1 Introduction

The financial sector, an industry historically steeped in tradition, is presently undergoing a paradigm shift characterized by an explosion of products, services, and information, posing significant challenges in processing and management. In a society that is increasingly structured and demanding, individuals are eager to evaluate and compare various products and services. This has led to a complex landscape where designing tools to gauge and enhance the service level and the number of served clients is becoming ever more intricate, especially with the introduction of intangible products where the delivery environment plays a pivotal role in adding value (PwC 2019; Maechler et al. 2018; Gómez 2008).

Traditionally, customer service in the banking sector relied heavily on physical branches, call centers, and ATMs, offering standardized solutions with a primary focus on metrics such as loyalty, retention, and referrals. The modern iteration, however, leverages digital technologies such as mobile applications, web platforms, chatbots, and social media, offering 24/7 accessibility and personalization, thus shifting the focus towards customer experience metrics such as ease, speed, and delight. This shift has not only introduced innovative business models but also intensified competition, with new entrants like FinTechs and Big Tech challenging the incumbents with lower costs and innovative solutions (Patel 2023; PwC 2019; Maechler et al. 2018).

Amidst this evolving landscape, it becomes imperative for financial institutions to adapt and innovate, aligning their services to understand better and anticipate customer needs and demands. Thus, establishing technological innovations within existing business models emerges as a crucial strategy to maintain a competitive edge (Aguirre 2015).

In this context, the Bass diffusion model, proposed by Frank Bass in 1969, presents itself as a potent tool to study the adoption patterns of new products or technologies within a population (Bass 1969). Based on Everett Rogers’ Diffusion of Innovations theory, this model delineates two types of adopters: innovators and imitators. The Bass diffusion model encapsulates the nuances of adoption patterns and facilitates the analysis of marketing strategies or environmental factors in the diffusion process.

The Bass diffusion model has found applications across diverse fields, including consumer durables, industrial goods, pharmaceuticals, telecommunications, energy, and social innovations (Baviskar et al. 2023; Horvat et al. 2020; Kusuma and Suryanegara 2019). Recent developments of the Bass diffusion model include explanatory parameters and extensions to incorporate system dynamics and feedback loops, which enable a more nuanced understanding of the diffusion process and the creation of effective interventions to promote adoption (Baviskar et al. 2023; Tani et al. 2022; Sokele and Moutinho 2018). Despite the extensive application of the Bass diffusion model in various fields, its application in understanding the evolutionary dynamics of customer service in the banking sector remains underexplored. Existing studies have primarily focused on adopting new products or technologies, leaving a gap in the literature concerning the analysis of customer service strategies and their impact on the service level, and the number of served clients in the banking sector.

This study addresses the identified research gap by dissecting the complex dynamics of customer service evolution in the banking sector and analyzing the factors that serve as drivers or barriers to diffusion. Therefore, this research aims to facilitate the formulation of strategies for more effective attraction and retention of clients. The study is developed on the following research question:

How the use and adoption of a mobile application impact the service times and the attraction of new clients in the financial sector?

To respond to the research question, we propose and implement a methodology supported by the agent-based version of the Bass diffusion model. In particular, the agent-based simulation has proven to be a tremendously helpful technique for modeling complex systems, particularly social systems (Zapata-Roldan and Sheikh 2020; Lee et al. 2019; Gilbert 2019; Ahrweiler 2017; Macal 2016; Macal and North 2014; Edmonds 2000). This type of simulation evaluates the actions and interactions of autonomous individuals within an environment, determining their effects on the system (Borshchev and Filippov 2004). Readers are referred to the works by Will et al. (2020) and Klein et al. (2018), as reviews of agent-based modeling for exploring social interactions. On the other side, the Bass diffusion model has demonstrated its simplicity and flexibility in examining the influence of information on consumers’ decisions to adopt a new product. In addition, this model perfectly captures how consumers change their choices over time (Wei et al. 2022).

The integration of both models, i.e., the agent-based model and the Bass diffusion model, allows us to determine the attraction of new clients due to better service levels. Thus, our proposed methodology deals with competing paradigms to forecast the demand effects and customer behaviors of the diffusion of innovative products or technologies (Rand and Stummer 2021; Abedi 2019; Massiani and Gohs 2015; Rand and Rust 2011). Unlike other demand prediction models, this model does not require historical sales information of the products (Rand and Rust 2004).

The main contributions of this paper are as follows:

  • We propose and implement a methodology to study customer behaviors and forecast the diffusion effect on customer adoption of a banking App due to better service levels.

  • An agent-based version of the Bass diffusion model was designed to study customer behaviors and forecast the diffusion effect on customer adoption of a banking App due to better service levels.

  • A novel methodology for service operations is developed. The proposed methodology is flexible for studying any service operation that involves interaction between entities within an artificial environment reflecting the real-world system.

  • Performance indicators, including served clients, service level, service time deviation, and average system occupancy rate, were analyzed.

  • A real-life problem with real-life data was considered for validating the proposed methodology.

The rest of this paper is structured as follows: Section 2 provides a review of the related literature. Section 3 describes the problem under study. Section 4 presents the proposed methodology, including an explanation of the mobile App, the descriptive formulation, and the simulation model employed to evaluate the effectiveness of the Application. Section 5 presents the analysis of the results, and Section 6 provides theoretical and managerial insights for academics and practitioners interested in the studied problem. Finally, Section 7 presents concluding remarks and opportunities for further research.

2 Literature review

Most real-world systems are too complex to be evaluated analytically (i.e., using mathematical methods to obtain exact information on the question of interest). Those systems must use simulation, where data are gathered to estimate the desired characteristics of the model (Law et al. 2015).

Simulation is applied to a wide range of problems and sectors, principally for the designing of manufacturing industries or modification of plants/production lines (Villanueva 2008). In the logistic sector, simulation has been used to design transportation flows of people or materials and optimization of platforms or terminals (Bernardi et al. 2020; Ozdemir and Kumral 2019; Rahimikelarijani et al. 2018; Solano-Charris and Paternina-Arboleda 2013). In addition, simulation has applications in scenarios such as hospitals, urban developments, police, banking sector, among others (Ataee Gortolmesh et al. 2020; Zapata-Roldan and Sheikh 2020; Samitas et al. 2018; Omogbai and Salonitis 2016; Krause et al. 2004).

In this context, simulation is a flexible tool that enables us to evaluate different alternatives of system configurations and operating strategies to support decision-making (Negahban and Smith 2014). The literature can distinguish several types of simulation models: discrete simulation or continuous simulation, deterministic or stochastic simulation, static or dynamic simulation, agent-based simulation, etc. da Silva Meireles et al. (2022) present a review of the most used methodologies to study customer behavior in the face of service innovations. This section presents the literature review dedicated to studies that have used simulation (1) to explore operational strategies to improve service quality and (2) to evaluate customer behaviors due to the inclusion of new products. The databases explored were Web of Science, Elsevier, and Scopus, which are considered one of the largest citation databases. The search equation was formed by the following keywords: service quality, customer behavior, acceptance of new products, and agent-based simulation. The literature review is limited to studies published between 2000 and 2022.

2.1 Methodologies for improving service quality

Generally, one of the most used tools to evaluate policies to improve service quality by considering social patterns is based on simulation. For example, Tian et al. (2022) proposed an agent-based simulation to model individual customers with different behaviors. The results showed that it is essential to assess the level of satisfaction of current and potential customers. In addition, the authors recognized the importance of developing hybrid models to capture industry dynamics and social patterns. In this context, multi-agent simulation can be used to model the behavior of the sector, considering competitors and customer behaviors.

Rand and Stummer (2021) proposed an agent-based simulation to model the behavior of customers and competitors when a new product is introduced in the market. Rand and Stummer (2021) used the Bass diffusion model to simulate the interaction between customers and competitors. The authors concluded that the integration of agent-based simulation and Bass diffusion modeling is helpful for critically evaluating the new product’s impact on the market and service quality.

Additionally, Malhotra and Mukherjee (2004) presented an analysis that demonstrated the positive impact on employee satisfaction that the inclusion of new technologies has. Findings indicated that job satisfaction and organizational commitment of employees have a significant impact on the service quality delivered. Following this research line, Jalilian et al. (2021) developed an agent-based simulation to optimize service provision. The model considered the interaction of various substitute products and how they affect service delivery. Jalilian et al. (2021) addressed a case study for a banking institution in Iran. The authors concluded that simulation allows representing the interaction between customers and new products for evaluating service quality.

2.2 Modeling of customer behaviors due to the inclusion of new products

Complementary to the previous subsection, this section focuses on modeling customer behaviors. For example, Dostov et al. (2022) developed a system dynamics model to evaluate the impact and acceptance of fast payment services. The authors recognized that the acceptance of any system depends on a maturation process in the market. Dostov et al. (2022) implemented the Bass diffusion model to model different customer behaviors. They used historical data from Great Britain to evaluate the acceptance of the fast payment service. The authors demonstrated that the proposed hybrid model can reproduce the behavior of fast payment system acceptance.

Sava and Marković (2017) developed an agent model to evaluate customer behaviors in an online marketplace. The study focused on representing the decision-making process of customers associated with the use of the online platform. The authors modeled the following parameters as a random variable: gender, customer age (integer random values between the range of 18 and 60 years), income levels, sensitivity to advertisements, information transfer between customers, and sensitivity to a specific product attribute. The proposed model was designed to evaluate the online sales system, customer behaviors, and the impact of customers on the performance of the system.

Considering the attributes of various products, Theerthaana and Sheik Manzoor (2020) studied a portfolio of investment projects offered by a financial organization. The authors presented a simulation model where they integrated the Bass diffusion model. The authors analyzed the behavior of customers when faced with a set of investment projects. The objective of the study was to evaluate the impact of the type of information transferred to customers and between customers. The results showed that investment projects have a better acceptance when the project provider shares the information risk associated with the project.

In particular, the combination of agent-based simulation and diffusion modeling has been frequently applied in the context of the supply chain (Backs et al. 2021), control and promotion of emerging firms to collaborative environments (Yu et al. 2020), product design (Zapata-Roldan and Sheikh 2020; Lee et al. 2019; Negahban and Smith 2018; Hu et al. 2018) and evaluating the impact of product innovations or operating policies on the market (Amini et al. 2012; Bohlmann et al. 2010). Despite the existing literature on agent-based simulation models, there remain opportunities for applying these models to new product diffusion (Rand and Stummer 2021).

The reviewed studies show that most studies focus on modeling clients’ decisions to purchase a product. One of the main objectives of these studies is to attract new clients and evaluate the quality of service. However, there is a lack of studies related to assessing the adoption of a complementary service that supports the provision of the business-leading service.

3 Generalities of the Colombian financial market and problem description

In the Colombian financial market, there are 27 banking establishments, 6 financial corporations, and 14 commercial financing companies, among other financial entities (Superintendencia de Industria y Comercio 2020). According to the degree of access to financial services, Bogotá is the city with the highest banking access in the country up to 98.7%, and the second one in terms of financial products up to 86.9% (Superintendencia Financiera de Colombia 2018).

As stated in the last Colombian financial inclusion report, the use of technological channels has increased over the years (Superintendencia Financiera de Colombia 2018). In fact, 2018 was the first year in which Internet operations exceeded those carried out in physical branches by handling 49.8% of all transactions (Superintendencia Financiera de Colombia 2018). Thus, ensuring digital and innovative solutions for the near future is of paramount importance for banking establishments.

Technological solutions are valuable tools for the financial inclusion of under-served populations and those groups whose needs are not being met by the traditional system (Superintendencia Financiera de Colombia 2018). However, the limited access to products and services offered by banks through virtual platforms makes bank offices necessary, for instance, the capacity of withdrawal operations and the provision of other banking services.

According to the statistics of financial institutions, client complaints, and claims are related to the insufficient capacity of bank tellers to perform a bank transaction safely and quickly. Additionally, due to the high perceived risk, building users’ initial trust is crucial to facilitating their adoption and usage of mobile banking (Zhou 2012).

Regarding this context, financial institutions must anticipate the behavior of the demand to enhance the service level and thus, retain and attract new clients, and motivate the inclusion of digital tools in the financial sector (Aguirre 2015).

3.1 Problem description

This study is carried out in a Colombian banking establishment. The bank has more than 70 years in the country and currently has 30 branches across Colombia. In this paper, the city of Bogotá is selected due to its highest level of banking activity. Here, the problem of long waiting times for clients at branch offices and the limited capacity of the system to cover all demands are addressed. The objective is to represent the current situation of three bank branches and evaluate whether or not the use of a mobile application impacts service levels for financial clients. The clients of the banking sector in Colombia feel safer making the transfers in the bank offices. As a result, there are many clients in the bank office, which generates long waiting times for clients.

We consider more than one financial service to represent the current system, i.e., payments (40%), deposits (35%), withdrawals (17%), check account balance (5.5%), transfers (2%), and international transfers (0.7%) (Superintendencia Financiera de Colombia 2014). These transactions have maintained stable behavior in the last few years in Colombia.

The service level is defined as the client’s waiting time before being served at the bank office. For the analysis, the bank branch offices in the downtown area of the city were strategically considered. The model considerations and parametrization of the case study are provided in the following sections.

3.2 Queue problem considerations

Bank branches classify clients into permanent clients and potential clients. Permanent clients are those with accounts/products opened in the same bank. Potential clients are the people who use the bank occasionally (e.g., for the payment of public services). The setting of system variables and model assumptions are presented below:

  • The service levels of bank branches are classified into general and preferential lines.

  • Each bank branch office has a client arrival rate (expected number of arrivals per unit of time).

  • Each bank branch office has an associated number of bank tellers and a capacity for attending the clients.

  • There are a limited number of transactions per client.

  • All bank tellers have the same service rate per transaction.

  • Each client has a bank branch office preference and a monthly frequency for doing the transactions.

  • Each transaction is associated with preferred days depending on a probability distribution.

  • Each client only performs one transaction at a time.

  • When a client cannot make a transaction or does not want to wait, the client exits the system.

4 Simulation-based methodology

We implement a discrete-event and agent-based simulation model to represent the current scenario. The simulation model compares the current system (Scenario A) and the one where the mobile App is implemented (Scenario B). The system represents the process of performing transactions by bank tellers. The process starts when a client enters the banking establishment to do a transaction and it ends after the transaction is done or when the client decides to exit the system without doing it.

The bank operation system is composed of a limited number of bank tellers, a general line, and a preferential line. The general line serves potential clients, while the preferential line serves permanent clients. A bank teller is assigned to the preferential line if at least one permanent client is in the system. When there are no permanent clients in the system, all bank tellers serve the general line. The service and waiting times depend on the number of transactions to be processed and follow the rule FIFO - first in first out (Larson 1987). Clients are considered agents to model the behavior between potential and permanent clients. Figure 1 describes the structure of our proposed methodology.

Fig. 1
figure 1

Framework of our methodology

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For scenarios A and B, the system operation is analyzed utilizing discrete events to determine the queues’ behavior and, afterward, the bank tellers’ behavior. The number of bank tellers changes according to the bank branch policy.

4.1 Scenario A: current scenario

Figure 2 describes the current system process. A truncated normal distribution for clients’ arrival time at branch offices is considered due to the statistical distribution fitting. Once a client is in the system, they evaluate whether to wait in line. If the number of clients in the row exceeds the occupancy perception (i.e., 40 clients in the queue), the client decides not to make the transaction and leaves the system (see Fig. 1).

Fig. 2
figure 2

Model structure for scenario A

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The following steps describe the logic process when a client decides to do a transaction in a bank office:

  1. 1.

    Is the client a permanent client?

    1. (a)

      Yes, the permanent client goes to the preferential client line.

    2. (b)

      No, the potential client goes to the general client line.

  2. 2.

    The client is placed in the waiting line.

  3. 3.

    The client remains in the waiting line.

  4. 4.

    Is the client served?

    1. (a)

      Yes, the client leaves the system.

    2. (b)

      No, evaluates how long the client has been in line. Is the waiting time greater than the maximum waiting time (parameter)?

      1. (i)

        Yes, the system is full, and the client exits the system without performing the bank transaction.

      2. (ii)

        No, the steps are repeated from step 2.

4.2 Scenario B: proposed scenario

Figure 3 presents scenario B. Scenario B considers the same parameterization as the current scenario. The process starts when a client enters the system. The client uses the mobile App to select a bank branch for doing a transaction.

Fig. 3
figure 3

Model structure for scenario B

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The following rules describe the selection process:

  1. 1.

    Each client and bank branch office have a different geographical location. When a client needs to make a transaction, the current location becomes the origin of a set of paths that connect each bank branch office. The sum of those path lengths defines the distance between the current position of the client and the bank branch office.

  2. 2.

    The client’s location is a random variable.

  3. 3.

    The distance is converted to time units assuming a walking speed of 4 km/h (Chi and Schmitt 2005).

  4. 4.

    The client searches in the mobile App for available bank offices.

  5. 5.

    Clients prefer the bank branch office with the shortest path.

  6. 6.

    Based on the list of available banks in step 4, the App evaluates the occupancy of all bank offices and suggests the ones with less occupancy.

  7. 7.

    The client selects the preferred bank office, and the App evaluates how much time the client takes to get to a bank office and complete a transaction.

  8. 8.

    The App generates a place in a digital queue for the client.

  9. 9.

    An attribute allows differentiating the type of client. Is the client a permanent client?

    1. (a)

      Yes, are there any potential clients in the system?

      1. (i)

        Yes, a permanent client is assigned to a preferred position (i.e., earlier placement in the digital queue) than the one assigned to potential clients.

      2. (ii)

        No, assign placement in the digital queue following the FIFO rule.

    2. (b)

      No, assign placement in digital queue following the FIFO rule.

Once the client selects a bank and has a place in a digital queue, the client arrives at the bank and does the transaction. The service performance depends on the number of bank tellers available to serve the client. The service time will be the sum of the exponential probability distributions by transaction type and quantity. The operating time is the sum of the service and travel time.

4.3 Behavior of financial clients

The Bass diffusion model is considered to identify how the use and adoption of a mobile Application are diffused among financial clients. The current process starts once the client arrives at the selected bank branch office. All clients entering the system require a bank transaction. A low service level is defined as the initial state for the client in the system. A potential client can become a permanent client for a bank branch office when its service level is high. Otherwise, the user must wait until someone refers to the bank’s services.

Clients will have a contact rate determined by the number of times a client refers a banking establishment to a random related contact (Dunbar 1992). The service level depends on the success of the transaction and the waiting time. When the transaction is completed with a waiting time less than or equal to the expected time by the client, the service level is high. On the contrary, if the waiting time exceeds the maximum expected waiting time, the client decides not to carry out the banking transaction at that bank branch office; thus, the client leaves the system, and the service level is low.

Figure 4 represents the status change of a client. A client can continue to be a permanent client if the service level is high. If the client receives a low service level, the permanent client will become a potential client. The transition from potential to permanent clients occurs through an information transfer between permanent and potential clients. Equation 1 formally describes the Bass diffusion model. The \({F}_{t}\) refers to the number of clients who have adopted the App at the time \(t\), the parameter \(p\) indicates the proportion of spontaneous adoptions given by external variables, and \(q\) represents the proportion of adoptions given by imitation (transfer information).

Fig. 4
figure 4

Exemplification of the service level in the bank, information transfer and transition between types of clients (Bass diffusion model)

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$${F}_{t=\frac{1-{e}^{-(p+q)\bullet t}}{1+\frac{q}{p}\bullet {e}^{-(p+q)\bullet t}}}$$
(1)

The client’s behavior depends on the bank occupation level (occupancy perception) and its proximity. Once the client is at a bank office, he decides to stay in line or leave the bank office according to the waiting time perception. For each permanent client, a variable is assigned that counts the number of times the client is served or not. When a permanent client has been served five consecutive times, they give references of the bank office to potential clients (i.e., information transfer). When a permanent client has not been served three times, they become a potential one. If the potential client completes six unattended services, the client decides not to visit the bank branch again.

5 Numerical experiments and results

The simulation model is implemented in the AnyLogic® simulation tool in an Intel Core i5, 2.5 GHz, 8 Gb RAM. The data were constructed from the Bank’s financial data reports. Three banking establishments denominated as bank 0, bank 1, and bank 2 are considered. The banking establishments are in the center of Bogotá, Colombia. Table 1 shows the parametrization of the simulation models:

Table 1 Parameters for simulation models
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In the computational experiments, the service level, and the number of served clients are provided as performance criteria for evaluating scenarios A and B. This section details the experimentation and results. The experimental phase is divided into two parts:

  1. 1.

    Analyze the effect on the average number of clients in the banking system. The analysis is done by testing the following research hypothesis:

    1. (a)

      There is no significant difference between the current and proposed scenario in terms of the average number of served clients in a day.

  2. 2.

    Analyze the effect on service times in the banking system. The analysis is done by testing the following research hypothesis:

    1. (a)

      There is no significant difference between the current and proposed scenarios in terms of service time deviation in a day.

5.1 Tuning parameters for the simulation model

This section defines the simulation horizon and the number of runs required to study the system. The simulation horizon of the model is 247 days (i.e., working days) and 10 runs. A total of 1,000 clients are assumed, considering the Colombian annual projection. Permanent clients constitute 4.2% of the total clients. Particularly, in scenario B, the system becomes saturated, and the proportion of permanent clients rises to 9.2% after 60 days. As a result, the planning horizon is defined in 60 days. Tables 2 and 3 show the results for Scenario A and B, respectively.

Table 2 Simulation results for scenario A
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Table 3 Simulation results for scenario B
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According to the results, the number of served clients in the three bank branches is greater in scenario B than in scenario A (see Fig. 5). This behavior is justified considering that clients select the preferred bank branch (the closer one with the shorter estimated waiting time).

Fig. 5
figure 5

Average total clients served and Average system occupancy rate overall bank branches

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Next, the number of simulations to run is determined statistically to capture the behavior of financial consumers (i.e., the Bass diffusion model). In this study, the number of clients served in the system follows a normal \(\sigma\) =103, with a 95% confidence level and an allowable error of 4 clients. Equation 2 describes the calculation of the number of runs in the system.

$$\begin{aligned}n&=\frac{{Z}^{2}\cdot {\sigma }^{2}\cdot N}{{Z}^{2}\cdot {\sigma }^{2}+{e}^{2}\cdot (N-1)}\\&=\frac{{1.94}^{2}\cdot 103\cdot 60}{{1.94}^{2}\cdot 103+{4}^{2}\cdot (60-1)}\approx 18\; runs\end{aligned}$$
(2)

where \(Z\) is the distance from the mean of a standard normal distribution, \(\sigma\) represents the standard deviation of the number of clients in the system, \(e\) shows the allowable error and \(N\) is the number of days considered for the simulation, i.e., 60 days. According to the results, 18 runs are needed for scenarios A and B.

5.2 Performance metric: impact on the total number of served clients

To analyze the impact on the total served clients, Table 4 shows the results of the simulation models for scenarios A and B. Figure 6 shows that in scenario B, the number of served clients in the system is greater than in scenario A.

Table 4 Simulation of the scenarios A and B: served clients
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Fig. 6
figure 6

System performance in terms of number of served clients for scenarios A and B

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A \(t\)-student test (see Eq. 3) is performed to statistically compare scenarios A and B and determine if there is a significant difference in the average number of clients entering a banking establishment and being served per day. Table 5 presents the parameterization of the simulation model for both scenarios.

$$t=\frac{(\overline{{\mu }_{scenarioB}}-\overline{{\mu }_{scenarioA}})}{sp\cdot \sqrt{\frac{1}{{n}_{scenarioB}}+\frac{1}{{n}_{scenarioA}}}}$$
(3)

where the \(\overline{{\mu }_{scenarioA}}\) and \(\overline{{\mu }_{scenarioB}}\) are the average of total clients in the system for scenarios A and B, respectively. The \(sp\) value is the standard deviation and \({n}_{scenarioA}\) and \({n}_{scenarioB}\) are the number of runs for each scenario. \({S}_{scenarioA}\) and \({S}_{scenarioB}\) are the standard deviation for each scenario. Equation 4 shows the calculation of \(sp\).

Table 5 Parameterization of the simulation model for both scenarios
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$$\begin{aligned}&sp= \sqrt{\frac{\left({n}_{scenarioB}-1\right)\cdot {{S}_{scenarioB}}^{2}+\left({n}_{scenarioA}-1\right)\cdot {{S}_{scenarioA}}^{2}}{{n}_{scenarioA}+{n}_{scenarioB}-2}}\\&sp= \sqrt{\frac{\left(18-1\right)\cdot {8.6}^{2}+\left(18-1\right)\cdot {9.2}^{2}}{18+18-2}}=8.957\end{aligned}$$
(4)

The value of the \(t\)-student test is -27.06, then \(p\) < 0.001. Results allow us to conclude that there is no evidence to affirm that the average number of clients entering the system is the same for both scenarios, i.e., there is a difference in the system performance of scenario A and scenario B. Figure 6 shows that in scenario B, the number of served clients in the system is greater than in scenario A. Additionally, we compared means to determine which is the best scenario (scenario with the most significant number of served clients). Equation 5 describes the difference between \({\mu }_{scenarioA}\) and \({\mu }_{scenarioB}\) with a confidence interval of 95%. \({v}_{1}\) and \({v}_{2}\) represent the degrees of freedom for scenario A and B, respectively.

$$\begin{aligned}{\mu }_{scenarioB}&-{\mu }_{scenarioA}\in \overline{{\mu }_{scenarioB}}-\overline{{\mu }_{scenarioA}}\pm {t}_{1-\frac{\alpha }{2},{v}_{1}+{v}_{2}-2}\cdot sp\cdot \sqrt{\frac{1}{{n}_{A}}+\frac{1}{{n}_{B}}}\\&{\mu }_{scenarioB}-{\mu }_{scenarioA}\in 198-279\pm 2.032\cdot 8.957\cdot \sqrt{\frac{1}{9}}\\&\qquad\quad-87{\le \mu }_{scenarioB}-{\mu }_{scenarioA}\le -75\end{aligned}$$
(5)

The results show that the difference between \({\mu }_{scenarioA}\) and \({\mu }_{scenarioB}\) is between -85 and -75. According to the results, scenario B generates a significantly higher number of served clients than Scenario A.

5.3 Performance metric: Impact on the service duration time

The impact on the service duration time for scenarios A and B are presented in Table 6.

Table 6 Total service time per bank, in scenarios A and B
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Considering these results, we evaluate whether there is no significant difference in the deviation of the service times of the financial clients served in a day for scenarios A and B. To do so, we evaluate if the variances of the system follow a Fisher – Snedecor distribution. The theoretical \(F\) value is \({F}_{0.025;17;17}\)=0.3740 and \({F}_{0.975;17;17}\)=2.673 assuming a confidence level of 95%. The calculated value is set to 2.49 \(({F}_{calculate=\frac{{S}_{scenarioB}^{2}}{{S}_{scenarioA}^{2}}}\)). Notice that \({F}_{0.025;17;17}\) ≤ \({F}_{calculate}\)  ≤ \({F}_{0.975;17;17}\). Thus, it allows us to conclude that there is sufficient evidence to affirm that the variance in service time in scenario B and the current scenario are significantly different. Considering Eq. 3 and a significance level of 0.05, the \(t\)-student test is performed to determine the difference between the average service time for both scenarios. The probability value is 476.57, then \(p\) < 0.001. Therefore, there is sufficient evidence to conclude that there is a significant difference in the average service duration time comparing scenarios A and B. Next, we compute the difference between \({\mu }_{scenarioA}\) and \({\mu }_{scenarioB}\) which represent the average service duration time (minutes) in scenarios A and B, respectively. The difference between \({\mu }_{scenarioA}\) and \({\mu }_{scenarioB}\) is between 3.095 min and 3.122 min. Thus, it is concluded that scenario B presents better performance. Figure 7 shows the system performance of scenarios A and B. It can be observed that the service duration times are better in scenario B.

Fig. 7
figure 7

System performance in terms of service time duration

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6 Theoretical and managerial implications

This study offers pivotal theoretical and managerial implications, underlining the potential extensive applicability of the proposed method in diverse contexts, for example, the adoption of tangible products such as food, clothing, cars, and beverages, among others, and the adoption of intangible products, such as consulting, transportation, insurance, etc. Besides, the proposed methodology offers a blueprint for future research to explore and expand upon this synergistic integration in other economic sectors, e.g., primary and secondary.

Furthermore, the studied problem revitalizes the discussion on the contrasting schemes of customer services in the banking sector, providing a contemporary lens to analyze and comprehend the dynamic of consumer behaviors in the digital era. The proposed model, validated with real-world data, offers a robust framework for future research to explore the variegated facets of technology adoption and diffusion, expanding the horizons of the traditional Bass diffusion model.

From a managerial standpoint, the findings from this research wield significant implications for practitioners in the banking sector and beyond. The developed mobile application serves as a harbinger of a transformative wave in customer service optimization, showcasing the immense potential of leveraging technological innovations to enhance the service level and the number of served clients. Financial institutions can glean invaluable insights from this study to fine-tune their service delivery mechanisms, fostering a seamless and enriched customer experience.

Moreover, the insights garnered from this study can be instrumental in guiding policy formulation and strategy development, aiding financial institutions to navigate the complex landscape of the digital revolution adeptly. The flexibility of the proposed methodology allows for its adaptation and implementation, serving as a versatile tool for organizations to optimize their operations and strategize their technological integration endeavors. This adaptability highlights the potential for a broader application, opening avenues for cross-industry collaborations and fostering innovation in service delivery mechanisms, heralding a new era of technology-driven transformation in service delivery and companies’ excellence.

In summary, this research contributes to the theoretical discourse and offers tangible and actionable insights for academics and practitioners, underscoring the extensive potential applicability of the proposed method. Future research can build upon this foundation, exploring the myriad avenues for innovation and optimization in the ever-evolving landscape of the digital era.

7 Conclusions

This article presented a methodology based on simulation to evaluate the impact of using a mobile App for reducing banking clients’ waiting time. The results provide evidence of the App impact on the service levels of the agents involved in the process (i.e., clients) and its effect on the attraction of new clients. Our study contributes a nuanced understanding of the existing literature, highlighting the potential of mobile applications to improve not only their operational performance but also the service experience. This fact is aligned with the identified research gap, where the application of technology in enhancing customer service experiences in the banking sector remains underexplored.

Despite the contributions of this work, future research lines include:

  • Identify and create forecast models for transactional loads by banking establishments, which allows them to identify the annual, monthly, and daily behavior of the entity.

  • To optimize the operation cost of adding bank tellers to accomplish the service promise.

  • To balance the number of bank tellers in bank branch offices.

  • Generate and evaluate campaigns or strategies so that the use of these transactional platforms and/or financial services (such as mobile banking) can achieve widespread implementation.

  • Expand the research to other service areas of the financial institution such as service platform, credit areas, etc.

  • Finally, implement the proposed methodology for tangible and intangible products and other economic sector.