Sustainable Integral Optimization of Service Queues: A Human-Centered Approach Using IAM and SMAA

García-Cáceres, Rafael Guillermo; Prado-Téllez, Angel Gabriel; Escobar-Velásquez, John Wilmer

doi:10.3390/su17188179

Open AccessArticle

Sustainable Integral Optimization of Service Queues: A Human-Centered Approach Using IAM and SMAA

by

Rafael Guillermo García-Cáceres

^1,*

,

Angel Gabriel Prado-Téllez

² and

John Wilmer Escobar-Velásquez

³

¹

School of Industrial Engineering, Universidad Pedagógica y Tecnológica de Colombia (UPTC), Sogamoso 152211, Colombia

²

General Management Department, Colombia Campamentos y Construcciones, Carrera 17a # 113-28, Bogotá 110110, Colombia

³

Department of Accounting and Finance, Universidad del Valle, Cali 760032, Colombia

^*

Author to whom correspondence should be addressed.

Sustainability 2025, 17(18), 8179; https://doi.org/10.3390/su17188179

Submission received: 15 July 2025 / Revised: 21 August 2025 / Accepted: 23 August 2025 / Published: 11 September 2025

(This article belongs to the Section Sustainable Management)

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Abstract

This paper contributes to queueing systems by introducing the concept of sustainable integral optimization, which integrates both quantitative and qualitative criteria, such as comfort, anxiety, and perceived service quality, into traditional performance metrics. We extend Kendall’s notation with a new element (G), focused on subjective sustainability indicators. The optimization process is conducted via the Integral Analysis Method (IAM), complemented by the SMAA framework for multicriteria decision-making. To demonstrate the applicability of the proposed approach, a case study in a high-end car dealership is developed. The goal is to address a persistent deficiency in queueing theory literature by incorporating human-centered sustainability principles, seeking a systemic balance between operational efficiency and user-oriented effectiveness in service delivery.

Keywords:

queueing theory; IAM; SMAA; user-centered sustainability; perceived service quality

1. Introduction

This study addresses the sustainability of service systems by integrating qualitative criteria, such as comfort, competition, and perceived service quality, into queueing theory models. These criteria have a direct impact on the user experience, resource efficiency, and service equity, all of which are central to the concept of sustainable management. By applying the Integral Analysis Method (IAM), we aim to support decision-making processes that balance operational performance with human-centered values. In this way, the study contributes to the sustainability of service delivery systems in sectors where user satisfaction and systemic resilience are key concerns [1].

Recent developments have integrated queueing models with human-centered criteria and multicriteria decision tools. For example, Ref. [2] proposed a task scheduling algorithm optimizing queue priority for platform integration. Moreover, techniques for queue disclosure under uncertain service quality settings [3] contribute to transparency and user-centered evaluation.

Other recent works include [4], incorporating secondary servers recruited dynamically from users in queue models, alongside broader sustainability-oriented multicriteria decision frameworks such as the Strong Sustainability-Paradigm AHP [5] and reviews on hybrid MCDM applied to organizational sustainability [6].

This study emphasizes the critical need to incorporate qualitative criteria—such as customer comfort, anxiety, and perception—into queuing analysis. Through the Integral Analysis Method (IAM), we provide a structured integration of these qualitative aspects into the traditional quantitative framework, offering a more comprehensive and context-sensitive approach to decision-making. Additionally, we propose a formal extension of Kendall’s classical notation by introducing a fourth element ‘G’, which captures the qualitative dimension of service quality. While this notation is not the core contribution of the study, it serves as a formal tool to support the integration of subjective variables within a standardized modelling paradigm.

Queuing theory seeks to improve the effectiveness of a system by balancing Effectiveness (i.e., providing the desired service level) and Efficiency (minimizing resource utilization and maximizing results). In 1953, Kendall defined the (A/B/C) notation to classify waiting systems, wherein A is the type of time probability distribution among consecutive client arrivals; B is the type of output time (also known as service or attention time) probability distribution; and C is the number of servers operating in parallel at the workstation. In 1966, Lee added two symbols to this notation: D and E, and in 1968, Taha added F as a sixth symbol [7]. D is the discipline or priority policy that defines customer service, which determines various aspects of service quality. E is the total capacity of the system (finite or infinite), i.e., the maximum allowable number of customers who are either queuing or being served, with E ≥ C. Finally, F is the size of the population or source from which customers are coming. Kendall’s notation has been repeatedly modified by many authors according to their needs [8,9]. Along these lines, the present work involves the inclusion of a new element (Q) in Kendall’s notation, associated with classical queueing systems but applicable to general queueing systems. This symbolic extension does not imply a structural change to Kendall’s notation, but it allows for a consistent representation of qualitative goals in human-centered service systems.

Despite important theoretical progress achieved over the decades, there is an unspoken deficiency in the literature regarding the lack of attention to the inclusion of qualitative aspects that are relevant to the analysis and performance of the system. One of the difficulties that is inherent to optimization contexts is the integration of qualitative and quantitative aspects. In this sense, the Integral Analysis Method—IAM [10] was presented as a pioneering work to overcome this problem by means of a technique known as integral optimization. It is worthwhile noting that the non-inclusion of qualitative aspects is not a particular condition of waiting line systems but an unspoken deficiency of optimization. While IAM and SMAA have been individually applied in areas such as industrial or financial planning or energy prioritization, their combined application to queueing systems—particularly those incorporating perceptual and comfort-based criteria—remains largely unexplored. This study differs by integrating both methods to support multi-perspective evaluation under uncertainty, while also embedding human-centered metrics as decision inputs. To our knowledge, no prior study has applied this hybrid framework to queue optimization with simultaneous consideration of service performance and user perception. It is important to note that this study does not aim to modify IAM or SMAA as methodologies. Instead, their established mechanisms are applied in a novel configuration that supports joint evaluation of human-centered qualitative indicators and classical queueing performance metrics within a strategic decision-making framework.

The current document is organized as follows: After a literature review, the relevant qualitative aspects of the queueing theory are highlighted. Subsequently, the application of integral optimization to the queueing theory is presented and then illustrated by means of a case study. Finally, conclusions and research perspectives are put forward.

2. Literature Review

The foundations of the queuing theory were set at the beginning of the 20th century, when its basic concepts and some practical applications were deployed, e.g., to satisfy the demand uncertainty that could be observed in telephone traffic systems [11]. In the 1970s, the theory focused on obtaining exact analytical expressions of indicators that described the performance of systems [12]. In the 1980s, a wave of solutions was generated, all of which contemplated time spent in the system as a strategic factor for companies, thus resulting in an increased number of applications [13].

The design and management of a system that is characterized by waiting lines or queues poses permanent challenges for any modern organization. This is so because this feature affects the costs and the service level, which, in turn, impacts competitiveness and business sustainability [14,15,16]. Applications based on the waiting line theory have been extended to a good number of industries and systems such as those aimed at maintenance service, quality inspections, service facilities for employees, computer centres, machinery installations [17,18,19], telephone communication [11,20], vehicle traffic and machine breakdowns [16,21], applications to judicial and legislative systems, hospital emergency rooms, real estate purchase subsidy allocation systems [21,22,23], car wash services [24], waste management [25], and airport terminals [26], among others [27].

Queues typically generate economic losses and social welfare detriment, thus threatening competitive priorities, among which quality and costs certainly stand out [28]. From a broad economic perspective, it can be said that waiting lines do not generate value and should, therefore, be reduced to their minimum possible dimensions [16,29]. However, they are actually unavoidable due to the capacity constraints of the systems, and, therefore, they must be properly managed to mitigate their negative effect on business performance. The theory of waiting lines constitutes a remarkable development in this sense, since it focuses on system performance [16]. Waiting line management is mainly aimed at service quality and customer satisfaction, which have been gaining importance in the current economic context, to the point of being often considered more important than costs [15,28].

From the literature review, it can be deduced that research on waiting lines has triggered multiple applications and application perspectives that seek to increase productivity by minimizing costs, maximizing service level, and optimizing queue size, service times, service utilization requests, and integrity in each one of the systems under study. It further shows that the applied techniques are particularly related to stochastic calculus and stochastic processes, which fundamentally link quantitative [16,22,23,30,31,32] and qualitative [33,34,35,36] considerations.

Qualitative approaches to the queuing theory have focused on aspects such as leisure [37], new trial [8,38], customer type (positive and negative) [23], computer attacks [34,39], service priorities [40], location of internet qualitative states [36], and banking entities [35], all of them treated quantitatively in the sense that expert preference information is not used. There is also evidence of the use of MCDM (Multicriteria Decision Making) techniques for the inclusion of quantitative and qualitative aspects in decision problems [34]. Nonetheless, the present work is the first one to treat preference information through a specialized method for the inclusion of qualitative and quantitative aspects, as is the case with IAM.

Recent contributions to queue optimization reinforce the importance of hybrid methods that integrate qualitative and quantitative indicators. Ref. [2] propose a queue-priority optimized algorithm for runtime systems, aligned with the IAM’s goal to manage multivariable constraints in real-time service environments. Similarly, Ref. [41] analyze queueing delay optimization for service-oriented networks, directly resonating with IAM’s integration of waiting time and server idleness. Additionally, Ref. [42] introduce a fairness-based waiting time model, which supports the inclusion of qualitative fairness and equity indicators within queue system designs. This perspective aligns with principles from human-centered design (HCD) and service operations literature, which emphasize the integration of user expectations and perceptual experience into system design and evaluation [43,44].

Recent studies have expanded upon traditional M/M/c models by incorporating time-varying arrival rates, strategic customer timing, and fairness-based service mechanisms. For instance, Ref. [45] present a survey on strategic arrival timing in queues, while Ref. [46] model multi-server systems with dynamically rate-dependent arrival processes. These developments highlight the increasing attention to adaptive and behavior-sensitive approaches in contemporary queueing theory.

3. Description of the Problem

Integral optimization is supported on the Integral Analysis Method (IAM), see Appendix A, proposed by [10], which consists of three mathematical steps: (i) Cardinal analysis, which in the present case resorts to the Queueing Theory to provide the solution method; (ii) Qualitative analysis, where the alternatives resulting from the quantitative analysis are evaluated through ordinal criteria by means of Stochastic Multicriteria Acceptability Analysis with Ordinal data (SMAA-O) [47]; and (iii) Integration analysis, where the alternatives are jointly analyzed using the deterministic version of SMAA [48,49] and probability elements. It is carried out using parameter (X), the expected number of idle servers, as central performance measure. The input of this integration analysis is the result of the cardinal and ordinal analyses [50]. The integral optimization of the queueing system discussed here seeks to determine the optimal number of servers in a queueing line, considering not only the typical quantitative aspects but also some other relevant qualitative variables.

To facilitate understanding for readers who may not be deeply familiar with queueing theory, Table 1 summarizes the key variables and parameters used throughout the manuscript. This also supports consistency in terminology and enhances the clarity of the mathematical formulation.

3.1. Cardinal Analysis

The queuing theory is the mathematical study of waiting lines within a given system, which, in turn, conditions the model developed to study it. Mathematical models usually employ notation systems that include parameters and variables, which, in the present case, correspond to λ = rate of arrivals to the system and µ = service rate of each service channel. On the other hand, the variables of the system are L = expected number of users in the system, Lq = expected number of users on the queue, W = expected time spent in the system by each user, and Wq = expected time spent by each user on the queue. These are the main performance indicators, which can be compared with industry parameters or to those resulting from the system’s own policies to assess its operation [17,18,19].

Little’s Law [51], which relates queue size to waiting time, is described as follows: L= λW; Lq = λWq; W = Wq + 1/µ, where L = and is the probability that there be n customers in the system [18,52,53]. The utilization factors are defined by where c is the number of s.

A classical queueing model (M/M/c: GD/∞/∞/Q) was applied to provide a clear example of the integral optimization process in queueing systems. This model characterizes the functioning of a dealership that sells high-end vehicles. It is assumed that the system is not capacity-constrained, as there is no evidence that customers refuse to come into the shop. The use of exponential interarrival and service times is inherent to the M/M/c formulation, where ‘M’ denotes a memoryless (Markovian) distribution. In this case study, the assumption was validated by the service provider using historical time records. The Poisson arrival processes range from 5 to 15 average customers per hour, depending on the day of the month. Exponentially distributed average service time was estimated to be 20 min. The service covers the following activities: identification of customer needs, orientation about possible vehicles to be purchased, showing the vehicles on display, and, finally, quoting and simulation of vehicle cost and delivery times.

The cardinal analysis focuses on the problem of minimizing the total cost, as indicated by the following objective function:

m i n C T (c) = c C_{c} + C_{w} L (c)

(1)

where CT(c) is the total cost, as a function of the number of servers; Cc is the cost per server; Cw is the cost of making a client wait; L(c) = L is the number of people waiting in the system as a function of the number of servers.

It is important to note that, for the purpose of analytical tractability and methodological clarity, cost parameters in Equation (1) are assumed to be constant. This simplification allows us to focus on the structure of the service queue model and the behavior of integrated decision criteria. However, the model can be generalized to incorporate variable cost functions or time-dependent costs, especially in dynamic or adaptive environments. This generalization is left for future extensions.

Although car dealers estimate that servers imply an elevated cost, the varying level of commercial competition usually observed in this trade has led them to favor commercial strategies that promote customer service quality [54]. In this case, the way the products are displayed is very important, together with an attentive attitude expressed in the way of being, having, and doing, always aimed at customer need satisfaction [55].

As a tool to support the decision process at this stage, we employed the parameter “aspiration level”, which resorts to the following variables of analysis: Average time spent in the system (Ws) and the percentage of inactivity of the servers (X), both of them as functions of the number of customers. The said percentage is calculated as the ratio between the average number of idle servers and the total number of servers, as shown in the following expression.

X = 1 - \frac{c - \bar{c}}{c} = (1 - \frac{λ_{e f}}{c μ})

(2)

Equation (2) follows the classic M/M/1 steady-state model and is derived in Appendix A, with detailed steps. The result is consistent with the formulation given in [17], which remains a standard reference in the field.

In order to determine an acceptance range that facilitates decision-making on the optimal number of servers, acceptance levels are defined for the variables Time spent in the system (α) and percentage of server inactivity (β), considering

W s \leq α \land X \leq β

(3)

The aspiration level model requires information about the acceptable levels of the two performance variables that express customer aspiration. Since they usually have antagonistic behaviors with respect to the number of servers (c), the latter has to be set at a value that optimizes both variables. Therefore, in the current case study, c is the decision variable of the cardinal analysis.

For the case study, the value of α was set at 25 min (α = 25 min), so that customers do not have to wait to be served for more than 5 min (Wq). This parameter was defined taking into account that customers are usually sensitive to waiting time [56]. In this competitive segment, the customer values exclusivity and product differentiation, and the service must be characterized by its high quality and immediacy, because the risk of customer loss is particularly high when dealing with high-end vehicles. On the other hand, 1 − β = 68% was defined as the maximum allowed level of idleness. This definition was based on the notion of “work pace valuation”, proposed by [57], in addition to the previously expressed quality considerations.

The following table presents the results of the simulation of the queueing system, which was carried out to determine the optimal number of servers (c) that satisfies the aspiration levels. This includes the typical performance variables of a queueing system, together with those contemplated in the aspiration function, in order to clearly determine the performance of the system. The data, rounded to two decimal places, include the following variables, in addition to those mentioned above: (λeff) Effective arrival rate, (p_n) Probability of the instance, and (p_i) Probability of the alternative. After obtaining the indicators, the percentage of server inactivity (X) was calculated through Equation (3).

The selected case study involves a high-end car dealership located in Bogotá, Colombia, which provides a representative environment to illustrate the proposed modelling approach. This dealership typically manages between five and fifteen clients per hour, offering complex advisory and sales processes that include needs identification, product demonstration, financial simulation, and negotiation. Historical records of arrival and service times over a two-year period were used to parameterize the quantitative model, ensuring consistency with the M/M/c assumptions. The project was led by the dealership’s general manager, who coordinated a multidisciplinary team composed of two senior service managers with more than ten years of experience in customer relationship management, two sales advisors specialized in premium vehicles, and one operations analyst responsible for monitoring service performance. Experts were selected to ensure independence across roles and to capture a balanced view of managerial, operational, and customer-facing perspectives. This configuration reflects the typical composition of decision-making teams in high-end service environments, where both strategic and operational criteria guide service quality assessments, and provides a transparent foundation for the Likert-based evaluations used in the ordinal analysis. Although illustrative, this real-world context offers a robust basis to demonstrate the practical integration of human-centered qualitative indicators into queue optimization. We acknowledge that no formal inter-rater reliability measure was applied in this exploratory study and therefore highlight this as a limitation. Future applications of the methodology should incorporate statistical reliability metrics (e.g., Cohen’s kappa or Cronbach’s alpha) or psychometrically validated surveys to enhance the robustness and reproducibility of qualitative evaluations.

The case study assumes moderate homogeneity among customers in terms of perception of service-related qualitative criteria. However, heterogeneity may be addressed in future research. The Likert-scale values for qualitative criteria were obtained through expert elicitation involving service managers and operational staff at the dealership. This approach is justified by the need for informed judgments when direct customer surveys are unavailable.

Based on historical data, 10 scenarios were evaluated in the 1 to 8 server range, the Poisson arrival rate varying from 6 to 15 customers per hour on average. Table 2 summarizes the results of the inspected scenarios. Table 2 presents the fundamental parameters of the M/M/c system used in this study, including Ls (expected number of users in the system), Ws (expected time in system), p_n (probability of having n customers in the system), and p_i (probability that the system is idle). These variables form the quantitative basis for assessing queue performance.

The results presented in the table above show that the studied range of servers conforms to the aspirational peaks. Probability p_i, which constitutes the outcome of IAM’s Cardinal Analysis, is the Marginal Cardinal Probability Distribution of variable i, which is discriminated by variable c, as can be seen in Table 1. In sum, p_i expresses the probability of the optimal values of the number of servers (c) resulting from the scenarios of the problem. In the present case, the optimal relation between them indicates that c = i + 3.

3.2. Ordinal Analysis

For illustrative purposes, the inclusion of the ‘G’ element in Kendall’s notation is exemplified through the evaluation of four qualitative criteria: consumer anxiety, noise, thermal load, and competition. Each of these aspects, typically assessed via expert-based Likert scoring, is formally represented within the extended notation rather than remaining external annotations. This symbolic integration ensures that qualitative indicators are treated as inherent system parameters, allowing decision-makers to interpret them alongside classical performance variables such as waiting time and server utilization. By embedding subjective service quality dimensions into the queueing notation, the ‘G’ extension provides a unified framework for integrating operational efficiency with human-centered sustainability considerations.

The qualitative factors that affect service queues were defined in such a way that they support decision-making by estimating indicators that help to measure the performance of the system (see Appendix B, e.g., the performance of the installed capacity or the percentage of idle servers) and those of interest to its customers (e.g., waiting time). These models also allow enhancing service quality by estimating and informing the customer how long they have to wait to be served, among other improvements [58]. This stage is essential for capturing sustainability-related subjective factors, such as user equity, perceived comfort, and service accessibility. These qualitative dimensions are critical for promoting human-centered sustainability in service systems, ensuring that operational improvements are aligned with user well-being and fairness.

Some qualitative aspects were included in the present study so as to provide an adequate example of the application of IAM to this type of information within the necessary decision-making process of queuing systems: (a) quality level, understood as the standard of a product or service, and related to the level of customer satisfaction; (b) comfort level, which is the well-being perceived by the interaction of the customer with the service environment; (c) competition, which is the level of monopoly that exists in the market wherein the company providing the service performs its activity [16,59].

Table 3, Table 4, Table 5 and Table 6 present the evaluation results across multiple criteria, classified into three performance levels: low, moderate, and high. These classifications were based on the distribution of each indicator across all alternatives, using tercile thresholds: low performance: values in the bottom third (≤33rd percentile); moderate performance: values between the 34th and 66th percentiles; and high performance: values in the top third (≥67th percentile). This approach ensures comparability across indicators with different scales and maintains consistency with multicriteria decision analysis practices. Specific threshold values used in each table are detailed in the respective captions. This approach is commonly adopted in early-stage sustainability assessments, where operational data are not yet available. The use of expert Likert ratings is considered a temporary proxy until psychometrically validated user data can be obtained. Although this study adopted a five-point Likert scale, the SMAA framework mitigates sensitivity to input granularity by operating on distributions rather than fixed weights. Informal tests using alternative scale resolutions produced consistent rankings. We recommend that future studies formally explore sensitivity to scale formats using simulation techniques or robustness metrics and consider structured surveys and validated user feedback tools to enhance representativeness.

Each of the alternatives is evaluated qualitatively. In this illustrative example, the additional element (G) is associated with four qualitative criteria that are evaluated through the Likert table shown in Table 3, which presents five ordinal categories.

Consumer Anxiety: This criterion is measured according to the Cognitive-Somatic Anxiety Questionnaire (CSAQ) [60], the Likert scale of which discriminates the level of customer anxiety. In the present case, it was determined that the higher the number of servers, the lower the level of anxiety within the optimal 3–8 server range, because the service time is correspondingly shorter. However, as the number of people in the system increases and the place gets crowded to its full capacity, density saturation is likely to generate more anxiety, this time not derived from waiting time but from the vital space decrease and the system’s entropy increase. The Likert scale, defined in the 1 to 5 range, identifies 1 as a very good level of anxiety and 5 as a very bad level.

Noise: This criterion is governed by Colombian regulations. Resolution 8321 of August 4, 1983, issued by the Ministry of Health, defines the maximum permissible levels of noise exposure, which depend on sound intensity and time spent under its influence. The Likert scale shown in Table 4 illustrates the impact of noise on human health, where the greater the number of servers, the higher the service and noise levels.

Thermal load: This criterion is proportionally related to the number of people, and its evaluation depends on the Wet-Bulb Globe Temperature (WBGT) index. The American Conference of Governmental Industrial Hygienists (ACGIH) defines its maximum allowed level at 38 °C. The following Likert scale (Table 5) is used for its estimation.

Although the temperature can be managed through air conditioning, the application of this resource involves cultural, economic, and environmental considerations. Therefore, the decision to use it requires analysis. In the case of Bogota, it is rarely employed.

Competition: Lerner’s Index, which ranges between 0 and 1, allows estimating competition, as expressed in Table 6.

It is commonly assumed that the higher the level of competition, the larger the number of servers and the concomitant investment. This analysis should take into account that the number of employees should not exceed the capacity of the company’s facilities.

According to IAM [10], the ordinal analysis supported by SMAA-O [47] is restricted to ordinal criteria. The following table rates the four ordinal criteria considered for the alternatives under study.

The ordinal stage of IAM makes it possible to identify the set of favorable weights that support each of the alternatives in a particular ranking, according to the ordinal evaluation criteria. The most important indicator of the analysis is the Ordinal Acceptability Index, which defines the probability of acceptance of each alternative and determines the ordinal ranking (r). The qualitative stage is supported by another indicator, the central weight vector, which represents the typical combination of weights that favors a given alternative in ordinal ranking 1. This vector represents the center of mass of the set of favorable weights. Table 7 presents the ordinal analysis for each one of the alternatives in question.

Ordinal ranking 1 favors alternatives 1 and 5,

b_{1}^{1} = b_{1}^{5} = 1 / 4

, which are supported by a higher volume of feasible weights. These alternatives not only attained the best and worst scores in two of the four criteria but also exhibit opposite ratings: Criteria 2 and 3 support Alternative 1, while criteria 1 and 4 support Alternative 5. Alternatives 3 to 5 are also well rated, with

b_{1}^{2} = b_{1}^{3} = b_{1}^{4} = 1 / 6

. Alternative 2 is supported by criteria 2 and 3, while Alternative 4 is supported by criteria 1 and 4. For its part, Alternative 3 is supported by an intermediate rating in all four criteria. The detailed central weight vectors supporting this ordinal analysis are presented in Appendix C (Table A5).

3.3. Integration Analysis

In the context of IAM, the integration analysis is supported by the deterministic version of SMAA [46,47]. Using as input the two output variables resulting from the prior cardinal and ordinal analyses (which are assumed to be independent), the integration analysis calculates the Joint Integral Index and the integration acceptability index, which assesses each alternative’s joint probability—both cardinal and ordinal—of obtaining a specific ranking. This integrative step embodies the principles of sustainability by balancing quantitative efficiency (e.g., productivity and waiting time reduction) with qualitative well-being (e.g., user comfort and anxiety mitigation). The ability to synthesize these dimensions within a unified decision-making framework reflects a commitment to sustainable service design, where operational goals do not compromise human-centered values.

p (e^{i}) = p_{i} b_{r}^{i}

(4)

Although the evaluated server counts range from c = 4 to c = 8, the analysis was extended to c = 9 based on SMAA recommendations. Thus, c* = 4 is reported as the optimal configuration considering both cardinal and ordinal criteria.

Table 8 presents the integration analysis. The two holistic indices mentioned above show that Alternative 1, with four servers, is the most supported one. Consequently, it is the one that integrally optimizes the problem in question.

Complementary evidence is provided in Appendix C (Table A6), where the central weight vectors confirm that only Alternative 1 consistently attains the first rank.

The optimization strategy used in this study is based on a discrete evaluation of server quantity scenarios (c values). For each scenario, performance metrics are calculated and then integrated using the SMAA multicriteria analysis. This enumeration-based approach enables practical decision-making without the use of continuous optimization or heuristic algorithms.

Sensitivity was examined at each modelling stage. In the cardinal IAM-based analysis, we also explored multiple service scenarios. These included changes in customer flow, capacity distribution, and constraints on operating conditions. The resulting rankings remained consistent, reinforcing the model’s reliability and generalizability. In the ordinal and integration analysis, SMAA inherently assesses robustness by computing acceptability indices across a wide spectrum of weight combinations, enabling probabilistic comparisons among alternatives. This approach reveals how variations in preferences influence rankings without requiring explicit deterministic simulations.

This case study, while grounded in a real decision environment, represents a controlled scenario. Future research should evaluate the proposed model across different sectors and service dynamics to assess its robustness and generalizability. Although the model is illustrated using a real operational context, further empirical validation in diverse field environments is recommended to test predictive performance and user response.

This modelling configuration reinforces the paradigm of service design that considers perceived time, comfort, and user expectations as critical inputs for evaluation—an approach consistent with established UX and HCD principles.

It is important to note that the ordinal stage of IAM is methodologically flexible, and techniques such as AHP, TOPSIS, or weighted-sum models could, in principle, be applied as alternatives. However, these methods on their own do not provide the integral perspective that characterizes IAM, nor do they embed qualitative and quantitative indicators within a unified optimization framework. The use of SMAA in this study is motivated by its capacity to incorporate uncertainty and probabilistic robustness, making it particularly suitable for human-centered service contexts. This configuration illustrates that IAM does not compete with MCDM methods per se but rather extends their applicability by treating decision problems from a systemic and integral viewpoint.

4. Conclusions and Future Perspectives

The present paper incorporates a new element (G) to the notation developed by Kendall (A/B/C), Lee (D/E), and Taha (F), among others [8]. This novel factor represents the qualitative aspects of the problem, which are the relevant criteria for the study of a queueing service system. This addition reflects a more realistic treatment of service environments where subjective perceptions (e.g., comfort, competition, anxiety) are key.

Although this study focuses on a vehicle service context, the proposed modelling framework is generalizable to other service environments involving queueing dynamics. Examples include outpatient clinics, financial institutions, and public administrative services, among others. The combination of IAM and SMAA allows flexible integration of context-specific qualitative and quantitative criteria, enabling adaptation to diverse operational and perceptual characteristics. Future applications may require adjustments in the selected variables and constraints, but the overall structure remains robust and scalable. Our results align with emerging literature that emphasizes both operational and human-centered optimization in queue systems [2,4] and support the efficacy of multicriteria integration as recommended by [5,6].

The use of a qualitative approach in the construction of models allows obtaining holistic solutions that are likely to come closer to reality. Including this type of aspect is an approach to the integral optimization of the queuing problem, which can contribute significantly to its theoretical and practical development.

In conclusion, the results demonstrate a well-structured and practical approach, particularly through the case application to high-end vehicle service management. The study effectively bridges queueing theory with multicriteria decision-making tools such as SMAA, offering a robust framework for incorporating both quantitative and qualitative dimensions in service system optimization.

In particular, future research could explore the integration of broader sustainability criteria within queueing models, such as environmental impacts of waiting environments, energy efficiency in service provision, and social equity in customer prioritization. These directions would further consolidate the role of queueing theory as a tool for achieving the Sustainable Development Goals (SDGs) in service contexts. Through the application of IAM, this paper presents a comprehensive optimization approach to queueing theory. Future research directions include not only extending the current study to other classical queueing systems but also applying the methodology to more complex queueing network systems. The work is well aligned with recent studies that address fairness, qualitative variables, and hybrid queueing models.

The robustness of the proposed approach is supported by Monte Carlo simulation at different stages of the IAM–SMAA framework: scenario exploration in the cardinal analysis and the stochastic generation of acceptability indices in the ordinal and integration analyses. These simulations provide probabilistic evidence of stability across alternatives under varying assumptions. Nevertheless, a variance-based global sensitivity analysis (GSA) using Sobol indices could complement the current design by quantifying the separate and joint contributions of uncertain inputs (arrival rate, service rate, server and waiting costs, aspiration thresholds, and Likert-derived criteria) to the probability of selecting each alternative and to the optimal number of servers. We identify this as an important avenue for future research and note related engineering applications where GSA has been successfully applied to complex modelling problems.

Author Contributions

Conceptualization, R.G.G.-C. and A.G.P.-T.; methodology, R.G.G.-C. and J.W.E.-V.; software, R.G.G.-C.; validation, R.G.G.-C. and J.W.E.-V.; formal analysis, R.G.G.-C. and A.G.P.-T.; investigation, R.G.G.-C. and A.G.P.-T.; resources, R.G.G.-C.; data curation, R.G.G.-C. and J.W.E.-V.; writing—original draft preparation, R.G.G.-C. and A.G.P.-T.; writing—review and editing, R.G.G.-C. and J.W.E.-V.; visualization, R.G.G.-C.; supervision, R.G.G.-C.; project administration, R.G.G.-C. All authors have read and agreed to the published version of the manuscript.

Funding

The authors gratefully acknowledge the financial support provided by the Universidad Pedagógica y Tecnológica de Colombia (UPTC), Grant SGI 3954.

Data Availability Statement

Data are contained within the article.

Conflicts of Interest

The authors declare no conflict of interest.

Appendix A. Integral Analysis Method (IAM)—Mathematics Stages

Cardinal Analysis—SMAA-O

The cardinal analysis is explicitly presented in the main body of the manuscript. The derivation of Equation (2) is presented below.

Expected Number of Idle Servers

The expected number of idle servers in an M/M/c queueing system under steady-state conditions is given by

\bar{c} = c - E [L s]

where

E[Ls] = is the expected number of busy servers.

Step 1: System Utilization

The system utilization (or traffic intensity) is

ρ = \frac{λ_{e f}}{c μ}, with ρ < 1

Here, λ_ef represents the effective arrival rate, that is, the average rate of customers that successfully enter the system. In systems with finite capacity (e.g., M/M/c/c), it is computed as

λ_{e f} = λ (1 - P_{blocking})

where P_blocking is the probability that all servers are occupied and an arriving customer is rejected

λ_{e f}

.

Step 2: Expected Busy Servers

Under steady-state conditions,

E [L s] = \frac{λ_{e f}}{μ}

This value represents the average number of servers that are occupied.

Step 3: Final Expression

Thus, the ratio between the average number of idle servers (

\bar{c}

) and the total number of servers as shown in the following expression:

X = 1 - \frac{c - \bar{c}}{c} = (1 - \frac{λ_{e f}}{c μ})

This expression is useful to adjust perceived service efficiency or availability in human-centered service models.

Ordinal Analysis—SMAA-O

The stage restricts the SMAA-O to ordinal criteria [47]. Each of the original variables of an optimization problem can have several ordinal variables associated that are defined by the decision makers. These values will be used to characterize the ordinal analysis, but before explaining the method, it is necessary to define the concept of class as a set of alternatives that have identical utilities for all ordinal variables associated. As a result of the analysis, a group of indicators is obtained that helps to determine the ordinal acceptability of each studied alternative.

Now the SMAA-O used by IAM is explained in more detail.

Constants, indexes, and sets:

m: Number of possibilities or alternatives

n: Number of ordinal variables

s: Number of class where s ≤ m

r: Ordinal ranking where r є {1, 2, …, N(r)} and N(r) ≤ m.

A: Classes where

a

є{1,2, …, s}

F(

a

): Alternatives concerning class (

a

)

C: Ordinal ranking

C_i: Ordinal ranking for alternative (i)

Other Symbols:

X_{a j}

: Ordinal criteria values of quantitative aspect or ordinal variable (j) for class (

a

)

w_j: Weights vector of ordinal variable (j)

u_{j} (X_{a j})

: Mapping from ordinal criteria values of ordinal variable (j) for class (

a

)

γ_jr: Random number from the uniform distribution (0,1)

ε_{a j}

: Stochastic cardinal criteria values of ordinal variable (j) for class (

a

)

W_{a}^{r}

: Favorable ordinal vector ranking weights for class (

a

)

W: Set of non-negative normalized weights

b_{a}^{r}

: Ordinal acceptability index for class (

a

)

b_{i}^{r}

: Ordinal acceptability index for alternative (i)

cⁱ: Number of ordinal rankings for alternative (i)

R^{i}

: Central ordinal value for alternative (i)

V^{i}

: Dispersion ordinal value for alternative (i)

The method is based on the idea of determining the utility values that support each class. The utility function can be additive, as in the following case:

u (X_{a}, w) = \sum_{j = 1}^{n} w_{j} u_{j} (X_{a j})

The process starts with the generation of random numbers from a uniform distribution in the interval (0,1) and sorting these numbers along with 1 and 0 in decreasing order, as shown below:

γ_{j 1} > γ_{j 1} > \dots, γ_{j j^{m a x}}

γ_{j 1} = 1, γ_{j j^{m a x}} = 0, γ_{j k} \leftarrow U (0,1), k \neq 1^j^{m a x}

This procedure ranks these random numbers The distinctness of (γ_jr) can be ensured by rejecting sets containing identical values. On the other hand (j^max) refers to the maximum number of different rank categories of possible solutions for each one of the selected variables. These numbers are used as a sample of stochastic cardinal criteria values (

ε_{a j}

) so that for each class (

a

), this is set equal to (γ_jr); thus:

u_{j} (X_{a j}) = ε_{a j}

(A1)

In which the elements of ε є X follow some distribution f (ε) such that

X = {ε \to f (ε)}

, where (X) is the set of stochastic cardinal utilities. All stochastic cardinal assignments must comply with the following requirements in order to find the weight sets:

ε_{a j} = γ_{j r}, r = X_{a j}

(A2)

Thus, the stochastic cardinal values and the convex weights accepted will be those that conform to the ordinal ranking of the alternative. In order to ensure that an adequate ranking for each alternative is assigned, the utility functions of each alternative are compared by means of the following function:

r a n k (ε_{a}, w) = 1 + \sum_{k = 1}^{s} ρ (ε_{k}, w > ε_{a}, w), w h e r e : ρ (t r u e) = 1, ρ (f a l s e) = 0

where each alternative is ranked from 1 to m. The ranking function’s objective is to obtain the set of feasible variable weights associated with each ranking by assigning cardinal utilities. The mathematical representation is the following:

W_{a}^{r} (ε) = \{w \in W : r a n k (ε_{a}, w) = r\}

W = \{w \in R^{n} : w \geq 0, \sum_{j = 1}^{n} w_{j} = 1\}

Finally, the ordinal acceptability index is found; it measures the weighted volume under the ranking (r) of each class (a). The index is calculated through a multidimensional integral on the distribution of stochastic cardinal utility and favorable weights:

b_{r}^{a} = \int_{X} f (ε) \int_{w_{a}^{r} (ε)} f (w) d_{w} d_{ε}

The ordinal ranking for each alternative can be easily obtained by assigning to it the ordinal ranking corresponding to the class to which it belongs, so that:

b_{r}^{i} = b_{r}^{a} \leftrightarrow i \in F (a)

The decision-making in the ordinal context is based on an agreement between the characteristics of feasible weights and acceptability indexes; these last ones support the ordinal ranking of each alternative.

In summary, as a result of the cardinal modelling, a solution set (F) of optimal possibilities for the combinatorial problem is obtained, which is made up of (m) alternatives that are the input for ordinal analysis. In the process, as a result of the ordinal analysis, a set of ordinal rankings associated with each alternative (C_i) is obtained:

C_{i} \in \{1, 2, \dots, m\}, w h e r e : N (C_{i}) = c_{i} \leq m

The number of ordinal rankings for each alternative is lower than or equal to (m), because there may be some alternatives with zero ordinal acceptability indexes. Finally, the number of ordinal rankings for the set of alternatives (C) is given by

N (C) = \sum_{i = 1}^{m} c_{i} \leq m^{2}

The central ordinal ranking value and the associated dispersed ranking value for each alternative are given by

R^{i} = \sum_{r = 1}^{m} {r b}_{r}^{i}

V^{i} = \sum_{r = 1}^{m} {{(r - R^{i})}^{2} b}_{r}^{i}

It is relevant to point out that these two values correspond to the expected ordinal ranking value and the variance ordinal ranking value for the class to which the alternative belongs.

Integration Analysis—SMAA

The stage restricts the SMAA deterministic version to two variables.

Indexes and sets

O_{r}^{i}

: Integral ranking in ordinal ranking r for alternative i, where i ϵ {1, 2, …, m}, and r ϵ {1, 2, …, m}

O_{r}

: Integral ranking in ordinal ranking r, where ϵ {1, 2, …, m}

Other symbols

u^{i, r}

: Overall utility in ordinal ranking r for alternative i

p (e^{i})

: Cardinal utility result variable for alternative i

b_{r}^{i}

: Utility of ordinal result variable in ranking r for alternative i

W_{r}

: Set of feasible weight vectors [

w_{p}^{r}, w_{p}^{r}

] in ordinal ranking r

w_{r}^{j}

: Weight of result variable j in ranking r

w_{r}^{i, b}

: Basic favorable weight vectors in ordinal ranking r for alternative i

w_{o}^{j, r, c}

: Central weight vector of integral ranking o in ordinal ranking r for alternative i

Set of favorable weight vectors in ordinal ranking r for alternative i

d_{o}^{i, r}

: Integral acceptability index o in ordinal ranking r for alternative i

o_{r}^{i}

: Number of integral states in ordinal ranking r for alternative i

p_{r} (e^{i})

: Joint integral index in ordinal ranking r for alternative i

The overall utility of each alternative is based on the typical relative values of its result (that is, the utility of cardinal and ordinal result variables). An additive utility function is used so that the overall utility of each alternative is the result of summing the products of the feasible convex weights of its variables and their associated utilities. The following is the corresponding mathematical description:

\begin{matrix} u_{r}^{i} = & w_{r}^{p} p (e^{i}) + w_{r}^{q} b_{r}^{i}, w ϵ W, r ϵ R, \\ W_{r} = & {w_{r} ϵ R^{2}, w_{r} \geq 0, w_{r}^{p} + w_{r}^{q} = 1} \end{matrix}

The set of favorable weight vectors W_r is a one-dimensional simplex in the bi-dimensional weight space. An alternative is dominant when its utility surpasses that of all other alternatives. The problem of finding a set of basic favorable weights for integral ranking o is solved for each ordinal ranking r by means of the following LP:

Max 0

Subjected to

w_{r}^{p} p (e^{i}) + w_{r}^{q} b_{r}^{i} \geq w_{r}^{p} p (e^{h}) + w_{r}^{q} b_{r}^{h},

\begin{matrix} h = 1,2, \dots, m, h \neq i, \\ w_{r}^{p} + w_{r}^{q} = 1, \\ w_{r}^{j} \geq 0, j ϵ {p, q} \end{matrix}

The set of basic favorable weights is a convex polytope that can be represented as a convex combination of its vertices:

w_{r}^{i} = {w_{r} ϵ R^{+} : w_{r} = \sum_{b} α_{r}^{b} w_{r}^{i, b}, \sum_{b} α_{r}^{b} = 1, α_{r}^{b} \geq 0}

The integral acceptability index of each alternative in ordinal ranking r is defined as the ratio between the alternative’s weight volume and its feasible weight volume, both in that same ranking, which gives the alternative an integration ranking o.

v o l (W_{o}^{i, r}) = \int_{w_{r}^{i}} {d w}_{r},

A low integral acceptability value of o (close to zero) for an alternative in any ordinal ranking r implies a low number of favorable weight combinations, which makes this (alternative) the dominant one, as presented below:

d_{o}^{i, r} = \frac{v o l (W_{o}^{i, r})}{v o l (W_{r}^{i})}, v o l (W_{r}^{i}) | > 0 .

Finally, it can be stated that without any prior knowledge of the decision-makers’ expertise, the central weight vector is the best representation of a typically non-biased decision-maker. The central weight vector in ordinal ranking r for integral ranking o is defined as the center of gravity of the polytope

w_{o}^{i, r, c} = \frac{\int_{w_{r}^{i}} w d w}{\int_{w_{r}^{i}} d w}

The set of integral rankings in ordinal ranking r for each alternative i

(O_{r}^{i})

is given by the following states:

O_{r}^{i} ϵ {1,2, \dots, m}

, which is made up of

O_{r}^{i}

states

N (O_{r}^{i}) = O_{r}^{i} \leq m .

Appendix B. Qualitative Aspects in Queuing Theory

Appendix B.1. Service Quality

This aspect concerns a comprehensive estimation of customer satisfaction with the waiting line service. For [61], perceived waiting time is as important as, or more important than, actual waiting time. Customer service is increasingly seen as the top business priority. It is predicted that in the future, most customers will not accept or tolerate inferior products or services [44].

The first research work on service quality attempted to provide conceptual models of this notion, analyzing what it is and how it can be measured. But this endeavor slowly evolved towards more complex models in which the main objective was not only associated with the conceptualization and measurement of service quality but also with the analysis of how it is related to other concepts, such as the satisfaction of the customer and their future behavioral intentions.

According to [61], perception is the sensory process by which subjects organize and interpret their physical impressions, thus giving meaning to the environment around them. For [62], perception is “the active process of perceiving reality and organizing it into sensible interpretations or visions”.

In psychological terms, the perception of time is subject to the “Pygmalion effect” or “self-fulfilling prophecy” [63], through which previous expectations regarding the occurrence of a certain phenomenon are likely to affect perception, because even the behavior of the perceiver contributes to the expected result. The present work considers the following facets of time perception: (1) Anxiety of the consumer, (2) Importance of the service for the client, (3) Service attentiveness.

Appendix B.1.1. Anxiety of the Consumer

The effect of anxiety on the perception of time (defined in terms of expected time and perceived time) in a given situation is a clear indicator of the impact of this factor on the tolerance of users to impatience in a waiting line [64]. Ref. [44], who is considered the father of psychological perception in queues, describes that the perception of time can generate user anxiety in the queue, giving the impression that the queue is longer than it actually is. This author mentions that the greater the anxiety, the greater the intolerance.

In order to measure the level of service anxiety, the following methodologies can be used:

The Magellan Anxiety Scale (EMANS) is a questionnaire containing 15 statements describing physiological sensations and involuntary movements related to tension, discomfort, and overwhelm, among others. The person being evaluated reports the frequency with which each of these sensations or movements has been experienced during the last two months [65].
Cognitive-Somatic Anxiety Questionnaire (CSAQ) by [60] consists of fourteen items, seven of which are cognitive (cognitive subscale), while seven are somatic (somatic subscale). When they feel nervous or anxious, subjects must answer the different items on a Likert-type scale graduated from 1 to 5, according to how they typically experience each of the symptoms.

Appendix B.1.2. Importance of Customer Service

Ref. [44] defines attention as a relevant aspect of service quality perception. It is in this context that the client estimates the value of the service, which to a large extent defines the time they are willing to wait [44]. This factor can be evaluated through a 1 to 5 Likert scale in the population that frequents the type of service to be analyzed. Likewise, the impact of the service with regard to the five hierarchical levels of basic human needs [55] is evaluated.

Appendix B.1.3. Service in Face of Customer’s Attention

Attention has often been conceived as an attribute of perception, through which we more effectively select the information that is relevant to us. The act of waiting (and expectation in general) focuses on the passing of time, which is therefore perceived as longer than usual. Therefore, in case it is not possible to reduce service time, it is important to create an environment that distracts people’s attention in order to improve the service experience [66]. These authors refer to the subjective perception of time in terms of attention given to it.

Attention can be measured by means of psychomotor reaction tests such as the Toulouse and Pieron Test or Attention Test, which handles the variables time, hits, and errors [67].

Appendix B.2. Comfort Level

The evaluation of time during service depends on the interaction of several factors, such as environmental comfort or discomfort, the individual characteristics of the subject, the internal attributes of the time period, or the cognitive tasks it requires, plus the time it actually takes for the person to carry out the activities imposed by the experimental or environmental requirements [68]. Ref. [69] point out that when the spatial distribution of a service environment promotes the perception of social justice, subjects express more positive evaluations of service provision.

A service provider must offer minimum environmental comfort conditions to their customers while they wait in the queue. These include temperature, lighting, seating space, and noise levels [43,70]. The following is a description of each of the components to be taken into account when assessing this factor.

Appendix B.2.1. Lighting

Its main purpose is to facilitate an adequate visualization of the workplace so that labor can be carried out in acceptable conditions of efficiency, comfort, and safety. This has a favorable impact on people, reducing fatigue and contributing to worker performance and work quality [71].

A synthesis of studies aimed at analyzing user comfort during service has been presented by [70]. The input data for calculating the lighting required in an area are (1) type of activity to be carried out and (2) dimensions and characteristics of the enclosure to be illuminated. This information influences the choice of lamp types, which depend on chromatic reproduction needs, lighting levels, and other conditions. To obtain it, measurements are taken with a luxmeter, and then the corresponding calculations are carried out to determine the necessary luminous flux, the power of the lamps, and their number and distribution [71]. According to [71], the following are the necessary indicators for the basic calculation of a lighting system:

Table A1. Lighting indicators.

Indicator	Calculation Formula
Total necessary luminous flux	$Φ_{t} = \frac{E_{m} \cdot S}{η \cdot f_{c}}$ , where $Φ_{t}$ $=$ Total necessary luminous flux (lumens) $E_{m} =$ Average illuminance (lux) $S =$ Area to be illuminated (m²) $η =$ Lighting performance $f_{c}$ = Maintenance factor of the lighting system
Average illuminance $E_{m}$	It is set according to the visual requirements of the tasks to be carried out, which are specified in the corresponding technical standards, such as Article 28 of Colombia’s General Ordinance on Safety and Hygiene at Work (Ordenanza General de Seguridad e Higiene en el Trabajo—OGSHT).
Lighting performance $(η)$	$(η) = η_{R} \cdot η_{L}$ where $η_{R} =$ Performance of the room $η_{L}$ = Luminaire performance
Maintenance factor of the lightning system $(f\| c)$	This factor ranges from 0.5 to 0.8. 0.5 corresponds to dusty rooms with poorly maintained lighting systems. 0.8 corresponds to lighting systems located in clean places, equipped with enclosed luminaires and low luminous depreciation lamps, where frequent cleaning and total or partial lamp replacements are systematically carried out. This factor is determined by loss of luminous flux, loss of reflection, or transmission of the lamps due to natural aging or dirt that is deposited on them.
Number of light points (N)	$N = \frac{Φ_{t}}{Φ_{n}},$ where $Φ_{t} =$ Total necessary luminous flux $Φ_{n}$ = Nominal luminous flux of the lamps contained in a luminaire If luminaires with high luminous flux are used, the same total flux is achieved with fewer light points (with a lower total cost of the system), but uniformity is directly affected because the space between luminaires is larger, which gives rise to intermediate zones with less illumination.
Average uniformity (f_um)	$f_{um} = \frac{Emed}{Emin}$
Height of luminaires above the working plane (h)	In order to achieve acceptable average uniformity and glare risk levels, the luminaires must be distributed at a certain height (h) above the working plane and a corresponding distance (d) between them. Minimum height: $h = \frac{2}{3} d$ Advisable height: $h = \frac{3}{4} d$ Optimum height: $h = \frac{4}{5} d$ In the case of indirect and semi-direct lighting, the optimum height must not be exceeded.
Distance between luminaires (d)	It is a function of (h) and the beam opening angle of the luminaire.
	Type of luminaire	Distance
	Intensive	D ≤ 1.2 h
	Semi-intensive	D ≤ 1.5 h
	Extensive	D ≤ 1.6 h
	Selection of luminaire type as a function of (h)
	Height of the room	Type of luminaire
	Up to 4 m	Extensive
	From 4 to 6 m	Semi-extensive
	From 6 to 10 m	Semi-intensive
	More than 10 m	Intensive

Appendix B.2.2. Noise

Noise is an acoustic phenomenon that produces unpleasant auditory sensations and interferes with or impedes some human activity. In the most unfavorable cases, it can lead to the appearance of significant psychological disabilities or limitations [71]. As presented by [71], the achievement of adequate sound levels is an issue that should be taken into account in the project phase of a new premise or enclosure. When this is not done, subsequent efforts are always more expensive and laborious and, sometimes, simply impossible.

The existing noise inside a room has two components: the noise received directly (direct acoustic wave) and the noise reflected on the different surfaces (reflected acoustic wave). The following are the necessary indicators to calculate the interior acoustic adequation of premises according to [71]:

Table A2. Noise indicators.

Indicator	Calculation Formula
Critical distance (r):	$r = 0.14 \sqrt{R \cdot Q}$ , where r: Critical distance in meters (within this distance, the acoustic conditioning of the walls is not appreciable because of the dominance of direct waves); R: Constant of the room, in square meters; Q: Directivity coefficient.
Absorption (A)	$A_{f} = \propto_{f} \cdot S,$ where A: Absorption of frequency f in m². It quantifies the energy extracted from the acoustic field when the sound wave passes through a given medium or collides with the boundary surfaces of the enclosure. $A_{m}$ : Average absorption in meters. $\propto_{f} :$ Absorption coefficient of the material. S: Surface of the material in m².
Reverberation time (T)	$T = 0.163 \frac{V}{A}$ , where V: Volume of the premises in m³ A: Absorption of the premises in m²

Appendix B.2.3. Thermal Load

It refers to the sum of the environmental thermal load, resulting from heat generated in metabolic processes. Its measurement consists of determining the Wet Bulb Globe Temperature (WBGT) Index [71]. According to [71], the following are the necessary indicators to calculate the indoor thermal load conditioning of premises:

Table A3. Thermal load indicators.

Indicator	Calculation Formula
Wet-Bulb Globe Temperature (WBGT)	The WBGT index consists of the fractional weighing of wet, balloon, and sometimes dry temperatures.
	(WBGT) outdoors (sun exposure)	(WBGT) indoors (in the shade)
	WBGT = 0.7 Tw + 0.2 Tg + 0.1 Ta	WBGT = 0.7 Tw + 0.3 Tg
	Where Tw: Natural temperature of wet bulb. Tg: Globe temperature (measured through radiation load on a thermometer inside a 6-inch diameter black copper sphere). Ta: Dry bulb temperature (basic ambient temperature; shaded thermometer shielded from radiation).

Appendix B.3. Marketing Factors

Marketing is the human activity directed at satisfying needs and wants through exchange [72]. It is also defined as the process of conceiving, planning, executing, pricing, promoting, and distributing ideas, goods, or services to create exchanges that satisfy the objectives of individuals and organizations” [73]. The most important marketing factors are price and competition.

Appendix B.4. Transaction Costs

The cost of a service is not only its monetary value but also all its associated costs. The time that users lose when they are waiting in a queue could be interpreted as an additional cost associated with the transaction. In economic theory, this is part of the transaction costs, which are necessary to carry out a given economic pursue but do not add value and should therefore be minimized [29,74,75]. The analysis of these costs is based on the assessment of their dimensions, namely asset specificity, transaction measurement uncertainty, and transaction measurement difficulty. As these dimensions rise, it is suggested to implement higher levels of integration between the agents carrying out the economic exchange. Since these dimensions cannot be directly measured, they have to be estimated through Likert tables.

Appendix B.5. Competition Level

When several companies deliver equal or substitute services to the market, their price tends to stabilize at a point where no competitor can reduce it anymore, since customers will always seek the lowest price, coupled with some quality standards, though. On the contrary, in a monopoly environment where a single company is the provider of a good or service, the price tends to rise at the will of the monopolist, a situation that can only be regulated by the state [76]. The following is an index of the level of monopoly in a market:

Table A4. Competition indicators.

Indicator

Calculation Formula

Lerner’s index (L)

In a market with perfect competition, the market price (P) would be equal to the marginal cost of production (MC). Based on this premise, the Lerner index (L) is defined by the difference between those parameters, divided by the market price (P), in order to establish a fractional measure. L represents the power of a monopoly in the market.

L = \frac{P - M C}{P}

This index ranges from 0 to 1. Higher values indicate greater market power. For a firm under perfect competition conditions (where P = CM), L = 0, which expresses that the firm has no market power. The higher the value of L, the greater the monopoly power.

L = \frac{- 1}{ped}

where
ped: Price elasticity of demand.

Appendix C. Central Weight Vectors

Appendix C.1. Ordinal Analysis (SMAA-O)

Table A5. Central weight vectors for the ordinal analysis (SMAA-O).

	w₁	w₂	w₃	w₄
wc₁	0.35	0.15	0.15	0.35
wc₂	0	0	0	0
wc₃	0	0	0	0
wc₄	0	0	0	0
wc₅	0.35	0.15	0.15	0.35

Table A5 presents the central weight vectors obtained from the ordinal analysis. The results indicate that alternatives 1 and 5 are those capable of achieving the first rank. This is consistent with the centroid configuration, which shows that the support for these alternatives arises mainly from criteria 1 and 4. In practical terms, this suggests that the relative importance of these two criteria drives the ordinal preference and that variations in the other criteria do not significantly alter the ranking outcome.

Appendix C.2. Integration Analysis (Deterministic SMAA)

Table A6 summarizes the central weight vectors under the deterministic SMAA integration analysis. The findings reveal that only Alternative 1 is capable of consistently attaining the first rank. Unlike the ordinal case, the central weights in this scenario indicate that the solution is robust across a wide range of possible weight configurations, since the vectors may assume practically any value. This reinforces the interpretation that Alternative 1 is structurally dominant under deterministic integration, regardless of the distribution of preferences across criteria.

Table A6. Central weight vectors for the integration analysis (SMAA deterministic).

	w₁	w₂
wc₁	0.5	0.5
wc₂	0	0
wc₃	0	0
wc₄	0	0
wc₅	0	0

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Table 1. Summary of variables used in the queuing model.

Symbol	Description	Units
λ	Arrival rate	customers/hour
μ	Service rate per server	customers/hour
c	Number of servers	count
L	Expected number of users	customers
Lq	Expected number in the queue	customers
W	Expected time in the system	minutes
Wq	Expected time in queue	minutes
X	% of server inactivity	percentage
α, β	Aspiration thresholds (W and X, respectively)	minutes/percentage

Table 2. Expected number of selected servers.

n	λ (Arrival Rate)	µ (Service Rate)	c (Number of Servers)	λeff	Ls	Ws (min)	Lq	Wq (min)	100 − X (%)	$p_{n}$	$p_{i}$
1	6	3	4	6	2.17	21.73	0.17	1.74	50.00	0.1	i = 1, 0.3
2	7	3	4	7	2.70	23.19	0.37	3.19	58.33	0.1
3	8	3	4	8	3.42	25.67	0.75	5.67	66.67	0.1
4	9	3	5	9	3.35	22.36	0.35	2.36	60.00	0.1	i = 2, 0.2
5	10	3	5	10	3.98	23.92	0.65	3.91	66.67	0.1	i = 2, 0.2
6	11	3	6	11	3.99	21.79	0.32	1.80	61.11	0.1	i = 3, 0.2
7	12	3	6	12	4.56	22.84	0.56	2.85	66.67	0.1	i = 3, 0.2
8	13	3	7	13	4.63	21.4	0.30	1.40	61.90	0.1	i = 4, 0.2
9	14	3	7	14	5.16	22.14	0.50	2.14	66.67	0.1	i = 4, 0.2
10	15	3	8	15	5.27	21.11	0.27	1.11	62.5	0.1	i = 5, 0.1

Note: The values of Lq and Wq were estimated based on standard approximations under the assumed arrival and service conditions. Slight deviations from Little’s Law may occur due to rounding and the integration of multicriteria adjustments within the IAM framework.

Table 3. Likert scale.

Range	Level
1	Very high
2	High
3	Medium
4	Low
5	Very low

Table 4. Results for the criterion noise.

dB	Likert Level	Level
≥80	5	Very bad
[70–80)	4	Bad
[60–70)	3	Medium
(50–60)	2	Good
50≤	1	Very good

Table 5. Results for the variable thermal load.

WBGT (°C)	Likert Level	Level
>38 o < 5	5	Very bad
[32–38) o [5–10)	4	Bad
[26–32) o [10–15)	3	Medium
[23–26)	2	Good
[15–23)	1	Very good

Table 6. Results of the variable competition.

Lerner’s Index	Likert Level	Level
[0–0.2)	1	Very good
[0.2–0.4)	2	Good
[0.4–0.6)	3	Medium
[0.6–0.8)	4	Bad
[0.8–1]	5	Very bad

Table 7. Ordinal analysis.

i	c	Frequency	$p_{i}$	j:1 Consumer’s Anxiety	j:2 Noise	j:3 Thermal Load	j:4 Competition	$b_{1}^{i}$
1	4	3	0.3	5	1	1	5	0.25
2	5	2	0.2	4	2	2	4	1/6
3	6	2	0.2	3	3	3	3	1/6
4	7	2	0.2	2	4	4	2	1/6
5	8	1	0.1	1	5	5	1	0.25

Table 8. Results of the integration analysis.

i	1	2	3	4	5
c * (optimal number of servers)	4	5	6	7	9
$p_{i}$	0.30	0.20	0.20	0.20	0.10
$b_{1}^{i}$	¼	1/6	1/6	1/6	¼
$p (e^{i})$	0.075	0.033	0.033	0.033	0.025
$a_{1}^{i}$	1	0	0	0	0

*: used as a superscript that emphasizes the optimal condition of the variable.

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García-Cáceres, R.G.; Prado-Téllez, A.G.; Escobar-Velásquez, J.W. Sustainable Integral Optimization of Service Queues: A Human-Centered Approach Using IAM and SMAA. Sustainability 2025, 17, 8179. https://doi.org/10.3390/su17188179

AMA Style

García-Cáceres RG, Prado-Téllez AG, Escobar-Velásquez JW. Sustainable Integral Optimization of Service Queues: A Human-Centered Approach Using IAM and SMAA. Sustainability. 2025; 17(18):8179. https://doi.org/10.3390/su17188179

Chicago/Turabian Style

García-Cáceres, Rafael Guillermo, Angel Gabriel Prado-Téllez, and John Wilmer Escobar-Velásquez. 2025. "Sustainable Integral Optimization of Service Queues: A Human-Centered Approach Using IAM and SMAA" Sustainability 17, no. 18: 8179. https://doi.org/10.3390/su17188179

APA Style

García-Cáceres, R. G., Prado-Téllez, A. G., & Escobar-Velásquez, J. W. (2025). Sustainable Integral Optimization of Service Queues: A Human-Centered Approach Using IAM and SMAA. Sustainability, 17(18), 8179. https://doi.org/10.3390/su17188179

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Sustainable Integral Optimization of Service Queues: A Human-Centered Approach Using IAM and SMAA

Abstract

1. Introduction

2. Literature Review

3. Description of the Problem

3.1. Cardinal Analysis

3.2. Ordinal Analysis

3.3. Integration Analysis

4. Conclusions and Future Perspectives

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

Appendix A. Integral Analysis Method (IAM)—Mathematics Stages

Appendix B. Qualitative Aspects in Queuing Theory

Appendix B.1. Service Quality

Appendix B.1.1. Anxiety of the Consumer

Appendix B.1.2. Importance of Customer Service

Appendix B.1.3. Service in Face of Customer’s Attention

Appendix B.2. Comfort Level

Appendix B.2.1. Lighting

Appendix B.2.2. Noise

Appendix B.2.3. Thermal Load

Appendix B.3. Marketing Factors

Appendix B.4. Transaction Costs

Appendix B.5. Competition Level

Appendix C. Central Weight Vectors

Appendix C.1. Ordinal Analysis (SMAA-O)

Appendix C.2. Integration Analysis (Deterministic SMAA)

References

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