## 4 Conclusions  
This study set out to optimize telemedicine queuing systems by developing a hybrid Artificial Neural Network-Particle Swarm Optimization (ANN-PSO) model aimed at improving system performance and patient experience.  
The first objective--simulating queuing scenarios to generate relevant data--was achieved using queuing theory-based simulations that captured key variables such as arrival rates, service rates, and system capacity. These simulations provided a realistic foundation for training predictive models.  
The second objective--training an ANN to forecast critical performance metrics--demonstrated that ANNs can effectively predict outputs such as queue intensity, system utilization, and delays with high accuracy. This predictive capability enables telemedicine administrators to anticipate demand and make proactive decisions.  
The third and central objective--optimizing the ANN model using PSO--was fulfilled by fine-tuning network parameters to enhance model generalization and minimize prediction error. Rather than replacing traditional analytical approaches, the ANN-PSO model serves as a complementary tool that offers flexible, data-driven predictions under dynamic conditions. By enabling real-time adaptation to varying patient loads, this hybrid model supports more responsive telemedicine queue management and contributes to efficient, patient-centered service delivery.  
The novelty of this study lies in its integrated use of ANN-PSO for simulating and optimizing M/M/s/N telemedicine queueing systems, with a unique focus on generating and evaluating optimal configurations for multiple performance metrics across a two-phase simulation. Unlike previous works, this study offers a practical framework tailored to address queue-related inefficiencies in dynamic telemedicine environments.  
The ANN-PSO framework complements closed-form analysis by functioning as a real-time predictive engine. While analytical formulas compute point estimates under strict assumptions, the ANN-PSO dynamically updates predictions based on streaming data and system fluctuations. This capability is critical in telemedicine platforms that operate in stochastic, multi-parameter environments where closed-form assumptions (e.g., constant \(\mu\), infinite queues) often fail to capture temporal variability. As demonstrated in Sections 3.1 and 3.2, ANN-PSO offers reliable forecasting across multiple metrics (utilization, queue delay, idle probability), making it suitable for decision support in adaptive scheduling and load balancing.  
Conceptualization, D.T. and L.T.; methodology, D.T.; software, D.T.; Validation, D.T. and L.T.; formal analysis, D.T.; investigation, D.T.; resources, L.T.; data curation, D.T.; writing--original draft preparation, D.T.; writing--review and editing, L.T.; visualization, D.T.; supervision, L.T.; project administration, L.T. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research received no external funding.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Data available in Appendix A.  
**Acknowledgments:** The authors gratefully acknowledge the use of the Revoleedu online tool for performing the queuing simulations required in this study. The optimization of the ANN-PSO model was conducted using existing MATLAB code available from academic resources. During the preparation of this manuscript, the author used ChatGPT (OpenAI, version March 2025) for the purposes of text restructuring, language improvement, and grammatical correction. The authors have carefully reviewed and edited the generated content and take full responsibility for the final version of this publication.  
**Conflicts of Interest:** The authors declare no conflict of interest.