times. The following Fig. 4 presents the call timeline of the ANGUS relay call center. Waiting time #1 corresponds to the deaf caller's time spent in ANGUS's waiting queue. Handling time #1 represents the initial discussion that takes place between the deaf caller and the ANGUS agent, in which they explain to the agent which third party they would like to connect with as well as the purpose of the call. Waiting time #2 corresponds to the deaf caller and ANGUS agent's time spent in the third party's waiting queue. Handling time #2 represents the discussion that takes place between the deaf caller and the third party employee, with the ANGUS agent playing the man-in-the-middle role by translating in real-time from the sign language to the spoken language and vice-versa.

Since the call termination is linked to the third party service, our performance evaluation focuses on the first ANGUS side of the call. We do however use handling time data for the entire call during our study due to the format of the available database. In addition, due to the possibility of calls not being answered by the third party, the resulting distribution of call handling times is a combination of relatively short calls for which the relay process was unsuccessful, and relatively long calls for which the relay process was successful as shown in Fig. 5. As the real-time translation process is heavily mentally-demanding, breaks with varying start times, end times and durations are taken by the ANGUS agents depending on their current workload. These non-standardized breaks can have severe impacts on the performance of planned intervals. All of the previously mentioned characteristics of the ANGUS study case make it difficult to choose a suitable queueing model and challenge the underlying assumptions of these models on the call center's functionality. We therefore decide to include the classical Erlang C model within our benchmarks in Sections 5 and 6 as it is one of the most commonly used call center staffing formulae.

Some additional characteristics can be seen on the planning level. Once the agent staffing is carried out, the planning supervisor has little to no control in terms of changing this staffing during the planned

month, despite new data coming in and overstaffing or understaffing being observed. It is therefore crucial for the assigned staffing to properly reflect the required number of agents throughout the planning period. Furthermore, as the ANGUS agents are chosen to be highly skilled interpreters to meet the requirements of the service they are proposing, the TP metric is to be highly valued in order to avoid any possible financial issues within the call center.

## 4. ML-based approach

With the purpose of potentially replacing classical queueing models and simulation models, our approach consists of an ML model that neither holds underlying assumptions on the arrivals of calls and handling times, nor requires extensive modeling of the complex internal workings of the call center. Learning directly from the available database, predictions on the performance evaluation of staffed planning intervals denoted as $QoS_{slot}$ are carried out and later used within the general planning process, typically within the framework of a staffing heuristic.

The model is trained on the real historical data of our study case call center, which inherently includes the natural variability of call arrival times and call handling times within each 15 min slot of available data. During the training process, the model attempts to learn these fluctuations in order to generalize its $QoS_{slot}$ predictions based on the observed set of model inputs.

As for the overall objective specified by the planning supervisor which our generated staffing is to achieve, a global $QoS_{day}$ target of 90% is assumed. Appropriately staffing the planning period must provide enough agents to achieve this global QoS goal, but not too many agents that would increase the TP metric beyond the limits of what the call center can provide. For the choice of ML models to benchmark, we decided to first include linear regression to serve as a simple data-driven model to which we can compare the other more complex models. We also include XGBoost due to its widespread success and high performance with respect to other machine learning-based models. Finally, we include the ANN model as our primary ML model to be benchmarked.

### 4.1. Model architecture and feature engineering

The ANN model architecture shown in Fig. 6 consists of two hidden layers. Taking inspiration from the classical Erlang C model, the number of agents, number of calls and average handling times per slot showed significant correlation to the predicted target, $QoS_{slot}$, and were consequently included in the inputs of the model. As shown in Deslauriers et al. (2007), neglecting transient effects from previous time slots can introduce errors into the performance evaluation process. We therefore decided to include the first lag values of each of the previously mentioned inputs, as well as the resulting QoS of the previous time slot within the model inputs.

During the feature engineering phase of our ML-based performance evaluation model, lag features were found to significantly improve the prediction performance of our model.

As different time slots of the day could exhibit different functionality patterns within the call center, we include the trigonometric transformations of the time slot value of the day being planned which further assists the model learning process. The ReLU activation function is chosen for both hidden layers in order to better capture the non-linear relationships that could exist between our model inputs and the target QoS output. As for the loss function, the mean squared error was chosen and minimized during the ANN training process using the Adam iterative optimization algorithm. Finally, a dropout of 20% is implemented within the two hidden layers in order to avoid overfitting on the training data.

A typical usage of the designed model within the context of agent staffing would be to estimate the QoS performance that would result from the assignment of a certain number of agents to a 15 min time slot
<figure><img src="image_5.png" /><figcaption>Fig. 3. Intra-day Average Call Arrivals based on data statistics of workdays between January 2020 and December 2023.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 4. ANGUS call timeline.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 5. Handling time distributions of relayed and Non-Relayed Calls based on data statistics of workdays between January 2020 and December 2023.</figcaption></figure>

of a day being planned. In a previous step, a demand forecasting phase is carried out in which the number of calls arriving during this time slot as well as their average handling times are predicted. Predictions of the number of call arrivals and their average handling times for the preceding time slot are also generated. These different generated predictions along with a chosen number of agents are then all inserted within the input layer of the model to estimate the previously mentioned QoS performance. In the event where the estimated QoS does not align with our desired performance criteria, the chosen number of agents is either increased or decreased until our target is reached. Based on a chosen optimization heuristic, this process is applied to all thirty-six 15 min time slots of the day until an agent staffing is obtained.

## 4.2. Model training and hyperparameter optimization

In order to train a generalized model using the available historical database, hyperparameters are optimized by dividing our data into

training and validation sets using backtesting, a form of time series cross-validation that ensures the temporal integrity of the problem, so that no future data is leaked into the training process. As the global $QoS_{day}$ objective is effected more by time slots with higher volumes of calls than ones with lower volumes, our hyperparameter optimization process runs with the objective of minimizing the Weighted Mean Absolute Error (WMAE) of predictions. In order to assign the appropriate weights, predictions for the volume of incoming calls are generated in a prior step to the ML model implementation. The weights are then computed as the ratio between calls arriving per time slot and the total number of calls throughout the day. As for the choice of optimization algorithm, we decide to adopt the Tree-structured Parzen Estimator (TPE) Bayesian optimization algorithm used in the optimization software Optuna, due to it is efficient and fast convergence towards a well-performing set of hyperparameters. The hyperparameters to be optimized within our ANN model are listed as follows:

* Neurons<sub>1</sub>: The number of nodes used within the first hidden layer.
<figure><img src="image_8.png" /><figcaption>Fig. 6. ANN model architecture.</figcaption></figure>

<table><caption>**Table 1**

Hyperparameter optimization results.</caption><thead><tr><td>Hyperparameter</td><td>Value range</td><td>Optimized value</td></tr></thead><tbody><tr><td>Neurons<sub>1</sub></td><td>(1–500)</td><td>389</td></tr><tr><td>Neurons<sub>2</sub></td><td>(1–500)</td><td>361</td></tr><tr><td>Batch Size</td><td>(32–256)</td><td>32</td></tr><tr><td>Learning Rate</td><td>1e-3, 1e-4, 1e-5, 1e-6</td><td>1e-5</td></tr><tr><td>Epochs</td><td>(10–100)</td><td>40</td></tr></tbody></table>

<table><caption>**Table 2**

$QoS_{day}$ 6-month prediction benchmark results.</caption><thead><tr><td>QoS evaluation model</td><td>Average $QoS_{day}$ MAPE (%)</td></tr></thead><tbody><tr><td>Erlang C</td><td>4.42 (baseline)</td></tr><tr><td>LR</td><td>3.36 (-24%)</td></tr><tr><td>XGB</td><td>2.66 (-40%)</td></tr><tr><td>ANN</td><td>2.55 (-42%)</td></tr></tbody></table>

* Neurons<sub>2</sub>: The number of nodes used within the second hidden layer.

* Batch Size: The number of samples processed simultaneously during the model training phase.

* Learning Rate: The step size that determines how much to adjust the model's internal weights.

* Epochs: The number of times the model iteratively trains on the entire training dataset.

The results of this optimization process can be seen in Table 1.

### 4.3. Staffing heuristic

Within our staffing heuristic, a global $QoS$ objective in the form of $QoS_{day}$ is assigned. This global $QoS$ value is computed as the weighted sum of $QoS_{slot}$ predictions, with the same weights used within the computation of the previously mentioned WMAE metric. The heuristic works by first computing a greedy agent staffing that satisfies the constraint of $QoS_{slot} = 100\%$ for all the slots of the day being planned. This constraint ensures that $QoS_{day} \ge 90\%$, with a certain degree of overstaffing taking place due to the nature of the greedy initial staffing. As it does not take into account the weight of each slot of the day, lower weighted slots are staffed beyond necessary, as we hold the possibility of reducing the number of agents within less important slots and compensating their resulting low-valued performance with that of more important slots with higher-valued performance. The heuristic proceeds to iteratively reduce the number of agents staffed within each slot of the day and therefore the TP metric, while ensuring our $QoS_{day}$ target remains achieved. We thus allow $QoS_{slot}$ to fluctuate throughout the slots of the day, with higher-weighted slots compensating lower-weighted slots in terms of performance. A minimum level of $QoS_{slot}$ is fixed however in order to avoid certain slots from being completely neglected during the staffing process due to their relatively low arrival rate.

In order to deal with the uncertainty represented by the prediction errors within our model inputs, a Monte Carlo uncertainty propagation approach is applied, where the ANN model is used to estimate the $QoS$ performance by running multiple replications of the generated agent staffing. During each of these replications, the inputs of the ANN model

are varied based on noise sampled from the residual error distribution of call arrival and average handling time. These residual errors are obtained by comparing previous demand forecasting predictions with what actually took place within the call center. Assuming these errors might reproduce within future planning periods, the Monte Carlo approach provides us with a possible distribution of the target daily $QoS$ metric rather than a point estimate. The 0.05 and 0.95 quantiles of this generated distribution are then used to obtain a 90% prediction interval for our objective metric. This interval can then be used to decide on which staffing to apply on the real-life system hereafter.

### 5. Model benchmark: $QoS_{day}$ prediction

Utilizing the previously constructed ML model for $QoS$ evaluation per planning time slot, we are able to generate predictions on the $QoS_{day}$ values based on our available database. Indeed, the objective within this section is to perform a benchmark in order to decide which model would best predict the performance metrics resulting from the application of a different agent staffing. As for the set of competing models, we decide to include the classical Erlang C model to serve as a baseline of comparison due to its mass usage within call centers today. We also include the simple Linear Regression (LR) model, the XGBoost (XGB) model and our previously designed ANN model. In order to validate the consistency of our found results, we perform this daily $QoS$ prediction benchmark on six different months, from January 2023 till June 2023. We decide to illustrate prediction errors in the form of a Mean Absolute Percentage Error (MAPE) as it is easy to comprehend and allows comparison between different months which could hold different volumes of incoming calls. Fig. 7 shows the $QoS_{day}$ MAPE for each of the tested models over each of the tested months. Table 2 presents the 6-month average MAPE benchmark results. The ANN model achieved the best performance having the lowest average MAPE, followed by the XGBoost model, the LR model and finally the Erlang C model. The Erlang C model performing the worst within this benchmark could be traced back to ANGUS's call center particularities specified in Section 3.3 and their possible rejection of the model's underlying assumptions.
<figure><img src="image_9.png" /><figcaption>Fig. 7. $QoS_{day}$ MAPE prediction monthly results.</figcaption></figure>

## 6. Model staffing capacity benchmark

As the previously tested models are to be used within the agent staffing problem, it is crucial to check whether their performance in terms of predicting the $QoS_{day}$ would translate into the proper construction of an agent staffing capable of achieving our global $QoS_{day}$ objective and reducing TP hours. We therefore assume the planning supervisor's role of staffing the month of May 2023 to test the Erlang C, LR, XGBoost and ANN models. Within this section, our initial goal is to examine the staffing capacity of the previously mentioned models when all information regarding the incoming volumes of calls and average handling times is given. Such a case would only occur if our call and handling time prediction models are flawless, a scenario that is most likely not to appear in a real-life application. The reasoning behind this experiment is that a model that is not capable of generating suitable agent staffing with perfectly predicted primitives would most likely not be able to do so with imperfectly predicted primitives. As for our choice of the planning month, May was chosen due to its abundance of national holidays. These holidays further complicate the staffing problem due to their impact on our call arrival predictions for periods before and after holidays. Applying the staffing heuristic mentioned in Section 4.3, each model generates an agent staffing for each business day of the month of May 2023. Within the applied heuristic, we do not include the uncertainty propagation method as no uncertainty exists within this experiment since we possess all the needed information. In order to evaluate the performance of the different plannings, we proceed to simulate these plannings on a validated discrete-event simulation tool which models ANGUS's internal structure and serves as our image of what would happen in reality if such a planning was executed.

Each simulation launched is that of a single business day at the study case call center starting at 9:00 AM and ending at 6:00 PM. As the month of May planned throughout this article consists of 19 business days, 19 consecutive daily simulations are launched. Within the inputs of this simulation model, we insert the real number of calls that actually took place within the month of May as well as the agent staffing to be tested. Due to the stochastic nature of the process being simulated, 20 replications are launched for each agent staffing. As for the reasoning behind our choice of 20 replications, the simple graphical method of Robinson (2014) was used to determine how many replications would be needed for the mean simulated daily $QoS$ to stabilize close to the estimated convergence line. During each of these replications, even though the number of incoming calls within each 15 min time slot remains the same, the arrival times of these calls are different as their inter-arrival times are sampled from an exponential distribution. Additionally, call handling times are also sampled from an empirical distribution of real data and therefore vary from one replication to another. Within the output of our simulation model, and in order to account for these variations, results are available in the form of 90% and 95% confidence intervals, minimum, maximum and mean values

<table><caption>Table 3
Model staffing capacity benchmark simulation Results.</caption><thead><tr><td>QoS evaluation model</td><td>Average simulated QoS<sub>day</sub> (%)</td><td>Total TP (hours)</td></tr></thead><tbody><tr><td>Erlang C</td><td>84</td><td>1344.00 (baseline)</td></tr><tr><td>LR</td><td>93</td><td>1720.50 (+28%)</td></tr><tr><td>XGB</td><td>90</td><td>1621.25 (+21%)</td></tr><tr><td>ANN</td><td>90</td><td>1572.00 (+17%)</td></tr></tbody></table>

of $QoS_{day}$ over the 20 simulated replications. The simulation model $QoS_{day}$ results reported within our study are the mean $QoS_{day}$ values.

Fig. 8 shows these mean $QoS_{day}$ simulation results for each of the different models' generated staffing. The average simulated mean $QoS_{day}$ results as well as the TP hours for the entire month of May are reported in Table 3.

As can be seen in the reported results, the Erlang C model tends to understaff the call center and therefore not acquire the 90% $QoS_{day}$ objective. The LR model tends to overstaff the call center, achieving the $QoS_{day}$ objective but possibly assigning too many TP hours. Both the XGBoost and ANN models manage to successfully attain our $QoS_{day}$ objective. However, the XGBoost model is found to employ 21% more TP hours than the Erlang C baseline model while the ANN model only employs 17% more TP hours. The superior performance of the ANN model in terms of $QoS_{day}$ prediction shown in the previous Section 6 has indeed translated into its superiority in terms of agent staffing when all information is given.

## 7. Model future planning

Within a real-life application of the agent staffing ML approach, the planning supervisor does not possess exact information regarding the future. Predictions are generated beforehand in order to assist within the planning process, but these predictions as previously mentioned are most likely imperfect. Within this section, we attempt to validate the agent staffing potential of the ANN model using predicted data regarding the future rather than real data. Based on the uncertainty propagation method applied within the staffing heuristic of Section 4.3, the generated staffing should result in mean $QoS_{day}$ values between the upper and lower 90% prediction interval bounds. As before within Section 6, our simulation model inputs include the real number of calls that arrived during the month of May 2023. Fig. 9 shows the $QoS_{day}$ values obtained from running the agent staffing within our simulation model, as well as the TP hours for the planned month. Our ML-based approach provides an average $QoS_{day}$ of 89.4%, with our global objective being 90%. During some days, the simulation model returned $QoS_{day}$ values which are lower than our predicted mean value, indicating the understaffing of the call center. On other days, the simulated value is above our predicted mean values, indicating the overstaffing of the call center. Over the entire month of May, most
<figure><img src="image_10.png" /><figcaption>Fig. 8. Simulated $QoS_{day}$ results for model-generated staffing of May 2023.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 9. Simulated $QoS_{day}$ results for ANN-generated staffing under uncertainty of May 2023.</figcaption></figure>

of the generated staffing have simulated performance values that lie within our prediction intervals. It must be noted that generating better call arrival predictions prior to the staffing process would yield more accurate prediction intervals. From a managerial point of view, based on the staffing performance evaluation, the planning supervisor may choose to apply the ML-generated staffing on the real-life system, taking into account the possible minimum and maximum $QoS_{day}$ values that could result.

## 8. Conclusion and future perspectives

In an effort to test the performance of an ML-based approach for agent staffing within an inbound call center for the deaf community, ANGUS, a series of benchmarks and tests have been carried out using the classical Erlang C model, a simple LR model, the XGBoost model and a more complex ANN model. Results indicated the superiority of the ANN model when it comes to $QoS_{day}$ prediction. Within an experiment where all future information is available, this superiority then translated into the ANN model's agent staffing superiority with respect to the other tested models. Finally, within a real-life scenario experiment using predicted future information and considering uncertainty, the ANN model managed to generate agent staffing which resulted in an average $QoS_{day}$ roughly equivalent to our global $QoS_{day}$ objective.

With the current technological advancements in the field of call center operations such as generative AI-based chatbots, one may question the relevance of employing and staffing human call center operators. Indeed, at a time where AI chatbots help in managing incoming volumes of calls, certain operator tasks remain non-automatable within certain call center sectors. One example is that of emergency call centers where the need for human operators to receive and manage incoming emergency calls is paramount. The proper planning of these operators would both ensure a high level of quality-of-service performance and

the proper management of the call center operational resources. We therefore realize the continued necessity to optimize the different steps of agent planning, including agent staffing.

After establishing the importance of optimized agent staffing, we may proceed to assess the value of adopting further machine learning-based models within the call center. Despite the ability of these models to automatically generate agent staffing levels without the intervention of a human planning supervisor, this planner's participation to validate and adjust these staffing levels is pivotal. Indeed, within the uncertain environment of call centers, certain events which are not accounted for by the models may affect the overall performance of the previously generated staffing levels. It is the role of this human planning supervisor to account for these events, with the automatically generated staffing serving as a starting point to both speed up and increase the efficiency of the planning process. The staffing process is then reformulated into a human-in-the-loop procedure, with our algorithms being implemented within a decision-support system. Such a system would assist the planning supervisor in carrying out future planning tasks and paving the way for more efficient call center operations to take place.

In terms of possible future work, one can imagine a resilience and robustness analysis taking place on our generated agent staffing. As our final decision of which agent staffing to choose based on the generated prediction intervals focused on their mean value being close to our global objective, further work on how these intervals can be used within the scope of decision-making for the planning supervisor can be carried out.

Furthermore, due to $QoS_{slot}$ as a function of agent staffing following a rather logistic relationship, an interesting inclusion within the list of benchmarked models for performance evaluation could be a logistic regression model. Such a model could carry interesting results to compare with those of our proposed ANN model. As previously mentioned throughout this study, simulation-based optimization could
be used to solve the agent staffing problem and potentially return some promising results. A future work attempting to build a simulation model using some fast-running coding language could also compete with our proposed approach. The Erlang A model of Palm (1957), an improvement to the Erlang C model which includes abandonment, could also be added to the list of models to be benchmarked. Despite the staffing heuristic implementation of our proposed ANN model successfully achieving our initial primary $QoS_{day}$ objective of 90% within the benchmark of Section 6 and being in close agreement with our target within the results of Section 7, further exploration of different staffing heuristics could result in the same obtained performance levels while reducing the total employed TP hours.

Finally, as this article deals with the agent staffing problem, a future work which considers the combined agent staffing and agent scheduling problem through ML-based techniques could be pursued.

## CRediT authorship contribution statement

**Samer Alsamadi:** Conceptualization, Methodology, Software, Writing – original draft. **Cléa Martinez:** Supervision, Writing – review & editing. **Canan Pehlivan:** Writing – review & editing. **Nicolas Cellier:** Writing – review & editing, Data Curation. **Oualid Jouini:** Writing – review & editing. **Yiping Fang:** Writing – review & editing. **Benjamin Legros:** Writing – review & editing. **Franck Fontanili:** Supervision, Writing – review & editing.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgments

This project is co-funded by the French call center Elioz and by the Occitanie region of France.

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