## 2 Materials and Methods  
This study employs the M/M/s/N queuing model, widely used to analyze and optimize systems characterized by random customer arrivals and service requirements, such as telecommunications networks, computer systems, and customer service centers [20]. The M/M/s/N queuing model is an extension of the classical M/M/s model, incorporating a finite system capacity (N). It is particularly useful for analyzing service systems where waiting space is limited, and customers may be denied entry if the system reaches its maximum capacity. The applicability of the M/M/s/N model to telemedicine is grounded in its alignment with core operational characteristics: in telemedicine platforms, patient consultation requests occur randomly, akin to a Poisson arrival process, and consultation durations vary, modeled effectively by exponential service times [21]. The parameters represent the number of available doctors conducting remote consultations in parallel, while N reflects real-world capacity constraints such as platform session limits, clinician availability, and bandwidth restrictions. These dynamics mirror the M/M/s/N model's treatment of arrivals blocked when capacity is reached. The assumptions used in the M/M/s/N queuing model were adopted based on its suitability for modeling stochastic systems like telemedicine, where arrival and service rates are random. These assumptions (Poisson arrivals, exponential service times, finite capacity, multiple servers) reflect the unpredictable and time-sensitive nature of patient requests in telemedicine, aligning well with real-world dynamics. This model consists of four key elements:  
* Arrival rate (number of customers/unit time), \(\lambda\). The arrival process follows a Poisson distribution with an average arrival rate; this means arrivals are random and independent of each other, making the interarrival times exponentially distributed. In the context of telemedicine, this reflects real-world patient demand, where consultations are requested unpredictably. * Service rate (number of customers/unit time), \(\mu\). The service process follows an exponential distribution with an average service rate, \(\mu\); this assumes that shorter service times occur more frequently than longer ones, which aligns with real-life consultations, where some cases require minimal interaction while others need extended discussions. * Number of servers (s). The model includes parallel servers, representing the number of healthcare providers available for telemedicine consultations. Each patient, upon arrival, will be immediately assigned to an available provider if one is free. If all providers are busy, the patient will either wait in a queue or be rejected, depending on system capacity. * Capacity of the system (N). Unlike the standard M/M/s model (which assumes an infinite queue), M/M/s/N introduces a finite limit (N) on the number of patients that can be in the system (including those waiting and being served). If a new patient arrives when the system is already at full capacity (N patients present), they are blocked and denied service. This is critical in telemedicine, where platform limitations, provider availability, or technological constraints may cap the number of simultaneous consultations. Additionally, the maximum queue size \(=\) N \(-\) s [20]. To understand system efficiency, the M/M/s/N model allows for the evaluation of several key performance indicators [20]:  
1. Utilization factor (U). The utilization factor represents the percentage of time that all servers are busy, given by the following:  
\[U=\frac{L-L_{q}}{s} \tag{1}\]  
where  
* U is the utilization factor;
* L is the average number of customers in the system (both waiting and being served);
* Lq is the average number of customers waiting in line for service;
* s is the number of service providers (servers). 2. Probability of an empty system (P\({}_{0}\)).